# Delta Characters and Crystalline Cohomology

Sudip Pandit and Arnab Saha<EMAIL_ADDRESS><EMAIL_ADDRESS>Indian Institute of Technology Gandhinagar, Gujarat 382355, India

###### Abstract.

The first part of the paper develops the theory of $m$-shifted $\pi$-typical
Witt vectors which can be viewed as subobjects of the usual $\pi$-typical Witt
vectors. We show that the shifted Witt vectors admit a delta structure that
satisfy a canonical identity with the delta structure of the usual
$\pi$-typical Witt vectors. Using this theory, we prove that the generalized
kernels of arithmetic jet spaces are jet spaces of the kernel at the first
level. This also allows us to interpret the arithmetic Picard-Fuchs operator
geometrically.

For a $\pi$-formal group scheme $G$, by a previous construction, one attaches
a canonical filtered isocrystal $\mathbf{H}_{\delta}(G)$ associated to the
arithmetic jet spaces of $G$. In the second half of our paper, we show that
$\mathbf{H}_{\delta}(A)$ is of finite rank if $A$ is an abelian scheme. We
also prove a strengthened version of a result of Buium on delta characters on
abelian schemes. As an application, for an elliptic curve $A$ defined over
$\mathbb{Z}_{p}$, we show that our canonical filtered isocrystal
$\mathbf{H}_{\delta}(A)\otimes\mathbb{Q}_{p}$ is weakly admissible. In
particular, if $A$ does not admit a lift of Frobenius, we show that that
$\mathbf{H}_{\delta}(A)\otimes\mathbb{Q}_{p}$ is canonically isomorphic to the
first crystalline cohomology
$\mathbf{H}^{1}_{\mathrm{cris}}(A)\otimes\mathbb{Q}_{p}$ in the category of
filtered isocrystals. On the other hand, if $A$ admits a lift of Frobenius,
then $\mathbf{H}_{\delta}(A)\otimes\mathbb{Q}_{p}$ is isomorphic to the sub-
isocrystal $H^{0}(A,\Omega_{A})\otimes\mathbb{Q}_{p}$ of
$\mathbf{H}^{1}_{\mathrm{cris}}(A)\otimes\mathbb{Q}_{p}$.

The above result can be viewed as a character theoretic interpretation of the
crystalline cohomology. The difference between the integral structures of
$\mathbf{H}_{\delta}(A)$ and $\mathbf{H}_{\mathrm{cris}}^{1}(A)$ is measured
by a delta modular form $f^{1}$ constructed by Buium.

###### Key words and phrases:

Witt vectors, lift of frobenius, arithmetic jet spaces, delta characters,
abelian schemes, deRham cohomology, crystalline cohomology, filtered
isocrystals

###### 2010 Mathematics Subject Classification:

Primary 11G07, 14F30, 14F40, 14G20, 14K15, 14L05, 14L15, 14B20.

##  1\. Introduction

Let us fix a Dedekind domain $\mathcal{O}$ and a nonzero prime ideal
$\mathfrak{p}$ in it. Let $k$ be the residue field with cardinality $q$ which
is a power of a prime $p$ and $\pi$ be a uniformizer of
$\mathfrak{p}A_{\mathfrak{p}}$. Let $R$ be an $\mathcal{O}$-algebra with a
$\pi$-derivation $\delta$ on it. Consider $X$ to be any scheme defined over
$\operatorname{\mathrm{Spec}}R$. In analogy with differential algebra, for all
$n$, one defines the $n$-th arithmetic jet functor as

$J^{n}X(B):=X(W_{n}(B))$

where $W_{n}(B)$ is the $\pi$-typical Witt vectors of length $n+1$ for any
$R$-algebra $B$ [5, 19, 20, 22]. By [2, 6], the functor $J^{n}X$ is
representable by an $R$-scheme, which we will continue to denote as $J^{n}X$.
In the category of $\pi$-formal schemes, $J^{n}X$ is precisely the arithmetic
jet space constructed by Buium [10].

Some of the applications of our main theorem will be in the category of
$\pi$-formal schemes. If $G$ is a $\pi$-formal smooth group scheme defined
over $\operatorname{\mathrm{Spf}}R$, then the natural projection map
$u:J^{n}G\rightarrow G$ is a surjection of group schemes [6, 13]. Let us
denote the kernel of $u$ as $N^{n}G$. Then they satisfy the following
canonical short exact sequence of $\pi$-formal group schemes

(1.1) $\displaystyle 0\rightarrow N^{n}G\rightarrow
J^{n}G\stackrel{{\scriptstyle u}}{{\rightarrow}}G\rightarrow 0.$

In the case when $A$ is an elliptic curve and $R$ is of characteristic 0 whose
ramification is bounded by $p-2$, Buium shows that
$N^{1}A\simeq\hat{\mathbb{G}}_{\mathrm{a}}$ and therefore one obtains a
canonical extension of the elliptic curve $A$ by
$\hat{\mathbb{G}}_{\mathrm{a}}$

(1.2) $\displaystyle 0\rightarrow\hat{\mathbb{G}}_{\mathrm{a}}\rightarrow
J^{1}A\rightarrow A\rightarrow 0.$

Hence $J^{1}A$ gives rise to a canonical extension class
$\eta_{\tiny{J^{1}A}}\in\mathrm{Ext}(A,\hat{\mathbb{G}}_{\mathrm{a}})\simeq
H^{1}(A,\mathcal{O}_{A})$. This class is trivial if and only if $A$ has a
canonical lift of Frobenius (which we will denote as CL). This leads to a
remarkable new $\delta$-modular function $f^{1}$ defined by Buium in [13] as

$f^{1}(A,\omega)=\langle\omega,\eta_{\tiny{J^{1}A}}\rangle,$

where $\omega$ is an invertible $1$-form on the elliptic curve $A$ and
$\langle~{},~{}\rangle$ is the pairing arising from Serre duality. Note that
as a (delta) modular function, $f^{1}$ has the property that it vanishes
whenever $A$ has a canonical lift of Frobenius. This is equivalent to having
the Serre-Tate parameter $q(A)=1$. Using the above property of $f^{1}$, Buium
and Poonen in [16] show that the intersection of the Heegner points with any
finite rank subgroup of a modular elliptic curve is finite.

The theory of $\delta$-geometry and $\delta$-modular forms is developed in a
series of articles such as [1, 7, 13, 14, 17, 18, 21]. In [11], Buium proved
an effective Manin-Mumford conjecture using $\delta$-geometry. In [9], Borger
and Saha construct canonical filtered isocrystals associated to delta
characters of a group scheme. We will use this construction in this article
and prove comparison results with the crystalline cohomology for elliptic
curves defined over ${\mathbb{Z}}_{p}$. The equal characteristic analogue of
the above construction was done in [8, 23].

In the first part of this paper, we construct $m$-shifted $\pi$-typical Witt
vectors $W_{[m]n}(B)$ of length $m+n+1$ for any $\mathcal{O}$-algebra $B$. In
[9], Borger and Saha introduced $0$-shifted Witt vectors. The shifted Witt
vectors should be thought of as certain subrings of the usual $\pi$-typical
Witt vectors. Interstingly we show that such shifted Witt vectors admit a
different $\delta$-structure than the usual ones. The lift of Frobenius
associated to this $\delta$-structure, called the Lateral Frobenius
$\tilde{F}:W_{[m]n}(B)\rightarrow W_{[m]n-1}(B)$ satisfies the following
canonical identity

$F^{m+2}\circ I=F^{m+1}\circ I\circ\tilde{F}$

where $F:W_{n}(B)\rightarrow W_{n-1}(B)$ is the usual Frobenius map of Witt
vectors and $I:W_{[m]n}(B)\rightarrow W_{m+n}(B)$ is a natural map between
rings.

Let $(X,P_{0})$ denote a scheme $X$ over $\operatorname{\mathrm{Spec}}R$ with
a marked $R$-point $P_{0}:\operatorname{\mathrm{Spec}}R\rightarrow X$. Then
composing with the map induced by $\exp_{\delta}$ (analogue of Hasse-Schmidt
differentiation map as in [2], Proposition $2.10$), $P_{0}$ induces an
$R$-point $P_{m}:\operatorname{\mathrm{Spec}}R\rightarrow J^{m}X$. Consider
the fiber product

$N^{[m]n}X:=J^{m+n}X\times_{J^{m}X,P_{m}}\operatorname{\mathrm{Spec}}R$

Here we would like to remark that the natural lift of Frobenius morphism
$\phi_{X}:J^{m+n}X\rightarrow J^{m+n-1}X$ in general does not restrict to a
morphism from $N^{[m]n}X$ to $N^{[m]n-1}X$. However using the lateral
Frobenius on $m$-shifted Witt vectors, we show that the system of schemes
$\\{N^{[m]n}X\\}_{n=0}^{\infty}$ naturally become a prolongation sequence when
$X$ is affine.

Note that, if $X$ is a group scheme over $\operatorname{\mathrm{Spec}}R$ with
$P_{0}$ as the identity section, then for all $m$, $J^{m}X$ is naturally a
group scheme with identity section $P_{m}$ as defined above. Then $N^{[m]n}X$
is the kernel of the natural projection map of group schemes
$u:J^{m+n}X\rightarrow J^{m}X$, that is, we have the following exact sequence
of group schemes

$0\rightarrow N^{[m]n}X\rightarrow J^{m+n}X\stackrel{{\scriptstyle
u}}{{\rightarrow}}J^{m}X$

for all $m$ and $n$. Observe that in the category of smooth $\pi$-formal group
schemes, the above exact sequence is also right exact by Corollary $1.5$ of
[10]. Our first main result is the following:

######  Theorem 1.1.

If $X$ is an affine scheme over $\operatorname{\mathrm{Spec}}R$, then for all
$n\geq 1$ we have

$N^{[m]n}X\simeq J^{n-1}(N^{[m]1}X).$

In particular $\\{N^{[m]n}\\}_{n=0}^{\infty}$ is the canonical prolongation
sequence attached to scheme $N^{[m]1}X$.

Now we will apply our result in the case of $\pi$-formal schemes. Assume that
$R$ is a $\pi$-adically complete discrete valuation ring and $l=R/\pi R$ be
its residue field. Let $X$ be a $\pi$-formal scheme over
$\operatorname{\mathrm{Spf}}R$ with a marked point
$P_{0}:\operatorname{\mathrm{Spf}}R\rightarrow X$. Then as a consequence of
Theorem 1.1, our next result characterizes $N^{[m]n}X$ for any $\pi$-formal
scheme $X$.

######  Theorem 1.2.

Let $X$ be a $\pi$-formal scheme over $\operatorname{\mathrm{Spf}}R$. Then for
all $n\geq 1$ we have

$N^{[m]n}X\simeq J^{n-1}(N^{[m]1}X).$

In particular $\\{N^{[m]n}\\}_{n=0}^{\infty}$ is the canonical prolongation
sequence attached to the $\pi$-formal scheme $N^{[m]1}X$.

Previously, the structure of the group scheme $N^{n}G(=N^{[0]n}G)$ in (1.1)
was unknown. In the case when $G$ is a smooth commutative $\pi$-formal group
scheme, Buium in [10] showed that $N^{n}G$ is a successive extension of the
additive group schemes $\hat{\mathbb{G}}_{\mathrm{a}}$. Let $\mathbb{W}_{n}$
denote the $\pi$-formal scheme ${\hat{\mathbb{A}}}^{n+1}$ endowed with the
group law of addition of Witt vectors. Then as an application of our Theorem
1.2, the canonical short exact sequence of (1.1) results in the following:

######  Theorem 1.3.

Let $R$ be of characteristic 0 satisfying $v_{\pi}(p)\leq p-2$. Let $G$ be a
commutative smooth $\pi$-formal group scheme of relative dimension $d$ over
$\operatorname{\mathrm{Spf}}R$. Then for all $n\geq 1$ we have

$0\rightarrow(\mathbb{W}_{n-1})^{d}\rightarrow J^{n}G\rightarrow G\rightarrow
0$

as an exact sequence of $\pi$-formal group schemes.

We now discuss an consequence of our above results. Let $L$ be any perfect
field extension of $k$ and consider the perfect $\mathcal{O}$-algebra
$B=W_{\infty}(L)$ and $B_{n}:=W_{n}(L)\simeq B/\pi^{n+1}B$. Then consider the
canonical exact sequence of groups

$\displaystyle 0\rightarrow G(\pi^{n+1}B)\rightarrow
G(B_{n})\stackrel{{\scriptstyle u}}{{\rightarrow}}G(L)$

where $u$ is the map induced from the quotient map of rings $B_{n}\rightarrow
L$ and $G(\pi^{n+1}B)$ is the kernel of $u$. Note that
$G(B_{n})=G(W_{n}(L))\simeq J^{n}G(L)$. Then our Theorem 1.2 implies that
$N^{n}G\simeq J^{n-1}(N^{1}G)$. This gives a geometric characterization of the
group $G(\pi^{n+1}B)$ as follows

$\displaystyle G(\pi^{n+1}B)\simeq J^{n-1}(N^{1}G)(L)\simeq N^{1}G(B_{n-1}).$

More so, if $R$ is of characteristic 0 such that $v_{\pi}(p)\leq p-2$ and $G$
is a commutative smooth group scheme of dimension $d$, then our Theorem 1.3
implies $G(\pi^{n+1}B)\simeq B_{n}^{d}$ where the group law is the one induced
from the additive structure of the ring $B_{n}$. As for example, if
$L={\mathbb{F}}_{p}$, then $B_{n}={\mathbb{Z}}/p^{n+1}{\mathbb{Z}}$. Hence for
any commutative smooth $\pi$-formal group scheme $G$ of dimension $d$, we have

$G(p^{n+1}{\mathbb{Z}}_{p})\simeq\left({\mathbb{Z}}/p^{n+1}{\mathbb{Z}}\right)^{d}$

as groups.

Given a $\pi$-formal group scheme $G$ over $\operatorname{\mathrm{Spf}}R$, our
Theorem 1.2 in the case $m=0$, gives for each $n$ the canonical lift of
Frobenius $\phi_{N^{1}G}:N^{n}G\simeq J^{n-1}(N^{1}G)\rightarrow
N^{n-1}G\simeq J^{n-2}(N^{1}G)$ which satisfies

(1.3) $\displaystyle\phi_{G}\circ i\circ\phi_{N^{1}G}=(\phi_{G})^{\circ
2}\circ i.$

Consider the $R$-module
$\varinjlim\mathrm{Hom}(N^{n}G,\hat{\mathbb{G}}_{\mathrm{a}})$. Then the pull-
back map $\phi_{N^{1}G}^{*}$ makes the above into an
$R\\{\phi_{N^{1}G}^{*}\\}$-module. Then we define as in [9],

${\bf
H}_{\delta}(G):=\varinjlim_{n}\frac{\mathrm{Hom}(N^{n}G,\hat{\mathbb{G}}_{\mathrm{a}})}{i^{*}\phi_{G}^{*}({\bf
X}_{n-1}(G)_{\phi})}.$

By (1.3), $\phi^{*}_{N^{1}G}$ on
$\varinjlim\mathrm{Hom}(N^{n}G,\hat{\mathbb{G}}_{\mathrm{a}})$ descends on
${\bf H}_{\delta}(G)$ and the resulting semilinear action on ${\bf
H}_{\delta}(G)$ will be denoted as $\mathfrak{f}^{*}$. Now consider the
$R$-module

${\bf X}_{\mathrm{prim}}(G):=\varinjlim~{}({\bf X}_{n}(G)/\phi^{*}_{G}({\bf
X}(G)_{\phi})).$

The above $R$-modules satisfy

(1.4) $\displaystyle 0\rightarrow{\bf X}_{\mathrm{prim}}(G)\rightarrow{\bf
H}_{\delta}(G)\rightarrow{\bf I}(G)\rightarrow 0$

where ${\bf I}(G)$ is an $R$-submodule of
$\mathrm{Ext}(G,\hat{\mathbb{G}}_{\mathrm{a}})$ defined as in (8.5). In the
case when $A$ is an elliptic curve, ${\bf I}(A)$ is the $R$-submodule
generated by the class $\eta_{J^{1}A}$ associated to (1.2). Then from Theorem
$9.4$ in [9], it follows that ${\bf
I}(A)=R\langle\eta_{J^{1}A}\rangle\subseteq H^{1}(A,\mathcal{O}_{A})$. The
above inclusion is an equality if and only if $\eta_{J^{1}A}$ is a basis for
$H^{1}(A,\mathcal{O}_{A})$, which means that $J^{1}A\simeq A^{\sharp}$ where
$A^{\sharp}$ is the universal vectorial extension of $A$. This is equivalent
to $f^{1}(A,\omega)\in R^{*}$ for all invertible $1$-forms $\omega$ on $A$.

Let $A$ a $\pi$-formal abelian scheme over $\operatorname{\mathrm{Spf}}R$.
Then we have the following map between the short exact sequences of
$R$-modules as in $(1.1)$ of [9]

(1.9)

Apriori, it is not obvious as to whether ${\bf X}_{\mathrm{prim}}(A)$ and
${\bf H}_{\delta}(A)$ are finite free $R$-modules. However it was shown in [9]
that the above $R$-modules when tensored with $K$ (the fraction field of $R$)
are finite dimensional vector spaces over $K$. Our next result shows that the
$R$-modules ${\bf X}_{\mathrm{prim}}(A)$ and ${\bf H}_{\delta}(A)$ are finite
free over $R$.

######  Theorem 1.4.

Let $A$ be a $\pi$-formal abelian scheme over $\operatorname{\mathrm{Spf}}R$
of dimension $g$. Then we have

  1. (1)

The $R$-module ${\bf X}_{\mathrm{prim}}(A)$ is free of rank $g$.

  2. (2)

The $R$-module ${\bf H}_{\delta}(A)$ is free satisfying
$g\leq\operatorname{\mathrm{rk}}_{R}{\bf H}_{\delta}(A)\leq 2g$.

As a consequence of the above, we obtain the following (in Section 8)

######  Theorem 1.5.

For an abelian scheme $A$ of dimension $g$, ${\bf X}_{\infty}(A)$ is freely
generated as an $R\\{\phi^{*}\\}$-module by $g$ $\delta$-characters of order
at most $g+1$.

Buium in [10] proved the above result for ${\bf X}_{\infty}(A)_{K}:={\bf
X}_{\infty}(A)\otimes_{R}K$. The positive characteristic analogue of the above
results in the case of Anderson modules were shown in [23].

Let us now assume $R=W(l)$ and $K$ be its fraction field. Let

$\mathrm{Iso}({\bf H}_{\delta}(A)):=({\bf
H}_{\delta}(A)_{K},\mathfrak{f}^{*},{\bf H}_{\delta}(A)_{K}^{\bullet})$

denote the object in the category of filtered isocrystals where
$\mathfrak{f}^{*}$ is the semilinear operator on ${\bf H}_{\delta}(A)_{K}$ and
${\bf H}_{\delta}(A)_{K}^{\bullet}$ denote the filtration given by ${\bf
H}_{\delta}(A)_{K}\supset{\bf X}_{\mathrm{prim}}(A)_{K}\supset\\{0\\}$.

Let $A_{0}$ denote the special fiber of $A$ over
$\operatorname{\mathrm{Spec}}l$. Let ${\bf H}^{1}_{\mathrm{cris}}(A)$ denote
the first crystalline cohomology of $A_{0}$ with coefficients in $R$. Then by
the comparison theorem in [3], ${\bf H}^{1}_{\mathrm{cris}}(A)\simeq{\bf
H}^{1}_{\mathrm{dR}}(A)$. Consider the filtered isocrystal

$\mathrm{Iso}({\bf H}^{1}_{\mathrm{cris}}(A)_{K}):=({\bf
H}^{1}_{\mathrm{cris}}(A)_{K},F_{\mathrm{cris}},{\bf
H}^{1}_{\mathrm{cris}}(A)_{K}^{\bullet})$

where $F_{\mathrm{cris}}$ is the semilinear operator on ${\bf
H}^{1}_{\mathrm{cris}}(A_{0},W(l))_{K}$ induced from the absolute Frobenius on
$A_{0}$ and ${\bf H}^{1}_{\mathrm{cris}}(A))^{\bullet}_{K}$ is the Hodge
filtration given by ${\bf H}^{1}_{\mathrm{cris}}(A)_{K}\supset
H^{0}(A,\Omega_{A})_{K}\supset\\{0\\}$. In the next result, our delta
geometric object ${\bf H}_{\delta}(A)$ gives a character theoretic
interpretation of the crystalline cohomology.

######  Theorem 1.6.

Let $A$ be an elliptic curve over ${\mathbb{Z}}_{p}$. Then $\mathrm{Iso}({\bf
H}_{\delta}(A)_{K})$ is a weakly admissible object in the category of filtered
isocrystals.

(1) If $A$ is a non-CL elliptic curve then

$\mathrm{Iso}({\bf H}_{\delta}(A)_{\mathbb{Q}_{p}})\simeq\mathrm{Iso}({\bf
H}^{1}_{\mathrm{cris}}(A)_{\mathbb{Q}_{p}})$

in the category of filtered isocrystals.

(2) If $A$ has CL then $\mathrm{Iso}({\bf
H}_{\delta}(A)_{\mathbb{Q}_{p}})\simeq\mathrm{Iso}(H^{0}(A,\Omega_{A})_{\mathbb{Q}_{p}})$
where $\mathrm{Iso}(H^{0}(A,\Omega_{A})_{\mathbb{Q}_{p}})$ is the one
dimensional sub-isocrystal of $\mathrm{Iso}({\bf
H}^{1}_{\mathrm{cris}}(A)_{\mathbb{Q}_{p}})$.

Here we would like to remark on the comparison between the integral structures
of the $R$-modules ${\bf H}_{\delta}(A)$ and ${\bf H}^{1}_{\mathrm{cris}}(A)$.
The proof of Theorem 1.6 implies that the canonical $R$-module map $\Phi:{\bf
H}_{\delta}(A)\rightarrow{\bf H}^{1}_{\mathrm{cris}}(A)$ is injective. Hence
$\Phi$ is an isomorphism if and only if ${\bf I}(A)=H^{1}(A,\mathcal{O}_{A})$.
In other words $\Phi:{\bf H}_{\delta}(A)\simeq{\bf H}^{1}_{\mathrm{cris}}(A)$
if and only if the delta modular form $f^{1}(A,\omega)$ is an invertible
element of $R$ for all invertible $1$-forms $\omega$ on $A$. Hence the
difference between the integral structures of ${\bf H}_{\delta}(A)$ and ${\bf
H}^{1}_{\mathrm{cris}}(A)$ is measured by the canonical class
$\eta_{J^{1}A}\in H^{1}(A,\mathcal{O}_{A})$ which determines the value of the
delta modular form $f^{1}$ at $(A,\omega)$.

Also in the light of the above results, one may ask for the true connection
between ${\bf H}_{\delta}(A)$ and the prismatic cohomology by Bhatt and
Scholze [4].

##  2\. Plan of the paper

In Section 4, we review the basic definitions of $\delta$-rings, prolongation
of schemes over a $\delta$-ring, and arithmetic jet spaces. In Section 5, we
introduce the notion of shifted Witt vectors. Next, we define the centered
polynomials in Section 6 and study their basic properties in Lemma 6.2 and
Proposition 6.4. In Section 7, we prove that the generalized $n$-th kernels
are, in fact, the $(n-1)$-th jet spaces of the first kernel. In Section 8, we
briefly review the construction of $R$-modules ${\bf X}_{\mathrm{prim}}(A)$
and ${\bf H}_{\delta}(A)$ from [9, 10]. We then prove the finiteness of the
$R$-modules ${\bf X}_{\mathrm{prim}}(A)$ and ${\bf H}_{\delta}(A)$ when $A$ is
an abelian scheme.

In Section 9, we interpret the arithmetic Picard Fuchs operator associated to
a delta character $\Theta$. Using this we prove Theorem 1.6 on the comparison
of ${\bf H}_{\delta}(A)$ with the crystalline cohomology ${\bf
H}^{1}_{\mathrm{cris}}(A)$ where $A$ is an elliptic curve over
${\mathbb{Z}}_{p}$.

##  3\. Notation

We collect here some notations fixed throughout the paper.

$\displaystyle p$ $\displaystyle=\text{a prime number}$
$\displaystyle\mathcal{O}$ $\displaystyle=\text{a Dedekind domain}$
$\displaystyle\mathfrak{p}$ $\displaystyle=\text{a fixed prime ideal
of}~{}\mathcal{O}$ $\displaystyle\pi$ $\displaystyle=\text{a generator of
}\mathfrak{p}\mathcal{O}_{\mathfrak{p}}$ $\displaystyle k$
$\displaystyle=\text{the residue field of $\mathcal{O}$ at $\pi$ with
cardinality $q$}$ $\displaystyle R$ $\displaystyle=\text{a fixed
$\mathcal{O}$-algebra}$ $\displaystyle\phi$ $\displaystyle=\text{an
endomorphism of $R$ satisfying $\phi(x)\equiv x^{q}\bmod\mathfrak{p}$, for all
$x\in R$}$ $\displaystyle S$ $\displaystyle=\operatorname{\mathrm{Spec}}R$
When $R$ is a $\pi$-adically complete discrete valuation ring then
$\displaystyle S$ $\displaystyle=\operatorname{\mathrm{Spf}}R$ $\displaystyle
M_{K}$ $\displaystyle=K\otimes_{R}M,\text{ for any $R$-module $M$ and
}K=\mathrm{Frac}(R)$ $\displaystyle\mathfrak{m}$ $\displaystyle=\text{the
maximal ideal of }R$ $\displaystyle v_{\pi}$ $\displaystyle=\text{the
valuation on $R$ normalized such that }v_{\pi}(\pi)=1$ $\displaystyle e$
$\displaystyle=\text{the absolute ramification index $v_{\pi}(p)\leq p-2$}$
$\displaystyle l$ $\displaystyle=\text{the residue field of $R$}$
$\displaystyle G$ $\displaystyle=\text{a commutative smooth $\pi$-formal group
scheme over $\operatorname{\mathrm{Spf}}R$}$ $\displaystyle A$
$\displaystyle=\text{a $\pi$-formal abelian scheme over
$\operatorname{\mathrm{Spf}}R$}$ $\displaystyle a_{p}$
$\displaystyle=p+1-\\#(A(\mathbb{F}_{p}))\text{ when $A/\mathbb{Z}_{p}$ is an
elliptic curve}$ $\displaystyle\omega$ $\displaystyle=\text{normalized
invariant differential 1-form of an elliptic curve.}$

##  4\. Preliminaries

Let $\mathcal{O}$ be a Dedekind domain and $\mathfrak{p}$ a non-zero prime
ideal with $k$ as the residue field and $q$ be the cardinality of $k$ where
$q$ is a power of a prime $p$. Let $\pi$ be one of the uniformizers of
$\mathfrak{p}$. For any $\mathcal{O}$-algebra $B$ and $B$-algebra $A$, we
define a $\pi$-derivation $\delta$ as a set-theoretic theoretic map
$\delta:B\rightarrow A$ that satisfies for all $x,y\in B$,

$(i)~{}\delta(x+y)=\delta x+\delta y+C_{\pi}(u(x),u(y))$

$(ii)~{}\delta(xy)=u(x)^{q}\delta y+u(y)^{q}\delta x+\pi\delta x\delta y$

where $u:B\rightarrow A$ is the structure map and

$C_{\pi}(X,Y)=\left\\{\begin{array}[]{ll}\frac{X^{q}+Y^{q}-(X+Y)^{q}}{\pi},&\mbox{if
}\mbox{char }\mathcal{O}=0\\\ 0,&\mbox{otherwise.}\end{array}\right.$

Since $\mathcal{O}$ is a domain, it easily follows that $\delta(1)=0$. Given
such a $\pi$-derivation $\delta$, define $\phi(x):=u(x)^{q}+\pi\delta x$ which
is then a ring homomorphism satisfying

$\phi(x)\equiv u(x)^{q}\bmod\mathfrak{p}.$

We will call such a $\phi$ a lift of Frobenius with respect to $u$. Fix an
$\mathcal{O}$-algebra $R$ with a $\pi$-derivation $\delta$ and call
$S=\operatorname{\mathrm{Spec}}R$. Let $X$ and $Y$ be schemes over $S$. We say
a pair $(u,\delta)$ is a prolongation and we write
$Y\stackrel{{\scriptstyle(u,\delta)}}{{\rightarrow}}X$, if $u:Y\rightarrow X$
is a map of $S$-schemes and $\delta:\mathcal{O}_{X}\rightarrow
u_{*}\mathcal{O}_{Y}$ is a $\pi$-derivation making the following diagram
commute:

$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{u_{*}\mathcal{O}_{Y}}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{\mathcal{O}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$

We now recall the notion of prolongation and arithmetic jet spaces over
$\pi$-formal schemes. Some of the definitions below were introduced by Buium
in the case of $\pi$-formal schemes. However their extension to the case of
schemes is natural and without any technical challenges. For a more detailed
treatment of this section we refer to [10, 9]. As in page 103 in [13], a
prolongation sequence is a sequence

$S\stackrel{{\scriptstyle(u,\delta)}}{{\longleftarrow}}T^{0}\stackrel{{\scriptstyle(u,\delta)}}{{\longleftarrow}}T^{1}\stackrel{{\scriptstyle(u,\delta)}}{{\longleftarrow}}\cdots,$

where $T^{i}\stackrel{{\scriptstyle(u,\delta)}}{{\longleftarrow}}T^{i+1}$ are
prolongations satisfying

$u^{*}\circ\delta=\delta\circ u^{*}$

where $u^{*}$ is the pull-back morphism of sheaves induced by $u$ for each
$i$. We will denote a prolongation sequence as $T^{*}$ or $\\{T^{n}\\}_{n\geq
0}$. Prolongation sequences naturally form a category $\mathcal{C}$. Define
$S^{*}$ the prolongation sequence defined by
$S^{i}=\operatorname{\mathrm{Spec}}R$ for all $i$, $u=\mathbbm{1}$ and
$\delta$ is the fixed $\pi$-derivation on $R$. Then let $\mathcal{C}_{S^{*}}$
denote the category of prolongation sequences defined over $S^{*}$. For any
$S$-scheme $X$ and for all $n\geq 0$ we define the $n$-th jet space functor
$J^{n}X$ as

$J^{n}X(B):=X(W_{n}(B))=\mathrm{Hom}_{S}(\operatorname{\mathrm{Spec}}(W_{n}(B)),X)$

for any $R$-algebra $B$. Then $J^{n}X$ is representable by a scheme over $S$
(This was shown in [6] for $S=\operatorname{\mathrm{Spec}}{\mathbb{Z}}$ and in
[2] for a general prolongation sequence $S^{*}$). Then
$J^{*}X:=\\{J^{n}X\\}_{n\geq 0}$ forms a prolongation sequence and is called
the canonical prolongation sequence as in [13] where
$\phi_{X}:J^{n+1}X\rightarrow J^{n}X$ denote the lift of Frobenius morphism
for each $n$. By Proposition 1.1 in [13], $J^{*}X$ satisfies the universal
property that for any $T^{*}\in\mathcal{C}_{S^{*}}$ and $X$ a scheme over $S$
we have

(4.1)
$\displaystyle\mathrm{Hom}_{S}(T^{0},X)=\mathrm{Hom}_{\mathcal{C}_{S^{*}}}(T^{*},J^{*}X).$

Given an $R$-algebra $B$, for any $n$, let $B^{\phi^{n}}$ denote the
$R$-algebra obtained by considering the structure map
$R\stackrel{{\scriptstyle\phi^{n}}}{{\rightarrow}}R\rightarrow B$. Then given
an $S$-scheme $X$ we define $X^{\phi^{n}}$ as
$X^{\phi^{n}}(B):=X(B^{\phi^{n}})$ for any $R$-algebra $B$. Then it is easy to
see that the above functor is represented by the base change of $X$ over the
map $\phi^{n}:S\rightarrow S$ given by $X^{\phi^{n}}=X\times_{S,\phi^{n}}S$.

##  5\. Shifted Witt Vectors

In this section we construct the general $m$-shifted $\pi$-typical Witt
vectors and describe its properties. Note that the ‘$(-1)$-shifted’ Witt
vectors are the usual $\pi$-typical Witt vectors. Given an $R$-algebra $B$,
for a fixed $m$ we define the set

$W_{[m]n}(B):=R^{m+1}\times B^{n}$

for all $n\geq 0$. Define

$\Pi_{[m]n}B=\left(R\times R^{\phi}\times\cdots
R^{\phi^{m}}\right)\times\left(B^{\phi^{m+1}}\times\cdots\times
B^{\phi^{m+n}}\right)$

to be the product of rings and is also naturally an $R$-algebra. For $n\geq
0$, set $\Pi_{n}B:=B\times B^{\phi}\times\cdots\times B^{\phi^{n}}$. Then we
have a natural map $\Pi_{[m]n}B\rightarrow\Pi_{m+n}B$. Consider the shifted
ghost map

$w:W_{[m]n}(B)\rightarrow\Pi_{[m]n}(B)$

given by

$(x_{0},\dots,x_{m},\dots,x_{m+n})\rightarrow\langle x_{0},x_{0}^{q}+\pi
x_{1},\dots,x_{0}^{q^{m+n}}+\pi
x_{1}^{q^{m+n-1}}+\cdots+\pi^{m+n}x_{m+n}\rangle$

Then $W_{[m]n}(B)$ is naturally endowed with the Witt ring structure of
addition and multiplication that makes $w$ a ring homomorphism. We now define
the following ring homomorphism on the ghost side:

  1. (1)

The restriction map $T_{w}:\Pi_{[m]n}(B)\rightarrow\Pi_{[m]n-1}(B)$ as

$T_{w}\langle z_{0},\dots,z_{m},z_{m+1},\dots,z_{n}\rangle=\langle
z_{0},\dots,z_{m},z_{m+1},\dots,z_{n-1}\rangle$

Clearly $T_{w}$ is a map of $R$-algebra.

  2. (2)

The Frobenius map $F_{w}:\Pi_{[m]n}(B)\rightarrow\Pi_{[m]n-1}(B^{\phi})$ as

$F_{w}\langle z_{0}\dots
z_{m-1},z_{m},\dots,z_{n}\rangle=\langle\phi(z_{0}),\dots,\phi(z_{m}),z_{m+2},\dots,z_{n}\rangle$

Note that following the similar arguments as in the case of usual Witt vectors
the $R$-algebra map $T:W_{[m]n}(B)\rightarrow W_{[m]n-1}(B)$ given by

$T(x_{0},\dots,x_{m},x_{m+1},\dots,x_{m+n})=(x_{0},\dots,x_{m},x_{m+1},\dots
x_{m+n-1}).$

######  Theorem 5.1.

There exists a ring homomorphism $\tilde{F}:W_{[m]n}(B)\rightarrow
W_{[m]n-1}(B)$ such that

$\textstyle{W_{[m]n}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{F}}$$\scriptstyle{w}$$\textstyle{\Pi_{[m]n}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{w}}$$\textstyle{W_{[m]n-1}(B^{\phi})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w}$$\textstyle{\Pi_{[m]n-1}(B^{\phi})}$

commutes. Moreover if
$\tilde{F}(x_{0},\dots,x_{m+n})=(\tilde{F}_{0},\dots,\tilde{F}_{m+n-1})$, then
for all $0\leq h\leq m+n-1$ we have

$\tilde{F}_{h}\equiv x_{h}^{q}\bmod\pi.$

###### Proof.

We will prove the result using induction. It is also sufficient to assume that
$B$ is $\pi$-torsion free. For $h=0$, we have $\tilde{F}_{0}=\phi(x_{0})$
which clearly satisfies the required condition. Assume that the result is true
for $h-1$. Then we have $\tilde{F}_{i}=x_{i}^{q}+\pi y_{i}$ where $y_{i}\in B$
for all $i=0,\dots,h-1$. Then comparing the ghost coordinates we get

$\tilde{F}_{0}^{q^{h}}+\pi\tilde{F}^{q^{h-1}}+\cdots+\pi^{h}\tilde{F}_{h}=x_{0}^{q^{h+1}}+\pi
x_{1}^{q^{h}}+\cdots+\pi^{h}x_{h}+\pi^{h+1}x_{h+1}.$

Grouping terms we obtain

$\tilde{F}_{h}=\sum_{i=0}^{h-1}\pi^{i-h}\left(x_{i}^{q^{(h+1)-i}}-\tilde{F}_{i}^{q^{h-i}}\right)+x_{h}^{q}+\pi
x_{h+1}$

and we would be done if we can show integrality of the expression on the right
hand side. Let
$L_{i}=\pi^{i-h}\left(x_{i}^{q^{(h+1)-i}}-\tilde{F}_{i}^{q^{h-i}}\right)$. By
the induction hypothesis for each $i=0,\dots,h-1$ we have

$L_{i}=\pi^{i-h}\sum_{j=1}^{q^{h-i}}\left(\begin{array}[]{l}q^{h-i}\\\
j\end{array}\right)\pi^{j}(x_{i}^{q})^{q^{h-i}-j}y_{i}^{j}.$

Then note that for each $i$ the $\pi$-valuation is

$\displaystyle v(L_{i})$ $\displaystyle\geq$ $\displaystyle
i-h+v(q^{h-i})+(j-v(j))$ $\displaystyle\geq$
$\displaystyle(h-i)v(q)+1-(h-i)\mbox{, since }j-v(j)\geq 1$ $\displaystyle=$
$\displaystyle(h-i)(v(q)-1)+1$ $\displaystyle\geq$ $\displaystyle 1$

and we are done. ∎

Consider the natural map $I:W_{[m]n}(B)\rightarrow W_{m+n}(B)$ given by

$I(x_{0},\dots,x_{m},x_{m+1},\dots,x_{m+n})=(x_{0},\dots,x_{m},x_{m+1},\dots,x_{m+n})$

where $x_{0},\dots,x_{m}\in R$. Note that $I$ is an injection if $B$ is flat
over $R$. Also denote $I_{w}:\Pi_{[m]n}(B)\rightarrow\Pi_{m+n}(B)$ the natural
map on the ghost side.

######  Proposition 5.2.

Let $I:W_{[m]n}(B)\rightarrow W_{n}(B)$ be the natural map. Then

$F^{m+2}\circ I=F^{m+1}\circ I\circ\tilde{F}.$

###### Proof.

It is sufficient to assume $B$ is $\pi$-torsion free. Then the ghost map $w$
is injective and hence it is enough to check the identity on the ghost
vectors. If $I_{w}:\Pi_{[m]n}(B)\rightarrow\Pi_{m+n}(B)$ denote the natural
map (which is inclusion since $B$ is $\pi$-torsion free), then it is
sufficient to show that

$F^{m+2}_{w}\circ I_{w}=F_{w}^{m+1}\circ I_{w}\circ\tilde{F}_{w}.$

Now we have

$\displaystyle(F_{w}^{m+1}\circ I_{w}\circ\tilde{F}_{w})\langle
z_{0},\dots,z_{m+n}\rangle$ $\displaystyle=$ $\displaystyle(F_{w}^{m+1}\circ
I_{w})\langle\phi(z_{0}),\dots,\phi(z_{m}),z_{m+2},\dots,z_{m+n}\rangle$
$\displaystyle=$ $\displaystyle
F_{w}^{m+1}\langle\phi(z_{0}),\dots,\phi(z_{m}),z_{m+2},\dots,z_{m+n}\rangle$
$\displaystyle=$ $\displaystyle\langle z_{m+2},\dots,z_{m+n}\rangle$

On the other hand,

$\displaystyle F_{w}^{m+2}\circ I_{w}\langle z_{0},\dots,z_{m+n}\rangle$
$\displaystyle=$ $\displaystyle F^{m+2}_{w}\langle z_{0},\dots,z_{m+n}\rangle$
$\displaystyle=$ $\displaystyle\langle z_{m+2},\dots,z_{m+n}\rangle$

and this completes the proof. ∎

Let the tuple $(X,P)$ denote the scheme $X$ with a marked $R$-point $P\in
X(R)$. Then by composing the universal map

$R\stackrel{{\scriptstyle\exp_{\delta}}}{{\longrightarrow}}W_{n}(R)$

with $P$, we naturally obtain an $R$-marked point $P^{n}$ on $J^{n}X$ which we
will denote by $(J^{n}X,P^{n})$. If $X=\operatorname{\mathrm{Spec}}A$ and
denote $i^{*}:A\rightarrow R$ be the ring map associated to $P$. If
$i^{*}_{n}:A\rightarrow W_{n}(R)$ denote the ring map of $P^{n}$ then we have
$i^{*}_{n}=\exp_{\delta}\circ i^{*}$. Then the composition $w\circ
i_{n}^{*}:A\rightarrow\Pi_{n}(B)$ is given by

(5.1) $\displaystyle a\mapsto\langle
i^{*}(a),\phi(i^{*}(a)),\dots,\phi^{n}(i^{*}(a))\rangle.$

For all $n\geq 0$, consider $N^{[m]n}X=J^{m+n}X\times_{J^{m}X}S$ which is the
following fiber product

$\textstyle{J^{m+n}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N^{[m]n}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{J^{m}X}$$\textstyle{S.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P^{m}}$

Then clearly $N^{[m]n}X=\operatorname{\mathrm{Spec}}N_{[m]n}A$ where
$N_{[m]n}A=J_{m+n}A\otimes_{J_{m}A}R$. Also functorially $N^{[m]n}X$ can be
described as

$\displaystyle N^{[m]n}X(B)$ $\displaystyle=\\{g:A\rightarrow
W_{[m]n}(B)|\mbox{ where if }g=(g_{0},\dots,g_{m},\dots,g_{m+n}),$ (5.2)
$\displaystyle\hskip 142.26378pt\mbox{ then }(g_{0},\dots,g_{m})=P^{*}_{m}\\}$

Also note that the usual projection map $u:J^{m+n+1}X\rightarrow J^{m+n}X$
induces $u:N^{[m]n+1}X\rightarrow N^{[m]n}X$ for all $n\geq 1$. Now we define
the the generalized lateral Frobenius as follows: for any $R$-algebra, define
$\mathfrak{f}:N^{[m]n}X(B)\rightarrow N^{[m]n-1}X(B)$ as

(5.3) $\displaystyle\mathfrak{f}(g)=\tilde{F}\circ g,\mbox{ for all }g\in
N^{[m]n}X(B).$

######  Theorem 5.3.

For each $n$, the lateral Frobenius $\mathfrak{f}:N^{[m]n}X\rightarrow
N^{[m]n-1}X$ is a lift of Frobenius and satisfies

$\phi^{m+n}\circ i=\phi^{m+n-1}\circ i\circ\mathfrak{f}$

for $n\geq 2$.

###### Proof.

The lateral Frobenius $\mathfrak{f}$ is a lift of Frobenius with respect to
$u$ follows from Proposition 5.1 and the compositional identity is an
immediate consequence of Proposition 5.2. ∎

Let $G$ be a group scheme and $P:S\rightarrow G$ be the identity section. Then
for each $n$, $N^{[m]n}G$ naturally forms a group scheme. Also the projection
map $u:J^{[m]n}G\rightarrow J^{[m]n-1}G$ induces the map (still denoted by
$u$) $u:N^{[m]n}G\rightarrow N^{[m]n-1}G$ of group schemes.

######  Theorem 5.4.

Let $(G,P)$ be as above. Then for each $n$, the lateral Frobenius
$\mathfrak{f}:N^{[m]n}G\rightarrow N^{[m]n-1}G$ is a morphism of group
schemes.

##  6\. Centered Polynomials

For any $B$-algebra $C$, given a lift of Frobenius $\Psi:B\rightarrow C$, we
say $\partial$ is an associated $\pi$-derivation to $\Psi$ if for all $b\in B$
we have

$\Psi(b)=u(b)^{q}+\pi\partial b.$

Fix a positive integer $d$. Let $B_{*}=\\{B_{n}\\}_{n=0}^{\infty}$ be a
prolongation sequence with $B_{n}=R[{\bf t}_{0},\dots,{\bf t}_{n}]$, where
${\bf t}_{i}$ for each $i$, denote the tuple of variables ${\bf
t}_{i}=\\{t_{1i},\dots,t_{di}\\}$. Consider the $\pi$-derivation
$\partial:B_{n}\rightarrow B_{n+1}$ attached to the prolongation sequence
$B_{*}$. By $\partial{\bf t}_{i}$ we will understand $\partial t_{hi}$ for any
chosen $h=1,\dots,d$.

######  Lemma 6.1.

Let $B_{*}$ be as above. If the coordinate functions ${\bf t}_{0},\dots,{\bf
t}_{n}$ satisfy

$\Psi^{n}({\bf t}_{0})={\bf t}_{0}^{q^{n}}+\pi{\bf
t}_{1}^{q^{n-1}}+\cdots+\pi^{n}{\bf t}_{n},$

for each $n$, then

${\bf t}_{n}=\partial{\bf
t}_{n-1}+\sum_{i=0}^{n-2}\left(\sum_{j=1}^{q^{n-1-i}}\pi^{i+j-n}\left(\begin{array}[]{l}q^{n-1-i}\\\
j\end{array}\right){\bf t}_{i}^{q(q^{n-1-i}-j)}(\partial{\bf
t}_{i})^{j}\right)$

###### Proof.

Follows from a similar computation as in Proposition 2.10 in [2]. ∎

We will define $G\in R[T_{1},\dots,T_{k}]$ to be centered if $G(0,\dots,0)=0$.
Given $k$-elements $a_{1},\dots,a_{k}\in B_{n}$ we define the subset:

$\displaystyle[a_{1},\dots,a_{k}]=\\{a\in
B_{n}~{}|~{}a=F(a_{1},\dots,a_{k})\mbox{ for some }F\in
R[T_{1},\dots,T_{k}]\mbox{ centered}\\}.$

######  Lemma 6.2.

If $a,b\in[a_{1},\dots,a_{k}]$ then

  1. (1)

$a+b,~{}ab\in[a_{1},\dots,a_{k}]$

  2. (2)

$\partial a\in[a_{1},\dots,a_{k},\partial a_{1},\dots,\partial a_{k}]$.

  3. (3)

$[a_{1},\dots,a_{k}]\subset(a_{1},\dots,a_{k})$.

  4. (4)

If $i^{*}:B_{n}\rightarrow R[\mathbf{y},\dots,\mathbf{y}^{(m)}]$ be a map of
$R$-algebras, if $a\in[a_{1},\dots,a_{k}]$ then

$i^{*}a=[i^{*}a_{1},\dots,i^{*}a_{k}].$

  5. (5)

If $a-b\in[a_{1},\dots,a_{k}]$ then $\partial a-\partial
b\in[b,a_{1},\dots,a_{k},\partial a_{1},\dots\partial a_{k}]$

###### Proof.

$(1)$ Let $a=F(a_{1},\dots,a_{k})$ and $b=G(a_{1},\dots,a_{k})$ where $F$ and
$G$ are centered. Then clearly $a+b=(F+G)(a_{1},\dots,a_{k})$ and
$ab=FG(a_{1},\dots,a_{k})$ where $F+G$ and $FG$ are both centered and we are
done.

$(2)$ Let $a=F_{1}(a_{1},\dots,a_{k})+\cdots+F_{n}(a_{1},\dots,a_{k})$ where
$F_{i}$s are centered monomials. Then

$\partial a=\partial F_{1}+\cdots+\partial F_{n}+C_{\pi}(F_{1},\dots,F_{n}).$

where $C_{\pi}$ is centered.

Claim: If $F=ca_{1}\dots a_{l}$, $c\in R$ and $a_{i}$s need not be distinct,
then

$\partial F=L(a_{1},\dots,a_{l},\partial a_{1},\dots,\partial a_{l})$

for some $L$ centered.

Proof of claim: We will prove this by induction on $l$. For $l=1$ it is clear.
Now assume true for $l-1$ and let $\partial(ca_{1}\dots a_{l})=L(a_{1}\dots
a_{l-1},\partial a_{1},\dots,\partial a_{l-1})$ with $L$ centered. Then

$\displaystyle\partial F$ $\displaystyle=$
$\displaystyle\partial(ca_{1},\dots,a_{l})$ $\displaystyle=$
$\displaystyle\partial((ca_{1},\dots,a_{l-1})a_{l})$ $\displaystyle=$
$\displaystyle(ca_{1},\dots,a_{k-1})^{q}\partial
a_{l}+a_{l}^{q}\partial(ca_{1},\dots,a_{l-1})+\pi\partial
a_{l}\partial(ca_{1}\dots a_{l-1})$ $\displaystyle=$
$\displaystyle(ca_{1}\dots a_{l-1})^{q}\partial
a_{l}+a_{l}^{q}L(a_{1},\dots,a_{l-1},\partial a_{1},\dots,\partial a_{l-1})$
$\displaystyle\hskip 28.45274pt+\pi\partial
a_{l}L(a_{1},\dots,a_{l-1},\partial a_{1},\dots,\partial a_{l-1})$
$\displaystyle\in$ $\displaystyle[a_{1},\dots,a_{l},\partial
a_{1},\dots,\partial a_{l}]$

and this proves the claim. Hence to complete the proof of (2), we have
$\partial a\in[a_{1},\dots,a_{k},\partial a_{1},\dots,\partial a_{k}]$ since
$C_{\pi}$ is a centered polynomial.

$(3)$ Clear from the definition.

$(4)$ Let $a\in[a_{1},\dots,a_{k}]$. Then $a=H(a_{1},\dots,a_{k})$ where $H$
is centered. Then $i^{*}a=H(i^{*}a_{1},\dots,i^{*}a_{k})$ and we are done.

$(5)$ Since $a-b\in[a_{1},\dots a_{k}]$ implies that
$a\in[b,a_{1},\dots,a_{k}]$. Then we have

$\displaystyle\partial(a-b)$ $\displaystyle\in$
$\displaystyle[a_{1},\dots,a_{k},\partial a_{1},\dots,\partial a_{k}]$
$\displaystyle\partial a-\partial b+C_{\pi}(a,-b)$ $\displaystyle\in$
$\displaystyle[a_{1},\dots,a_{k},\partial a_{1},\dots,\partial a_{k}]$

and the result follows since $a\in[b,a_{1},\dots,a_{k}]$. ∎

######  Lemma 6.3.

For all $m\geq 1$ we have

$\Psi^{m}(a)=\pi^{m}\partial^{m}(a)+P_{m-1}(a,\partial
a,\dots,\partial^{m-1}a).$

where $P_{m-1}$ is centered.

###### Proof.

For $m=1$ we have $\Psi(a)=a^{q}+\pi\partial a$ for all $a$ and hence we are
done. We will prove using induction. Let the result be true for $m-1$.
Therefore we have

$\Psi^{m-1}(a)=\pi^{m-1}\partial^{m-1}a+P_{m-2}(a,\dots,\partial^{m-2}a)$

where $P_{m-2}$ is centered. Then we have

$\displaystyle\Psi^{m}(a)$ $\displaystyle=$ $\displaystyle\Psi(\Psi^{m-1}(a))$
$\displaystyle=$
$\displaystyle\Psi(\pi^{m-1}\partial^{m-1}a+P_{m-2}(a,\dots,\partial^{m-2}a))$
$\displaystyle=$
$\displaystyle\Psi(\pi^{m-1}\partial^{m-1}a)+\Psi(P_{m-2}(a,\dots,\partial^{m-2}a))$
$\displaystyle=$
$\displaystyle(\pi^{m-1}\partial^{m-1}a)^{q}+\pi\partial(\pi^{m-1}\partial^{m-1}a)+(P_{m-2}^{q}+\pi\partial
P_{m-2})$

Then $\pi\partial P_{m-2}\in[a,\partial a,\dots\partial^{m-1}a]$ by Lemma 6.2
(2). Hence

$(\pi^{m-1}\partial^{m-1}a)^{q}+P^{q}_{m-2}+\pi\partial P_{m-2}\in[a,\partial
a,\dots,\partial^{m-1}a].$

Now for the other term

$\displaystyle\partial(\pi^{m-1}\partial^{m-1}a)$ $\displaystyle=$
$\displaystyle\phi(\pi^{m-1})\partial^{m}a+(\partial^{m-1}a)^{q}\partial(\pi^{m-1})$
$\displaystyle=$
$\displaystyle\pi^{m-1}\partial^{m}a+\pi^{m-2}(1-\pi^{(m-1)(q-1)})(\partial^{m-1}a)^{q}$

Hence substituting the above in the equation for $\Psi^{m}(a)$ we get

$\Psi^{m}(a)=\pi^{m}\partial^{m}a+P_{m-1}$

where

$P_{m-1}=(\pi^{m-1}\partial^{m-1}a)^{q}+P^{q}_{m-2}+\pi\partial
P_{m-2}+\pi^{m-1}(1-\pi^{(m-1)(q-1)})(\partial^{m-1}a)^{q}$

which satisfies $P_{m-1}\in[a,\partial a,\dots,\partial^{m-1}a]$ and we are
done. ∎

######  Proposition 6.4.

Let $B_{*}$ be as above. If ${\bf t}_{0},\dots,{\bf t}_{m}$ satisfies

$\Psi^{m}({\bf t}_{0})={\bf t}_{0}^{q^{m}}+\pi{\bf
t}_{1}^{q^{m-1}}+\cdots+\pi^{m}{\bf t}_{m}$

for all $0\leq m\leq n$, then we have

  1. (1)

${\bf t}_{m}-\partial^{m}{\bf t}_{0}\in[{\bf t}_{0},\dots,\partial^{m-1}{\bf
t}_{0}]=[{\bf t}_{0},{\bf t}_{1},\dots,{\bf t}_{m-1}].$

  2. (2)

$B_{m}\simeq J_{m}B_{0}$.

###### Proof.

We will prove this by induction on $m$. For $m=0$ then the result is clear.
Now assume the result is true for $m-1$. Then for all $0\leq i\leq m-1$ we
have

${\bf t}_{i}-\partial^{i}{\bf t}_{0}\in[{\bf t}_{0},\dots,\partial^{i-1}{\bf
t}_{0}]=[{\bf t}_{0},{\bf t}_{1},\dots,{\bf t}_{i-1}].$

Now by induction hypothesis we have

${\bf t}_{m-1}-\partial^{m-1}{\bf t}_{0}\in[{\bf
t}_{0},\dots,\partial^{m-2}{\bf t}_{0}]=[{\bf t}_{0},\dots,{\bf t}_{m-2}]$

Hence by Lemma 6.2 (5) we have

(6.1) $\displaystyle\partial{\bf t}_{m-1}-\partial^{m}{\bf t}_{0}\in[{\bf
t}_{0},\dots,\partial^{m-2}{\bf t}_{0},\partial^{m-1}{\bf t}_{0}]=[{\bf
t}_{0},{\bf t}_{1},\dots,{\bf t}_{m-1}]$

Then by Lemma 6.1 along with the induction hypothesis we have

(6.2) $\displaystyle{\bf t}_{m}-\partial{\bf t}_{m-1}\in[{\bf
t}_{0},\dots,\partial^{m-1}{\bf t}_{0}]=[{\bf t}_{0},\dots,{\bf t}_{m-1}]$

Hence combining (6.1) and (6.2) we get

${\bf t}_{m}-\partial^{m}{\bf t}_{0}\in[{\bf t}_{0},\dots,\partial^{m-1}{\bf
t}_{0}]=[{\bf t}_{0},\dots,{\bf t}_{m-1}]$

and we are done.

$(2)$ From $(1)$ we get $B_{m}\simeq R[{\bf t}_{0},\dots,\partial^{m}{\bf
t}_{0}]$ and this completes the proof. ∎

##  7\. Kernel of Jet spaces

As before we will consider the category of schemes over
$S=\operatorname{\mathrm{Spec}}R$ where $R$ is a ring with a fixed
$\pi$-derivation $\delta$ on it. Let $X=\operatorname{\mathrm{Spec}}A$ such
that $A$ is an $R$-algebra with presentation $A=R[\mathbf{x}]/I$ where
$\mathbf{x}$ is a collection of indeterminates and $I$ is an ideal inside
$R[\mathbf{x}]$. Consider the representing scheme
$J^{n}X=\operatorname{\mathrm{Spec}}J_{n}A$ where

$J_{n}A=\frac{R[\mathbf{x},\mathbf{x}^{\prime},\cdots,\mathbf{x}^{(n)}]}{(I,\delta
I,\cdots,\delta^{n}I)}\simeq\frac{R[\mathbf{x},\mathbf{x}_{1},\cdots,\mathbf{x}_{n}]}{(I,P_{1}(I),\cdots,P_{n}(I))}.$

and the above isomorphism is as in [2]. Define

(7.1) $\displaystyle N^{[m]n}X:=J^{m+n}X\times_{J^{m}X}P^{n}$

Then

$N^{[m]n}X=\operatorname{\mathrm{Spec}}~{}N_{[m]n}A$

where $N_{[m]n}A=J_{m+n}A\otimes_{J_{m}A}R$ obtained from the following base-
change

$\textstyle{J_{m+n}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}_{m}}$$\textstyle{N_{[m]n}A}$$\textstyle{J_{m}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}_{m}}$$\textstyle{R.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

###  7.1. The Affine $n$-plane case

Let $X=\operatorname{\mathrm{Spec}}A$ where $A=R[\mathbf{x}]$ and $\mathbf{x}$
is as before. We will do explicit computation based on the chosen coordinate
function $\mathbf{x}$. Now for any $n$ we have
$J^{n}X=\operatorname{\mathrm{Spec}}J_{n}A$ where
$J_{n}A=R[\mathbf{x},\mathbf{x}_{1},\mathbf{x}_{2},\dots,\mathbf{x}_{n}]\simeq
R[\mathbf{x},\mathbf{x}^{\prime},\dots,\mathbf{x}^{(n)}]$ where on $J^{n}X$ we
have $(\mathbf{x},\mathbf{x}_{1},\dots)$ to be the induced Witt coordinates
and $(\mathbf{x},\mathbf{x}^{\prime},\dots)$ are the $\delta$-coordinates.
Then an $R$-point of $X$ corresponds to an $R$-algebra map $i^{*}:A\rightarrow
R$ and the induced point $i^{*}_{m}:J_{m}A\rightarrow R$ correspond to the
point $P_{m}:\operatorname{\mathrm{Spec}}R\rightarrow J^{m}X$. Then we have

$\displaystyle N_{[m]n}A$ $\displaystyle=$ $\displaystyle
J_{m+n}A\otimes_{J_{m}A}R$ $\displaystyle=$ $\displaystyle
R[\mathbf{x},\mathbf{x}^{\prime},\dots,\mathbf{x}^{(m+n)}]\otimes_{R[\mathbf{x},\dots,\mathbf{x}^{(m)}]}R$
$\displaystyle\simeq$ $\displaystyle
R[\mathbf{x}^{(m+1)},\dots,\mathbf{x}^{(m+n)}]$

satisfying

$\textstyle{J_{m+n}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}_{m}}$$\textstyle{N_{[m]n}A}$$\textstyle{J_{m}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}_{m}}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where

$i^{*}_{m}(\mathbf{x}^{(j)})=\left\\{\begin{array}[]{ll}i^{*}_{m}(\mathbf{x}^{(j)})&\mbox{
if }j\leq m\\\ \mathbf{x}^{(j)}&\mbox{ if }j\geq m+1.\end{array}\right.$

For a fixed $m$, consider the lateral Frobenius
$\mathfrak{f}:N_{[m]n}A\rightarrow N_{[m]n+1}A$ for $n\geq 0$ as in Theorem
5.3. The above theorem also implies

$\mathfrak{f}i^{*}\phi^{m+1}=i^{*}\phi^{m+2}.$

Let $\Delta$ be the unique $\pi$-derivation associated to $\mathfrak{f}$, that
is $\mathfrak{f}(a)=a^{q}+\pi\Delta(a)$. Now the adjunction property proved in
Theorem 1.3 of [2] implies

(7.2) $\displaystyle
J^{n}X(B)\simeq\mathrm{Hom}_{R}(J_{n}A,B)\simeq\mathrm{Hom}_{R}(A,W_{n}(B))$

for all $R$-algebra $B$. In this case when
$J_{n}A=R[\mathbf{x},\mathbf{x}_{1},\dots,\mathbf{x}_{n}]$, giving an
$R$-algebra homomorphism from $J_{n}A$ to $C$ is equivalent to specifying the
image of the generators $\\{\mathbf{x},\dots,\mathbf{x}_{n}\\}$ in $C$. Hence
with respect to the Witt coordinates, the above isomorphism in (7.2) gives the
following correspondence:

$\left(\\{\mathbf{x}_{0},\mathbf{x}_{1},\dots,\mathbf{x}_{n}\\}\mapsto\\{{\bf
b}_{0},{\bf b}_{1},\dots,{\bf
b}_{n}\\}\right)\longleftrightarrow\left(\mathbf{x}\mapsto({\bf
b}_{0},\dots,{\bf b}_{n})\right)$

where ${\bf b}_{i}\in B$ for all $i$. Then $N^{[m]n}X$ is a closed subfunctor
of $J^{m+n}X$ satisfying

$\textstyle{J^{m+n}X(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Hom}_{R}(J_{m+n}A,B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Hom}_{R}(A,W_{m+n}(B))}$$\textstyle{N^{[m]n}X(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Hom}_{R}(N_{[m]n}A,B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\mathrm{Hom}}_{R}(A,W_{[m]n}(B))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $\widetilde{\mathrm{Hom}}_{R}(A,W_{[m]n}(B))$ consists of elements
$g\in\mathrm{Hom}(A,W_{[m]n}(B))$ such that

$g(a)=(i^{*}_{m}(a),g_{m+1}(a),\dots,g_{m+n}(a))$

where $i^{*}_{m}(a)=(b_{0},\dots,b_{m})\in W_{m}(R)$ and
$g_{m+1}(a),\dots,g_{m+n}(a)\in B$. Then giving an element $g\in N^{[m]n}X(B)$
corresponds to giving a map sending the generators of the algebra $N_{[m]n}A$
to elements in $B$ as follows:

$\\{\mathbf{x}_{m+1},\dots,\mathbf{x}_{m+n}\\}\mapsto\\{{\bf
b}_{m+1},\dots,{\bf b}_{m+n}\\}.$

The above map then under the identification in the diagram corresponds to an
element in $\widetilde{\mathrm{Hom}}(A,W_{[m]n}(B))$ which is given by (still
denoted by $g$):

$g:\mathbf{x}\mapsto(i^{*}_{m}(\mathbf{x}),{\bf b}_{m+1},\dots,{\bf
b}_{m+n}).$

We may choose the coordinate $\mathbf{x}$ such that $i^{*}(\mathbf{x})=0$.
This induces $i_{m}^{*}(\mathbf{x})=(0,\dots,0)\in W_{m}(R)$. Then the above
map $g$ is given by

$\mathbf{x}\mapsto(0,\dots,0,{\bf b}_{m+1},\dots,{\bf b}_{m+n}).$

Consider the composition of $g$ with the ghost map $w$

$A\stackrel{{\scriptstyle g}}{{\rightarrow}}W_{[m]n}(B)\stackrel{{\scriptstyle
w}}{{\rightarrow}}\Pi_{[m]n}(B)$

given by

$\displaystyle\mathbf{x}$ $\displaystyle\stackrel{{\scriptstyle
g}}{{\longmapsto}}$ $\displaystyle(0,\dots,0,{\bf b}_{m+1},\dots,{\bf
b}_{m+n})\stackrel{{\scriptstyle w}}{{\longmapsto}}$ $\displaystyle\langle
0,\dots,0,\pi^{m+1}{\bf b}_{m+1},\pi^{m+1}{\bf b}_{m+1}^{q}+\pi^{m+2}{\bf
b}_{m+2},\dots,\pi^{m+1}{\bf b}_{m+1}^{q^{n}}+\cdots+\pi^{m+n}{\bf
b}_{m+n}\rangle.$

Now choose $B=N_{[m]n}A$. Then the identity map $\mathbbm{1}\in
N^{[m]n}X(N_{[m]n}A)$ corresponds to the element
$(\mathbf{x}\mapsto(0,\dots,0,\mathbf{x}_{m+1},\dots,\mathbf{x}_{m+n}))\in\widetilde{\mathrm{Hom}}(A,W_{[m]n}(N_{[m]n}A))$.
Note that if $\mathfrak{f}:N_{[m]n}A\rightarrow N_{[m]n+1}A$ is the associated
lift of Frobenius (associated to the generalised lateral Frobenius defined in
(5.3)), then we have the following commutative diagram of rings

$\textstyle{W_{[m]n}(N_{[m]n}A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{f}^{i}}$$\scriptstyle{w}$$\textstyle{\Pi_{[m]n}(N_{[m]n}A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{w}^{\circ
i}}$$\textstyle{W_{[m]n-i}(N_{[m]n+i}A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w}$$\textstyle{\Pi_{[m]n-i}(N_{[m]n+i}A)}$

where the element $(0,\dots,0,\mathbf{x}_{m+1},\dots,\mathbf{x}_{m+n})\in
W_{[m]n}(N_{[m]n}A)$ traces the following images

$\textstyle{(0,\dots,0,\mathbf{x}_{m+1},\dots,\mathbf{x}_{m+n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{f}^{i}}$$\scriptstyle{w}$$\textstyle{\langle
0,\dots,0,\pi^{m+1}\mathbf{x}_{m+1},\dots\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{w}^{\circ
i}}$$\textstyle{(0,\dots,0,\mathfrak{f}^{i}(\mathbf{x}_{m+1}),\dots)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w}$$\textstyle{\langle
0,\dots,0,\pi^{m+1}\mathbf{x}_{m+1}^{q^{i-1}}+\cdots+\pi^{m+i}\mathbf{x}_{m+i},\dots\rangle}$

Hence comparing the $(m+1)$-th ghost vector component in the above diagram we
obtain

$\displaystyle\pi^{m+1}\mathfrak{f}^{i}(\mathbf{x}_{m+1})$ $\displaystyle=$
$\displaystyle\pi^{m+1}\mathbf{x}_{m+1}^{q^{i-1}}+\pi^{m+2}\mathbf{x}_{m+2}^{q^{i-2}}+\cdots+\pi^{m+i}\mathbf{x}_{m+i}$
$\displaystyle\mathfrak{f}^{i}(\mathbf{x}_{m+1})$ $\displaystyle=$
$\displaystyle\mathbf{x}_{m+1}^{q^{i-1}}+\pi\mathbf{x}_{m+2}^{q^{i-2}}+\cdots+\pi^{i-1}\mathbf{x}_{m+i}.$

######  Corollary 7.1.

If $X={\mathbb{A}}^{N}=\operatorname{\mathrm{Spec}}R[\mathbf{x}]$, then for
all $n$ we have

$N^{[m]n}X\simeq J^{n-1}N^{[m]1}X.$

###### Proof.

We have $N^{[m]n}X=\operatorname{\mathrm{Spec}}N_{[m]n}A$ where
$N_{[m]n}A\simeq R[\mathbf{x}_{m+1},\dots,\mathbf{x}_{m+n}]$ such that the
coordinate functions satisfy

$\mathfrak{f}^{i}(\mathbf{x}_{m+1})=\mathbf{x}_{m+1}^{q^{i-1}}+\pi\mathbf{x}_{m+2}^{q^{i-2}}+\cdots+\pi^{i-1}\mathbf{x}_{m+i}$

for all $i=0,\dots,n$. Hence by Proposition 6.4 we have $N_{[m]n}A\simeq
J_{n-1}N_{[m]1}A$ and we are done. ∎

###  7.2. The case for Affine Schemes

Let $S=\operatorname{\mathrm{Spec}}R$. Consider $I\subset R[\mathbf{x}]$ to be
an ideal and let $(I,\delta I,\dots,\delta^{n}I)$ denote the ideal generated
by $\delta^{i}f,f\in I,i=0,\dots,n$. Let $i^{*}:R[\mathbf{x}]\rightarrow R$ be
a ring homomorphism such that $I\subset\ker i^{*}$, that is $i^{*}$
corresponds to an element
$P:\operatorname{\mathrm{Spec}}R\rightarrow\operatorname{\mathrm{Spec}}~{}(R[\mathbf{x}]/I)$.

Our next aim is to show that the above Corollary 7.1 is true for any affine
$X=\operatorname{\mathrm{Spec}}R[\mathbf{x}]/I$. Note that any $R$-point $P\in
X(R)$ corresponds to a surjective map $i^{*}:R[\mathbf{x}]\rightarrow R$ such
that $I\subset\ker(i^{*})$. By the universal property of jet rings the induced
$R$-point $P_{m}\in J^{m}X$ corresponds to the $R$-algebra map
$i^{*}_{m}:R[\mathbf{x},\dots,\mathbf{x}^{(m)}]\rightarrow R$ given by
$i^{*}_{m}(\mathbf{x}^{j})=\delta^{j}(i^{*}(\mathbf{x}))$ for all
$j=0,\dots,m$. Clearly we have $(I,\delta
I,\dots,\delta^{m}I)\subset\ker(i^{*}_{m})$. Set $A=R[\mathbf{x}]/I$. Then
$J_{m}A=\frac{R[\mathbf{x},\mathbf{x}^{\prime},\dots,\mathbf{x}^{(m)}]}{(I,\delta
I,\dots,\delta^{m}I)}$ and we have the induced morphism

$i^{*}_{m}:J_{m}A\rightarrow R.$

Recall the following tensor product of algebras

$\textstyle{J_{m+n}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}_{m}}$$\textstyle{N_{[m]n}A}$$\textstyle{J_{m}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}_{m}}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $N_{[m]n}A:=J_{m+n}A\otimes_{J_{m}A}R$ and
$N^{[m]n}X:=\operatorname{\mathrm{Spec}}N_{[m]n}A$. Then
$N^{[m]n}X=J^{m+n}X\times_{J^{m}X}S$.

######  Lemma 7.2.

For all $m$ and $n$ we have

$N_{[m]n}A\simeq\frac{R[\mathbf{x}^{(m+1)},\dots,\mathbf{x}^{(m+n)}]}{(i^{*}\delta^{m+1}I,\dots,i^{*}\delta^{m+n}I)}.$

###### Proof.

Directly follows from above. ∎

Consider $N_{[m]1}A\simeq R[\mathbf{x}^{(m+1)}]/(i^{*}\delta^{m+1}I)$. Then
consider the jet ring

$J_{n-1}(N_{[m]1}A)\simeq\frac{R[\mathbf{x}^{(m+1)},\dots,\mathbf{x}^{(m+n)}]}{(i^{*}\delta^{m+1}I,\Delta
i^{*}\delta^{m+1}I,\dots,\Delta^{n-1}i^{*}\delta^{m+1}I)}$

where $\Delta:R[\mathbf{x}^{(m)},\dots,\mathbf{x}^{(m+n)}]\rightarrow
R[\mathbf{x}^{(m)},\dots,\mathbf{x}^{(m+n+1)}]$ is given by
$\Delta(\mathbf{x}^{(m+h)})\rightarrow\mathbf{x}^{(m+h+1)}$ satisfying
$\mathfrak{f}(a)=a^{q}+\pi\Delta a$ for all $a\in J_{n}(N_{[m]1}A)\simeq
J_{n}(R[\mathbf{x}^{(m+1)}])=R[\mathbf{x}^{(m+1)},\dots,\mathbf{x}^{(m+n)}]$.
We will abbreviate $i^{*}_{m}$ as $i^{*}$ for the next proposition.

######  Proposition 7.3.

Let $b\in J_{m}A$ be such that $i^{*}b=i^{*}\delta
b=\cdots=i^{*}\delta^{m}b=0$. Then for all $n\geq 1$ we have

$\Delta^{n}i^{*}\delta^{m+1}b-i^{*}\delta^{m+n+1}b\in[i^{*}\delta^{m+1}b,\dots,i^{*}\delta^{m+n}b].$

More so the above implies that

$[i^{*}\delta^{m+1}b,\dots,i^{*}\delta^{m+n+1}b]=[i^{*}\delta^{m+1}b,\Delta
i^{*}\delta^{m+1}b,\dots,\Delta^{n}i^{*}\delta^{m+1}b].$

###### Proof.

For all $a\in J_{n}A$ by Theorem 5.3 we have

$i^{*}\phi^{m+2}(a)=\mathfrak{f}i^{*}\phi^{m+1}(a).$

By Lemma 6.3 we have

$\phi^{m+2}(a)=\pi^{m+2}\delta^{m+2}(a)+P_{m+1}(a,\delta
a,\dots,\delta^{m+1}a).$

for some centered polynomial $P_{m+1}(T_{0},\cdots T_{m+1})$ defined over $R$.
Applying $i^{*}$ to the above relation we obtain

$\displaystyle i^{*}\phi^{m+2}(a)$ $\displaystyle=$
$\displaystyle\pi^{m+2}i^{*}\delta^{m+2}(a)+i^{*}P_{m+1}(a,\delta
a,\dots\delta^{m+1}a)$ $\displaystyle=$
$\displaystyle\pi^{m+2}i^{*}\delta^{m+2}(a)+P_{m+1}(i^{*}a,i^{*}\delta a,\dots
i^{*}\delta^{m+1}a)$

Now note that applying Lemma 6.3 again we get

$\displaystyle\mathfrak{f}i^{*}\phi^{m+1}(a)$ $\displaystyle=$
$\displaystyle\mathfrak{f}i^{*}(\pi^{m+1}\delta^{m+1}a+P_{m}(a,\delta
a,\dots,\delta^{m}a))$ $\displaystyle=$
$\displaystyle\pi^{m+1}\mathfrak{f}(i^{*}\delta^{m+1}a)+\mathfrak{f}(P_{m}(i^{*}a,\dots,i^{*}\delta^{m}a))$
$\displaystyle=$ $\displaystyle\pi^{m+1}((i^{*}\delta^{m+1}a)^{q}+\pi\Delta
i^{*}\delta^{m+1}(a))+P_{m}(i^{*}a,\dots,i^{*}\delta^{m}a)^{q}$
$\displaystyle\hskip 128.0374pt+\pi\Delta
P_{m}(i^{*}a,\dots,i^{*}\delta^{m}a)$ $\displaystyle=$
$\displaystyle\pi^{m+2}\Delta
i^{*}\delta^{m+1}(a)+\pi^{m+1}(i^{*}\delta^{m+1}a)^{q}+P_{m}(i^{*}a,\dots,i^{*}\delta^{m}a)^{q}$
$\displaystyle\hskip 128.0374pt+\pi\Delta
P_{m}(i^{*}a,\dots,i^{*}\delta^{m}a)$

Hence combining the above two we get

$\displaystyle\pi^{m+2}i^{*}\delta^{m+2}(a)$ $\displaystyle=$
$\displaystyle\pi^{m+2}\Delta
i^{*}\delta^{m+1}(a)+\pi^{m+1}(i^{*}\delta^{m+1}a)^{q}$
$\displaystyle+P_{m+1}(i^{*}a,i^{*}\delta a,\dots i^{*}\delta^{m+1}a)$
$\displaystyle+P_{m}(i^{*}a,\dots,i^{*}\delta^{m}a)^{q}+\pi\Delta
P_{m}(i^{*}a,\dots,i^{*}\delta^{m}a)$

This gives us

$\displaystyle i^{*}\delta^{m+2}a$ $\displaystyle=$ $\displaystyle\Delta
i^{*}\delta^{m+1}a+\frac{1}{\pi^{m+2}}(P_{m}(i^{*}a,\dots
i^{*}\delta^{m}a)^{q}+\pi^{m+1}(i^{*}\delta^{m+1}a)^{q}$ $\displaystyle\hskip
42.67912pt+\pi\Delta
P_{m}(i^{*}a,\dots,i^{*}\delta^{m}a)-P_{m+1}(i^{*}a,\dots,i^{*}\delta^{m+1}a))\hskip
14.22636pt\cdots~{}(*)$

Substituting $a$ for $b$ and $i^{*}b=i^{*}\delta b=\cdots=i^{*}\delta^{m}b=0$
in the above we obtain

$\displaystyle i^{*}\delta^{m+2}b$ $\displaystyle=$ $\displaystyle\Delta
i^{*}\delta^{m+1}b+\frac{1}{\pi^{m+2}}(\pi^{m+1}(i^{*}\delta^{m+1}b)^{q}-P_{m+1}(0,\dots,0,i^{*}\delta^{m+1}b))$
$\displaystyle=$ $\displaystyle\Delta
i^{*}\delta^{m+1}b+H_{1}(i^{*}\delta^{m+1}b)$

where $H_{1}(t)=\frac{1}{\pi^{m+2}}(\pi^{m+1}t^{q}-P_{m+1}(0,\dots,0,t))$ and
this shows the result is true for $n=1$. Now we will use induction to prove
our result and assume that it holds true for $n-1$. Then we have

$\Delta^{n-1}i^{*}\delta^{m+1}b-i^{*}\delta^{m+n}b\in[i^{*}\delta^{m+1}b,i^{*}\delta^{m+2}b,\dots,i^{*}\delta^{m+n-1}b]$

which in turn implies

(7.3)
$\displaystyle[i^{*}\delta^{m+1}b,\dots,i^{*}\delta^{m+n}b]=[i^{*}\delta^{m+1}b,\Delta
i^{*}\delta^{m+1}b,\dots,\Delta^{n-1}i^{*}\delta^{m+1}b],$

since
$[i^{*}\delta^{m+1}b,i^{*}\delta^{m+2}b,\dots,i^{*}\delta^{m+n-1}b]=[i^{*}\delta^{m+1}b,\Delta
i^{*}\delta^{m+1}b,\dots,\Delta^{n-2}i^{*}\delta^{m+1}b]$ by the induction
hypothesis. Let

(7.4)
$\displaystyle\Delta^{n-1}i^{*}\delta^{m+1}b-i^{*}\delta^{m+n}b=L(i^{*}\delta^{m+1}b,\Delta
i^{*}\delta^{m+1}b,\dots,\Delta^{n-2}i^{*}\delta^{m+1}b)$

for some centered $L$. Then applying $\Delta$ to (7.4) gives us

$\displaystyle\Delta^{n}i^{*}\delta^{m+1}b-\Delta i^{*}\delta^{m+n}b+$
$\displaystyle C_{\pi}(\Delta^{n-1}i^{*}\delta^{m+1}b,-i^{*}\delta^{m+n}b)$
$\displaystyle=$ $\displaystyle\Delta L(i^{*}\delta^{m+1}b,\Delta
i^{*}\delta^{m+1}b,\dots,\Delta^{n-2}i^{*}\delta^{m+1}b)$ $\displaystyle=$
$\displaystyle
M(i^{*}\delta^{m+1}b,\dots,\Delta^{n-1}i^{*}\delta^{m+1}b)\mbox{ by Lemma
\ref{sideal}(2)}$

for some centered $M$ of degree $\geq 1$. Therefore we obtain

$\displaystyle\Delta^{n}i^{*}\delta^{m+1}b-\Delta i^{*}\delta^{m+n}b$
$\displaystyle=$ $\displaystyle
M(i^{*}\delta^{m+1}b,\dots,\Delta^{n-1}i^{*}\delta^{m+1}b)$ $\displaystyle-
C_{\pi}(\Delta^{n-1}i^{*}\delta^{m+1}b,-i^{*}\delta^{m+n}b)$

Note that substituting $i^{*}\delta^{m+n}b$ by
$\Delta^{n-1}i^{*}\delta^{m+1}b-L(i^{*}\delta^{m+1}b,\dots,\Delta^{n-2}i^{*}\delta^{m+1}b)$
as in (7.4) we observe that

$C_{\pi}(\Delta^{n-1}i^{*}\delta^{m+1}b,-i^{*}\delta^{m+n}b)=D(i^{*}\delta^{m+1}b,\dots,\Delta^{n-1}i^{*}\delta^{m+1}b)$

for some centered $D$ of degree $\geq 1$. Hence we obtain

(7.5) $\displaystyle\Delta^{n}i^{*}\delta^{m+1}b-\Delta
i^{*}\delta^{m+n}b=H(i^{*}\delta^{m+1}b,\dots,\Delta^{n-1}i^{*}\delta^{m+1}b)$

where $H=M-D$. Now substitute $a=\delta^{n-1}b$ in $(*)$. Then we have

$\displaystyle i^{*}\delta^{m+n+1}b$ $\displaystyle=$ $\displaystyle\Delta
i^{*}\delta^{m+n}b+\frac{1}{\pi^{m+2}}(P_{m}(i^{*}\delta^{n-1}b,\dots,i^{*}\delta^{m+n-1}b)^{q}+\pi^{m+1}(i^{*}\delta^{m+n}b)^{q}$
$\displaystyle+\pi\Delta
P_{m}(i^{*}\delta^{n-1}b,\dots,i^{*}\delta^{m+n-1}b)-P_{m+1}(i^{*}\delta^{n-1}b,\dots,i^{*}\delta^{m+n}b))$

Then setting

$i^{*}b=i^{*}\delta b=\cdots=i^{*}\delta^{m}b=0$

in the above relation we obtain

(7.6) $\displaystyle\Delta
i^{*}\delta^{m+n}b-i^{*}\delta^{m+n+1}b=\tilde{K}(i^{*}\delta^{m+1}b,\dots,i^{*}\delta^{m+n}b)$

for some centered $\tilde{K}$. But since
$[i^{*}\delta^{m+1}b,\dots,i^{*}\delta^{m+n}b]=[i^{*}\delta^{m+1}b,\dots,\Delta^{n-1}i^{*}\delta^{m+1}b]$
as in (7.3) we must have

(7.7)
$\displaystyle\tilde{K}(i^{*}\delta^{m+1}b,\dots,i^{*}\delta^{m+n}b)=K(i^{*}\delta^{m+1}b,\dots,\Delta^{n-1}i^{*}\delta^{m+1}b)$

for some centered $K$. Therefore adding (7.5) and (7.6) along with
substituting (7.7) in it we obtain

$\displaystyle\Delta^{n}i^{*}\delta^{m+1}b-i^{*}\delta^{m+n+1}b$
$\displaystyle=$ $\displaystyle
K(i^{*}\delta^{m+1}b,\dots,\Delta^{n-1}i^{*}\delta^{m+1}b)$
$\displaystyle\hskip
56.9055pt+H(i^{*}\delta^{m+1}b,\dots,\Delta^{n-1}i^{*}\delta^{m+1}b)$

Hence we have

$\Delta^{n}i^{*}\delta^{m+1}b-i^{*}\delta^{m+n+1}b\in[i^{*}\delta^{m+1}b,\dots,\Delta^{n-1}i^{*}\delta^{m+1}b]$

and therefore we also have

$\Delta^{n}i^{*}\delta^{m+1}b-i^{*}\delta^{m+n+1}b\in[i^{*}\delta^{m+1}b,\dots,i^{*}\delta^{m+n}b].$

Then the above clearly implies

$[i^{*}\delta^{m+1}b,\dots,\Delta^{n}i^{*}\delta^{m+1}b]=[i^{*}\delta^{m+1}b,\dots
i^{*}\delta^{m+n+1}b]$

and we are done. ∎

######  Corollary 7.4.

For all $m$ and $n$ we have

$(i^{*}\delta^{m+1}b,\dots,\Delta^{n}i^{*}\delta^{m+1}b)=(i^{*}\delta^{m+1}b,\dots
i^{*}\delta^{m+n+1}b).$

###### Proof.

This follows immediately from the fact that for all $m$ and $n$ by Proposition
7.3 we have

$[i^{*}\delta^{m+1}b,\dots,\Delta^{n}i^{*}\delta^{m+1}b]=[i^{*}\delta^{m+1}b,\dots,i^{*}\delta^{m+n+1}b].$

∎

######  Corollary 7.5.

For $A$ as above, for all $n\geq 1$ we have

$N_{[m]n}A=J_{n-1}(N_{[m]1}A).$

###### Proof.

Since $N_{[m]1}A=R[\mathbf{x}^{(m+1)}]/(i^{*}_{m}\delta^{m+1}I)$ we have

$\displaystyle J_{n-1}(N_{[m]1}A)$ $\displaystyle=$
$\displaystyle\frac{R[\mathbf{x}^{(m+1)},\dots,\mathbf{x}^{(m+n)}]}{(i^{*}_{m}\delta^{m+1}I,\dots,\Delta^{n-1}i^{*}_{m}\delta^{m+1}I)}$
$\displaystyle\simeq$
$\displaystyle\frac{R[\mathbf{x}^{m+1},\dots,\mathbf{x}^{(m+n)}]}{(i^{*}_{m}\delta^{m+1}I,\dots,i^{*}_{m}\delta^{m+1}I)}\hskip
14.22636pt\mbox{(by Corollary \ref{sameideal})}$

and we are done. ∎

######  Theorem 7.6.

If $X/S$ is affine, then for all $n\geq m$ we have

$N^{[m]n}X\simeq J^{n-1}(N^{[m]1}X).$

###### Proof.

The result follows directly from corollary 7.5. ∎

###  7.3. The $\pi$-formal Schemes case

Let $\mathcal{O}$ is a Dedekind domain of characteristic $0$ and $R$ is a
$\pi$-adically complete discrete valuation ring. Let $l=R/\pi R$ be its
residue field. Denote $S=\operatorname{\mathrm{Spf}}R$. Let $X$ be a
$\pi$-formal scheme over $\operatorname{\mathrm{Spf}}R$ where $R$ is now
$\pi$-adically complete and $P:\operatorname{\mathrm{Spf}}R\rightarrow X$ be
an $R$-marked point. For all $n\geq 1$, let $u:J^{n}X\rightarrow X$ be the
natural projection morphism.

######  Lemma 7.7.

For any open $\pi$-formal subscheme $U\hookrightarrow X$ we have

$u^{-1}U:=U\times_{X}J^{n}X\simeq J^{n}U.$

###### Proof.

This is Proposition 1.7 of [13]. ∎

######  Theorem 7.8.

If $X$ is a $\pi$-formal scheme over $\operatorname{\mathrm{Spf}}R$, then for
all $n\geq m$ we have

$N^{[m]n}X\simeq J^{n-1}(N^{[m]1}X).$

###### Proof.

Let $U\hookrightarrow X$ be an open affine $\pi$-formal scheme such that the
marked point $P:\operatorname{\mathrm{Spf}}R\rightarrow X$ factors through
$U$. Hence by lemma 7.7 we have that the morphism
$P^{m}:\operatorname{\mathrm{Spf}}R\rightarrow J^{m}X$ factors through the
open affine $\pi$-formal scheme $J^{m}U$. Then for each $n$ we have

$N^{[m]n}X:=J^{m+n}X\times_{J^{m}X}P^{m}=J^{m+n}U\times_{J^{m}U}P^{m}.$

Hence it is sufficient to assume $X$ is affine and then the result follows
from Theorem 7.6.

∎

Let $\mathbb{W}_{n}$ denote the Witt vectors of length $n+1$ as a group (in
fact ring) object in the category of $\pi$-formal schemes. Then
$\mathbb{W}_{n}\simeq\operatorname{\mathrm{Spf}}R[x_{0},\dots,x_{n}]\hat{\ }$
and the ring $R[x_{0},\dots,x_{n}]\hat{\ }$ has a coalgebra structure induced
from the Witt vector addition. For each $n$ let us denote $N^{n}G:=N^{[0]n}G$.

######  Lemma 7.9.

For all $n$ we have $J^{n}\hat{\mathbb{G}}_{\mathrm{a}}\simeq\mathbb{W}_{n}$.

###### Proof.

Let $x$ be a coordinate at the origin of $\hat{\mathbb{G}}_{\mathrm{a}}$. Then
$\hat{\mathbb{G}}_{\mathrm{a}}\simeq\operatorname{\mathrm{Spf}}R[x]\hat{\ }$.
Now as a functor of points $J^{n}G$, given any $R$-algebra $B$ we have,

$J^{n}G(B)=\hat{\mathbb{G}}_{\mathrm{a}}(W_{n}(B))=\mathrm{Hom}_{R}(R[x]\hat{\
},W_{n}(B))\simeq\mathbb{W}_{n}(B)$

and we are done. ∎

######  Theorem 7.10.

Let $R$ be of characteristic 0 satisfying $v_{\pi}(p)\leq p-2$. Let $G$ be a
commutative smooth $\pi$-formal group scheme of relative dimension $d$ over
$\operatorname{\mathrm{Spf}}R$. Then
$N^{[m]n}G\simeq\left(\mathbb{W}_{n-m-1}\right)^{d}$ as $\pi$-formal group
schemes. In particular we have the following short exact sequence of
$\pi$-formal group schemes

$0\rightarrow\left(\mathbb{W}_{n-m-1}\right)^{d}\rightarrow
J^{m+n}G\rightarrow J^{m}G\rightarrow 0$

###### Proof.

Since $G$ is a smooth $\pi$-formal group scheme, we have the following short
exact sequence

$0\rightarrow N^{[m]n}G\rightarrow J^{m+n}G\rightarrow J^{m}G\rightarrow 0$

of $\pi$-formal group schemes. By Proposition 2.2 and Lemma $2.3$ in [10] we
have $N^{[m]1}G\simeq(\hat{\mathbb{G}}_{\mathrm{a}})^{d}$. Hence by Theorem
7.8 and Lemma 7.9 we get

$N^{[m]n}G\simeq
J^{n-1}(N^{[m]1}G)\simeq(J^{n-1}\hat{\mathbb{G}}_{\mathrm{a}})^{d}\simeq(\mathbb{W}_{n-1})^{d}$

and we are done. ∎

######  Corollary 7.11.

Let $R$ and $G$ be as above. Then
$N^{n}G\simeq\left(\mathbb{W}_{n-1}\right)^{d}$ and $J^{n}G$ satisfies the
following short exact sequence of $\pi$-formal group schemes

$0\rightarrow\left(\mathbb{W}_{n-1}\right)^{d}\rightarrow J^{n}G\rightarrow
G\rightarrow 0.$

###### Proof.

It follows directly from Theorem 7.10 for $m=0$. ∎

##  8\. Delta Isocrystals and their Integral Models

From now on, $\mathcal{O}$ is a Dedekind domain of characteristic $0$ and $R$
is a $\pi$-adically complete discrete valuation ring with a $\pi$-derivation
$\delta$ lifting the one on $\mathcal{O}$. Let $l=R/\pi R$ be its residue
field. Also assume $v_{\pi}(p)\leq p-2$. Denote
$S=\operatorname{\mathrm{Spf}}R$. Let $G$ be a $\pi$-formal group scheme over
$S$ of relative dimension $g$. From this section onwards, we will restrict
ourselves to the case $m=0$. For any group scheme $G$, let us denote
$J^{n}G:=J^{[0]n}G$ and $N^{n}G=N^{[0]n}G$.

Hence by Theorem 7.8, we have $N^{n}G\simeq J^{n-1}(N^{1}G)$ and hence
$\mathfrak{f}=\phi_{N^{1}G}$.

###  8.1. Delta Characters of Group Schemes

We recall some basic results on $\delta$-characters from Sections $7,8$ of [9]
that led to the construction of a natural isocrystal ${\bf H}_{\delta}(A)$
over $K$, which we call the $\delta$-isocrystal associated to an abelian
scheme $A$. We will now show that ${\bf H}_{\delta}(A)$ is a non-degenerate
isocrystal in the case when $A$ is an elliptic curve over $\mathbb{Z}_{p}$.

Let $T^{*}$ be a prolongation sequence. For any $s\geq 0$, define the shifted
prolongation sequence to be $T^{*+s}=\\{T^{s+n}\\}_{n=0}^{\infty}$. Then a
morphism $\Theta:J^{n}G\rightarrow\hat{\mathbb{G}}_{\mathrm{a}}$ is called a
$\delta$-character of $G$ of order $\leq n$. By the universal property of jet
spaces, such a $\Theta$ induces a morphism of prolongation sequences
$\Theta:J^{*+n}G\rightarrow\hat{\mathbb{G}}_{\mathrm{a}}$.

Define a $\delta$-character of order $n$,
$\Theta:G\rightarrow\hat{\mathbb{G}}_{\mathrm{a}}$ to be a $\delta$-morphism
of order $n$ from $G$ to $\hat{\mathbb{G}}_{\mathrm{a}}$, which is also a
group homomorphism of $\pi$-formal group schemes. By the universal property of
jet schemes as in Proposition 4.1, an order $n$ $\delta$-character is
equivalent to a homomorphism
$\Theta:J^{n}G\rightarrow\hat{\mathbb{G}}_{\mathrm{a}}$ of $\pi$-formal group
schemes over $S$. We denote the group of $\delta$-characters of order $n$ by
${\bf X}_{n}(G)$:

${\bf X}_{n}(G)=\mathrm{Hom}_{S}(J^{n}G,\hat{\mathbb{G}}_{\mathrm{a}}).$

Note that ${\bf X}_{n}(G)$ comes with an $R$-module structure via
$\hat{\mathbb{G}}_{\mathrm{a}}$. For any $R$-module $M$, let us denote

$M_{\phi}=R\otimes_{\phi,R}M.$

Then recall that the $\phi$-linear map $\phi^{*}:{\bf X}_{n-1}(G)\to{\bf
X}_{n}(G)$ induces a linear map ${\bf X}_{n-1}(G)_{\phi}\to{\bf X}_{n}(G)$,
which we will abusively also denote $\phi^{*}$. We then define

${\bf
H}_{n}(G)=\frac{\mathrm{Hom}(N^{n}G,\hat{\mathbb{G}}_{\mathrm{a}})}{i^{*}\phi^{*}({\bf
X}_{n-1}(G)_{\phi})}.$

Note that $u:N^{n+1}G\rightarrow N^{n}G$ induces
$u^{*}:\mathrm{Hom}(N^{n}G,\hat{\mathbb{G}}_{\mathrm{a}})\rightarrow\mathrm{Hom}(N^{n+1}G,\hat{\mathbb{G}}_{\mathrm{a}})$.
Moreover, since $u$ commutes with both $i$ and $\phi$, we have

$u^{*}i^{*}\phi^{*}({\bf X}_{n}(G))=i^{*}\phi^{*}u^{*}({\bf X}_{n}(G))\subset
i^{*}\phi^{*}({\bf X}_{n+1}(G)),$

and hence $u$ also induces a map $u^{*}:{\bf H}_{n}(G)\rightarrow{\bf
H}_{n+1}(G)$. Define

(8.1) ${\bf H}_{\delta}(G)=\varinjlim{\bf H}_{n}(G)$

where the limit is taken in the category of $R$-modules. Similarly,
$\mathfrak{f}:N^{n+1}G\rightarrow N^{n}G$ induces
$\mathfrak{f}^{*}:\mathrm{Hom}(N^{n}G,\hat{\mathbb{G}}_{\mathrm{a}})\rightarrow\mathrm{Hom}(N^{n+1}G,\hat{\mathbb{G}}_{\mathrm{a}})$,
which descends to a $\phi$-semilinear morphism of $R$-modules

(8.2) $\mathfrak{f}^{*}:{\bf H}_{n}(G)\rightarrow{\bf H}_{n+1}(G)$

because we have $\mathfrak{f}^{*}i^{*}\phi^{*}({\bf
X}_{n-1}(G))=i^{*}\phi^{*}\phi^{*}({\bf X}_{n-1}(G))\subset i^{*}\phi^{*}{\bf
X}_{n}(G)$. This in turn induces a $\phi$-semilinear endomorphism
$\mathfrak{f}^{*}:{\bf H}_{\delta}(G)\rightarrow{\bf H}_{\delta}(G)$.

######  Proposition 8.1.

For any character $\Theta$ in ${\bf X}_{n}(G)$, let the derivative at the
identity with respect to our chosen coordinates be
$D\Theta=(A_{0},\cdots,A_{n})$ where $A_{j}\in\mbox{Mat}_{1\times g}(R)$.

  1. (1)

We have

$i^{*}\phi^{*}\Theta=\mathfrak{f}^{*}(i^{*}\Theta)+\gamma.\mathbf{\Psi}_{1},$

where $\gamma=\pi A_{0}$.

  2. (2)

For $n\geq 2$, we have

$i^{*}(\phi^{\circ n})^{*}\Theta=(\mathfrak{f}^{n-1})^{*}i^{*}\phi^{*}\Theta.$

###### Proof.

See Proposition $6.3$ in [9]. ∎

###  8.2. Finiteness of the $R$-module ${\bf X}_{\mathrm{prim}}(A)$ and ${\bf
H}_{\delta}(A)$

Let $A$ be a $\pi$-formal abelian scheme of relative dimension $g$ over
$\operatorname{\mathrm{Spf}}R$. For every $n$ we have the following short
exact sequence of $\pi$-formal schemes

(8.3) $0\rightarrow N^{n}\rightarrow J^{n}A\rightarrow A\rightarrow 0.$

Applying $\mathrm{Hom}(-,\hat{\mathbb{G}}_{\mathrm{a}})$ to the above short
exact sequence gives us

(8.4) $0\rightarrow{\bf
X}_{n}(A)\rightarrow\mathrm{Hom}(N^{n},\hat{\mathbb{G}}_{\mathrm{a}})\stackrel{{\scriptstyle\partial}}{{\rightarrow}}\mathrm{Ext}(A,\hat{\mathbb{G}}_{\mathrm{a}}).$

Then by the theory of extensions of groups that admit rational sections (see
[24], page 185, Theorem 7) we have
$\mathrm{Ext}(A,\hat{\mathbb{G}}_{\mathrm{a}})\simeq
H^{1}(A,\mathcal{O}_{A})\simeq R^{g}$. Let ${\bf
I}_{n}(A):=\mbox{image}(\partial)$. Note that since for all $n$, there are
maps $\mathrm{Hom}(N^{n},\hat{\mathbb{G}}_{\mathrm{a}})\stackrel{{\scriptstyle
u^{*}}}{{\hookrightarrow}}\mathrm{Hom}(N^{n+1},\hat{\mathbb{G}}_{\mathrm{a}})$,
we have ${\bf I}_{n}(A)\subset{\bf I}_{n+1}(A)$. Define

(8.5) ${\bf I}(A):=\varinjlim{\bf I}_{n}(A)$

and

$h_{i}=\operatorname{\mathrm{rk}}{\bf I}_{i}(A)-\operatorname{\mathrm{rk}}{\bf
I}_{i-1}(A)$

for all $i\geq 1$. We define the upper splitting number to be the smallest
number $m_{\mathrm{u}}\geq 1$ such that $h_{n}=0$ for all $n\geq
m_{\mathrm{u}}$. Note that $m_{\mathrm{u}}$ exists since

${\bf I}_{0}(A)\subset{\bf
I}_{1}(A)\subset\cdots\subset\mathrm{Ext}(A,\hat{\mathbb{G}}_{\mathrm{a}})\cong
R^{g}.$

We define the lower splitting number to be the unique $m_{\mathrm{l}}$
satisfying ${\bf X}_{m_{\mathrm{l}}}(A)\neq\\{0\\}$ and ${\bf
X}_{m_{\mathrm{l}}-1}(A)=\\{0\\}$. We say a $\delta$-character $\Theta\in{\bf
X}_{n}(A)_{K}$ is primitive if

$\Theta\notin u^{*}{\bf X}_{n-1}(A)_{K}+\phi^{*}({\bf
X}_{n-1}(A)_{K})_{\phi}.$

######  Theorem 8.2 (Buium, Borger-Saha).

For any abelian scheme $A$ of dimension $g$, ${\bf X}_{\infty}(A)_{K}$ is
freely $K\\{\phi^{*}\\}$-generated by $g$ $\delta$-characters of order at most
$g+1$.

###### Proof.

See Theorem $1.1$ in [9] and Theorem B in [10]. ∎

Now define

${\bf X}_{\mathrm{prim}}(A):=\varinjlim{\bf X}_{n}(A)/\phi^{*}{\bf
X}_{n-1}(A)_{\phi}.$

######  Corollary 8.3.

We have dimension of ${\bf X}_{\mathrm{prim}}(A)_{K}$ as a $K$-vector space to
be $g$, and

${\bf X}_{\mathrm{prim}}(A)_{K}\simeq{\bf
X}_{m_{\mathrm{u}}}(A)_{K}/\phi^{*}({\bf
X}_{m_{\mathrm{u}}-1}(A)_{K})_{\phi}.$

###### Proof.

See Corollary $7.7$ in [9]. ∎

######  Corollary 8.4.

If $g=1$, then $m_{\mathrm{l}}=m_{\mathrm{u}}=:m$ and ${\bf
X}_{\mathrm{prim}}(A)_{K}\simeq{\bf X}_{m}(A)_{K}$.

###### Proof.

See Corollary $7.8$ in [9]. ∎

We will, in fact, show that the module of $\delta$-characters is finitely
generated as an $R\\{\phi^{*}\\}$-module. As a consequence, it gives an
integral lattice of the $\delta$-isocrystal.

######  Theorem 8.5.

For $n\geq m_{u}$, we have the following commutative diagram

$\textstyle{{\bf X}_{n}(A)/{\bf
X}_{n-1}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}}$$\scriptstyle{\phi^{*}}$$\scriptstyle{\sim}$$\textstyle{{\bf
X}_{n+1}(A)/{\bf
X}_{n}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}}$$\textstyle{\mathrm{Hom}(N^{n},\hat{\mathbb{G}}_{\mathrm{a}})/\mathrm{Hom}(N^{n-1},\hat{\mathbb{G}}_{\mathrm{a}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{f}^{*}}$$\scriptstyle{\sim}$$\textstyle{\mathrm{Hom}(N^{n+1},\hat{\mathbb{G}}_{\mathrm{a}})/\mathrm{Hom}(N^{n},\hat{\mathbb{G}}_{\mathrm{a}})}$

with all the maps bijective .

###### Proof.

From Proposition 6.4 in [9] we already have $i^{*}$ and $\phi^{*}$ are
injective and $\mathfrak{f}^{*}$ is bijective. Consider the exact sequence of
finite free $R$-modules

$0\longrightarrow{\bf
X}_{n}(A)\longrightarrow\mathrm{Hom}(N^{n},\hat{\mathbb{G}}_{\mathrm{a}})\xrightarrow{\partial}{\bf
I}_{n}(A)\longrightarrow 0.$

Since ${\bf I}_{n}(A)$ is free, there is a section $s_{n}:{\bf
I}_{n}(A)\longrightarrow\mathrm{Hom}(N^{n},\hat{\mathbb{G}}_{\mathrm{a}})$.
Moreover, this section implies that the following exact sequence of
$R$-modules

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf
X}_{n}(A)/{\bf
X}_{n-1}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}}$$\textstyle{\frac{\mathrm{Hom}(N^{n},\hat{\mathbb{G}}_{\mathrm{a}})}{\mathrm{Hom}(N^{n-1},\hat{\mathbb{G}}_{\mathrm{a}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{{\bf
I}_{n}A/{\bf
I}_{n-1}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{0}$

splits. By Proposition 5.2 in [9] we have
$\mathrm{Hom}(N^{n},\hat{\mathbb{G}}_{\mathrm{a}})/\mathrm{Hom}(N^{n-1},\hat{\mathbb{G}}_{\mathrm{a}})$
is a free module of rank $g$. Hence ${\bf I}_{n}A/{\bf I}_{n-1}A$ is also free
being a sub-module of a free module over a discrete valuation ring. On the
other hand,

$\mathrm{rk}\left({\bf I}_{n}A/{\bf I}_{n-1}A\right)=\mathrm{rk}({\bf
I}_{n}A)-\mathrm{rk}({\bf I}_{n-1}A)=h_{n}.$

Since $h_{n}=0$ for all $n\geq m_{u}$, we have ${\bf I}_{n}A/{\bf I}_{n-1}A=0$
and $i^{*}$ is bijective. Now all the maps are bijective, that will make
$\phi^{*}$ also bijective. ∎

Let us define a finite subset $\mathbb{D}_{i}\subset{\bf X}_{i}(A)$ such that
the image of its elements in

$\dfrac{{\bf X}_{i}(A)}{u^{*}{\bf X}_{i-1}(A)+\phi^{*}({\bf
X}_{i-1}(A))_{\phi}}$

are distinct and form a minimal generating set. Since $R$ is a discrete
valuation ring and the modules are finite, the cardinality of such a minimal
generating set is well-defined. Such a $\mathbb{D}_{i}$ is called a _primitive
generating set_ of ${\bf X}_{i}(A)$. Let us define the subset

$S_{n}(\mathbb{D}_{i})=\\{{\phi^{*}}^{h}\Theta~{}|~{}\Theta\in\mathbb{D}_{i},0\leq
h\leq n-i\\}.$

######  Lemma 8.6.

The $R$-module ${\bf X}_{n}(A)$ can be generated by

$S_{n}(\mathbb{D}_{1})\cup S_{n}(\mathbb{D}_{2})\cup\ldots\cup
S_{n}(\mathbb{D}_{n}).$

###### Proof.

We will proceed by induction on $n$. For $n=1$ is clear from the definition.
Let us denote

$\mathbb{S}_{i}=S_{i}(\mathbb{D}_{1})\cup S_{i}(\mathbb{D}_{2})\cup\ldots\cup
S_{i}(\mathbb{D}_{i}).$

We assume that $\mathbb{S}_{n-1}$ generates ${\bf X}_{n-1}(A)$ as $R$-module.
Consider the short exact sequence of $R$-modules

$0\longrightarrow u^{*}{\bf X}_{n-1}(A)+\phi^{*}({\bf
X}_{n-1}(A))_{\phi}\longrightarrow{\bf X}_{n}(A)\longrightarrow\dfrac{{\bf
X}_{n}(A)}{u^{*}{\bf X}_{n-1}(A)+\phi^{*}({\bf
X}_{n-1}(A))_{\phi}}\longrightarrow 0.$

Invoking induction hypothesis, we have ${\bf X}_{n}(A)$ is generated by

$\mathbb{D}_{n}\cup\mathbb{S}_{n-1}\cup\phi^{*}(\mathbb{S}_{n-1})\subset\mathbb{S}_{n}.$

∎

######  Theorem 8.7.

The $R$-module ${\bf X}_{\mathrm{prim}}(A)$ is free of rank $g$.

###### Proof.

Since $R$ is a discrete valuation ring, it is enough to prove that ${\bf
X}_{\mathrm{prim}}(A)$ is finitely generated and $\pi$-torsion free. We will
firstly show ${\bf X}_{\mathrm{prim}}(A)$ is a finitely generated $R$-module.

By Theorem 8.5, for $n\geq m_{u}$, we have the following commutative diagram

$\textstyle{{\bf X}_{n}(A)/{\bf
X}_{n-1}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{n}}$$\scriptstyle{\phi^{*}}$$\scriptstyle{\sim}$$\textstyle{{\bf
X}_{n+1}(A)/{\bf
X}_{n}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{n+1}}$$\textstyle{\dfrac{{\bf
X}_{n}(A)}{u^{*}{\bf X}_{n-1}(A)+\phi^{*}({\bf
X}_{n-1}(A))_{\phi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\phi}^{*}}$$\textstyle{\dfrac{{\bf
X}_{n+1}(A)}{u^{*}{\bf X}_{n}(A)+\phi^{*}({\bf X}_{n}(A))_{\phi}}}$

Since, the vertical maps are surjective and $\phi^{*}$ is bijective, we have
$\bar{\phi}^{*}$ also surjective. But note that $\bar{\phi}^{*}$ is a zero
map. Hence this implies $\dfrac{{\bf X}_{n+1}(A)}{u^{*}{\bf
X}_{n}(A)+\phi^{*}({\bf X}_{n}(A))_{\phi}}$ is zero for $n\geq m_{u}$.
Therefore by the above lemma, ${\bf X}_{\infty}(A)$ is generated by

$\mathbb{D}_{1}\cup\mathbb{D}_{2}\cup\ldots\cup\mathbb{D}_{m_{u}}$

as $R\\{\phi^{*}\\}$-module, hence its image generates ${\bf
X}_{\mathrm{prim}}(A)$ as $R$-module.

Now we will show that ${\bf X}_{\mathrm{prim}}(A)$ is $\pi$-torsion free.

Let $\pi[f]=0$ in ${\bf X}_{\mathrm{prim}}(A)$, i.e there exists a
$g=\displaystyle{\sum_{I}}a_{I}\mathbf{x}^{I}\in{\bf X}_{\infty}(A)$ such that

$\pi
f(\mathbf{x})=\phi^{*}g(\mathbf{x})=g(\phi(\mathbf{x}))=\displaystyle{\sum_{I}}a_{I}\mathbf{x}^{qI}+\pi
h(\mathbf{x}).$

The above equation shows that each $a_{I}$ is divisible by $\pi$, hence
$\frac{g(\mathbf{x})}{\pi}\in{\bf X}_{\infty}(A)$. Since ${\bf X}_{\infty}(A)$
is free and hence $\pi$-torsion free, we get $f=\phi^{*}\frac{g}{\pi}$. Hence
${\bf X}_{\mathrm{prim}}(A)$ is free and by Lemma 8.3 the rank has to be $g$.∎

Recall from Section $8$ of [9] that the following diagram of short exact
sequences of $R$-modules commutes.

(8.6)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf
X}_{\mathrm{prim}}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Upsilon}$$\textstyle{{\bf
H}_{\delta}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{{\bf
I}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{\mathrm{Lie}}(A)}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Ext}^{\sharp}(A,\hat{\mathbb{G}}_{\mathrm{a}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Ext}(A,\hat{\mathbb{G}}_{\mathrm{a}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

Hence as a direct consequence we get:

######  Theorem 8.8.

Let $A$ be an abelian scheme of dimension $g$ over $R$. Then ${\bf
H}_{\delta}(A)$ is a free $R$-module with
$g\leq\operatorname{\mathrm{rk}}_{R}{\bf H}_{\delta}(A)\leq 2g$.

######  Corollary 8.9.

For an abelian scheme $A$ of dimension $g$, ${\bf X}_{\infty}(A)$ is freely
generated as $R\\{\phi^{*}\\}$-module by $g$ $\delta$-characters of order at
most $g+1$.

###### Proof.

Since $\mathbb{D}_{1}\cup\mathbb{D}_{2}\cup\ldots\cup\mathbb{D}_{m_{u}}$
generates ${\bf X}_{\mathrm{prim}}(A)$ as $R$-module, we can choose a minimal
generating subset and its cardinality has to be $d$, because ${\bf
X}_{\mathrm{prim}}(A)$ has rank $g$ and $R$ is a discrete valuation ring.
Hence ${\bf X}_{\infty}(A)$ will be freely generated by $g$
$\delta$-characters of order upto $m_{u}\leq g+1$ as in Theorem 7.6 in [9]. ∎

##  9\. Delta Isocrystal and Crystalline Cohomology of Elliptic Curves

In this section, we prove the comparison isomorphism between the
$\delta$-isocrystal and the crystalline cohomology of an elliptic curve over
$\mathbb{Q}_{p}$ and recover the usual Hodge filtration in terms of
$\delta$-characters. Let $A$ be an elliptic curve over $R$ and fix an
invariant differential 1-form $\omega$ of $A$. Recall from (8.4) that we have
the following exact sequence of $R$-modules:

$0\rightarrow{\bf
X}_{n}(A)\rightarrow\mathrm{Hom}(N^{n},\hat{\mathbb{G}}_{\mathrm{a}})\stackrel{{\scriptstyle\partial}}{{\rightarrow}}\mathrm{Ext}(A,\hat{\mathbb{G}}_{\mathrm{a}}).$

Further, recall from Section $9$ in [9], that for an elliptic curve $A$, the
character group ${\bf X}_{\infty}(A)=R\\{\phi^{*}\\}\langle\Theta_{m}\rangle$.
Then we have two possible cases:

  * (i)

$A$ has a canonical lift iff $m=1$, then $i^{*}\Theta_{1}=\Psi_{1}$.

  * (ii)

Otherwise, we have $m=2$ and $i^{*}\Theta_{2}=\Psi_{2}-\lambda\Psi_{1}$, where
$\lambda\in R$. In this case $\partial\Psi_{1}$ is nonzero and since
$i^{*}\Theta_{2}$ lies in the kernel of $\partial$, we get
$\lambda=\partial\Psi_{2}/\partial\Psi_{1}$ (fixing a basis element depending
on $\omega$).

Thus pulling back $\Theta_{1},\Theta_{2}$ by $\phi$ and $i$, we have

$\displaystyle i^{*}\phi^{*}\Theta_{1}$
$\displaystyle=\mathfrak{f}^{*}i^{*}\Theta_{1}+\gamma\Psi_{1}=\Psi_{2}+\gamma\Psi_{1}$
$\displaystyle i^{*}\phi^{*}\Theta_{2}$
$\displaystyle=\mathfrak{f}^{*}i^{*}\Theta_{2}+\gamma\Psi_{1}=\Psi_{3}-\phi(\lambda)\Psi_{2}+\gamma\Psi_{1}.$

###  9.1. Geometric Interpretation of the Arithmetic Picard-Fuchs Operator

The main aim of this subsection is to prove Proposition 9.2 that will play an
important role in proving Theorem 1.6. Recall from Lemma 2.8 in [10], we have
the morphism $u:\hat{\mathbb{G}}_{\mathrm{a}}\longrightarrow A$, which can
also be described as the composition of the following maps below:

$\displaystyle\hat{\mathbb{G}}_{\mathrm{a}}\stackrel{{\scriptstyle{\frac{1}{\pi}\mathrm{exp}_{\mathcal{F}}}(\pi
x)}}{{\longrightarrow}}N^{1}A\stackrel{{\scriptstyle
i}}{{\rightarrow}}J^{1}A\stackrel{{\scriptstyle\phi_{A}}}{{\longrightarrow}}A\hskip
113.81102pt$ $\displaystyle x\mapsto 1/\pi~{}\mathrm{exp}_{\mathcal{F}}(\pi
x)\mapsto(0,1/\pi~{}\mathrm{exp}_{\mathcal{F}}(\pi
x))\mapsto\mathrm{exp}_{\mathcal{F}}(\pi x)\hskip 56.9055pt$

where $\mathcal{F}$ denotes the formal group associated to the group law of
$A$. Therefore as in Page $324$ in [10], this induces an injective pullback
map

$\displaystyle u^{*}:{\bf X}_{\infty}(A)\longrightarrow{\bf
X}_{\infty}(\hat{\mathbb{G}}_{\mathrm{a}})=R[\phi]~{}\mathrm{such~{}that}$
$\displaystyle u^{*}({\bf X}_{n}(A))\subset{\bf
X}_{n}(\hat{\mathbb{G}}_{\mathrm{a}})=R+R\langle\phi\rangle+\ldots+R\langle\phi^{n}\rangle.$

Let $B$ be a $\pi$-adically complete $R$ algebra which has a $\pi$-derivation
$\delta$ that lifts the fixed derivation on $R$. For each $n$, by the
universal property of Witt vectors (cf. Section 1 in [5]) the $\pi$-derivation
induces the canonical map
$B\stackrel{{\scriptstyle\exp_{\delta}}}{{\longrightarrow}}W_{n}(B)$. Hence
given a $B$-point of a $\pi$-formal scheme $X$ induces $\nabla:X(B)\rightarrow
X(W_{n}(B))=J^{n}X(B)$.

Consider the morphism $\phi_{A}\circ i:N^{1}A\rightarrow A$. Then by the
universal property of jet spaces, Proposition 4.1, this induces a unique map
of prolongation sequences $N^{*+1}A\rightarrow J^{*}A$. In the case of $m=0$
in Theorem 5.3, it is easy to see that the morphism between the prolongation
sequences at each level is given by $\phi_{A}\circ i:N^{n+1}A\rightarrow
J^{n}A$. Also by Theorem 7.8 in the case $m=0$, for all $n\geq 0$ we have
$N^{n+1}A\simeq J^{n}(N^{1}A)$.

In the case when $A$ is an elliptic curve over $\operatorname{\mathrm{Spf}}R$,
we have an isomorphism
$\vartheta^{-1}:\hat{\mathbb{G}}_{\mathrm{a}}\rightarrow N^{1}A$ and hence
induces an isomorphism
$J^{n}(\vartheta^{-1}):J^{n}\hat{\mathbb{G}}_{\mathrm{a}}\simeq\mathbb{W}_{n}\rightarrow
N^{n+1}A$.

######  Lemma 9.1.

Let $\Psi:N^{n}A\rightarrow\hat{\mathbb{G}}_{\mathrm{a}}$ be given by
$\Psi=b_{1}\Psi_{1}+\cdots+b_{n}\Psi_{n}$ where $b_{i}\in R$ for all
$i=1,\dots,n$. Then

$\Psi\circ
J^{n-1}(\vartheta^{-1})=b_{1}\mathbbm{1}_{\hat{\mathbb{G}}_{\mathrm{a}}}+b_{2}\phi_{\hat{\mathbb{G}}_{\mathrm{a}}}+\cdots+b_{n}\phi^{n-1}_{\hat{\mathbb{G}}_{\mathrm{a}}}.$

###### Proof.

It is enough to prove the result for $\Psi=\Psi_{i}$ for all $i$. Recall
$\Psi_{i}=\vartheta\circ\phi_{\tiny N^{1}A}^{\circ(i-1)}$. By Theorem 7.8, we
have for all $j\leq n-1$

$\vartheta^{-1}\circ\phi_{\hat{\mathbb{G}}_{\mathrm{a}}}^{\circ
j}=\phi_{N^{1}A}^{\circ j}\circ J^{n-1}(\vartheta^{-1}).$

Then we have

$\displaystyle\Psi\circ J^{n-1}(\vartheta^{-1})$ $\displaystyle=$
$\displaystyle\vartheta\circ\phi_{N^{1}A}^{\circ(i-1)}\circ
J^{n-1}(\vartheta^{-1})$ $\displaystyle=$
$\displaystyle\vartheta\circ\vartheta^{-1}\circ\phi^{(i-1)}$ $\displaystyle=$
$\displaystyle\phi^{(i-1)}$

and this proves our result. ∎

Let $\Theta\in{\bf X}_{r}(A)$ be a delta character of order $r$ of $A$. Then
$\Theta:J^{r}A\rightarrow\hat{\mathbb{G}}_{\mathrm{a}}$ is a morphism of
$\pi$-formal group schemes. Hence again by the universal property of jet
spaces we have for each $n\geq 0$ the following compatible system of morphisms

(9.7)

Consider the morphism
$\Lambda_{\theta}:J^{r}\hat{\mathbb{G}}_{\mathrm{a}}\rightarrow\hat{\mathbb{G}}_{\mathrm{a}}$
given by the following composition

(9.8) $\displaystyle\Lambda_{\Theta}:=\Theta\circ(\phi_{A}\circ i)\circ
J^{r}(\vartheta^{-1}).$

Note that $\Lambda_{\Theta}$ is a delta character of
$\hat{\mathbb{G}}_{\mathrm{a}}$ of order $r$, in other words,
$\Lambda_{\Theta}\in{\bf X}_{r}(\hat{\mathbb{G}}_{\mathrm{a}})$. This is known
as arithmetic Picard-Fuchs operator. It is also called the symbol of $\Theta$
in [15]. Hence for all $i=0,\dots,r$ there exist $b_{i}\in R$ such that

(9.9) $\displaystyle\Lambda_{\Theta}$
$\displaystyle=b_{r}\phi^{r}_{\hat{\mathbb{G}}_{\mathrm{a}}}+\cdots+b_{1}\phi_{\hat{\mathbb{G}}_{\mathrm{a}}}+b_{0}$

For any $\pi$-formal scheme $G$, by Theorem 7.8 we have $N^{r+1}G\simeq
J^{r}(N^{1}G)$. Hence for any $R$-point of $N^{1}G$, we can consider the
canonical lift $\nabla:N^{1}G(R)\rightarrow J^{r}(N^{1}G)(R)\simeq
N^{r+1}G(R)$. Then evaluating diagram (9.7) on $R$-points we obtain

(9.14)

When $A$ does not have CL, then $r=2$ and the $\phi$-linear endomorphism of
$R$ in Page 325 of [10] is obtained by the following composition

$\Theta\circ\nabla\circ(\phi_{A}\circ i)\circ\vartheta^{-1}$

Then by (9.14) the above map is the same as

$\Theta\circ(\phi_{A}\circ i)\circ
J^{2}(\vartheta^{-1})\circ\nabla=\Lambda_{\Theta}\circ\nabla$

The following result relates $\Lambda_{\Theta}$ with $i^{*}\Theta$
geometrically:

######  Proposition 9.2.

Let $A$ be an ellitpic curve over $R$, and $\Theta\in{\bf X}_{r}(A)$ such that
$i^{*}\Theta=a_{r}\Psi_{r}+\cdots+a_{1}\Psi_{2}++a_{1}\Psi_{1}$. Then

$\Lambda_{\Theta}=\phi(a_{r})\phi^{r}_{\hat{\mathbb{G}}_{\mathrm{a}}}+\cdots+\phi(a_{1})\phi_{\hat{\mathbb{G}}_{\mathrm{a}}}+\gamma$

where $\gamma\in R$ is as given in Proposition 8.1.

###### Proof.

By Proposition 8.1(1) we have

$(\phi\circ
i)^{*}\Theta=\phi(a_{r})\Psi_{r+1}+\cdots+\phi(a_{1})\Psi_{2}+\gamma\Psi_{1}$

Then $\Lambda_{\Theta}=(\phi\circ i)^{*}\Theta\circ J^{r}(\vartheta^{-1})$ and
the result follows from applying Lemma 9.1 to the above. ∎

###  9.2. Proof of Theorem 1.6

We have the following two cases depending on whether $A$ admits a lift of
Frobenius or not. In both cases, we will first show that $\mathrm{Iso}({\bf
H}_{\delta}(A))$ is a weakly admissible isocrystal over $\mathbb{Q}_{p}$ and
then prove part (1) and part (2) of the theorem respectively.

(1) Non-CL case: Let $A$ be a non-CL elliptic curve over $\mathbb{Z}_{p}$.
Then by Theorem 9.7(b) in [9], we have ${\bf
X}_{\infty}(A)=R\\{\phi^{*}\\}\langle\Theta_{2}\rangle$ and ${\bf
H}_{\delta}(A)=R\langle\Psi_{1},\Psi_{2}\rangle.$ Recall from Section $9$ in
[9] that the matrix of $\mathfrak{f}^{*}$ with respect to the basis
$\\{\Psi_{1},\Psi_{2}\\}$ is given by

$[\mathfrak{f}^{*}]=\begin{bmatrix}0&-\gamma\\\ 1&\lambda&\end{bmatrix}.$

Note that $\phi$ is identity on ${\mathbb{Z}}_{p}$ and hence
$\mathfrak{f}^{*}$ becomes linear. Note that, since
$i^{*}\Theta_{2}=\Psi_{2}-\lambda\Psi_{1}$ and $\phi(\lambda)=\lambda$ because
$\lambda\in{\mathbb{Z}}_{p}$, by Proposition 9.2, we obtain that the
arithmetic Picard-Fuchs operator of $\Theta_{2}$ as

$\displaystyle\Lambda_{\Theta_{2}}=\phi^{2}-\lambda\phi+\gamma\in R[\phi].$

On the other hand, from Theorem $1.10$ in [12], we have

$\Lambda_{\Theta_{2}}=\phi^{2}-a_{p}\phi+p\in R[\phi]={\bf
X}_{\infty}(\hat{\mathbb{G}}_{\mathrm{a}}),$

where $a_{p}=p+1-\\#A_{0}(\mathbb{F}_{p})$. Therefore under this
identification, it follows that $\lambda=a_{p}$ and $\gamma=p$. Combining
Theorem 9.7 in [9], this proves that $\mathrm{Iso}({\bf H}_{\delta}(A))$ is
weakly admissible filtered isocrystal.

Now recall diagram 8.6 of short exact sequences of $R$-modules, which induce
the following diagram of $\mathbb{Q}_{p}$-vector spaces:

(9.15)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf
X}_{\mathrm{prim}}(A)_{\mathbb{Q}_{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Upsilon}$$\textstyle{{\bf
H}_{\delta}(A)_{\mathbb{Q}_{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{{\bf
I}(A)_{\mathbb{Q}_{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{\mathrm{Lie}}(A)}^{\vee}_{\mathbb{Q}_{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Ext}^{\sharp}(A,\hat{\mathbb{G}}_{\mathrm{a}})_{\mathbb{Q}_{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Ext}(A,\hat{\mathbb{G}}_{\mathrm{a}})_{\mathbb{Q}_{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(A,\Omega_{A})_{\mathbb{Q}_{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf
H}^{1}_{\mathrm{dR}}(A)_{\mathbb{Q}_{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(A,\mathcal{O}_{A})_{\mathbb{Q}_{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

Now note that from Proposition 6.3 and 8.1 in [9], we get the map $\Upsilon$
sends $\Theta_{2}$ to $\gamma/p$ (fixing an invariant differential form
$\omega$). In other words, $\Upsilon$ sends $\Theta_{2}$ to $\omega$ in this
case. On the other hand, ${\bf
I}(A)_{\mathbb{Q}_{p}}=\mathrm{Ext}(A,\hat{\mathbb{G}}_{\mathrm{a}})_{\mathbb{Q}_{p}}$
as both are one dimensional. Therefore by the commutative diagram, $\Phi$
becomes a bijective map. Moreover, it is well known that the short exact
sequences in the last two rows are functorially isomorphic. Hence there is a
functorial isomorphism between all the short exact sequences above. This shows
that ${\bf H}_{\delta}(A)_{\mathbb{Q}_{p}}\supset{\bf
X}_{\mathrm{prim}}(A)_{\mathbb{Q}_{p}}\supset 0$ realizes the Hodge
filtration.

Define the operator
$\tilde{\mathfrak{f}^{*}}:=\Phi\circ\mathfrak{f}^{*}\circ\Phi^{-1}$ on ${\bf
H}^{1}_{\mathrm{dR}}(A)_{\mathbb{Q}_{p}}$. Then we also have that the
characteristic polynomial

$P_{\tilde{\mathfrak{f}^{*}}}(T)=P_{\mathfrak{f}^{*}}(T)=T^{2}-a_{p}T+p.$

We now identify ${\bf H}^{1}_{\mathrm{cris}}(A_{0},{\mathbb{Z}}_{p})$ with
${\bf H}^{1}_{\mathrm{dR}}(A)$ by the isomorphism given in [3]. Let
$F_{\mathrm{cris}}$ be the operator on ${\bf
H}^{1}_{\mathrm{cris}}({A_{0}},{\mathbb{Z}}_{p})$ arising from the absolute
Frobenius on $A_{0}$. Then the characteristic polynomial of
$F_{\mathrm{cris}}$ is

$P_{F_{\mathrm{cris}}}(T)=T^{2}-a_{p}T+p.$

It is clear that the characteristic polynomial have two distinct roots. Hence
the two operators $\tilde{\mathfrak{f}^{*}}$ and $F_{\mathrm{cris}}$ are
conjugate on the two dimensional vector space ${\bf
H}^{1}_{\mathrm{dR}}(A)_{\mathbb{Q}_{p}}$. This proves that the two weakly
admissible isocrystals $\mathrm{Iso}({\bf H}_{\delta}(A)_{\mathbb{Q}_{p}})$
and $\mathrm{Iso}({\bf H}^{1}_{\mathrm{cris}}(A)_{\mathbb{Q}_{p}})$ are
isomorphic.

(2) CL case: Let $\tilde{F}:A\longrightarrow A$ be a canonical lift of
Frobenius on $A$. In this scenario, by Theorem 9.7(b) in [9], we have ${\bf
X}_{\infty}(A)=R\\{\phi^{*}\\}\langle\Theta_{1}\rangle$ and ${\bf
H}_{\delta}(A)=R\langle\Psi_{1}\rangle.$ Therefore the semilinear operator act
as

$\mathfrak{f}^{*}(\Psi_{1})=\gamma\Psi_{1}.$

Here we have $i^{*}\Theta_{1}=\Psi_{1}$, hence by Proposition 9.2, we obtain
the arithmetic Picard-Fuchs operator of $\Theta_{1}$ as

$\displaystyle\Lambda_{\Theta_{1}}=\phi+\gamma\in R[\phi].$

From Theorem $1.10$ in [12], we have

$\Lambda_{\Theta_{1}}=\phi-\beta\in R[\phi]={\bf
X}_{\infty}(\hat{\mathbb{G}}_{\mathrm{a}}),$

where $\beta$ is the nonunit root of the polynomial $T^{2}-a_{p}T+p$.
Therefore we have $\gamma=-\beta$ in this case. Hence by Theorem 9.7 of [9] we
have that $({\bf H}_{\delta}(A),\mathfrak{f}^{*})$ is a weakly admissible
filtered isocrystal as it is one dimensional. Similarly, considering the left
vertical maps in diagram (9.15), by Proposition 6.1 in [9] gives an
isomorphism

$\Upsilon=-\gamma/\pi:{{\bf H}_{\delta}}(A)_{\mathbb{Q}_{p}}={{\bf
X}_{\mathrm{prim}}}(A)_{\mathbb{Q}_{p}}\longrightarrow{\operatorname{\mathrm{Lie}}(A)}^{\vee}_{\mathbb{Q}_{p}}\cong
H^{0}(A,\Omega_{A})_{\mathbb{Q}_{p}}.$

Note that $\tilde{F}$ functorially induces an operator $\tilde{F}^{*}$ on
${\bf H}^{1}_{\mathrm{dR}}(A)_{\mathbb{Q}_{p}}$, which restricts to
$H^{0}(A,\Omega_{A})_{\mathbb{Q}_{p}}$. On the other hand, the comparison
isomorphism

${\bf H}^{1}_{\mathrm{dR}}(A)_{\mathbb{Q}_{p}}\cong{\bf
H}^{1}_{\mathrm{cris}}(A_{0},\mathbb{Z}_{p})_{\mathbb{Q}_{p}}$

endow another semilinear operator $F_{\mathrm{cris}}$ on ${\bf
H}^{1}_{\mathrm{dR}}(A)_{\mathbb{Q}_{p}}$. Then $\tilde{F}^{*}$ and
$F_{\mathrm{cris}}$ both determine the same isocrystal structure on ${\bf
H}^{1}_{\mathrm{dR}}(A)_{\mathbb{Q}_{p}}$ by Theorem 2.4 in [3]. This implies
that $F_{\mathrm{cris}}$ also restricts to
$H^{0}(A,\Omega_{A})_{\mathbb{Q}_{p}}$ when $A$ has a canonical lift. Since
$\beta$ is a non-unit root of $P_{F_{\mathrm{cris}}}$, we must have that
$F_{\mathrm{cris}}$ restricted to $H^{0}(A,\Omega_{A})_{\mathbb{Q}_{p}}$ is
multiplication by $\beta$. Hence $F_{\mathrm{cris}}$ and $\mathfrak{f}^{*}$ on
$H^{0}(A,\Omega_{A})_{\mathbb{Q}_{p}}$ determine the same isocrystal
structure. ∎

Acknowledgements. The second author would like to thank James Borger for many
insightful discussions. He would also like to thank Alessandra Bertapelle and
Nicola Mazzari for stimulating conversations. The author was partially
supported by the SERB grant SRG/2020/002248.

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