# A user’s guide to Beilinson–Kato’s zeta elements

Chan-Ho Kim  June E Huh Center for Mathematical Challenges, Korea Institute
for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract.

In his ground-breaking work [Kat04], K. Kato constructed the Euler system of
Beilinson–Kato’s zeta elements and proved spectacular results on the Iwasawa
main conjecture for elliptic curves and the classical and $p$-adic Birch and
Swinnerton-Dyer conjectures by using these elements. The goal of this
expository lecture note is to explain how Kato’s Euler systems fit into the
framework of the arithmetic of elliptic curves and their Iwasawa theory, and
we hope that this approach eventually helps the reader to read Kato’s original
paper _more easily and with less pain_. It is written for the proceedings of
the program _Elliptic curves and the special values of $L$-functions (ONLINE)_
in ICTS in August 2021.

###### Key words and phrases:

elliptic curves, Birch and Swinnerton-Dyer conjecture, Iwasawa theory, Kato’s
Euler systems

###### 2010 Mathematics Subject Classification:

11R23 (Primary); 11F33 (Secondary)

This research was partially supported by a KIAS Individual Grant (SP054103)
via the Center for Mathematical Challenges at Korea Institute for Advanced
Study, by the National Research Foundation of Korea(NRF) grant funded by the
Korea government(MSIT) (No. 2018R1C1B6007009), and by the International Centre
for Theoretical Sciences (ICTS) for the online program - Elliptic curves and
the special values of $L$-functions (code: ICTS/ECL2021/8).

###### Contents

  1. 1 Lecture 1: Overview and Birch and Swinnerton-Dyer conjecture
  2. 2 Lecture 2: Cyclotomic Iwasawa theory for elliptic curves
  3. 3 Lecture 3: The Euler and Kolyvagin system machinery
  4. 4 Lecture 4: A sketch of the construction of Beilinson–Kato’s zeta elements

## Prologue

### 0.1.

It does not seem terribly bad to start with my personal experience. From
around 2014, I have started to read Kato’s beautiful work on his Euler systems
[Kat04] and have failed so many times due to many reasons, mainly the lack of
my mathematical maturity. I still learn a lot from this paper when I read it
again. My initial motivation for studying this paper was to develop the mod
$p$ numerical criterion for verifying the main conjecture [KKS20]. From this
perspective, I did not care about the construction of the Euler system too
much for the first time, but I tried hard to understand the meaning of the
final output of the construction. Even only for such a “small” purpose, it
turns out that I needed to understand the precise meaning of

${}_{c,d}z^{(p)}_{m}(f,r,r^{\prime},\xi,S)\in\mathrm{H}^{1}(\mathbb{Q}(\zeta_{m}),T_{f}(k-r))$

in [Kat04, (8.1.3) and Ex. 13.3] and

$\mathbf{z}^{(p)}_{\gamma}\in\varprojlim_{n}\mathrm{H}^{1}(\mathbb{Q}(\zeta_{p^{n}}),T_{f})$

in [Kat04, Thm. 12.5]. Ugh, I should go back to the construction to some
extent and try to understand the behavior of these elements under Bloch–Kato’s
dual exponential map at least. It required a considerable amount of time and
energy for me at that time. My aim is for this lecture note to make diving
into Kato’s original paper a breeze, minimizing the struggle and maximizing
understanding.

### 0.2.

Although [Kat04] was published in 2004, Kato’s work was announced and was
accepted by experts in early 1990’s. See [Kat92c, Kat92b, Kat92a, Kat95] for
his announcement in Japanese111These are all hand-written.. Also, [Sch98,
Rub98] were written for the proceeding of the Durham conference held at 1996.

### 0.3.

There are now various expository papers and books on Kato’s work from
different viewpoints. See [Rub98, Sch98, Col04, Del08, Wan13] for example.

### 0.4.

As prerequisites, we expect that the reader knows some algebraic number theory
[Neu99], Galois cohomology [Rub00, §1], elliptic curves [Sil09] and modular
forms [DS05]. The most important prerequisite would be a certain amount of
experiences to try to read [Kat04].

### 0.5.

Of course, we are not able to explain every detail of Kato’s paper here. As we
mentioned above, our goal is to explain how Kato’s Euler systems apply to the
arithmetic of elliptic curves. We explain relatively small parts of his paper
in details _only for achieving this goal_. In this sense, our exposition can
be seen as employing a reverse-engineering approach. Since it grew out from a
_lecture note_ , this article contains some _intentional_ repetitions. We hope
this structure make the reader more comfortable.

##  1\. Lecture 1: Overview and Birch and Swinnerton-Dyer conjecture

In the first lecture, we illustrate how Kato’s Euler systems work in the
arithmetic of elliptic curves and to sketch how they are constructed briefly.
Our focus lies on elliptic curves, and we will not cover higher weight modular
forms in this lecture note.

###  1.1. Rational points on elliptic curves

How do we understand rational points on elliptic curves?

######  Example 1.1.

Consider two elliptic curves:

  * •

$E_{1}:y^{2}=x^{3}-x$ (32.a.3)

  * •

$E_{2}:y^{2}+y=x^{3}-x$ (37.a.1)

Although the defining equations of these two curves do not look very
different, their Diophantine properties are significantly different:

$\textstyle{E_{1}(\mathbb{Q})\simeq\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z},}$$\textstyle{E_{2}(\mathbb{Q})\simeq\mathbb{Z}.}$

How can we detect this difference?

Let $E$ be an elliptic curve over $\mathbb{Q}$. As in Example 1.1, Human
beings do not know whether $E(\mathbb{Q})$ is infinite or not, but
$L$-functions know. This is called Birch and Swinnerton-Dyer conjecture. In
this sense, we observe an important inequality222I learned this inequality
from the lecture series of Masato Kurihara at Postech, January 2012.

$\textrm{human beings}<\textrm{$L$-functions}.$

The following modularity theorem is essential for the statement of Birch and
Swinnerton-Dyer conjecture as well as the applications of Kato’s Euler systems
to elliptic curves.

######  Theorem 1.2 (Wiles, Taylor–Wiles, $\cdots$,
Breuil–Conrad–Diamond–Taylor).

There exists a one-to-one correspondence between

  * •

$\mathbb{Q}$-isogeny classes of rational elliptic curves of conductor $N$ and

  * •

weight two cuspidal newforms with rational coefficients of level
$\Gamma_{0}(N)$

satisfying $L(E,s)=\sum_{n\geq 1}\dfrac{a_{n}}{n^{s}}$ with
$\mathrm{Re}(s)>3/2$ where $f_{E}=\sum_{n\geq 1}a_{n}q^{n}\in
S_{2}(\Gamma_{0}(N))$. In particular, $L(E,s)$ is entire.

It is known that the complex $L$-function $L(E,s)$ is an isogeny invariant.

###### Proof.

See [Wil95, TW95, BCDT01] ∎

The following conjecture is the most important problems in the arithmetic of
elliptic curves [BSD63, BSD65, Tat74, Gol82].

######  Conjecture 1.3 (Birch–Swinnerton-Dyer).

Let $E$ be an elliptic curve over $\mathbb{Q}$.

  1. (1)

$\mathrm{rk}_{\mathbb{Z}}E(\mathbb{Q})=\mathrm{ord}_{s=1}L(E,s)(=r)$.

  2. (2)

If (1) holds and Tate–Shafarevich group $\textrm{{SH}}(E/\mathbb{Q})$ is
finite, then

$\lim_{s\to
1}\dfrac{L(E,s)}{(s-1)^{r}\cdot\Omega^{+}_{E}\cdot\mathrm{Reg}_{E}}=\dfrac{\\#\textrm{{SH}}(E/\mathbb{Q})\cdot\mathrm{Tam}(E)}{(\\#E(\mathbb{Q})_{\mathrm{tors}})^{2}}$

where

  * •

$\mathrm{Reg}_{E}=\mathrm{det}\left((\langle
x_{i},x_{j}\rangle_{\mathrm{NT}})_{i,j}\right)$ with
$E(\mathbb{Q})/E(\mathbb{Q})_{\mathrm{tors}}=\oplus^{r}_{i=1}\mathbb{Z}x_{i}$,

  * •

$\langle-,-\rangle_{\mathrm{NT}}$ is the Néron–Tate height pairing,

  * •

$\mathrm{Tam}(E)=\prod_{\ell|N}[E(\mathbb{Q}_{\ell}):E_{0}(\mathbb{Q}_{\ell})]$
with the preimage of the non-singular points $E_{0}(\mathbb{Q}_{\ell})$ under
the mod $\ell$ reduction, and

  * •

$\Omega^{+}_{E}=\int_{E(\mathbb{R})}|\omega_{E}|$,
$\omega\in\Omega^{1}_{\mathcal{E}/\mathbb{Z}}$ with the Néron model
$\mathcal{E}$ of $E$.

###  1.2. Beilinson–Kato elements and the finiteness of rational points

We first illustrate a simple application of (the bottom of) Beilinson–Kato
elements to the arithmetic of elliptic curves.

#### 1.2.1.

Let $p$ be a prime and $E$ an elliptic curve over $\mathbb{Q}$ without complex
multiplication. Let
$T=\mathrm{Ta}_{p}E=\varprojlim_{n}E(\overline{\mathbb{Q}})[p^{k}]$ be the
$p$-adic Tate module and $V=V_{p}E=T\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$ be
the 2-dimensional $\mathbb{Q}_{p}$-vector space endowed with the continuous
action of $G_{\mathbb{Q}}=\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.
Denote the corresponding Galois representation by
$\rho:G_{\mathbb{Q}}\to\mathrm{Aut}_{\mathbb{Q}_{p}}(V)\simeq\mathrm{GL}_{2}(\mathbb{Q}_{p})$.
Let $\Sigma$ be a finite set of places of $\mathbb{Q}$ containing $p$,
$\infty$, and bad reduction primes for $E$, and denote by
$\mathbb{Q}_{\Sigma}$ the maximal extension of $\mathbb{Q}$ unramified outside
$\Sigma$. Then the information of $E(\mathbb{Q})$ can be detected in Galois
cohomology group
$\mathrm{H}^{1}(\mathbb{Q},V)=\mathrm{H}^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q},V)$
via Kummer map

$E(\mathbb{Q})\otimes\mathbb{Q}_{p}\to\mathrm{H}^{1}(\mathbb{Q},V)$

which makes the connection between geometry and cohomology. The same rule
applies to the local case.

#### 1.2.2.

We first investigate the local nature of Galois cohomology at $p$.

######  Exercise 1.4.

Show that $\mathrm{H}^{1}(\mathbb{Q}_{p},V)$ is a 2-dimensional
$\mathbb{Q}_{p}$-vector space. (Hint: Use the local Euler characteristic
formula.)

The local Kummer map
$E(\mathbb{Q}_{p})\otimes\mathbb{Q}_{p}\hookrightarrow\mathrm{H}^{1}(\mathbb{Q}_{p},V)$
embeds a 1-dimensional geometric object and a 2-dimensional cohomological one.
The Weil pairing

$V\times V\to\mathbb{Q}_{p}(1)$

induces a non-degenerate cup product pairing

$\langle-,-\rangle_{p}:\mathrm{H}^{1}(\mathbb{Q}_{p},V)\times\mathrm{H}^{1}(\mathbb{Q}_{p},V)\overset{\cup}{\to}\mathrm{H}^{2}(\mathbb{Q}_{p},\mathbb{Q}_{p}(1))\simeq\mathbb{Q}_{p}.$

Under this pairing, we have the following orthogonality

$E(\mathbb{Q}_{p})\otimes\mathbb{Q}_{p}\perp
E(\mathbb{Q}_{p})\otimes\mathbb{Q}_{p}$

due to local Tate duality. Consider the commutative diagram

|  |
---|---|---
|  |
$\textstyle{\mathrm{H}^{1}(\mathbb{Q}_{p},V)}$$\textstyle{\times}$$\textstyle{\mathrm{H}^{1}(\mathbb{Q}_{p},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{exp}^{*}_{\omega_{E}}}$$\textstyle{\mathbb{Q}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E(\mathbb{Q}_{p})\otimes\mathbb{Q}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Q}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\mathrm{exp}_{\widehat{E}}}$$\textstyle{\times}$$\textstyle{\mathbb{Q}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Q}_{p}}$$\textstyle{\mathrm{H}^{0}(E/\mathbb{Q}_{p},\Omega^{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$

where

  * •

the map $\mathrm{exp}_{\widehat{E}}:\mathbb{Q}_{p}\to
E(\mathbb{Q}_{p})\otimes\mathbb{Q}_{p}$ extends the formal exponential map
$\mathrm{exp}_{\widehat{E}}:p\mathbb{Z}_{p}\to\widehat{E}(p\mathbb{Z}_{p})$
and the source $\mathbb{Q}_{p}$ is the tangent space of $E/\mathbb{Q}_{p}$,

  * •

the latter $\mathbb{Q}_{p}$ is isomorphic to the space of global 1-forms
$\mathrm{H}^{0}(E/\mathbb{Q}_{p},\Omega^{1})=\mathbb{Q}_{p}\omega_{E}$ and
$\frac{L(E,1)}{\Omega^{+}_{E}}\cdot\omega_{E}\in\mathbb{Q}_{p}\omega_{E}$,

  * •

$\mathrm{exp}^{*}_{\omega_{E}}:\mathrm{H}^{1}(\mathbb{Q}_{p},V)\to\mathbb{Q}_{p}$
is the composition of
$\mathrm{exp}^{*}:\mathrm{H}^{1}(\mathbb{Q}_{p},V)\to\mathrm{H}^{0}(E/\mathbb{Q}_{p},\Omega^{1})$
and isomorphism
$\mathrm{H}^{0}(E/\mathbb{Q}_{p},\Omega^{1})\simeq\mathbb{Q}_{p}$, and

  * •

the bottom pairing is given by multiplication: $(a,b)\mapsto a\cdot b$.

The characterization of the kernel of the dual exponential map is important
for us.

(1.1)
$\mathrm{ker}(\mathrm{exp}^{*}_{\omega_{E}})=E(\mathbb{Q}_{p})\otimes\mathbb{Q}_{p}\subseteq\mathrm{H}^{1}(\mathbb{Q}_{p},V).$

We now see the simplest form of Kato’s work and feel its power for the first
time.

######  Theorem 1.5 (Kato).

There exists a global Galois cohomology class
$z_{\mathbb{Q}}\in\mathrm{H}^{1}(\mathbb{Q},V)$ such that

$\textstyle{\mathrm{H}^{1}(\mathbb{Q},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{loc}_{p}}$$\textstyle{\mathrm{H}^{1}(\mathbb{Q}_{p},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{exp}^{*}}$$\textstyle{\mathbb{Q}_{p}\omega_{E}}$$\textstyle{z_{\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{exp}^{*}(\mathrm{loc}_{p}(z_{\mathbb{Q}}))}$

and

$\mathrm{exp}^{*}(\mathrm{loc}_{p}(z_{\mathbb{Q}}))=\dfrac{L^{(p)}(E,1)}{\Omega^{+}_{E}}\cdot\omega_{E}$

where $L^{(p)}(E,1)$ is the $L$-value of $E$ at $s=1$ removing the Euler
factor at $p$.

######  Corollary 1.6 (Kato).

If $\mathrm{rk}_{\mathbb{Z}}E(\mathbb{Q})>0$, then $L(E,1)=0$.

###### Proof.

Let $P\in E(\mathbb{Q})$ be a point of infinite order. Under the natural map
$E(\mathbb{Q})\otimes\mathbb{Q}_{p}\to
E(\mathbb{Q}_{p})\otimes\mathbb{Q}_{p}$, the image of $P$ generates
$E(\mathbb{Q}_{p})\otimes\mathbb{Q}_{p}$ Since
$z_{\mathbb{Q}}\in\mathrm{H}^{1}(\mathbb{Q},V)$ and $P$ are global, the Artin
reciprocity law implies that

$\sum_{\ell\leq\infty}\langle\mathrm{loc}_{\ell}(z_{\mathbb{Q}}),P\rangle_{\ell}=0.$

Since $\mathrm{H}^{1}(\mathbb{Q}_{\ell},V)=0$ for every place $\ell\neq p$
(including the infinite place), we have
$\langle\mathrm{loc}_{\ell}(z_{\mathbb{Q}}),P\rangle_{p}=0$. By the self-
orthogonality of $E(\mathbb{Q}_{p})\otimes\mathbb{Q}_{p}$, we have
$\mathrm{loc}_{p}(z_{\mathbb{Q}})\in E(\mathbb{Q}_{p})\otimes\mathbb{Q}_{p}$.
By (1.1), $\mathrm{exp}^{*}\circ\mathrm{loc}_{p}(z_{\mathbb{Q}})=0$. Thus,
$L(E,1)=0$ by Theorem 1.5. ∎

The same result was also obtained by Kolyvagin via the use of Heegner points
[Kol90].

This is the very starting point of Kato’s Euler systems, and the cohomology
class $z_{\mathbb{Q}}$ is just a part of a much deeper object.

###  1.3. The finiteness of Selmer groups

We recall the Selmer groups of $p$-power torsion points
$E[p^{\infty}]=\bigcup_{n\geq 0}E[p^{n}]$ of $E$. Then for $n\geq 1$, the
Selmer group of $E[p^{n}]$ over $\mathbb{Q}$ is defined by

$\mathrm{Sel}(\mathbb{Q},E[p^{n}]):=\mathrm{ker}\left(\mathrm{H}^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q},E[p^{n}])\to\prod_{\ell\in\Sigma}\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{\ell},E[p^{n}])}{E(\mathbb{Q}_{\ell})\otimes\mathbb{Z}/p^{n}\mathbb{Z}}\right)$

and the Selmer group of $E[p^{\infty}]$ over $\mathbb{Q}$ is defined by
$\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])=\varinjlim_{n}\mathrm{Sel}(\mathbb{Q},E[p^{n}])$
via the restriction maps.

######  Exercise 1.7.

Check that the definition of Selmer groups is independent of the choice of
$\Sigma$. (The answer can be found in [Mil06, Cor. I.6.6].)

The Selmer group $\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])$ fits into the
fundamental exact sequence

${0}$${E(\mathbb{Q})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}}$${\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])}$${\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]}$${0,}$

and $\mathbb{Q}$ can be replaced by any algebraic extension of $\mathbb{Q}$.

######  Remark 1.8.

The notion of Selmer groups can be generalized by imposing different local
conditions (instead of the image of the local Kummer map). This variation is
very important for the Euler/Kolyvagin system argument.

Kato’s result on the finiteness of rational points (Corollary 1.6) can be
strengthened including the finiteness of Tate–Shafarevich groups.

######  Theorem 1.9 (Kato).

Let $E$ be an elliptic curve without complex multiplication and $p$ be any
prime.

  1. (1)

If $L(E,1)\neq 0$, then $\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])$ is finite, so
$E(\mathbb{Q})$ and $\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]$ are finite.

  2. (2)

Let $F/\mathbb{Q}$ be a finite abelian extension and
$\chi:\mathrm{Gal}(F/\mathbb{Q})\to\mathbb{C}^{\times}$ be a character. If the
twisted $L$-value $L(E,\chi,1)\neq 0$, then the $\chi$-isotypic components of
$E(F)$ and $\textrm{{SH}}(E/F)[p^{\infty}]$ are finite, i.e.

$\displaystyle E(F)^{\chi}$
$\displaystyle=E(F)\otimes_{\mathbb{Z}[\mathrm{Gal}(F/\mathbb{Q})],\chi}\mathbb{Z}[\chi],$
$\displaystyle\textrm{{SH}}(E/F)[p^{\infty}]^{\chi}$
$\displaystyle=\textrm{{SH}}(E/F)[p^{\infty}]\otimes_{\mathbb{Z}[\mathrm{Gal}(F/\mathbb{Q})],\chi}\mathbb{Z}[\chi]$

are finite where $\mathbb{Z}[\chi]$ is the ring generated by the values of
$\chi$ over $\mathbb{Z}$.

Theorem 1.9.(1) was also obtained by Kolyvagin [Kol90] in the same manner.

We discuss a refined version of Theorem 1.9.(1) for all but finitely many
primes in Theorem 1.14. Theorem 1.9 certainly needs _more than_ one element
$z_{\mathbb{Q}}$ and now the theory of Euler systems comes in more seriously.

######  Remark 1.10.

Euler systems can be viewed as the cohomological shape of $L$-functions. By
definition, Euler systems consist of families of Galois cohomology classes
$z_{\mathbb{Q}(\zeta_{mp^{n}})}$ over abelian extensions of $\mathbb{Q}$ with
norm compatibility. This norm compatibility remember all the Euler factors of
$L$-functions333This is where the name ‘Euler system’ comes from..
Furthermore, Euler systems recover $L$-values via explicit reciprocity laws,
which are very deep results in general. On the other hand, Kolyvagin systems
can be viewed as the cohomological shape of the Taylor expansion of
$L$-functions.

###  1.4. Iwasawa theory for elliptic curves

We briefly introduce Iwasawa theory for elliptic curves _with good ordinary
reduction_ to some extent to illustrate the strength of Kato’s Euler systems.
A more detailed discussion will follow in the second lecture. See also [Maz72,
Gre01, Gre99].

One of the mottoes of Iwasawa theory is the following counter-intuitive
principle:

> “Infinite extensions are easier than finite extensions to deal with.”

The Iwasawa main conjecture444John Coates first called the corresponding
statement for totally real fields the “main conjecture” [Coa77, Coa20]. for
elliptic curves can be viewed as the cyclotomic deformation of the Birch and
Swinnerton-Dyer conjecture.

#### 1.4.1.

Let $p$ be an odd prime and $E$ an elliptic curve with good ordinary reduction
at $p$, i.e. $p\nmid N\cdot a_{p}(E)$. Consider the tower of cyclotomic fields

|
---|---
$\textstyle{\mathbb{Q}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{Z}_{p}\simeq}$$\textstyle{\mathbb{Q}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq\mathbb{Z}/p^{n}\mathbb{Z}}$
and  let
$\Lambda=\mathbb{Z}_{p}\llbracket\mathrm{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q})\rrbracket=\varprojlim_{n}\mathbb{Z}_{p}[\mathrm{Gal}(\mathbb{Q}_{n}/\mathbb{Q})]\simeq\mathbb{Z}_{p}\llbracket
T\rrbracket$ be the cyclotomic Iwasawa algebra, which is isomorphic to the
one-variable power series ring over $\mathbb{Z}_{p}$. $\textstyle{\mathbb{Q}}$

Here, the isomorphism sends a topological generator
$\gamma\in\mathrm{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q})$ to $1+T$.

#### 1.4.2.

We recall the notion of characteristic ideals in (one-variable) Iwasawa
theory. When $M$ be a finitely generated torsion $\Lambda$-module, $M$ is
isomorphic to $\bigoplus_{i}\Lambda/(f_{i})$ where $f_{i}\in\Lambda$ _up to
finite kernel and cokernel_. More details will be given in the second lecture
($\S$2.2). Then the characteristic ideal of $M$ is defined by

$\mathrm{char}_{\Lambda}(M)=\left(\prod_{i}f_{i}\right)\subseteq\Lambda.$

The $p$-adic $L$-function $L_{p}(E)\in\Lambda$ _remembers_ all twisted
$L$-values $L(E,\chi,1)$ by the interpolation formula

$\chi(L_{p}(E))\doteq\tau(\overline{\chi})\cdot\dfrac{L(E,\chi,1)}{\Omega^{\chi(-1)}_{E}}$

for every finite order character
$\chi:\mathrm{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q})\to\mathbb{C}^{\times}$. In
order to describe the interpolation formula more precisely, we need to fix
embeddings
$\overline{\mathbb{Q}}_{p}\hookleftarrow\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$.

#### 1.4.3.

The following non-vanishing result is fundamental [Roh84, Roh88].

######  Theorem 1.11 (Rohrlich).

The $p$-adic $L$-function $L_{p}(E)$ is non-zero.

We now state the Iwasawa main conjecture à la Greenberg–Mazur [Maz72, Gre99].

######  Conjecture 1.12 (Iwasawa main conjecture).

The Selmer group $\mathrm{Sel}(\mathbb{Q}_{\infty},E[p^{\infty}])$ is
$\Lambda$-cotorsion, and

(1.2)
$\left(L_{p}(E)\right)=\mathrm{char}_{\Lambda}(\mathrm{Sel}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee})$

as ideals of $\Lambda$.

By using his Euler systems, Kato proved the one-sided divisibility of
Conjecture 1.12 [Kat04, Thm. 17.4].

######  Theorem 1.13 (Kato).

If the image of $\rho$ contains a conjugate of
$\mathrm{SL}_{2}(\mathbb{Z}_{p})$, then
$\mathrm{Sel}(\mathbb{Q}_{\infty},E[p^{\infty}])$ is $\Lambda$-cotorsion and

(1.3)
$\left(L_{p}(E)\right)\subseteq\mathrm{char}_{\Lambda}(\mathrm{Sel}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee}).$

The opposite divisibility is proved by Skinner–Urban for a large class of
elliptic curves [SU14] (Theorem 2.10). Theorem 1.13 yields an upper bound of
Selmer groups.

######  Theorem 1.14.

Let $E$ be an elliptic curve over $\mathbb{Q}$ without complex multiplication.
Suppose that $L(E,1)\neq 0$. For all but finitely many primes $p$, we have
divisibility

$\\#\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]\Big{|}\dfrac{L(E,1)}{\Omega^{+}_{E}}.$

#### 1.4.4.

We explain how a proof of Theorem 1.14 goes following [Rub00, Thm. 3.5.11 and
Cor. 3.5.19]. By Serre’s open image theorem [Ser72], we have the implication

> $E$ has no CM $\Rightarrow$
> $\rho_{E,p}:G_{\mathbb{Q}}\to\mathrm{Aut}_{\mathbb{Z}_{p}}(\mathrm{Ta}_{p}E)$
> is surjective for $p\gg 0$.

In addition, we remove the following finitely many primes:

  1. (1)

all bad reduction primes,

  2. (2)

all primes dividing Euler factors at $s=1$ at bad reduction primes,

  3. (3)

2,

  4. (4)

all the primes dividing some constant $c_{E}$ (independent of $p$) satisfying

$``\mathrm{exp}^{*}\circ\mathrm{loc}_{p}z_{\mathbb{Q}}=c_{E}\cdot\dfrac{L^{(p)}(E,1)}{\Omega^{+}_{E}}",$

and

  5. (5)

some constant $M_{E}$ such that if $E$ has good ordinary reduction at $p$ and
$p$ divides $M_{E}$ then
$\mathrm{Sel}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee}$ has non-trivial
finite $\Lambda$-submodule. For example, if $p$ does not divide
$\prod_{q|N}\\#E(\mathbb{Q}_{q})_{\mathrm{tors}}$, then
$\mathrm{Sel}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee}$ has no non-trivial
finite $\Lambda$-submodule [Rub00, Cor. 3.5.18].

When $p$ does not divide $\\#E(\mathbb{F}_{p})$, the standard Euler system
argument over $\mathbb{Q}$ is enough. When $p$ divides $\\#E(\mathbb{F}_{p})$,
we need to work over $\mathbb{Q}_{\infty}$. By using Theorem 1.13 and the non-
existence of non-trivial finite $\Lambda$-submodules, Kato’s divisibility
(1.3) implies that

$\\#\left|\mathrm{Sel}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\mathrm{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q})}\right|\textrm{
divides }\textbf{1}(L_{p}(E)).$

The conclusion follows from the interpolation formula of the $p$-adic
$L$-function (2.1) and Mazur’s control theorem (Proposition 2.7).

#### 1.4.5.

We have the following application.

######  Corollary 1.15.

Let $E$ be an elliptic curve over $\mathbb{Q}$ without complex multiplication.
If $L(E,1)\neq 0$, then $E(\mathbb{Q})$ and $\textrm{{SH}}(E/\mathbb{Q})$ are
finite.

###### Proof.

By Theorem 1.9.(1), $E(\mathbb{Q})$ is finite and
$\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]$ is finite for every prime $p$. By
Theorem 1.14.(1), $\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]$ is trivial for all
but finitely many prime $p$. ∎

From Theorem 1.9.(2), we also deduce the following corollary.

######  Corollary 1.16.

Let $E$ be an elliptic curve over $\mathbb{Q}$ without complex multiplication.
Then $E(\mathbb{Q}_{\infty})$ is (still!) finitely generated.

We can understand Corollary 1.16 as a behavior of the “function $E$”; in other
words, the assignment

$\mathbb{Q}_{n}\mapsto E(\mathbb{Q}_{n})$

does not grow too fast. This growth behavior shows the philosophy of Iwasawa
theory very well.

#### 1.4.6.

How does Kato’s Euler system play the role in Theorem 1.13? Let

$z_{\mathbb{Q}_{\infty}}=\varprojlim_{n}z_{\mathbb{Q}_{n}}\in\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q},T)=\varprojlim_{n}\mathrm{H}^{1}(\mathbb{Q}_{n},T)$

be Beilinson–Kato’s zeta element over $\mathbb{Q}_{\infty}$ where the inverse
limit is taken with respect to the corestriction, and the norm compatibility
$\varprojlim_{n}z_{\mathbb{Q}_{n}}$ is a part of the Euler system relation.
The punchline is that we can realize the $p$-adic $L$-function $L_{p}(E)$ by
using $z_{\mathbb{Q}_{\infty}}$. More precisely, there exists a Coleman map

$\mathrm{Col}:\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T)\to\Lambda$

sending $\mathrm{loc}_{p}(z_{\mathbb{Q}_{\infty}})$ to $L_{p}(E)$. If $E$ has
good supersingular reduction, the target of the Coleman map becomes larger
than $\Lambda$, but the signed Coleman maps work as in the ordinary case. By
using the Coleman map, we have the equivalence between

  * •

Kato’s main conjecture for elliptic curves (Conjecture 2.19), and

  * •

Mazur–Greenberg’s main conjecture for elliptic curves with good ordinary
reduction (Conjecture 1.12).

There is no assumption on the reduction type in Kato’s formulation. We discuss
this equivalence in the second lecture ($\S$2.5).

The notion of Coleman map is purely local and the generalizations of the
Coleman map is an important part of $p$-adic Hodge theory. See [PR94, Nak14]
for example.

###  1.5. Refining the Euler system argument

In the Euler system argument [Rub00], the _derivative process_ plays the
crucial role to produce the _ramified_ classes over $\mathbb{Q}$ from the
unramified cohomology classes over ramified abelian extensions of
$\mathbb{Q}$. We can bound Selmer groups with these ramified classes via the
global Poitou–Tate duality. A more careful organization of the derivative
process leads to Mazur–Rubin’s theory of Kolyvagin systems [MR04], which has
the following advantages:

  1. (1)

the theory of Kolyvagin systems provides a mod $p$ numerical criterion to have
equality, not divisibility. See [KKS20].

  2. (2)

the theory of Kolyvagin systems provides the structure theorem of Selmer
groups. See also [Kol91].

In Mazur’s diagram below

$\textstyle{{\begin{subarray}{c}\textrm{$L$-values or}\\\ \textrm{derivatives
of
$L$-values}\end{subarray}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$DIFFICULT$\textstyle{{\begin{subarray}{c}\textrm{algebraic
treasures}\\\ \textrm{(Euler
systems)}\end{subarray}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\begin{subarray}{c}\textrm{explicit
reciprocity law/}\\\ \textrm{Gross--Zagier type
formula}\end{subarray}}}$derivative process
$+\epsilon$$\scriptstyle{(1)}$$\textstyle{{\begin{subarray}{c}\textrm{arithmetically
interesting}\\\ \textrm{modules (Selmer groups, class groups,
$\cdots$)}\end{subarray}}}$$\textstyle{\textrm{Kolyvagin
systems},\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\begin{subarray}{c}\textrm{control
by}\\\ \textrm{global duality}\end{subarray}}}$$\scriptstyle{(2)}$

the mod $p$ criterion (1) concerns the surjectivity of the map from Euler
systems to Kolyvagin systems and the structure theorem (2) is a refined
version of the global duality argument.

###  1.6. Where do Euler systems come from?

One big question is:

######  Question 1.17.

Where do Euler systems come from?

This seems the most non-trivial question in the theory of Euler systems. There
was a handful of Euler systems a while ago (e.g. [BCD+14]), but a general
strategy to construct Euler systems has been developed thanks to the series of
the important works of Loeffler–Zerbes and their collaborators. See [LZ23] for
the progress of Loeffler–Zerbes’ program on the construction of Euler systems
for automorphic Galois representations. Also, M. Sangiovanni and C. Skinner
have recently announced another general strategy to construct Euler systems.

###  1.7. A very2 rough picture

In order to describe ${}_{c,d}z_{m}(f,r,r^{\prime},\xi,S)$, we draw the
following very rough picture. We hope it provides a guideline to read [Kat04].

$\textstyle{\textrm{2 Siegel units
}{}_{c}g_{\alpha,\beta},{}_{d}g_{\gamma,\delta}\in\mathcal{O}\left({\begin{subarray}{c}\textrm{an
open modular curve}\\\ \textrm{with certain level
structure}\end{subarray}}\right)^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\begin{subarray}{c}\textrm{pairing
via Steinberg symbols}\\\ \textrm{$+$ certain
twist}\end{subarray}}}$$\textstyle{K_{2}(Y_{1}(N)_{\mathbb{Q}(\zeta_{m})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\begin{subarray}{c}\textrm{Chern
character}\\\ \textrm{\cite[cite]{[\@@bibref{}{kato-euler-systems}{}{},
(8.4.3)]}}\end{subarray}}}$$\textstyle{{\begin{subarray}{c}{}_{c}\theta_{E},{}_{d}\theta_{E}\\\
\textrm{certain functions on}\\\ \textrm{the universal elliptic
curve}\end{subarray}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\mathrm{dlog}$
involved$\textstyle{\mathrm{H}^{2}(Y_{1}(N)_{\mathbb{Q}(\zeta_{m})},\mathbb{Z}_{p}(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\begin{subarray}{c}\textrm{Hochschild
--Serre}\\\ \textrm{spectral sequence}\\\
\textrm{(HSSS)}\end{subarray}}}$$\textstyle{\mathrm{H}^{1}(\mathbb{Q}(\zeta_{m}),\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(Y_{1}(N)_{\overline{\mathbb{Q}}},\mathbb{Z}_{p}(2)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(\mathbb{Q}(\zeta_{m}),\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(Y_{1}(N)_{\overline{\mathbb{Q}}},\mathbb{Z}_{p}(1))(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$“$\otimes(\zeta_{p^{n}})^{\otimes-1}$”$\textstyle{\mathrm{H}^{1}(\mathbb{Q}(\zeta_{m}),\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(Y_{1}(N)_{\overline{\mathbb{Q}}},\mathbb{Z}_{p}(1)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textrm{zeta
modular forms}\in\textrm{the space of modular forms}}$

Here, the zeta modular form can be viewed as a product of two weight 1
Eisenstein series. The projection to an elliptic curve $E$ yields the
$L$-value formula for $E$ via the Rankin–Selberg method. When we project the
zeta modular form to $E$, we need to invert $p$, so the integrality issue
becomes subtle in general [Kat04, $\S$6.3]. When we use the Chern character
map, the parameters $(r,r^{\prime})$ in ${}_{c,d}z_{m}(f,r,r^{\prime},\xi,S)$
are determined. (Since we deal with elliptic curves only here, $f$ is a weight
2 modular form.) We will elaborate this picture more in the last lecture.

##  2\. Lecture 2: Cyclotomic Iwasawa theory for elliptic curves

Let $\mathbb{Q}_{\infty}$ be the cyclotomic $\mathbb{Z}_{p}$-extension of
$\mathbb{Q}$. The following picture shows how Kato’s Euler systems, Kato’s
Kolyvagin systems, and the Iwasawa main conjecture are related.

$p$-adic $L$-functionsIwasawa main conjecture (with $p$-adic
$L$-functions)$\textstyle{{\begin{subarray}{c}\textrm{Selmer groups}\\\
\textrm{over
}\mathbb{Q}_{\infty}\end{subarray}}}$$\textstyle{{\begin{subarray}{c}\textrm{modular
symbols/$L$-values}\\\
\textrm{(``Betti")}\end{subarray}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\begin{subarray}{c}\textrm{Stickelberger
type}\\\
\textrm{construction}\end{subarray}}}$$\textstyle{{\begin{subarray}{c}\textrm{Kato's
Euler systems}\\\
\textrm{(``\'{e}tale")}\end{subarray}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{exp}^{*}}$$\scriptstyle{{\begin{subarray}{c}\textrm{derivative}\\\
\textrm{construction}\end{subarray}}}$Coleman mapKato’s Kolyvagin
systems$\scriptstyle{{\begin{subarray}{c}\textrm{control via}\\\
\textrm{global
duality}\end{subarray}}}$$\textstyle{{\begin{subarray}{c}\textrm{Selmer
groups}\\\ \textrm{over
}\mathbb{Q}\end{subarray}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varinjlim}$

The goal of the second lecture is to explain some part of the above picture in
detail. One may ask that the “de Rham” aspect is missing in this picture. This
nature is more or less hidden in the dual exponential map since the
Rankin–Selberg method is used in the proof of Kato’s zeta value formula
[Kat04, $\S$7]. See also §2.4.

###  2.1. $p$-adic $L$-functions and modular symbols: Betti construction

We review the modular symbol construction of $p$-adic $L$-functions following
[MTT86, Pol14, Bel21].

####  2.1.1. $L$-values

How to extract $L$-values from (modular) elliptic curves?

Let $f_{E}(z)=\sum_{n\geq 1}a_{n}\cdot q^{n}$ where $q=e^{2\pi iz}$. We can
express the $L$-values in terms of integrals _very simply if we ignore the
convergence issue_. Recall the $\Gamma$-function

$\displaystyle\Gamma(s)$ $\displaystyle=\int^{\infty}_{0}e^{-x}\cdot
x^{s-1}dx$ $\displaystyle=(2\pi i)^{s}\cdot\int^{\infty}_{0}e^{-2\pi nx}\cdot
x^{s-1}dx.$

Then

$\displaystyle L(E,s)\cdot\Gamma(s)$ $\displaystyle=\sum_{n\geq
1}a_{n}\cdot\dfrac{\Gamma(s)}{n^{s}}$
$\displaystyle=(2\pi)^{s}\cdot\int^{\infty}_{0}\left(\sum_{n\geq 1}a_{n}\cdot
e^{2\pi nx}\right)\cdot x^{s-1}dx.$

and $f(ix)=\sum_{n\geq 1}a_{n}\cdot e^{2\pi nx}$. Therefore, we have

$\displaystyle L(E,1)$ $\displaystyle=2\pi\cdot\int^{\infty}_{0}f(iy)dy$
$\displaystyle=2\pi i\cdot\int^{0}_{i\infty}f(z)dz.$

######  Exercise 2.1.

Where did we ignore the convergence issue?

####  2.1.2. Twists

Let $\chi:(\mathbb{Z}/d\mathbb{Z})^{\times}\to\mathbb{C}^{\times}$ be a
Dirichlet character of conductor $d$ with $(d,N)=1$. Then how can we express
the twisted $L$-value $L(E,\chi,1)$ in terms of integrals? Consider

$\tau(n,\chi)=\sum_{a\in(\mathbb{Z}/n\mathbb{Z})^{\times}}\chi(a)\cdot e^{2\pi
in\cdot(a/d)}$

where the sum should be viewed as the sum over $a=1,\cdots,d$ with $(a,d)=1$.
Then $\tau(\chi)=\tau(1,\chi)$ is the Gauss sum and
$\tau(n,\chi)=\overline{\chi(n)}\cdot\tau(\chi)$. Consider the twisted modular
form

$\displaystyle f_{\overline{\chi}}(z)$ $\displaystyle=\sum_{n\geq
1}\overline{\chi}(n)\cdot a_{n}\cdot e^{2\pi inz}$
$\displaystyle=\dfrac{1}{\tau(\chi)}\cdot\sum_{n\geq 1}\tau(n,\chi)\cdot
a_{n}\cdot e^{2\pi inz}$ $\displaystyle=\dfrac{1}{\tau(\chi)}\cdot\sum_{n\geq
1}\sum_{a\in(\mathbb{Z}/n\mathbb{Z})^{\times}}\chi(a)\cdot a_{n}\cdot e^{2\pi
in(z+a/d)}.$

By Birch’s lemma [MTT86, (8.3)], we have

$\displaystyle L(E,\overline{\chi},1)$ $\displaystyle=\dfrac{2\pi
i}{\tau(\chi)}\cdot\sum_{a\in(\mathbb{Z}/n\mathbb{Z})^{\times}}\chi(a)\cdot\int^{a/d}_{i\infty}f(z)dz$
$\displaystyle=\dfrac{\tau(\overline{\chi})}{d}\cdot\sum_{a\in(\mathbb{Z}/n\mathbb{Z})^{\times}}\chi(a)\cdot
2\pi i\int^{-a/d}_{i\infty}f(z)dz$

These computations lead to the following concept. Let
$X_{0}(N)(\mathbb{C})=\Gamma_{0}(N)\backslash(\mathfrak{h}\cup\mathbb{P}^{1}(\mathbb{Q}))$
be the complex points of compactified modular curve $X_{0}(N)$. For a pair
$\alpha,\beta\in\mathbb{P}^{1}(\mathbb{Q}),$ we define

$\left\\{\alpha,\beta\right\\}_{f}:=2\pi i\cdot\int^{\beta}_{\alpha}f(z)dz$

and it is called a (non-normalized) modular symbol.

####  2.1.3. Modular elements of Mazur–Tate

Fix an isomorphism

$(\mathbb{Z}/d\mathbb{Z})^{\times}\simeq\mathrm{Gal}(\mathbb{Q}(\zeta_{d})/\mathbb{Q})$

by sending $a$ to $(\sigma_{a}:\zeta_{d}\mapsto\zeta^{a}_{d})$. Let
$\chi:(\mathbb{Z}/d\mathbb{Z})^{\times}\to\mathbb{C}^{\times}$ be a Dirichlet
character and we extend it linearly

$\chi:\mathbb{C}[\mathrm{Gal}(\mathbb{Q}(\zeta_{d})/\mathbb{Q})]\to\mathbb{C}$

by sending $\sigma\in\mathrm{Gal}(\mathbb{Q}(\zeta_{d})/\mathbb{Q})$ to
$\chi(\sigma)$. Under this map, we have

$\sum_{a\in(\mathbb{Z}/d\mathbb{Z})^{\times}}\left\\{i\infty,a/d\right\\}_{f}\cdot\sigma_{a}\mapsto\tau(\chi)\cdot
L(E,\overline{\chi},1).$

In other words, (non-normalized) Mazur–Tate elements know twisted $L$-values.

####  2.1.4. Some hints on $p$-adic $L$-functions

Let $\chi:(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}\to\mathbb{C}^{\times}$ be a
Dirichlet character of modulus $p^{n}$. Then we have

$L(E,\overline{\chi},1)=\dfrac{2\pi
i}{\tau(\chi)}\cdot\sum_{a\in(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}}\chi(a)\cdot\int^{a/p^{n}}_{i\infty}f(z)dz$

and the summation looks like a Riemann sum since

$\mathbb{Z}^{\times}_{p}=\coprod_{a\in(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}}(a+p^{n}\mathbb{Z}_{p}),$

if $\chi$ is a locally constant function on $\mathbb{Z}^{\times}_{p}$, then
the twisted $L$-value can be understood as a $p$-adic integration.

######  Remark 2.2.

Where does the distribution relation come from? Hecke operators!

####  2.1.5. Normalizations

Write $\\{r\\}_{f}=\\{i\infty,r\\}_{f}$ for $r\in\mathbb{P}^{1}(\mathbb{Q})$
and then we symmetrize integrals:

$\\{r\\}^{\pm}_{f}=2\pi
i\cdot\left(\int^{r}_{i\infty}f(z)dz\pm\int^{-r}_{i\infty}f(z)dz\right)$

We recall the algebraicity result of Manin and Shimura.

######  Theorem 2.3 (Manin, Shimura).

There exist a pair of complex periods $\Omega^{\pm}_{f}\in\mathbb{C}^{\times}$
such that

$\\{r\\}^{\pm}_{f}\in\mathbb{Q}_{f}\cdot\Omega^{\pm}_{f}$

where $\mathbb{Q}_{f}=\mathbb{Q}(a_{n}(f);n)$ is the number field generated by
Fourier coefficients of $f$.

Write

$[r]^{\pm}_{f}=\\{r\\}^{\pm}_{f}/\Omega^{\pm}_{f}\in\mathbb{Q}_{f}\subseteq\overline{\mathbb{Q}}$

and

$[r]_{f}=[r]^{+}_{f}+[r]^{-}_{f}.$

####  2.1.6. Hecke action and distribution relations

For $\alpha,\beta\in\mathbb{P}^{1}(\mathbb{Q})$, consider the homology class
$\\{\alpha,\beta\\}\in\mathrm{H}_{1}(X_{0}(N),\mathbb{Q})$. Here, having the
$\mathbb{Q}$-coefficients in the homology is due to Manin [Man72]. For a prime
$\ell$ not dividing $N$, we have

$T_{\ell}\\{\alpha,\beta\\}=\\{\ell\alpha,\ell\beta\\}+\sum^{\ell-1}_{r=0}\\{\dfrac{\alpha+r}{\ell},\dfrac{\beta+r}{\ell}\\}.$

When $\ell=p$, we want to remove $\\{p\alpha,p\beta\\}$ in the above sum in
order to have the correct Riemann sum relation. This is why we need the
$p$-stabilization process.

Let

$\theta_{\mathbb{Q}(\zeta_{M})}=\sum_{a\in(\mathbb{Z}/M\mathbb{Z})^{\times}}[a/M]_{f}\cdot\sigma_{a}\in\mathbb{Q}[\mathrm{Gal}(\mathbb{Q}(\zeta_{M})/\mathbb{Q})]$

be the modular element, which is the Stickelberger element for elliptic
curves. Varying $M$, we have two natural maps among them. Let

$\pi:\mathbb{Q}[\mathrm{Gal}(\mathbb{Q}(\zeta_{M\ell})/\mathbb{Q})]\to\mathbb{Q}[\mathrm{Gal}(\mathbb{Q}(\zeta_{M})/\mathbb{Q})]$

be the natural projection and

$\nu:\mathbb{Q}[\mathrm{Gal}(\mathbb{Q}(\zeta_{M})/\mathbb{Q})]\to\mathbb{Q}[\mathrm{Gal}(\mathbb{Q}(\zeta_{M\ell})/\mathbb{Q})]$

be the norm map sending $\sigma\in\mathbb{Q}(\zeta_{M})/\mathbb{Q})$ to the
sum of liftings $\sum_{\pi(\tau)=\sigma}\tau$.

######  Proposition 2.4.

$\pi\left(\theta_{\mathbb{Q}(\zeta_{M})}\right)=\left\\{\begin{array}[]{ll}a_{\ell}\cdot\theta_{\mathbb{Q}(\zeta_{M})}-\nu\left(\theta_{\mathbb{Q}(\zeta_{M/\ell})}\right)&\textrm{
if }\ell|M,\\\
(a_{\ell}-\sigma_{\ell}-\sigma^{-1}_{\ell})\cdot\theta_{\mathbb{Q}(\zeta_{M})}&\textrm{
if }\ell\not|M.\end{array}\right.$

###### Proof.

We left it to the reader as an exercise. (Hint: use Hecke action.) ∎

In favorable situations, the modular elements are integral with Néron periods.

######  Remark 2.5.

If $E[p]$ is irreducible and the Manin constant is prime to $p$, then

$\theta_{\mathbb{Q}(\zeta_{p^{n}})}\in\mathbb{Z}_{p}[\mathrm{Gal}(\mathbb{Q}(\zeta_{p^{n}})/\mathbb{Q})]$

for every $n\geq 1$.

####  2.1.7. $p$-adic $L$-functions

We modify the three term relation (the first one in Proposition 2.4 with
$\ell=p$) of modular elements in order to have the norm compatible sequence.
This is where the $p$-stabilization plays the role. Take $\alpha$ a root of
$X^{2}-a_{p}X+p=0$ and put

$\theta^{\alpha}_{\mathbb{Q}(\zeta_{p^{n}})}=\dfrac{1}{\alpha^{n}}\cdot\left(\theta_{\mathbb{Q}(\zeta_{p^{n}})}-\dfrac{1}{\alpha}\cdot\theta_{\mathbb{Q}(\zeta_{p^{n-1}})}\right).$

Then $\left\\{\theta^{\alpha}_{\mathbb{Q}(\zeta_{p^{n}})}\right\\}$ forms a
projective system, i.e. for $n\geq 1$, we have

$\pi\left(\theta^{\alpha}_{\mathbb{Q}(\zeta_{p^{n+1}})}\right)=\theta^{\alpha}_{\mathbb{Q}(\zeta_{p^{n}})}.$

Suppose that $\alpha$ is a $p$-adic unit (i.e. $E$ has good ordinary reduction
at $p$). Then the $p$-adic $L$-function $L_{p}(E)$ of $E$ is defined by

$\textstyle{\varprojlim_{n}\mathbb{Z}_{p}\llbracket\mathrm{Gal}(\mathbb{Q}(\zeta_{p^{n}})/\mathbb{Q})\rrbracket\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}_{p}\llbracket\mathrm{Gal}(\mathbb{Q}(\zeta_{p^{\infty}})/\mathbb{Q})\rrbracket\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}_{p}\llbracket\mathrm{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q})\rrbracket}$$\textstyle{\varprojlim_{n}\theta^{\alpha}_{\mathbb{Q}(\zeta_{p^{n}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\theta^{\alpha}_{\mathbb{Q}(\zeta_{p^{\infty}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L_{p}(E).}$

We have the following interpolation formulas:

(2.1)
$\displaystyle\begin{split}\mathbf{1}\left(\theta^{\alpha}_{\mathbb{Q}(\zeta_{p^{\infty}})}\right)&=\left(1-\dfrac{1}{\alpha}\right)^{2}\cdot\theta_{\mathbb{Q}}\\\
&=\left(1-\dfrac{1}{\alpha}\right)^{2}\cdot\dfrac{L(E,1)}{\Omega^{+}_{E}}.\end{split}$

If $\chi$ is a Dirichlet character of conductor $p^{n}$ ($\neq$ 1), then

$\chi\left(\theta^{\alpha}_{\mathbb{Q}(\zeta_{p^{\infty}})}\right)=\dfrac{\tau(\chi)}{\alpha^{n}}\cdot\dfrac{L(E,\overline{\chi},1)}{\Omega^{\chi(-1)}_{E}}.$

These interpolation formulas also characterize
$\theta^{\alpha}_{\mathbb{Q}(\zeta_{p^{\infty}})}$555This is related to being
of the non-critical slope..

###  2.2. Selmer groups over $\mathbb{Q}_{\infty}$ and the main conjecture

#### 2.2.1.

Let $M$ be any finitely generated torsion $\Lambda$-module. Then there exists
a $\Lambda$-homomorphism

$M\to\Lambda/f^{e_{1}}_{1}\oplus\cdots\oplus\Lambda/f^{e_{r}}_{r}$

with finite kernel and cokernel. The map between finitely generated
$\Lambda$-modules with finite kernel and cokernel is called a pseudo-
isomorphism. Then $(f^{e_{1}}_{1}),\cdots,(f^{e_{r}}_{r})$ are uniquely
determined by $M$, and the characteristic ideal of $M$ is defined by

$\mathrm{char}_{\Lambda}(M)=\left(\prod^{r}_{i=1}f^{e_{i}}_{i}\right)\subseteq\Lambda.$

This notion of characteristic ideals is an analogue of characteristic
polynomials for linear algebra over $\Lambda$. See [Was97, $\S$13.2] for the
basic of linear algebra over $\Lambda$.

######  Theorem 2.6 (Kato).

Assume $E$ has good ordinary reduction at $p$. Then
$\mathrm{Sel}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee}=\mathrm{Hom}_{\mathbb{Z}_{p}}(\mathrm{Sel}(\mathbb{Q}_{\infty},E[p^{\infty}]),\mathbb{Q}_{p}/\mathbb{Z}_{p})$
is a finitely generated torsion $\Lambda$-module.

###### Proof.

This follows from Theorem 1.11 and Theorem 1.13. ∎

We recall Mazur’s control theorem [Maz72, Gre99].

######  Proposition 2.7 (Mazur).

Assume $E$ has good ordinary reduction at $p$. Then the natural restriction
map

$\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])\to\mathrm{Sel}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\Gamma}$

has finite kernel and cokernel where
$\Gamma=\mathrm{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q})$.

#### 2.2.2.

We recall Conjecture 1.12 and Theorem 1.13.

######  Conjecture 2.8 (Mazur, Greenberg).

Assume $E$ has good ordinary reduction at $p$. Then

$(L_{p}(E))=\mathrm{char}_{\Lambda}(\mathrm{Sel}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee})$

as ideals of $\Lambda$.

######  Theorem 2.9 (Kato).

Assume $E$ has good ordinary reduction at $p$. If the image of $\rho$ contains
a conjugate of $\mathrm{SL}_{2}(\mathbb{Z}_{p})$, then

$(L_{p}(E))\subseteq\mathrm{char}_{\Lambda}(\mathrm{Sel}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee})$

as ideals of $\Lambda$. Without the image assumption, we still have the same
divisibility as ideals of $\Lambda\otimes\mathbb{Q}_{p}$.

By using a completely different idea (Eisenstein congruences), Skinner–Urban
proved the opposite divisibility [SU14].

######  Theorem 2.10 (Skinner–Urban).

Assume $E$ has good ordinary reduction at $p$. If $E[p]$ is irreducible and
there exists a prime $\ell$ exactly dividing $N$ such that $E[p]$ is ramified
at $\ell$, then

$(L_{p}(E))\supseteq\mathrm{char}_{\Lambda}(\mathrm{Sel}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee})$

as ideals of $\Lambda$.

Therefore, Conjecture 1.12 is confirmed for a large class of elliptic curves
with good ordinary reduction, and we now illustrate its application to Birch
and Swinnerton-Dyer conjecture.

$\displaystyle L(E,1)=0$ $\displaystyle\Leftrightarrow\mathbf{1}(L_{p}(E))=0$
(2.1)
$\displaystyle\Leftrightarrow\mathrm{length}\left(\mathrm{Sel}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee}\right)_{\Gamma}=\infty$
(1.2)
$\displaystyle\Leftrightarrow\mathrm{length}\left(\mathrm{Sel}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\Gamma}\right)^{\vee}=\infty$
$\displaystyle\Leftrightarrow\mathrm{length}\left(\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])\right)^{\vee}=\infty$
Prop. 2.7

This equivalence strengthens Theorem 1.9 to the if and only if statement.

######  Remark 2.11.

We actually need the Iwasawa main conjecture _localized at the augmentation
ideal_ only. This type of idea generalizes the Selmer rank one case, which is
called the rank one $p$-converse to the theorem of Gross–Zagier and Kolyvagin.
See [Kim23] for example.

#### 2.2.3.

We briefly discuss the $p$-adic Birch and Swinnerton-Dyer conjecture following
[MTT86], [Kat04, $\S$18], and [Kat02].

######  Conjecture 2.12 ($p$-adic BSD).

Assume $E$ has good ordinary reduction at $p$. Then

$\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])=\mathrm{ord}_{\chi=\mathbf{1}}L_{p}(E).$

Kato proved the following inequality which bounds Selmer ranks (so
Mordell–Weil ranks) [Kat04, Thm. 18.4].

######  Corollary 2.13 (Kato).

$\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])\leq\mathrm{ord}_{\chi=\mathbf{1}}L_{p}(E).$

In fact, the formulation and the result both extend to the finite slope case,
but the exceptional zero occurs for the multiplicative reduction case. Unlike
the results on the classical Birch and Swinnerton-Dyer conjecture, this result
does not have any low rank restriction.

###  2.3. From Kato’s zeta elements to modular elements: étale construction I

In this subsection, we study the dotted arrow in the diagram below

Kato’s zeta
elements$\scriptstyle{\mathrm{exp}^{*}}$$\scriptstyle{\mathrm{Col}}$$p$-adic
$L$-functionsmodular symbols/$L$-valuesStickelbergermodular
elements$\scriptstyle{\textrm{$p$-stabilization}+\varprojlim}$

following the work of Kurihara [Kur02].

#### 2.3.1.

We discuss a little bit of $p$-adic Hodge theory. Fix a compatible system
$(\zeta_{p^{n}})_{n}\in\mathbb{Z}_{p}(1)$ of primitive $p$-power roots of
unity. Let
$D=\mathbf{D}_{\mathrm{dR}}(V)=(V\otimes\mathbf{B}_{\mathrm{dR}})^{G_{\mathbb{Q}_{p}}}$
be the filtered $\varphi$-module of rank two associated to $V$ and
$D^{0}=\mathbf{D}^{0}_{\mathrm{dR}}(V)=\mathbb{Q}_{p}\cdot\omega_{E}\subseteq
D$ the $\varphi$-stable submodule of rank one where $\mathbf{B}_{\mathrm{dR}}$
is the de Rham period ring and $\omega_{E}$ is the invariant Néron
differential of $E$. Then $D^{0}$ is the cotangent space of $E$ (at the
identity) and $D/D^{0}$ is the tangent space of $E$. Furthermore, the map

$\mathrm{exp}:D\otimes\mathbb{Q}_{p}(\zeta_{p^{n+1}})\to
D/D^{0}\otimes\mathbb{Q}_{p}(\zeta_{p^{n+1}})\to\mathrm{H}^{1}(\mathbb{Q}_{p}(\zeta_{p^{n+1}}),V)$

extends the formal exponential map

$\widehat{\mathbb{G}}_{a}(\mathbb{Q}_{p}(\zeta_{p^{n+1}}))\otimes\mathbb{Q}_{p}\to\widehat{E}(\mathbb{Q}_{p}(\zeta_{p^{n+1}}))\otimes\mathbb{Q}_{p}\to\mathrm{H}^{1}(\mathbb{Q}_{p}(\zeta_{p^{n+1}}),V)$

where the last map is the Kummer map.

#### 2.3.2.

Consider an element

$\displaystyle\gamma_{n}$
$\displaystyle=\sum^{n}_{i=0}\varphi^{i+1}(\omega_{E})\otimes\zeta_{n+1-i}+(1-\varphi)^{-1}\varphi^{n+2}(\omega_{E})$
$\displaystyle\in D\otimes\mathbb{Q}_{p}(\zeta_{p^{n+1}})$

and a homomorphism

$P_{n}:\mathrm{H}^{1}(\mathbb{Q}_{p}(\zeta_{p^{n+1}}),T)\to\mathbb{Q}_{p}[G]=\mathbb{Q}_{p}[\mathrm{Gal}(\mathbb{Q}_{p}(\zeta_{p^{n+1}})/\mathbb{Q}_{p})]\simeq\mathbb{Q}_{p}[\mathrm{Gal}(\mathbb{Q}(\zeta_{p^{n+1}})/\mathbb{Q})]$

sending $a$ to

$\dfrac{1}{[\varphi(\omega_{E}),\omega_{E}]_{\mathrm{dR}}}\cdot\sum_{\sigma\in
G}\langle\mathrm{exp}(\gamma^{\sigma}_{n}),a\rangle_{p}\cdot\sigma$

where

$\textstyle{\gamma_{n}\in
D\otimes\mathbb{Q}_{p}(\zeta_{p^{n+1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{exp}}$$\textstyle{\times}$$\textstyle{D\otimes\mathbb{Q}_{p}(\zeta_{p^{n+1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[-,-]_{\mathrm{dR}}}$$\textstyle{\mathbb{Q}_{p}(\zeta_{p^{n+1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Tr}}$$\textstyle{\mathrm{exp}(\gamma_{n})\in\mathrm{H}^{1}(\mathbb{Q}_{p}(\zeta_{p^{n+1}}),V)}$$\textstyle{\times}$$\textstyle{a\in\mathrm{H}^{1}(\mathbb{Q}_{p}(\zeta_{p^{n+1}}),V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{exp}^{*}}$$\scriptstyle{\langle-,-\rangle_{p}}$$\textstyle{\mathbb{Q}_{p}.}$

######  Proposition 2.14.

If $E[p]$ is irreducible, then

$\mathrm{Im}(P_{n})\subseteq\mathbb{Z}_{p}[\mathrm{Gal}(\mathbb{Q}(\zeta_{p^{n+1}})/\mathbb{Q})].$

######  Theorem 2.15 (Kurihara).

Suppose that $E[p]$ is irreducible. Then

$P_{n}(z_{\mathbb{Q}(\zeta_{p^{n+1}})})=\theta_{\mathbb{Q}(\zeta_{p^{n+1}})}$

and $\theta_{\mathbb{Q}_{n}}$ is the image of
$P_{n}(z_{\mathbb{Q}(\zeta_{p^{n+1}})})$ in
$\mathbb{Z}_{p}[\mathrm{Gal}(\mathbb{Q}_{n}/\mathbb{Q})]$.

The above theorem should be considered as the _étale_ construction of
Mazur–Tate’s modular elements.

###  2.4. Coleman maps: étale construction II

Various constructions of $p$-adic $L$-functions have similarities with the
realizations of motives666I first learned this picture from Shanwen Wang’s
talk at Sendai International Conference on Arithmetic Geometry in 2016.

---
Kato’s zeta elements (étale)Coleman mapselliptic curvesmodular symbols
(Betti)$\S$2.1$p$-adic $L$-functionsRankin–Selberg method (de Rham)

and the Coleman maps play the central role in the étale construction. The
Rankin–Selberg method construction of $p$-adic $L$-functions can be found in
the work of Hida, Panchishkin, and T. Fukaya [Hid81, Pan03, Fuk03].

Let
$\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T)=\varprojlim_{n}\mathrm{H}^{1}(\mathbb{Q}_{n,p},T)$
be the local Iwasawa cohomology of $T$.

####  2.4.1. The ordinary case

Assume that $E$ has good ordinary reduction at $p$.

######  Theorem 2.16.

There exists the Coleman map

$\mathrm{Col}:\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T)\to\Lambda$

satisfying $\mathrm{Col}(z_{\mathbb{Q}_{\infty}})=L_{p}(E)$.

###### Proof.

See [Rub98, Appendix] for the explicit construction. ∎

####  2.4.2. The supersingular case

Assume that $E$ has good supersingular reduction at $p\geq 5$. See Sprung
[Spr12] for the $p\leq 3$ case.

######  Theorem 2.17 (Kobayashi).

There exists the signed Coleman maps

$\mathrm{Col}^{\pm}:\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T)\to\Lambda$

satisfying $\mathrm{Col}^{\pm}(z_{\mathbb{Q}_{\infty}})=L^{\pm}_{p}(E)$

###### Proof.

See [Kob03]. ∎

####  2.4.3. Remarks on the construction

In order to construct Coleman maps for elliptic curves (or modular abelian
varieties), we use explicit local points in the formal groups like
$\gamma_{n}$ in Kurihara’s construction of modular elements. For the case of
higher weight modular forms, we certainly need $p$-adic Hodge theory following
Perrin-Riou’s strategy [PR94]. See [Kat04, Thm. 16.4]. Kurihara’s construction
of modular elements also generalizes to modular forms of higher weight [Kima,
EPW].

###  2.5. The global duality argument and Kato’s main conjecture

The global Poitou–Tate duality yields the exact sequence

|
---|---
$\textstyle{\mathrm{H}^{1}(\mathbb{Q},T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{loc}^{s}_{p}}$$\textstyle{\mathrm{H}^{1}_{/f}(\mathbb{Q}_{p},T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{exp}^{*}}$$\textstyle{\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sel}_{0}(\mathbb{Q},E[p^{\infty}])^{\vee}}$$\textstyle{\mathrm{H}^{0}(E/\mathbb{Q}_{p},\Omega^{1})}$

and Kato’s zeta element $z_{\mathbb{Q}}\in\mathrm{H}^{1}(\mathbb{Q},T)$ maps
to

$\mathrm{exp}^{*}\circ\mathrm{loc}^{s}_{p}(z_{\mathbb{Q}})=\dfrac{L^{(p)}(E,1)}{\Omega^{+}_{E}}\cdot\omega_{E}\in\mathbb{Q}_{p}\omega_{E}=D^{0}=\mathrm{H}^{0}(E/\mathbb{Q}_{p},\Omega^{1}).$

Here, $\mathrm{Sel}_{0}$ means the $p$-strict Selmer group. We interpolate
this picture over the cyclotomic tower.

####  2.5.1. Kato’s main conjecture

Recall that

$z_{\mathbb{Q}_{\infty}}=\varprojlim_{n}z_{\mathbb{Q}_{n}}\in\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q},T)=\varprojlim_{n}\mathrm{H}^{1}(\mathbb{Q}_{n},T)$

is Beilinson–Kato’s zeta element over $\mathbb{Q}_{\infty}$. The formulation
of Kato’s main conjecture [Kat04, Conj. 12.10] and the one-sided divisibility
result [Kat04, Thm. 12.5.(4)] are insensitive to the reduction type.

######  Theorem 2.18 (Kato).

Assume that the image of $\rho$ contains a conjugate of
$\mathrm{SL}_{2}(\mathbb{Z}_{p})$.

  1. (1)

$\mathrm{Sel}_{0}(\mathbb{Q}_{\infty},E[p^{\infty}])$ is $\Lambda$-cotorsion.

  2. (2)

$\mathrm{char}_{\Lambda}\dfrac{\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q},T)}{\Lambda
z_{\mathbb{Q}_{\infty}}}\subseteq\mathrm{char}_{\Lambda}\mathrm{Sel}_{0}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee}$.

######  Conjecture 2.19 (Kato’s main conjecture).

Assume that the image of $\rho$ contains a conjugate of
$\mathrm{SL}_{2}(\mathbb{Z}_{p})$. Then

$\mathrm{char}_{\Lambda}\dfrac{\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q},T)}{\Lambda
z_{\mathbb{Q}_{\infty}}}=\mathrm{char}_{\Lambda}\mathrm{Sel}_{0}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee}.$

####  2.5.2. The good ordinary case

When $E$ has good ordinary reduction at $p$, the local Galois representation
at $p$ is reducible, so we have exact sequence of $G_{\mathbb{Q}_{p}}$-modules

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Then the following exact sequence follows from the global duality argument

|
---|---
$\textstyle{\dfrac{\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q},T)}{\Lambda
z_{\mathbb{Q}_{\infty}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{loc}^{s}_{p}}$$\textstyle{\dfrac{\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T)}{\langle\mathrm{loc}^{s}_{p}z_{\mathbb{Q}_{\infty}}\rangle+\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T^{0})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Col}}$$\scriptstyle{\simeq}$$\textstyle{\mathrm{Sel}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sel}_{0}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee}}$$\textstyle{\Lambda/L_{p}(E)}$

Therefore, Conjecture 1.12 and Conjecture 2.19 are equivalent. See [Kat04,
$\S$17.13] for the detail.

####  2.5.3. The good supersingular case

When $E$ has good supersingular reduction at p and $a_{p}(E)=0$ (e.g. $p>3$),
we use Kobayashi–Pollack’s $\pm$-Iwasawa theory [Kob03, Pol03]. Then the
following exact sequence follows from the global duality argument

|
---|---
$\textstyle{\dfrac{\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q},T)}{\Lambda
z_{\mathbb{Q}_{\infty}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{loc}^{s}_{p}}$$\textstyle{\dfrac{\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T)}{\langle\mathrm{loc}^{s}_{p}z_{\mathbb{Q}_{\infty}}\rangle+\mathrm{ker}(\mathrm{Col}^{\pm})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Col}^{\pm}}$$\scriptstyle{\simeq}$$\textstyle{\mathrm{Sel}^{\pm}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sel}_{0}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee}}$$\textstyle{\Lambda/L^{\pm}_{p}(E)}$

Therefore, Kobayashi’s signed main conjecture and Conjecture 2.19 are
equivalent. See [Kob03, Thm. 7.4] for the detail.

##  3\. Lecture 3: The Euler and Kolyvagin system machinery

The main theme of the third lecture is to explain how to develop the
connection between the size/structure of arithmetically interesting modules
(ideal class groups, Selmer groups, Tate–Shafarevich groups, $\cdots$) and the
special values of $L$-functions. See [Rub00, MR04, LZ] for the detail.

###  3.1. A rough description

#### 3.1.1.

We start with a very informal and imprecise definition of Euler systems. In
fact, this shows what we expect to see from Euler systems.

######  Definition 3.1 (informal and imprecise).

An Euler system attached to a $p$-adic representation $V$ is a cohomology
class

$c=c_{\mathbb{Q}}\in\mathrm{H}^{1}(\mathbb{Q},V)$

with properties:

  1. (1)

it can be related to $L(V^{*}(1),0)$ or to $p$-adic $L$-functions (i.e.
$L$-values in families)

  2. (2)

it can be “deformed” $p$-adically or adelically.

The former condition is called the _explicit reciprocity law_ and the latter
one is called the _Euler system norm relation_. All the Euler factors appear
in the second one.

Then the Euler system _machinery_ yields the following theorem.

######  Theorem 3.2 (Euler system machinery).

If there exists a non-zero Euler system and the image of the Galois
representation is large enough, then the Selmer group of the dual
representation is finite.

###### Proof.

See [Rub00, Kat99, PR98]. ∎

More precise versions of the above statements will follow (Definition 3.4 and
Theorem 3.5).

#### 3.1.2.

There are three layers of the applications of Euler systems:

  1. (1)

Bounding the exponent of Selmer groups (also known as the annihilation
result). This ensures the finiteness but it does not give a precise upper
bound.

  2. (2)

Bounding the size of Selmer groups. This gives an upper bound of the Selmer
group by the bottom class of the Euler system.

  3. (3)

Describing the structure of Selmer groups. This determines the structure of
the Selmer group by the associated Kolyvagin system.

In order to have better applications, more strict conditions on the Galois
image are needed. This is essentially related to the use of Chebotarev density
theorem.

We describe the corresponding Galois image assumptions for the case of
elliptic curves.

  1. ($\mathrm{Hyp}(\mathbb{Q},V)$)

$V$ is irreducible, and there exists
$\tau\in\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}(\zeta_{p^{\infty}}))$
such that $V/(\tau-1)V$ is one-dimensional [Rub00, page 37]. This is enough
for bounding the exponent of Selmer groups.

  2. ($\mathrm{Hyp}(\mathbb{Q},T)$)

$p>2$, $E[p]$ is irreducible, and there exists
$\tau\in\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}(\zeta_{p^{\infty}}))$
such that $T/(\tau-1)T$ is one-dimensional [Rub00, page 37]. This is enough
for bounding the size of Selmer groups.

  3. (KS)

$p>5$, $E[p]$ is absolutely irreducible, there exists
$\tau\in\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}(\zeta_{p^{\infty}}))$
such that $T/(\tau-1)T$ is one-dimensional, and
$\mathrm{H}^{1}(\mathbb{Q}_{T}(\zeta_{p^{\infty}})/\mathbb{Q},E[p])=0$ where
$\mathbb{Q}_{T}$ is the minimal field extension of $\mathbb{Q}$ such that
$\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}_{T})$ acts trivially on $T$
[MR04, page 27]. This is enough for describing the structure of Selmer groups.

Recently, the $p>5$ condition in (KS) is weakened to $p>2$ by the work of
Sakamoto [Sak]. Our focus lies in the latter two statements.

###  3.2. Setup

Let $p>2$ be a prime and $F$ be a finite extension of $\mathbb{Q}_{p}$. Let
$\mathcal{O}=\mathcal{O}_{F}$, and denote by $\mathfrak{m}_{F}=(\varpi)$ the
maximal ideal of $\mathcal{O}$. Let $T$ be a free $\mathcal{O}$-module of
finite rank $n$ with continuous action of $G_{\mathbb{Q}}$. Let
$V=T\otimes_{\mathcal{O}}F$. We assume that $V$ is geometric, i.e.

  * •

$V$ is unramified outside a finite set of primes $\Sigma$

  * •

$V$ is de Rham at $p$ in the sense of Fontaine’s $p$-adic Hodge theory.

This means that $V$ is “reasonably good”. Then
$\rho:G_{\mathbb{Q}}\to\mathrm{Aut}_{\mathcal{O}}(V)$ factors through
$G_{\mathbb{Q},\Sigma}=\mathrm{Gal}(\mathbb{Q}_{\Sigma}/\mathbb{Q})$ where
$\mathbb{Q}_{\Sigma}$ is the maximal extension of $\mathbb{Q}$ unramified
outside $\Sigma$. We also write $W=V/T$. When $T=\mathrm{Ta}_{p}E$, we have
$W\simeq E[p^{\infty}]$.

For $m\geq 1$, we define $W_{m}=\varpi^{-m}T/T=W[\varpi^{m}]\simeq
T/\varpi^{m}T$ Define their Cartier duals by

$\textstyle{T^{*}(1)=\mathrm{Hom}_{\mathcal{O}}(T,\mathcal{O}(1)),}$$\textstyle{V^{*}(1)=\mathrm{Hom}_{F}(V,F(1)),}$$\textstyle{W^{*}(1)=V^{*}(1)/T^{*}(1).}$

###  3.3. $L$-functions and Selmer groups

For $\ell\in\Sigma$, let

$P_{\ell}(V,x)=\mathrm{det}(I_{n}-x\cdot\rho(\mathrm{Frob}_{\ell})|_{V})$

where $\mathrm{Frob}_{\ell}$ is the arithmetic Frobenius777Kato follows the
_cohomological_ convention of modular Galois representations but uses
_arithmetic_ Frobenius elements [Kat04, $\S$14.10]. at $\ell$. Set

$L^{\Sigma}(V,s)=\prod_{\ell\not\in\Sigma}P(V,\ell^{-s})^{-1}$

which converges for $\mathrm{Re}(s)\gg 0$.

We recall the notion of Selmer structures/local conditions.

#### 3.3.1.

For every prime $\ell$ except $p$ and $\infty$, define
$\mathrm{H}^{1}_{f}(\mathbb{Q}_{\ell},V)=\mathrm{ker}\left(\mathrm{H}^{1}(\mathbb{Q}_{\ell},V)\to\mathrm{H}^{1}(I_{\ell},V)\right)$
where
$I_{\ell}\subseteq\mathrm{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})$
is the inertia subgroup at $\ell$.

#### 3.3.2.

For $\ell=\infty$, we have $\mathrm{H}^{1}_{f}(\mathbb{R},V)=0$ since $p>2$.

#### 3.3.3.

For $\ell=p$, we consider two different structures.

  * •

$\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},V)=0$ for the p-strict Selmer groups.

  * •

$\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},V)=\mathrm{ker}\left(\mathrm{H}^{1}(\mathbb{Q}_{p},V)\to\mathrm{H}^{1}(\mathbb{Q}_{p},V\otimes\mathbf{B}_{\mathrm{dR}})\right)$
for the Bloch–Kato Selmer groups.

#### 3.3.4.

In any case, $\mathrm{H}^{1}_{f}(\mathbb{Q}_{\ell},T)$ and
$\mathrm{H}^{1}_{f}(\mathbb{Q}_{\ell},W)$ are defined as the preimage and the
image of $\mathrm{H}^{1}_{f}(\mathbb{Q}_{\ell},V)$ with respect to $T\to V\to
W$, respectively. Write
$\mathrm{H}^{1}_{/f}=\dfrac{\mathrm{H}^{1}}{\mathrm{H}^{1}_{f}}$.

#### 3.3.5.

Let $\Sigma^{\prime}$ be a finite set of primes and $M$ be a Galois module.
Then the $\Sigma^{\prime}$-relaxed Selmer group of $M$ is defined by

$\displaystyle\mathrm{Sel}^{\Sigma^{\prime}}(\mathbb{Q},M)$
$\displaystyle=\mathrm{ker}\left(\mathrm{H}^{1}(\mathbb{Q},M)\to\prod_{\ell\not\in\Sigma^{\prime}}\mathrm{H}^{1}_{/f}(\mathbb{Q}_{\ell},M)\right)$
$\displaystyle=\mathrm{ker}\left(\mathrm{H}^{1}(\mathbb{Q}_{\Sigma\cup\Sigma^{\prime}},M)\to\prod_{\ell\not\in(\Sigma\cup\Sigma^{\prime})\setminus\Sigma^{\prime}}\mathrm{H}^{1}_{/f}(\mathbb{Q}_{\ell},M)\right)$

and the $\Sigma^{\prime}$-strict Selmer group of $M$ is defined by

$\mathrm{Sel}_{\Sigma^{\prime}}(\mathbb{Q},M)=\mathrm{ker}\left(\mathrm{Sel}^{\Sigma^{\prime}}(\mathbb{Q},M)\to\prod_{\ell\in\Sigma^{\prime}}\mathrm{H}^{1}(\mathbb{Q}_{\ell},M)\right).$

#### 3.3.6.

The weak form of the Bloch–Kato conjecture can be stated as follows [Bel]:

######  Conjecture 3.3 (Bloch–Kato).

$\mathrm{ord}_{s=0}L(V,s)=\mathrm{dim}_{F}\mathrm{Sel}(\mathbb{Q},V^{*}(1))-\mathrm{dim}_{F}\mathrm{H}^{0}(\mathbb{Q},V^{*}(1)).$

###  3.4. The Euler system argument: a sketch of the idea

Let $\mathbb{Q}^{\mathrm{ab}}$ be the maximal abelian extension of
$\mathbb{Q}$. For a prime $q$ with $q\equiv 1\pmod{p}$, denote by
$\mathbb{Q}(q)$ the maximal $p$-subextension of $\mathbb{Q}$ in
$\mathbb{Q}(\zeta_{q})$. Let $\mathcal{K}$ be a subextension of $\mathbb{Q}$
in $\mathbb{Q}^{\mathrm{ab}}$ such that
$\mathbb{Q}_{\infty}\subseteq\mathcal{K}$ and
$\mathbb{Q}(q)\subseteq\mathcal{K}$ for infinitely many primes $q$ with
$q\equiv 1\pmod{p}$.

######  Definition 3.4.

An Euler system for $(T,\mathcal{K})$ is a collection of cohomology classes

$\mathbf{z}=\left\\{z_{K}\in\mathrm{H}^{1}(K,T):\mathbb{Q}\subseteq
K\subseteq\mathcal{K}\right\\}$

where $K$ runs over _finite_ extensions of $\mathbb{Q}$ in $\mathcal{K}$ such
that

$\mathrm{cores}_{\mathbb{Q}(\zeta_{n\ell})/\mathbb{Q}(\zeta_{n})}(z_{\mathbb{Q}(\zeta_{n\ell})})=\left\\{\begin{array}[]{ll}z_{\mathbb{Q}(\zeta_{n})}&\textrm{
if }\ell|n\textrm{ or }\ell\in\Sigma\\\
P_{\ell}(V^{*}(1),\mathrm{Fr}^{-1}_{\ell})\cdot
z_{\mathbb{Q}(\zeta_{n})}&\textrm{ otherwise.}\end{array}\right.$

As a rough picture, $z_{\mathbb{Q}(\zeta_{n})}$ is related to
$L^{\Sigma_{n}}(V,0)$ where $\Sigma_{n}=\Sigma\cup\\{\ell|n\\}$.

######  Theorem 3.5 (Rubin).

Let $\mathbf{z}$ be an Euler system for $(T,\mathcal{K})$. Assume
$\mathrm{Hyp}(\mathbb{Q},T)$ in $\S$3.1.2 holds. If $z_{\mathbb{Q}}$ is not a
torsion, then $\mathrm{Sel}_{\\{p\\}}(\mathbb{Q},W^{*}(1))$ is finite.

We sketch the idea of the proof of Theorem 3.5. We bound
$\mathrm{Sel}_{\\{p\\}}(\mathbb{Q},W^{*}_{m}(1))$ independently of $m$, and
the main tools are local and global dualities in Galois cohomology.

Let

$\langle-,-\rangle_{\ell}:\mathrm{H}^{1}(\mathbb{Q}_{\ell},W_{m})\times\mathrm{H}^{1}(\mathbb{Q}_{\ell},W^{*}_{m}(1))\to\mathcal{O}/\varpi^{m}\mathcal{O}$

be the local Tate pairing. Then the local duality says
$\mathrm{H}^{1}_{f}(\mathbb{Q}_{\ell},W_{m})^{\perp}=\mathrm{H}^{1}_{f}(\mathbb{Q}_{\ell},W^{*}_{m}(1))$.
Fix a finite set of primes $\Sigma^{\prime}$ not including $p$. Consider the
diagram

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172.08755pt\raise-109.3674pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.24995pt\hbox{$\scriptstyle{\sum_{\ell\in\Sigma^{\prime}}\langle-,-\rangle_{\ell}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
209.63426pt\raise-119.77681pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
101.33539pt\raise-130.27681pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
188.64629pt\raise-130.27681pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{O}/\varpi^{m}\mathcal{O}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$

where

$\displaystyle\mathrm{loc}^{s}_{\Sigma^{\prime}}$
$\displaystyle=\bigoplus_{\ell\in\Sigma^{\prime}}\mathrm{loc}^{s}_{\ell}:\mathrm{Sel}^{\Sigma^{\prime}\cup\\{p\\}}(\mathbb{Q},W_{m})\to\bigoplus_{\ell\in\Sigma^{\prime}}\mathrm{H}^{1}(\mathbb{Q}_{\ell},W_{m})\to\bigoplus_{\ell\in\Sigma^{\prime}}\mathrm{H}^{1}_{/f}(\mathbb{Q}_{\ell},W_{m}),$
$\displaystyle\mathrm{loc}^{f}_{\Sigma^{\prime}}$
$\displaystyle=\bigoplus_{\ell\in\Sigma^{\prime}}\mathrm{loc}^{f}_{\ell}:\mathrm{Sel}_{\\{p\\}}(\mathbb{Q},W^{*}_{m}(1))\to\bigoplus_{\ell\in\Sigma^{\prime}}\mathrm{H}^{1}_{f}(\mathbb{Q}_{\ell},W^{*}_{m}(1)).$

The global duality implies
$\mathrm{im}(\mathrm{loc}^{s}_{\Sigma^{\prime}})^{\perp}=\mathrm{im}(\mathrm{loc}^{f}_{\Sigma^{\prime}})$.
One of the key intuitions of the Euler system argument is the following
behavior.

> If $\mathrm{im}(\mathrm{loc}^{s}_{\Sigma^{\prime}})$ gets larger, then
> $\mathrm{im}(\mathrm{loc}^{f}_{\Sigma^{\prime}})$ gets smaller.

Therefore, it suffices to construct “relevant” $\Sigma^{\prime}$ and elements
$\kappa_{\Sigma^{\prime}}\in\mathrm{Sel}^{\Sigma^{\prime}\cup\\{p\\}}(\mathbb{Q},W_{m})$
from $\mathbf{z}$ such that
$\mathrm{length}(\mathrm{coker}(\mathrm{loc}^{s}_{\Sigma^{\prime}}))$ and
$\mathrm{length}(\mathrm{Sel}_{\Sigma^{\prime}\cup\\{p\\}}(\mathbb{Q},W^{*}_{m}(1)))$
are bounded independently of $m$. Here, $\kappa_{\Sigma^{\prime}}$’s are
called Kolyvagin derivative classes.

######  Remark 3.6.

Also, “relevant” $\Sigma^{\prime}$ means that the image of the arithmetic
Frobenius $\mathrm{Fr}_{\ell}$ at $\ell\in\Sigma^{\prime}$ is equivalent to
the image of
$\tau\in\mathrm{Gal}(\mathbb{Q}(W_{m})(\zeta_{p^{m}})/\mathbb{Q})$. Chebotarev
density theorem plays the key role to construct such a $\Sigma^{\prime}$.

Therefore, the finiteness of the $p$-strict Selmer group
$\mathrm{Sel}_{\\{p\\}}(\mathbb{Q},W^{*}(1))$ can be proved.

######  Assumption 3.7.

From now on, we consider the case of elliptic curves only.

Then how to obtain Theorem 1.9 from the finiteness of
$\mathrm{Sel}_{\\{p\\}}(\mathbb{Q},W^{*}(1))$? We now compare the difference
between $\mathrm{Sel}_{\\{p\\}}(\mathbb{Q},W^{*}(1))$ and
$\mathrm{Sel}(\mathbb{Q},W^{*}(1))$. We consider the following variant of
(3.1) with the Bloch–Kato local condition at $p$. In other words, we have

$\textstyle{\mathrm{Sel}(\mathbb{Q},T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sel}^{\\{p\\}}(\mathbb{Q},T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{loc}^{s}_{p}}$$\textstyle{\mathrm{H}^{1}_{/f}(\mathbb{Q}_{p},T)}$$\textstyle{\times}$$\textstyle{\mathrm{Sel}_{\\{p\\}}(\mathbb{Q},W^{*}(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sel}(\mathbb{Q},W^{*}(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{loc}^{f}_{p}}$$\textstyle{\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},W^{*}(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\langle-,-\rangle_{p}}$$\textstyle{F/\mathcal{O}}$

with
$\mathrm{im}(\mathrm{loc}^{s}_{p})^{\perp}=\mathrm{im}(\mathrm{loc}^{f}_{p})$.
Note that
$\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T)^{\perp}=\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},W^{*}(1))$.
Therefore, the proof of the finiteness of $\mathrm{Sel}(\mathbb{Q},W^{*}(1))$
reduces to showing that the rational restriction map

$\mathrm{loc}^{s}_{p}:\mathrm{Sel}^{\\{p\\}}(\mathbb{Q},V)\to\mathrm{H}^{1}_{/f}(\mathbb{Q}_{p},V)$

is surjective. By Theorem 1.5, we have

$\textstyle{\mathrm{Sel}^{\\{p\\}}(\mathbb{Q},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{loc}^{s}_{p}}$$\textstyle{\mathrm{H}^{1}_{/f}(\mathbb{Q}_{p},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{exp}^{*}}$$\scriptstyle{\simeq}$$\textstyle{\mathrm{Fil}^{0}(\mathbf{D}_{\mathrm{cris}}(V))}$$\textstyle{z_{\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{exp}^{*}\circ\mathrm{loc}^{s}_{p}(z_{\mathbb{Q}})=\dfrac{L^{(p)}(E,1)}{\Omega^{+}_{E}}\cdot\omega_{E}\neq
0,}$

so $\mathrm{loc}^{s}_{p}$ is surjective. Theorem 1.9 follows.

###  3.5. The Kolyvagin system argument

We move to _Kolyvagin systems_.

#### 3.5.1.

First, why Kolyvagin systems? As we mentioned before, there are at least two
advantages.

  1. (1)

The sharp bound via the primitivity, which gives a mod $p$ criterion for
verifying the exact formula and the main conjecture.

  2. (2)

The structure theorem, not only the size (e.g. [Kol91]).

We focus on the second advantage in this article. By utilizing the Kolyvagin
system argument, the following statement (Theorem 3.5)

> If $\mathbf{z}=\\{z_{K}\\}_{K}$ is an Euler system and $z_{\mathbb{Q}}$ is
> not a torsion, then $\mathrm{Sel}_{\\{p\\}}(\mathbb{Q},W^{*}(1))$ is finite.

can be refined as follows:

> Let $\boldsymbol{\kappa}=\\{\kappa_{n}\\}_{n}$ be the Kolyvagin system
> attached to the above Euler system $\mathbf{z}$. If
> $\kappa_{1}=z_{\mathbb{Q}}$ is not a torsion, then the structure of
> $\mathrm{Sel}_{\\{p\\}}(\mathbb{Q},W^{*}(1))$ is described in terms of all
> $\kappa_{n}$’s.

What’s the cost for this refinement? Let $z_{F}$ be an Euler system class and
$\Sigma^{\prime}$ be the finite set of primes $\ell$ where $\ell$ is ramified
in $F/\mathbb{Q}$. Applying the Kolyvagin derivative $D_{F}$ (e.g. $\S$3.5.2)
to an Euler system class $z_{F}$, we have _a priori_

$\kappa_{\Sigma^{\prime}}:=D_{F}z_{F}\pmod{\varpi^{m}}\in\mathrm{Sel}^{\\{p\\}\cup\Sigma^{\prime}}(\mathbb{Q},W_{m}).$

Following the theory of Kolyvagin systems, we can organize
$\kappa_{\Sigma^{\prime}}$ with more controlled local conditions at primes in
$\Sigma^{\prime}$. Let $\ell$ be a prime with $\ell\equiv 1\pmod{\varpi^{m}}$.
Then the transverse local condition at $\ell$ is defined by

$\mathrm{H}^{1}_{\mathrm{tr}}(\mathbb{Q}_{\ell},W_{m})=\mathrm{ker}\left(\mathrm{H}^{1}(\mathbb{Q}_{\ell},W_{m})\to\mathrm{H}^{1}(\mathbb{Q}_{\ell}(\zeta_{\ell}),W_{m})\right),$

and it can be viewed as the complement of the finite local condition
$\mathrm{H}^{1}_{f}$ at $\ell$. There exists the finite-singular comparison
map

$\varphi^{\mathrm{fs}}_{\ell}:\mathrm{H}^{1}_{f}(\mathbb{Q}_{\ell},W_{m})\to\mathrm{H}^{1}_{/f}(\mathbb{Q}_{\ell},W_{m})=\mathrm{H}^{1}_{\mathrm{tr}}(\mathbb{Q}_{\ell},W_{m})$

obtained from the Euler factor at $\ell$. Let
$n=\prod_{\ell\in\Sigma^{\prime}}\ell$, and we would like to construct

$\kappa_{n}\in\mathrm{Sel}^{\\{p\\}}_{n\textrm{-}\mathrm{tr}}(\mathbb{Q},W_{m}),$

i.e.
$\mathrm{loc}_{\ell}(\kappa_{n})\in\mathrm{H}^{1}_{\mathrm{tr}}(\mathbb{Q}_{\ell},W_{m})$
for every $\ell\in\Sigma^{\prime}$. Following [MR04, Appendix A], there exists
an explicit (but complicated) formula for $\kappa_{n}$ in terms of
$\kappa_{\Sigma^{\prime\prime}}$’s where $\Sigma^{\prime\prime}$ runs over
subsets of $\Sigma^{\prime}$. The axiom for Kolyvagin systems is the following
local relation

$\mathrm{loc}_{\ell}(\kappa_{n\ell})=\varphi^{\mathrm{fs}}_{\ell}(\mathrm{loc}_{\ell}(\kappa_{n})).$

######  Proposition 3.8.

  1. (1)

$\mathrm{H}^{1}_{f}(\mathbb{Q}_{\ell},W_{m})$ and
$\mathrm{H}^{1}_{f}(\mathbb{Q}_{\ell},W^{*}_{m}(1))$ are orthogonal to each
other with respect to the local Tate pairing $\langle-,-\rangle_{\ell}$.

  2. (2)

$\mathrm{H}^{1}_{\mathrm{tr}}(\mathbb{Q}_{\ell},W_{m})$ and
$\mathrm{H}^{1}_{\mathrm{tr}}(\mathbb{Q}_{\ell},W^{*}_{m}(1))$ are orthogonal
to each other with respect to the local Tate pairing
$\langle-,-\rangle_{\ell}$.

###### Proof.

See [MR04, Prop. 1.3.2]. ∎

######  Remark 3.9.

  1. (1)

We consider the core rank one case only. There are the theories of higher
(core) rank Kolyvagin systems and Stark systems [MR16, Sak18, BS21, BSS21,
BSS].

  2. (2)

The image of the Galois representation is expected to be sufficiently large as
described in $\S$3.1.2.

#### 3.5.2.

From now on, we restrict ourselves to the case of elliptic curves again. Let
$\mathbf{z}=\left\\{z_{F}\right\\}_{F}$ be Kato’s Euler system. Then
$z_{F}\in\mathrm{H}^{1}(F,T)$ is characterized by

$\sum_{\sigma\in\mathrm{Gal}(K/\mathbb{Q})}\left(\mathrm{exp}^{*}\circ\mathrm{loc}^{s}_{p}z^{\sigma}_{F}\right)\cdot\chi(\sigma)=\dfrac{L^{(Sp)}(E,\chi,1)}{\Omega^{\chi(-1)}_{E}}\cdot\omega_{E}$

where

  * •

$\chi$ is an even character of $\mathrm{Gal}(F/\mathbb{Q})$, and

  * •

$S$ is the product of the ramified primes of $F/\mathbb{Q}$.

For an integer $m\geq 1$, denote by $\mathcal{P}_{m}$ the set of primes $\ell$
such that $(\ell,Np)=1$, $a_{\ell}\equiv\ell+1\pmod{p^{m}}$ and $\ell\equiv
1\pmod{p^{m}}$. For each $\ell\in\mathcal{P}_{m}$, fix a primitive root
$\eta_{\ell}$ mod $\ell$ and write

$\textstyle{\mathrm{Gal}(\mathbb{Q}(\zeta_{\ell})/\mathbb{Q})}$$\textstyle{(\mathbb{Z}/\ell\mathbb{Z})^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\sigma_{\eta_{\ell}}}$$\textstyle{\eta_{\ell}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

the Kolyvagin derivative operator $D_{\ell}$ at $\ell\in\mathcal{P}_{m}$ is
defined by

$D_{\ell}=\sum i\cdot\sigma^{i}_{\eta_{\ell}}$

Let $\mathcal{N}_{m}$ be the set of square-free products of primes in
$\mathcal{P}_{m}$ and $n\in\mathcal{N}_{m}$. Then we define
$D_{n}=\prod_{\ell|n}D_{\ell}$. Write
$G_{n}=\mathrm{Gal}(\mathbb{Q}(\zeta_{n})/\mathbb{Q})$ for convenience. Let
$\mathrm{ES}(\mathbb{Q}(\zeta_{n}),T)\subseteq\mathrm{H}^{1}(\mathbb{Q}(\zeta_{n}),T)$
be the
$\mathbb{Z}_{p}[\mathrm{Gal}(\mathbb{Q}(\zeta_{n})/\mathbb{Q})]$-submodule
generated by Euler systems over $\mathbb{Q}(\zeta_{n})$. See also [Rub00,
$\S$4.2]. Then we have the commutative diagram

$\textstyle{\mathrm{ES}(\mathbb{Q}(\zeta_{n}),T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D_{n}}$$\textstyle{D_{n}\left(\mathrm{ES}(\mathbb{Q}(\zeta_{n}),T)\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bmod{p^{m}}}$$\textstyle{\left(\mathrm{H}^{1}(\mathbb{Q}(\zeta_{n}),T)/p^{m}\right)^{G_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(\mathbb{Q}(\zeta_{n}),T/p^{m}T)^{G_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\mathrm{res}^{-1}}$$\textstyle{\mathrm{H}^{1}(\mathbb{Q},T/p^{m}T)}$

and the Euler system class $z_{\mathbb{Q}(\zeta_{n})}$ maps to the Kolyvagin
system class $\kappa_{n}$ following the above diagram

$z_{\mathbb{Q}(\zeta_{n})}\mapsto D_{n}z_{\mathbb{Q}(\zeta_{n})}\mapsto
D_{n}z_{\mathbb{Q}(\zeta_{n})}\pmod{p^{m}}\mapsto\kappa_{n}.$

######  Remark 3.10.

Although there is an explicit formula for $\kappa_{n}$ in terms of
$D_{n^{\prime}}z_{\mathbb{Q}(\zeta_{n^{\prime}})}\pmod{p^{m}}$ for
$n^{\prime}$ dividing $n$, we can just identify
$D_{n}z_{\mathbb{Q}(\zeta_{n})}\pmod{p^{m}}$ with $\kappa_{n}$ for this case.
It is essentially because Euler factors are quadratic.

#### 3.5.3.

We explain how the Kolyvagin system argument works [MR04, Prop. 4.5.8]. Let
$\boldsymbol{\kappa}^{(m)}=(\kappa^{(m)}_{n})_{n\in\mathcal{N}_{m}}$ be the
mod $p^{m}$ reduction of Kato’s Kolyvagin system $\boldsymbol{\kappa}$, and

$\displaystyle\lambda(n,E[p^{m}])$
$\displaystyle=\mathrm{length}_{\mathbb{Z}_{p}}\left(\mathrm{Sel}_{\\{p\\},n\textrm{-}\mathrm{tr}}(\mathbb{Q},E[p^{m}])\right),$
$\displaystyle\partial^{(r)}(\boldsymbol{\kappa}^{(m)})$
$\displaystyle=\mathrm{min}\left\\{m-\mathrm{length}_{\mathbb{Z}_{p}}\left(\mathbb{Z}/p^{m}\mathbb{Z}\cdot\kappa^{(m)}_{n}\right):n\in\mathcal{N}_{m},\nu(n)=r\right\\}.$

######  Theorem 3.11 (Mazur–Rubin).

Suppose that $\boldsymbol{\kappa}^{(m)}$ is non-trivial. Write

$\mathrm{Sel}_{\\{p\\}}(\mathbb{Q},E[p^{m}])\simeq\bigoplus_{i\geq
1}\mathbb{Z}/p^{d_{i}}\mathbb{Z}$

with $d_{1}\geq d_{2}\geq\cdots$. Then for every $r\geq 0$, we have

$\partial^{(r)}(\boldsymbol{\kappa}^{(m)})=\mathrm{min}\left\\{m,j+\sum_{i>r}d_{i}\right\\}$

where $j\geq 0$ is an integer independent of $n$ and satisfies

$\mathbb{Z}/p^{m}\mathbb{Z}\cdot\kappa^{(m)}_{n}=p^{j+\lambda(n,E[p^{m}])}\cdot\mathrm{Sel}^{\\{p\\}}_{n\textrm{-}\mathrm{tr}}(\mathbb{Q},E[p^{m}]).$

###### Proof.

This is easy to get the equality when $r=0$. When $r>0$, consider the map

$\bigoplus_{\ell|n}\mathrm{loc}_{\ell}:\mathrm{Sel}_{\\{p\\}}(\mathbb{Q},E[p^{m}])\to\bigoplus_{\ell|n}E(\mathbb{Q}_{\ell})\otimes\mathbb{Z}/p^{m}\mathbb{Z}\simeq(\mathbb{Z}/p^{m}\mathbb{Z})^{\oplus\nu(n)}.$

Then the kernel of this map is contained in

$\mathrm{Sel}_{\\{p\\},n\textrm{-}\mathrm{tr}}(\mathbb{Q},E[p^{m}]).$

Therefore, $\geq$ is done, so it suffices to prove the opposite inequality.
Using Chebotarev density theorem, we can choose a prime
$\ell\in\mathcal{P}_{m}$ such that

  * •

$\mathrm{Sel}_{\\{p\\},\ell\textrm{-}\mathrm{str}}(\mathbb{Q},E[p^{m}])\simeq\bigoplus_{i\geq
2}\mathbb{Z}/p^{d_{i}}\mathbb{Z}$, and

  * •

$\mathrm{Sel}_{\\{p\\},\ell\textrm{-}\mathrm{str}}(\mathbb{Q},E[p^{m}])=\mathrm{Sel}_{\\{p\\},\ell\textrm{-}\mathrm{tr}}(\mathbb{Q},E[p^{m}])$.

Then use induction on $\nu(n)$. It is remarkable that this process kills a
generator in each step. ∎

##  4\. Lecture 4: A sketch of the construction of Beilinson–Kato’s zeta
elements

This last lecture on the construction may be the most painful part of this
lecture series although we give a sketch only, give no proof, and do not check
the norm compatibility.

  * •

It would not be very important to you if you mainly concern the _applications_
of Kato’s Euler systems.

  * •

It must be very important to you if you are interested in the construction of
_new_ Euler systems based on Kato’s construction (e.g. Loeffler–Zerbes’
program).

When I first try to read [Kat04], I was quite confused with the notation

$\textstyle{{}_{c,d}z^{(p)}_{m}(f,r,r^{\prime},\xi,S),}$$\textstyle{\mathbf{z}^{(p)}_{\gamma}.}$

Of course, $f$ comes from the modular form and $m$ comes from
$\mathbb{Q}(\zeta_{m})$. However, where do all the other “variables” come
from? We need to understand the geometry of modular curves and specific
functions on them.

###  4.1. Siegel units and zeta elements

####  4.1.1. Modular forms and elliptic curves

We follow [Kat04, $\S$1.4]. Let $N\geq 3$ be an integer and $Y(N)$ the modular
curve over $\mathbb{Q}$ of level $N$ without cusps representing the functor

$\textstyle{\underline{\mathrm{Sch}}_{/\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underline{\mathrm{Sets}}}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textrm{the
isomorphism classes of }(E,e_{1},e_{2})}$

where

  * •

$\underline{\mathrm{Sch}}_{/\mathbb{Q}}$ is the category of schemes over
$\mathbb{Q}$ and $\underline{\mathrm{Sets}}$ is the category of sets, and

  * •

$E$ is an elliptic curve over $S$, and $(e_{1},e_{2})$ is a pair of sections
of $E/S$ and it forms a $\mathbb{Z}/N\mathbb{Z}$-basis of $E[N]$.

This is a smooth irreducible affine curve, and the constant field of $Y(N)$ is
$\mathbb{Q}(\zeta_{N})$, not $\mathbb{Q}$. Denote by $X(N)\supseteq Y(N)$ the
smooth compactification of $Y(N)$.

Let $n$ be an integer with $(n,N)=1$. By using the finite étale surjective map

$\textstyle{Y(Nn)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y(N)}$$\textstyle{(E,e_{1},e_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(E,n\cdot
e_{1},n\cdot e_{2}),}$

we have inclusion $\mathcal{O}(Y(N))\subseteq\mathcal{O}(Y(Nn))$ via the pull-
back of this map.

####  4.1.2. Theta functions

The _theta function_ in the following proposition is the very starting point
of the construction.

######  Proposition 4.1.

Let $E$ be an elliptic curve over a scheme $S$. Let $c$ be an integer with
$(c,6)=1$.

  1. (1)

There exists a unique ${}_{c}\theta_{E}\in\mathcal{O}(E\setminus
E[c])^{\times}$ such that

    1. (a)

$\mathrm{Div}({}_{c}\theta_{E})=c^{2}\cdot(0)-E[c]$ on $E$

    2. (b)

$\mathrm{N}_{a}({}_{c}\theta_{E})={}_{c}\theta_{E}$ for an integer $a$ with
$(a,c)=1$ where $\mathrm{N}_{a}$ is the norm map
$\mathrm{N}_{a}:\mathcal{O}(E\setminus
E[ac])^{\times}\to\mathcal{O}(E\setminus E[c])^{\times}$ associated to the
pull-back homomorphism by the multiplication by $a$.

  2. (2)

Let $d$ be another integer with $(d,6)=1.$ Then

$({}_{d}\theta_{E})^{c^{2}}\cdot(c^{*}({}_{d}\theta_{E}))^{-1}=({}_{c}\theta_{E})^{d^{2}}\cdot(d^{*}({}_{c}\theta_{E}))^{-1}$

in $\mathcal{O}(E\setminus E[ac])^{\times}$ where

  3. (3)

Over $\mathbb{C}$, ${}_{c}\theta_{E}$ can be written more explicitly. Let
$\mathfrak{h}$ be the upper-half plane. Let $\tau\in\mathfrak{h}$,
$z\in\mathbb{C}\setminus c^{-1}(\mathbb{Z}\tau+\mathbb{Z})$. Then
${}_{c}\theta(\tau,z)$ is the value at $z$ of
${}_{c}\theta_{\mathbb{C}/(\mathbb{Z}\tau+\mathbb{Z})}$ over $\mathbb{C}$.
Therefore, we have the explicit formula

${}_{c}\theta(\tau,z)=q^{\frac{1}{12}\cdot(c^{2}-1)}\cdot(-t)^{\frac{1}{2}\cdot(c-c^{2})}\cdot\gamma_{q}(t)^{c^{2}}\cdot\gamma_{q}(t^{c})^{-1}$

where $q=e^{2\pi i\tau}$, $t=e^{2\pi iz}$, and

$\gamma_{q}(t)=\prod_{n\geq 0}(1-q^{n}\cdot t)\cdot\prod_{n\geq
1}(1-q^{n}\cdot t^{-1}).$

  4. (4)

If $h:E\to E^{\prime}$ be an isogeny of elliptic curve of degree prime to $c$,
then we have the corresponding norm map
$h_{*}:{}_{c}\theta_{E}\to{}_{c}\theta_{E^{\prime}}$.

###### Proof.

See [Kat04, Prop. 1.3 and $\S$1.10] and [Sch98, $\S$1.2]. ∎

####  4.1.3. Siegel units from theta functions

Following [Kat04, $\S$1.4], we review the construction of the Siegel units
from the theta functions above

$\textstyle{{}_{c}g_{\alpha,\beta}\in\bigcup_{N}\mathcal{O}(Y(N))^{\times},}$$\textstyle{g_{\alpha,\beta}\in\bigcup_{N}\mathcal{O}(Y(N))^{\times}\otimes\mathbb{Q}}$

where $(\alpha,\beta)\in(\mathbb{Q}/\mathbb{Z})^{\oplus
2}\setminus\\{(0,0)\\}$ and
$(c,6\cdot\mathrm{ord}(\alpha)\cdot\mathrm{ord}(\beta))=1$. Note that the
former one is integral, but the latter allows denominators. If
$N\cdot\alpha=N\cdot\beta=0$, then

$\textstyle{{}_{c}g_{\alpha,\beta}\in\mathcal{O}(Y(N))^{\times},}$$\textstyle{g_{\alpha,\beta}\in\mathcal{O}(Y(N))^{\times}\otimes\mathbb{Q}.}$

In this situation, we have relation

${}_{c}g_{\alpha,\beta}=(g_{\alpha,\beta})^{c^{2}}\cdot(g_{c\alpha,c\beta})^{-1}\in\mathcal{O}(Y(N))^{\times}\otimes\mathbb{Q}.$

How to produce these elements? Consider the universal elliptic curve

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{1},e_{2}}$

where $N\geq 3$ is an integer satisfying $N\cdot\alpha=N\cdot\beta=0$. Write

$(\alpha,\beta)=(a/N,b/N)\in(\frac{1}{N}\mathbb{Z}/\mathbb{Z})^{\oplus
2}\setminus\\{(0,0)\\}$

with $a,b\in\mathbb{Z}$. Consider the map

$\iota_{\alpha,\beta}:=a\cdot e_{1}+b\cdot e_{2}:Y(N)\to E\setminus
E[c]\subseteq E.$

Note that $\mathrm{im}(\iota_{\alpha,\beta})\cap E[c]=\emptyset$ since
$(c,\mathrm{ord}(\alpha)\cdot\mathrm{ord}(\beta))=1$ and
$(\alpha,\beta)\neq(0,0)$.

Then we define

${}_{c}g_{\alpha,\beta}=\iota^{*}_{\alpha,\beta}({}_{c}\theta_{E})\in\mathcal{O}(Y(N))^{\times}$

and we have

$g_{\alpha,\beta}={}_{c}g_{\alpha,\beta}\otimes(c^{2}-1)^{-1}\in\mathcal{O}(Y(N))^{\times}\otimes\mathbb{Q}$

by taking an integer $c$ satisfying

  * •

$(c,6)=1$,

  * •

$c\equiv 1\pmod{N}$, and

  * •

$c\neq\pm 1$.

The resulting $g_{\alpha,\beta}$ is independent of such $c$ and is also
independent of $N$.

######  Remark 4.2.

The reader should check [Kat04, Rem. 1.5] for the important intuition on
Siegel units and theta functions, which is not true logically.

####  4.1.4. Group action on $Y(N)$

We follow [Kat04, $\S$1.6]. Consider the following (left) action
$\mathrm{GL}_{2}(\mathbb{Z}/N\mathbb{Z})$ on $Y(N)$.

$\sigma=\begin{pmatrix}a&c\\\
b&d\end{pmatrix}:(E,e_{1},e_{2})\mapsto(E,e^{\prime}_{1},e^{\prime}_{2})$

where

$\begin{pmatrix}e^{\prime}_{1}\\\
e^{\prime}_{2}\end{pmatrix}=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\cdot\begin{pmatrix}e_{1}\\\ e_{2}\end{pmatrix}$

and $\sigma(\zeta_{N})=\zeta^{\mathrm{det}(\sigma)}_{N}$ on the total constant
field.

####  4.1.5. A little bit of $K$-theory

Let $X$ be a regular separate Noetherian scheme. Then Quillen’s $K$-groups
$K_{i}(X)$ are defined for $i\geq 0$ with cup product

$\cup:K_{i}(X)\times K_{j}(X)\to K_{i+j}(X).$

Two properties are important for our purpose.

  1. (1)

there exists a canonical homomorphism

$\mathcal{O}(X)^{\times}\to K_{1}(X).$

  2. (2)

there exists the universal (Steinberg) symbol map via cup product

$\mathcal{O}(X)^{\times}\times\mathcal{O}(X)^{\times}\to K_{2}(X)$

defined by sending a pair $(u,v)$ to the symbol $\\{u,v\\}$.

######  Remark 4.3.

If $X=\mathrm{Spec}(k)$ for a field $k$, then

$K_{2}(X)=K_{2}(k)=(\wedge^{2}k^{\times})/(\\{u,1-u\\}:u\in k^{\times}).$

####  4.1.6. More modular curves

We follow [Kat04, $\S$2.1]. Let $M,N\geq 1$ be integers, and we now define
$Y(M,N)$. Choose a sufficiently big $L\geq 3$ such that $M$ divides $L$ and
$N$ divides $L$. Then we define

$Y(M,N)=G\backslash Y(L),$

which is independent of $L$, where

$G=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{GL}_{2}(\mathbb{Z}/L\mathbb{Z}):\gamma\equiv\begin{pmatrix}1&0\\\
*&*\end{pmatrix}\pmod{M},\gamma\equiv\begin{pmatrix}*&*\\\
0&1\end{pmatrix}\pmod{N}\right\\}.$

Let $X(M,N)\supseteq Y(M,N)$ be the smooth compactification of $Y(M,N)$.
Suppose that $M+N\geq 5$. Then $Y(M,N)$ represents the functor

$\textstyle{\underline{\mathrm{Sch}}_{/\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underline{\mathrm{Sets}}}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textrm{the
isomorphism classes of }(E,e_{1},e_{2})}$

where

  * •

$E$ is an elliptic curve over $S$, and

  * •

$(e_{1},e_{2})$ is a pair of sections of $E/S$ such that $M\cdot e_{1}=N\cdot
e_{2}=0$ and the map $\mathbb{Z}/M\mathbb{Z}\times\mathbb{Z}/N\mathbb{Z}\to E$
defined by sending $(a,b)\mapsto a\cdot e_{1}+b\cdot e_{2}$ is injective.

Since $M$ divides $L$ and $N$ divides $L$, we have natural map

$\textstyle{Y(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y(M,N)}$$\textstyle{(E,e_{1},e_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(E,(L/M)\cdot
e_{1},(L/N)\cdot e_{2}).}$

####  4.1.7. Zeta elements in $K_{2}$, zeta functions, and Beilinson’s theorem

We follow [Kat04, $\S$2.2]. We produce one zeta element in $K_{2}$ of $Y(M,N)$
from two Siegel units as follows.

Let $M,N\geq 2$ be integers such that $M+N\geq 5$. Choose
$c,d\in\mathbb{Z}_{\geq 1}$ such that $(c,6M)=1$ and $(d,6N)=1$. The zeta
elements are defined by

$\displaystyle{}_{c,d}z_{M,N}$
$\displaystyle:=\\{{}_{c}g_{1/M,0},{}_{d}g_{0,1/N}\\}\in K_{2}(Y(M,N)),$
$\displaystyle z_{M,N}$ $\displaystyle:=\\{g_{1/M,0},g_{0,1/N}\\}\in
K_{2}(Y(M,N))\otimes\mathbb{Q}$

satisfying certain norm compatibilities [Kat04, Props. 2.3 and 2.4].

Then we can define the corresponding zeta function [Kat04, $\S$2.5], which is
the Hecke operator version of the zeta function associated to the zeta element
$z_{M,N}$

$Z_{M,N}(s)=\sum_{n\geq 1,(n,M)=1}T^{\prime}(n)\cdot\begin{pmatrix}1/n&0\\\
0&1\end{pmatrix}^{*}\cdot n^{-s}$

where $T^{\prime}(n)$ is the dual Hecke operator acting on
$\mathrm{H}^{1}(Y(M,N)(\mathbb{C}),\mathbb{C})$ and $\begin{pmatrix}1/n&0\\\
0&1\end{pmatrix}^{*}$ is the pull-back by the action of
$\begin{pmatrix}1/n&0\\\ 0&1\end{pmatrix}$. Recall the isomorphism coming from
Poincaré duality

$\textstyle{\mathrm{H}_{1}(X(M,N)(\mathbb{C}),\\{\mathrm{cusps}\\},\mathbb{Z})\otimes\mathbb{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(Y(M,N)(\mathbb{C}),\mathbb{C})}$$\textstyle{(0,\infty)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\delta_{M,N}.}$

See [Kat04, $\S$2.7].

######  Theorem 4.4 (Beilinson).

There exists the regulator map

$\mathrm{reg}_{M,N}:K_{2}(Y(M,N))\to\mathrm{H}^{1}(Y(M,N)(\mathbb{C}),\mathbb{R}\cdot
i)$

sending $z_{M,N}$ to

$\lim_{s\to 0}\dfrac{1}{s}\cdot Z_{M,N}(s)\cdot 2\pi i\cdot\delta^{-}_{M,N}$

###### Proof.

See [Kat04, Thm. 2.6 and $\S$7] and [Bei85, $\S$5]. ∎

Up to now, we concern the elements on $Y(M,N)$ over $\mathbb{Q}$. However, in
order to have Euler systems in the sense of the third lecture, we need to
construct certain elements on $Y_{1}(N)$ over $\mathbb{Q}(\zeta_{m})$.

###  4.2. Euler systems on $X_{1}(N)$ over $\mathbb{Q}(\zeta_{m})$: $(\xi,S)$

We briefly summarize how the elements in $K_{2}(Y(M,N))$ map to the elements
in $K_{2}(Y_{1}(N)\otimes\mathbb{Q}(\zeta_{m}))$. For example, there exists a
surjective map

$Y(m,L)\twoheadrightarrow Y_{1}(N)\otimes\mathbb{Q}(\zeta_{m})$

for every integer $L\geq 1$ with $m|L$ and $N|L$.

There are two types of zeta elements in terms of $(\xi,S)$:

  1. (1)

$\xi=a(A)$ and $S=\\{\ell:\ell|mA\\}$ where $a(A)$ is just a symbol with
$a\in\mathbb{Z}$ and $A\in\mathbb{Z}_{\geq 1}$.

  2. (2)

$\xi\in\mathrm{SL}_{2}(\mathbb{Z})$ and $S=\\{\ell:\ell|mN\\}$.

######  Remark 4.5.

The zeta elements coming from (1) recover the bad Euler factors, and the zeta
elements coming from (2) are useful for the integrality.

From $(\xi,S)$, we sketch the construction of the zeta element (the “without
$(c,d)$” version)

$z_{1,N,m}(\xi,S)\in
K_{2}(Y_{1}(N)\otimes\mathbb{Q}(\zeta_{m}))\otimes\mathbb{Q}$

as follows.

####  4.2.1. The first case

Choose $M\geq 1$, $L\geq 4$ such that $mA|M$, $N|L$, and $M|L$. Let
$S=\\{\ell:\ell|M\\}$, and $L=\\{\ell:\ell\in S\textrm{ or }\ell|N\\}$.
Consider the map on the upper-half plane $\mathfrak{h}\to\mathfrak{h}$ defined
by sending $\tau$ to $A^{-1}(\tau+a)$ This map is compatible with the
following natural map

$Y(M,L)\to Y_{1}(N)\otimes\mathbb{Q}(\zeta_{m})$

The corresponding norm map on $K$-groups

$t_{m,a(A)}:K_{2}(Y(M,L))\otimes\mathbb{Q}\to
K_{2}(Y_{1}(N)\otimes\mathbb{Q}(\zeta_{m}))\otimes\mathbb{Q}$

is defined by sending $z_{M,L}$ to $z_{1,N,m}(a(A),S)$, and
$z_{1,N,m}(a(A),S)$ is independent of $M$ and $L$.

####  4.2.2. The second case

Choose $L\geq 3$ such that $m|L$, $N|L$, and let $S=\\{\ell:\ell|L\\}$.
Consider the canonical projection $Y(L)\to
Y_{1}(N)\otimes\mathbb{Q}(\zeta_{m})$, and the corresponding norm map on
$K$-groups

$K_{2}(Y(L))\otimes\mathbb{Q}\to
K_{2}(Y_{1}(N)\otimes\mathbb{Q}(\zeta_{m}))\otimes\mathbb{Q}$

is defined by sending $\xi^{*}(z_{L,L})$ to $z_{1,N,m}(\xi,S)$, and
$z_{1,N,m}(\xi,S)$ is also independent of $L$ again.

####  4.2.3. How about their integral versions?

There exist the “with $(c,d)$” versions of the above construction and these
are integral, so $\otimes\mathbb{Q}$ can be removed.

###  4.3. A summary of the construction and so what?

#### 4.3.1.

We recall some parts of $\S$1.7.

---
$\textstyle{{}_{c}\theta_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$Siegel
units$\scriptstyle{\cup}$$\textstyle{K_{2}(Y(M,N))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(*)}$Eisenstein
series$\textstyle{\mathrm{H}^{1}(\mathbb{Q},\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(Y(M,N),\mathbb{Z}_{p})(k-r))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\begin{subarray}{c}\textrm{dual
exponential}\\\ \textrm{\cite[cite]{[\@@bibref{}{kato-euler-systems}{}{}, Thm.
9.7]}}\end{subarray}}}$zeta modular formsRankin–Selbergzeta value formula

where $(*)$ involves the Chern character map
$\mathrm{Ch}_{M,N}(k,r,r^{\prime})$ [Kat04, (8.4.3)], Hochschild–Serre
spectral sequence, and Soulé’s twist. The parameters $(r,r^{\prime})$ are
chosen in this process. In order to have a genuine Euler system, we need to
rewrite everything with $Y(M,N)$ by that with
$Y_{1}(N)\otimes\mathbb{Q}(\zeta_{m})$. The projection to an eigenform $f$ can
be taken by the quotient by the height one prime ideal associated to $f$ of
the Hecke algebra and it yields the zeta modular form for $f$.

#### 4.3.2.

We are now able to explain the relations among
${}_{c,d}z^{(p)}_{m}(f,r,r^{\prime},\xi,S)$, $\mathbf{z}^{(p)}_{\gamma}$, and
$z_{\mathbb{Q}(\zeta_{n})}$.

In the element ${}_{c,d}z^{(p)}_{m}(f,r,r^{\prime},\xi,S)$, the parameters
come from the following choices.

  * •

the choice of $(c,d)$ comes from the choice of two theta functions
(Proposition 4.1),

  * •

the choice of $(r,r^{\prime})$ comes from the Chern character map
$\mathrm{Ch}_{M,N}(k,r,r^{\prime})$ in [Kat04, (8.4.3)], and

  * •

the choice of $(\xi,S)$ comes from the maps from $Y(M,N)$ to
$Y_{1}(N)\otimes\mathbb{Q}(\zeta_{m})$ in $\S$4.2.

The element $\mathbf{z}^{(p)}_{\gamma}$ is defined over
$\mathbb{Q}(\zeta_{p^{\infty}})$ and is explicitly built out from

${}_{c,d}\mathbf{z}^{(p)}_{p^{n}}(f,r,r^{\prime},\xi,S)=\varprojlim_{n}{}_{c,d}z^{(p)}_{p^{n}}(f,r,r^{\prime},\xi,S).$

See [Kat04, $\S$13.9] for the explicit formula. Here, $\gamma$ is an element
of $V_{f}=V_{p}E(-1)$. We now suppose that the image of $\rho$ contains the
conjugate of $\mathrm{SL}_{2}(\mathbb{Z}_{p})$. Then
$\mathrm{H}^{1}_{\mathcal{I}w}(\mathbb{Q},T_{f})=\varprojlim_{n}\mathrm{H}^{1}(\mathbb{Q}(\zeta_{p^{n}}),T_{f})$
is free of rank one over
$\mathbb{Z}_{p}\llbracket\mathrm{Gal}(\mathbb{Q}(\zeta_{p^{\infty}})/\mathbb{Q})\rrbracket$.
Furthermore, $\varprojlim_{n}\mathrm{H}^{1}(\mathbb{Q}(\zeta_{mp^{n}}),T_{f})$
is still free over
$\mathbb{Z}_{p}\llbracket\mathrm{Gal}(\mathbb{Q}(\zeta_{p^{\infty}})/\mathbb{Q})\rrbracket$
for $m\geq 1$. By using the argument of [Kat04, $\S$13.14], we obtain an
_integral_ Euler system and this construction with a suitable choice of
$\gamma\in T_{f}$ gives us $z_{\mathbb{Q}(\zeta_{n})}$ in the second lecture.
See [KN20, Appendix A] for the detail. Note that the choice of $\gamma$ is
equivalent to the choice of periods.

###  4.4. Eisenstein series

The motto is

$``\mathrm{dlog}(\textrm{Siegel units})=\textrm{Eisenstein series}."$

####  4.4.1. Setup

Consider the diagram

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X(N)}$

where $\overline{E}$ is smooth Néron model of $E$, and consider an invertible
$\mathcal{O}_{X(N)}$-module

$\mathrm{coLie}(\overline{E})=\pi_{*}(\Omega^{1}_{\overline{E}/X(N)})(``=\omega").$

Then the spaces of modular forms/cuspforms have the following geometric
descriptions, respectively:

$\displaystyle M_{k}(X(N))$
$\displaystyle=\Gamma(X(N),\mathrm{coLie}(\overline{E})^{\otimes k})$
$\displaystyle=\Gamma(X(N),\mathrm{coLie}(\overline{E})^{\otimes
k-2}\otimes_{\mathcal{O}(X(N))}\Omega^{1}_{X(N)/\mathbb{Q}}(\mathrm{log}(\mathrm{cusps})))$

where $\Omega^{1}_{X(N)/\mathbb{Q}}(\mathrm{log}(\mathrm{cusps}))$ is the
space of differential 1 forms with logarithmic poles at cusps, and

$S_{k}(X(N))=\Gamma(X(N),\mathrm{coLie}(\overline{E})^{\otimes
k-2}\otimes_{\mathcal{O}(X(N))}\Omega^{1}_{X(N)/\mathbb{Q}}).$

For a positive integer $c$ with $(c,6N)=1$, we define Eisenstein series
${}_{c}E^{(k)}_{\alpha,\beta}\in M_{k}(X(N))$ where $k\geq 1$ is an integer
and $(\alpha,\beta)\in(\mathbb{Z}/N\mathbb{Z})^{\oplus 2}$.

We now construct the product of Eisenstein series explicitly. Let
$r\in\mathbb{Z}$. Consider the map

$D:\pi^{*}\mathrm{coLie}(E)^{\otimes r}\to\pi^{*}\mathrm{coLie}(E)^{\otimes
r+1}$

defined by sending $f\otimes\omega^{\otimes r}$ to
$\dfrac{df}{\omega}\otimes\omega^{\otimes r+1}$ where
$\mathrm{coLie}(E)=\pi_{*}(\Omega^{1}_{E/Y(N)})$ $\omega\in\mathrm{coLie}(E)$
is a local basis $f\in\mathcal{O}_{E}$, and
$\dfrac{df}{\omega}\in\mathcal{O}_{E}$ is determined by

$df=\dfrac{df}{\omega}\cdot\omega\in\Omega^{1}_{E/Y(N)}.$

Since

$\displaystyle\mathrm{dlog}({}_{c}\theta_{E})$
$\displaystyle\in\Gamma(E\setminus E[c],\Omega^{1}_{E/Y(N)})$
$\displaystyle=\Gamma(E\setminus E[c],\pi^{*}\mathrm{coLie}(E)),$

we have

$D^{k}(\mathrm{dlog}({}_{c}\theta_{E}))\in\Gamma(E\setminus
E[c],\pi^{*}\mathrm{coLie}(E)^{\otimes k}).$

Let $(\alpha,\beta)\in(1/N\mathbb{Z}/\mathbb{Z})^{\oplus
2}\setminus\\{(0,0)\\}$, and $(c,6N)=1$. Then we have Eisenstein series

${}_{c}E^{(k)}_{\alpha,\beta}=\iota^{*}_{\alpha,\beta}\left(D^{k-1}\mathrm{dlog}({}_{c}\theta_{E})\right)\in\Gamma(Y(N),\mathrm{coLie}(E)^{\otimes
k})$

where $\iota_{\alpha,\beta}=a\cdot e_{1}+b\cdot e_{2}:Y(N)\to E\setminus E[c]$
with

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{1},e_{2}}$

and $(\alpha,\beta)=(a/N,b/N)$. In fact, ${}_{c}E^{(k)}_{\alpha,\beta}\in
M_{k}(X(N))$.

#### 4.4.2.

What if $(\alpha,\beta)=(0,0)$? We define

${}_{c}E^{(k)}_{0,0}=(a^{k}-1)^{-1}\cdot\sum^{\prime}_{\alpha,\beta}{}_{c}E^{(k)}_{\alpha,\beta}$

where $\sum^{{}^{\prime}}$ is the sum runs over all non-zero elements
$(\alpha,\beta)$ of $(\frac{1}{a}\mathbb{Z}/\mathbb{Z})^{\oplus 2}$ Then
${}_{c}E^{(k)}_{0,0}$ is independent of $a$ if $a\neq\pm 1$ and is prime to
$c$.

#### 4.4.3.

When $k=2$? There exists a variant

${}_{c}\widetilde{E}^{(2)}_{\alpha,\beta}\in M_{2}(X(N))$

which can be described in terms of an algebraic version of Weierstrass
$\wp$-function.

#### 4.4.4.

There is another Eisenstein series (without $(c,d)$-version)

$F^{(k)}_{\alpha,\beta}=\left\\{\begin{array}[]{ll}N^{-k}\cdot\sum_{x,y\in\mathbb{Z}/N\mathbb{Z}}E^{(k)}_{x/N,y/N}\cdot\zeta^{bx-
ay}_{N}&\textrm{ if }k\neq 2,\\\
N^{-2}\cdot\sum_{x,y\in\mathbb{Z}/N\mathbb{Z}}\widetilde{E}^{(k)}_{x/N,y/N}\cdot\zeta^{bx-
ay}_{N}&\textrm{ if }k=2\textrm{ and
}(\alpha,\beta)\neq(0,0).\end{array}\right.$

#### 4.4.5.

Let

  * •

$1\leq r,r^{\prime}\leq k-1$ and one of them is $k-1$,

  * •

if $r=k-2$ and $r^{\prime}=k-1$, then $M\geq 2$, and

  * •

$(r,r^{\prime})\neq(2,k-1)$, $(k-1,2)$, or $(k-1,k-2)$.

We are now able to define zeta modular forms

$z_{M,N}(k,r,r^{\prime})\in M_{k}(X(M,N))$

by

$z_{M,N}(k,r,r^{\prime}):=\left\\{\begin{array}[]{ll}\textrm{(some
constant)}\cdot F^{(k-r)}_{1/M,0}\cdot E^{(r)}_{0,1/N}&\textrm{ if
}r^{\prime}=k-1,\\\ \textrm{(some constant)}\cdot
E^{(k-r^{\prime})}_{1/M,0}\cdot E^{(r^{\prime})}_{0,1/N}&\textrm{ if
}r=k-1.\end{array}\right.$

The explicit formulas for zeta modular forms can be found in [Kat04, Prop.
5.8]. The notion of zeta modular forms is the starting point for the proof of
the zeta value formula. See [Kat04, $\S$7].

## Brief remarks on further developments

We conclude this article by mentioning two recent developments of Kato’s Euler
systems very briefly.

### Structural refinements

As we mentioned in the third lecture, the language of Kolyvagin systems
provides Mazur–Rubin’s structure theorem for $p$-strict Selmer groups as well
as a mod $p$ criterion of the Iwasawa main conjecture. Here, we briefly
explain how to extend Mazur–Rubin’s structure theorem to Kurihara’s structure
theorem for standard Selmer groups [Kur14b, Kur14a, Kur24].

Recall that $\ell\in\mathcal{P}_{k}$ means $\ell\equiv 1\pmod{p^{k}}$ and
$a_{\ell}\equiv\ell+1\pmod{p^{k}}$. and $n\in\mathcal{N}_{k}$ is a square-free
product of the primes in $\mathcal{P}_{k}$. and for $n\in\mathcal{N}_{k}$, let
$I_{n}=\prod_{\ell|n}(\ell-1,1-a_{\ell}+\ell)\subseteq p^{k}\mathbb{Z}_{p}$.
The Kurihara number at $n$ is defined by

$\widetilde{\delta}_{n}=\sum_{a\in(\mathbb{Z}/n\mathbb{Z})^{\times}}\overline{\left[\dfrac{a}{n}\right]^{+}}\cdot\prod_{\ell|n}\overline{\mathrm{log}_{\eta_{\ell}}(a)}\in\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}.$

The collection of Kurihara numbers can be viewed as the “Betti realization” of
Kato’s _Kolyvagin_ systems, and it is also an elliptic curve analogue of
higher annihilators in [Aok05]. Under the non-triviality of the collection of
Kurihara numbers, we are able to describe the structure of
$\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])$ in terms of
$\widetilde{\delta}_{n}$’s. Also, the non-triviality of Kurihara numbers is
directly related to Kato’s main conjecture. These results sometimes give us
more arithmetic information than the Birch and Swinnerton-Dyer conjecture. See
[Sak22, Kimb] for details.

### Variation in families

Beilinson–Kato’s zeta elements live in the étale cohomology of modular curves
(before $\mathrm{exp}^{*}$) or the space of modular forms (after
$\mathrm{exp}^{*}$). In order to deform Beilinson–Kato’s zeta elements in
$p$-adic families, we evidently need to put the cohomology of modular curves
and the space of modular forms into their $p$-adic families first.

There are a variety of works at different levels of generalities under
different technical assumptions. We list some of them.

  1. (1)

Hida families (slope zero): [Och05, Del08, FK24].

  2. (2)

Coleman families (finite slope): [Büy16, Wan14, Wan21, BB22, BB23, BB24, Han].

  3. (3)

Deformation rings (no slope condition): [Nak23, CW].

The deformation of Beilinson–Kato’s zeta elements essentially generalizes the
“zeta morphism” (not the zeta element!)

$T_{f}\to\mathrm{H}^{1}_{\mathcal{I}w}(\mathbb{Q},T_{f})$

sending $\gamma$ to $\mathbf{z}^{(p)}_{\gamma}$. More precisely, the Galois
representation $T_{f}$ in the map should be generalized to its deformation and
then making the coherent and canonical choice of a suitable “$\gamma$” in its
deformation becomes a highly non-trivial task.

## Acknowledgement

This article is based on the online lecture series given by the author which
is a part of the program “Elliptic curves and the special values of
$L$-functions (ONLINE)” (2 – 7 August 2021). The lecture videos are available
on Youtube.

The author deeply thanks the organizers Ashay Burungale, Haruzo Hida, Somnath
Jha and Ye Tian, the enthusiastic and active participants, as well as the
staff members of ICTS.

The author learned a lot about Kato’s Euler systems from the lecture series
given by Masato Kurihara at “Postech Winter School 2012: The Birch and
Swinnerton-Dyer conjecture” (January 2012), and from the lecture series given
by Kentaro Nakamura and Shanwen Wang at “An explicit week: on the recent
development of Kato’s Euler systems and explicit reciprocity laws” (February
2016). The author is very thankful to them all for teaching him this beautiful
subject.

## References

  * [Aok05] M. Aoki, _Notes on the structure of the ideal class groups of abelian number fields_ , Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 5, 69–74.
  * [BB22] D. Benois and K. Büyükboduk, _Interpolation of Beilinson–Kato elements and $p$-adic $L$-functions_, Ann. Math. Québec 46 (2022), no. 2, 231–287.
  * [BB23] by same author, _On the exceptional zeros of $p$-non-ordinary $p$-adic $L$-functions and a conjecture of Perrin-Riou_, Trans. Amer. Math. Soc. 376 (2023), no. 1, 231–284.
  * [BB24] by same author, _Arithmetic of critical $p$-adic $L$-functions_, Mem. Amer. Math. Soc., American Mathematical Society, 2024, to appear.
  * [BCD+14] Massimo Bertolini, Francesc Castella, Henri Darmon, Samit Dasgupta, Kartik Prasanna, and Victor Rotger, _$p$ -adic $L$-functions and Euler systems: a tale in two trilogies_, Automorphic forms and Galois representations. Vol. 1, London Math. Soc. Lecture Note Ser., vol. 414, Cambridge Univ. Press, Cambridge, 2014, pp. 52–101. MR 3444223
  * [BCDT01] C. Breuil, B. Conrad, F. Diamond, and R. Taylor, _On the modularity of elliptic curves over $\mathbb{Q}$: wild 3-adic exercises_, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939.
  * [Bei85] A.A. Beilinson, _Higher regulators and values of $L$-functions_, J. Soviet Math. (1985), 2036–2070.
  * [Bel] J. Bellaïche, _Introduction to the conjecture of Bloch–Kato_ , notes from Hawaii CMI summer school. http://people.brandeis.edu/ jbellaic/.
  * [Bel21] by same author, _The eigenbook–eigenvarieties, families of Galois representations, $p$-adic $L$-functions_, Pathw. Math., Birkhäuser Cham, 2021.
  * [BS21] D. Burns and T. Sano, _On the theory of higher rank Euler, Kolyvagin and Stark systems_ , Int. Math. Res. Not. IMRN 2021 (2021), no. 13, 10118–10206.
  * [BSD63] B. J. Birch and H. P. F. Swinnerton-Dyer, _Notes on elliptic curves. I_ , J. Reine Angew. Math. 212 (1963), 7–25.
  * [BSD65] by same author, _Notes on elliptic curves. II_ , J. Reine Angew. Math. 218 (1965), 79–108.
  * [BSS] D. Burns, R. Sakamoto, and T. Sano, _On the theory of higher rank Euler, Kolyvagin and Stark systems, III: applications_ , preprint, arXiv:1902.07002.
  * [BSS21] by same author, _On the theory of higher rank Euler, Kolyvagin and Stark systems, II_ , Int. Math. Res. Not. IMRN 2021 (2021), no. 13, 10118–10206.
  * [Büy16] K. Büyükboduk, _Deformations of Kolyvagin systems_ , Ann. Math. Québec 40 (2016), no. 2, 251–302.
  * [Coa77] John Coates, _$p$ -adic $L$-functions and Iwasawa’s theory_, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), Academic Press, London-New York, 1977, pp. 269–353.
  * [Coa20] by same author, _Memories of Kenkichi Iwasawa_ , Development of Iwasawa theory—the centennial of K. Iwasawa’s birth, Adv. Stud. Pure Math., vol. 86, Math. Soc. Japan, Tokyo, [2020] ©2020, pp. i–viii.
  * [Col04] P. Colmez, _La conjecture de Birch et Swinnerton-Dyer $p$-adique_, Astérisque 294 (2004), no. ix, 251–319, Séminaire BOURBAKI, 55ème année, 2002–2003, No. 919.
  * [CW] P. Colmez and S. Wang, _Une factorisation de la cohomologie complétée et du système de Beilinson–Kato_ , preprint.
  * [Del08] D. Delbourgo, _Elliptic curves and big Galois representations_ , London Math. Soc. Lecture Note Ser., vol. 356, Cambridge University Press, 2008.
  * [DS05] F. Diamond and J. Shurman, _A first course in modular forms_ , Grad. Texts in Math., vol. 228, Springer-Verlag, 2005.
  * [EPW] M. Emerton, R. Pollack, and T. Weston, _Explicit reciprocity laws and Iwasawa theory for modular forms_ , preprint, arXiv:2210.02013.
  * [FK24] T. Fukaya and K. Kato, _On conjectures of Sharifi_ , Kyoto J. Math. (2024), to appear.
  * [Fuk03] T. Fukaya, _Coleman power series for $K_{2}$ and $p$-adic zeta functions of modular forms_, Doc. Math. (2003), 387–442, Extra Volume: Kazuya Kato’s Fiftieth Birthday.
  * [Gol82] D. Goldfeld, _Sur les produits partiels eulériens attachés aux courbes elliptiques_ , C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 14, 471–474.
  * [Gre99] R. Greenberg, _Iwasawa theory for elliptic curves_ , Arithmetic theory of elliptic curves (Cetraro, 1997) (Berlin) (C. Viola, ed.), Lecture Notes in Math., vol. 1716, Centro Internazionale Matematico Estivo (C.I.M.E.), Florence, Springer-Verlag, 1999, Lectures from the 3rd C.I.M.E. Session held in Cetraro, July 12-–19, 1997, pp. 51––144.
  * [Gre01] by same author, _Introduction to Iwasawa theory for elliptic curves_ , Arithmetic Algebraic Geometry (B. Conrad and K. Rubin, eds.), IAS/Park City Math. Ser., vol. 9, AMS, 2001, pp. 407–464.
  * [Han] D. Hansen, _Iwasawa theory of overconvergent modular forms, I: Critical-slope $p$-adic $L$-functions_, preprint, June 15, 2016.
  * [Hid81] H. Hida, _Congruence of cusp forms and special values of their zeta functions_ , Invent. Math. 63 (1981), no. 2, 225––261.
  * [Kat92a] K. Kato, _The explicit reciprocity law and zeta values (Japanese)_ , Sūrikaisekikenkyūsho Kūkyūroku 810 (1992), 264–279, http://hdl.handle.net/2433/82999.
  * [Kat92b] by same author, _$K_{2}$ of modular curves and “$L(E,1)\neq 0\Rightarrow E(\mathbb{Q})<\infty$” (Japanese)_, Sūrikaisekikenkyūsho Kūkyūroku 805 (1992), 238–248, http://hdl.handle.net/2433/82917.
  * [Kat92c] by same author, _A plan of Iwasawa theory for Hasse–Weil $L$-functions (Japanese)_, Sūrikaisekikenkyūsho Kūkyūroku 797 (1992), 88–101, http://hdl.handle.net/2433/82778.
  * [Kat95] by same author, _Elliptic curves: Iwasawa theory for modular forms (Japanese)_ , Sūrikaisekikenkyūsho Kūkyūroku 925 (1995), 66–76, http://hdl.handle.net/2433/59808.
  * [Kat99] by same author, _Euler systems, Iwasawa theory, and Selmer groups_ , Kodai Math. J. 22 (1999), no. 3, 313–372.
  * [Kat02] by same author, _Tamagawa number conjecture for zeta values_ , Proceedings of the International Congress of Mathematicians (Beijing), vol. 2, Higher Ed. Press, 2002, pp. 163–171.
  * [Kat04] by same author, _$p$ -adic Hodge theory and values of zeta functions of modular forms_, Astérisque 295 (2004), 117–290.
  * [Kima] C.-H. Kim, _Refined applications of Kato’s Euler systems for modular forms_ , submitted, arXiv:2203.12157.
  * [Kimb] by same author, _The structure of Selmer groups and the Iwasawa main conjecture for elliptic curves_ , submitted, arXiv:2203.12159.
  * [Kim23] by same author, _On the soft $p$-converse to a theorem of Gross–Zagier and Kolyvagin_, Math. Ann. 387 (2023), 1961–1968.
  * [KKS20] C.-H. Kim, M. Kim, and H.-S. Sun, _On the indivisibility of derived Kato’s Euler systems and the main conjecture for modular forms_ , Selecta Math. (N.S.) 26 (2020), no. 31, 47 pages.
  * [KN20] C.-H. Kim and K. Nakamura, _Remarks on Kato’s Euler systems for elliptic curves with additive reduction_ , J. Number Theory 210 (2020), 249–279.
  * [Kob03] S. Kobayashi, _Iwasawa theory for elliptic curves at supersingular primes_ , Invent. Math. 152 (2003), no. 1, 1–36.
  * [Kol90] V. Kolyvagin, _Euler systems_ , The Grothendieck Festschrift Volume II (P. Cartier, L. Illusie, N. M. Katz, G. Laumon, Y. Manin, and K. A. Ribet, eds.), Progr. Math., vol. 87, Birkhäuser Boston, 1990, pp. 435–483.
  * [Kol91] by same author, _On the structure of Selmer groups_ , Math. Ann. 291 (1991), no. 2, 253–259.
  * [Kur02] M. Kurihara, _On the Tate Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction I_ , Invent. Math. 149 (2002), 195–224.
  * [Kur14a] by same author, _Refined Iwasawa theory for $p$-adic representations and the structure of Selmer groups_, Münster J. of Math. 7 (2014), no. 1, 149–223.
  * [Kur14b] by same author, _The structure of Selmer groups of elliptic curves and modular symbols_ , Iwasawa Theory 2012: State of the Art and Recent Advances (T. Bouganis and O. Venjakob, eds.), Contrib. Math. Comput. Sci., vol. 7, Springer, 2014, pp. 317–356.
  * [Kur24] by same author, _Some anayltic quantities yielding arithemtic information about elliptic curves_ , Arithmetic Geometry—Proceedings of the International Colloquium, Mumbai, 2020, Tata Inst. Fundam. Res. Stud. Math., vol. 24, Hindustan Book Agency, New Delhi, 2024, pp. 345–384.
  * [LZ] D. Loeffler and S. L. Zerbes, _Euler systems_ , Arizona Winter School 2018 notes.
  * [LZ23] by same author, _Euler systems and the Bloch–Kato conjecture for automorphic Galois representations_ , ICM—International Congress of Mathematicians. Vol. III. Sections 1–4, EMS Press, Berlin, [2023] ©2023, pp. 1918–1939. MR 4680305
  * [Man72] J. I. Manin, _Parabolic points and zeta functions of modular curves_ , Math. USSR-Izv. 6 (1972), no. 1, 19–64, translated by N. Koblitz.
  * [Maz72] B. Mazur, _Rational points of abelian varieties with values in towers of number fields_ , Invent. Math. 18 (1972), 183–266.
  * [Mil06] J. Milne, _Arithmetic duality theorems_ , second ed., BookSurge, LLC, 2006.
  * [MR04] B. Mazur and K. Rubin, _Kolyvagin Systems_ , Mem. Amer. Math. Soc., vol. 168, American Mathematical Society, March 2004.
  * [MR16] by same author, _Controlling Selmer groups in the higher core rank case_ , J. Théor. Nombres Bordeaux 28 (2016), no. 1, 145–183.
  * [MTT86] B. Mazur, J. Tate, and J. Teitelbaum, _On $p$-adic analogues of the conjectures of Birch and Swinnerton-Dyer_, Invent. Math. 84 (1986), no. 1, 1–48.
  * [Nak14] K. Nakamura, _Iwasawa theory of de Rham $(\varphi,\Gamma)$-modules over the Robba ring_, J. Inst. Math. Jussieu 13 (2014), no. 1, 65–118.
  * [Nak23] by same author, _Zeta morphisms for rank two universal deformations_ , Invent. Math. 234 (2023), 171–290.
  * [Neu99] J. Neukirch, _Algebraic Number Theory_ , Grundlehren Math. Wiss., vol. 322, Springer-Verlag Berlin Heidelberg, 1999, Translated from the German by N. Schappacher.
  * [Och05] T. Ochiai, _Euler system for Galois deformations_ , Ann. Inst. Fourier (Grenoble) 55 (2005), no. 1, 113–146.
  * [Pan03] A.A. Panchishkin, _Two variable $p$-adic $L$-functions attached to eigenfamilies of positive slope_, Invent. math. 154 (2003), 551–615.
  * [Pol03] R. Pollack, _On the $p$-adic $L$-function of a modular form at a supersingular prime_, Duke Math. J. 118 (2003), no. 3, 523–558.
  * [Pol14] by same author, _Overconvergent modular symbols_ , Computations with Modular Forms (Heidelberg 2011), Contrib. Math. Comput. Sci., vol. 6, Springer, 2014\.
  * [PR94] B. Perrin-Riou, _Théorie d’Iwasawa des représentations $p$-adiques sur un corps local_, Invent. Math. 115 (1994), 81–149, with an appendix by J.-M. Fontaine.
  * [PR98] by same author, _Systèmes d’Euler $p$-adiques et théorie d’Iwasawa_, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 5, 1231–1307.
  * [Roh84] D. Rohrlich, _On $L$-functions of elliptic curves and cyclotomic towers_, Invent. Math. 75 (1984), 409–423.
  * [Roh88] by same author, _$L$ -functions and division towers_, Math. Ann. 281 (1988), 611–632.
  * [Rub98] K. Rubin, _Euler systems and modular elliptic curves_ , Galois representations in arithmetic algebraic geometry (Durham, 1996) (A. Scholl and R. Taylor, eds.), London Math. Soc. Lecture Note Ser., vol. 254, Cambridge University Press, 1998, pp. 351–367.
  * [Rub00] by same author, _Euler Systems_ , Ann. of Math. Stud., vol. 147, Princeton University Press, 2000.
  * [Sak] R. Sakamoto, _The theory of Kolyvagin systems for $p=3$_, preprint.
  * [Sak18] by same author, _Stark systems over Gorenstein local rings_ , Algebra Number Theory 12 (2018), no. 10, 2295–2326.
  * [Sak22] by same author, _$p$ -Selmer groups and modular symbols_, Doc. Math. 27 (2022), 1891–1922.
  * [Sch98] A. Scholl, _An introduction to Kato’s Euler systems_ , Galois representations in arithmetic algebraic geometry (Durham, 1996) (A. Scholl and R. Taylor, eds.), London Math. Soc. Lecture Note Ser., vol. 254, Cambridge University Press, 1998.
  * [Ser72] J.-P. Serre, _Propriétés galoisiennes des points d’ordre fini des courbes elliptiques_ , Invent. Math. 15 (1972), no. 4, 259–331.
  * [Sil09] J. Silverman, _The Arithmetic of Elliptic Curves_ , 2nd ed., Grad. Texts in Math., vol. 106, Springer-Verlag, 2009.
  * [Spr12] F. Sprung, _Iwasawa theory for elliptic curves at supersingular primes: A pair of main conjectures_ , J. Number Theory 132 (2012), no. 7, 1483–1506.
  * [SU14] C. Skinner and E. Urban, _The Iwasawa main conjectures for $\mathrm{GL}_{2}$_, Invent. Math. 195 (2014), no. 1, 1–277.
  * [Tat74] J. Tate, _The arithmetic of elliptic curves_ , Invent. Math. 23 (1974), 179–206.
  * [TW95] R. Taylor and A. Wiles, _Ring theoretic properties of certain Hecke algebras_ , Ann. of Math. (2) 141 (1995), 553–572.
  * [Wan13] S. Wang, _Le système d’Euler de Kato_ , J. Théor. Nombres Bordeaux 25 (2013), no. 3, 677–758.
  * [Wan14] by same author, _Le système d’Euler de Kato en famille (I)_ , Comment. Math. Helv. 89 (2014), 819–865.
  * [Wan21] by same author, _Le système d’Euler de Kato en famille (II)_ , Acta Math. Sin. (Engl. Ser.) 37 (2021), no. 1, 173–204.
  * [Was97] L. Washington, _Introduction to Cyclotomic Fields_ , second edition ed., Grad. Texts in Math., vol. 83, Springer-Verlag, 1997.
  * [Wil95] A. Wiles, _Modular elliptic curves and Fermat’s last theorem_ , Ann. of Math. (2) 141 (1995), 443–551.