# On the $p$-torsion of the Tate-Shafarevich group
of abelian varieties
over higher dimensional bases over finite fields

Timo Keller  Timo Keller
Universität Bayreuth
Lehrstuhl Mathematik II (Computeralgebra)
Universitätsstraße 30
95440 Bayreuth
Germany<EMAIL_ADDRESS>https://www.timo-keller.de

###### Abstract.

We prove a finiteness theorem for the first flat cohomology group of finite
flat group schemes over integral normal proper varieties over finite fields.
As a consequence, we can prove the invariance of the finiteness of the Tate-
Shafarevich group of Abelian schemes over higher dimensional bases under
isogenies and alterations over/of such bases for the $p$-part. Along the way,
we generalize previous results on the Tate-Shafarevich group in this
situation.

###### Key words and phrases:

Tate-Shafarevich groups of abelian varieties over higher dimensional bases
over finite fields, $p$-torsion in characteristic $p>0$; Abelian varieties of
dimension $>1$; Étale and other Grothendieck topologies and cohomologies;
Arithmetic ground fields for abelian varieties

###### 2020 Mathematics Subject Classification:

11G10, 14F20, 14K15

I thank Jean-Louis Colliot-Thélène, Aise Johan de Jong, Ulrich Görtz, Uwe
Jannsen and Moritz Kerz, and Ariyan Javanpeykar, darx and David Loeffler from
mathoverflow. I also thank the anonymous referee for improving the exposition.
T. K. is supported by the Deutsche Forschungsgemeinschaft (DFG), Projektnummer
STO 299/18-1, AOBJ: 667349 while working on this article.

##  1\. Introduction

The Tate-Shafarevich group $\Sha(\mathscr{A}/X)$ of an Abelian scheme
$\mathscr{A}$ over a base scheme $X$ is of great importance for the arithmetic
of $\mathscr{A}$. It classifies everywhere locally trivial
$\mathscr{A}$-torsors. Its finiteness is sufficient to establish our analogue
of the conjecture of Birch and Swinnerton-Dyer [Kel19] over higher dimensional
bases over finite fields.

In [Kel16, section 4.3], we showed that finiteness of an $\ell$-primary
component of the Tate-Shafarevich group descends under generically étale
alterations of generical degree prime to $\ell$ for $\ell$ invertible on the
base scheme. This is used in [Kel19, Corollary 5.11] to prove the finiteness
of the Tate-Shafarevich group and an analogue of the Birch-Swinnerton-Dyer
conjecture for certain Abelian schemes over higher dimensional bases over
finite fields under mild conditions. In [Kel16, section 4.4], we showed that
finiteness of an $\ell$-primary component of the Tate-Shafarevich group is
invariant under étale isogenies. In this article, we prove these results also
for the $p^{\infty}$-torsion.

Recall that we defined the Tate-Shafarevich group of $\mathscr{A}/X$ for
$\dim{X}>1$ and $\mathscr{A}$ of good reduction as
$\mathrm{H}_{\mathrm{\acute{e}t}}^{1}(X,\mathscr{A})$. In [Kel16, Lemma 4.15]
we proved as a hypothesis in [Kel16, Theorem 4.5]:

######  Lemma 1.1.

Let $X/k$ be a smooth variety and $\mathscr{C}/X$ a smooth proper relative
curve. Assume $\dim{X}\leq 2$. Let $Z\hookrightarrow X$ be a reduced closed
subscheme of codimension $\geq 2$. Then

$\mathrm{H}^{i}_{Z}(X,\mathbf{Pic}^{0}_{\mathscr{C}/X})=0\quad\text{for $i\leq
2$}.$

If $\dim{X}>2$, this holds at least up to $p$-torsion.

We weaken the hypothesis that $\mathscr{A}$ is a Jacobian in [Kel16, Lemma
4.15] to arbitrary Abelian schemes:

######  Theorem.

(See Corollary 2.5) Let $X$ be a regular integral Noetherian separated scheme
and $\mathscr{A}/X$ be an Abelian scheme. Let $Z\hookrightarrow X$ be a closed
subscheme of codimension $\geq 2$. Then the vanishing condition [Kel16, (4.4)]
holds for $\mathscr{A}/X$: $\mathrm{H}^{i}_{Z}(X,\mathscr{A})$ is torsion for
all $i$. Furthermore, $\mathrm{H}^{0}_{Z}(X,\mathscr{A})=0$, and for $i=1,2$,
the only possible torsion is $p$-torsion for $p$ not invertible on $X$.

Our main results are now as follows:

######  Theorem.

(See Theorem 3.14) Let $X$ be a proper integral normal variety over a finite
field and $G/X$ be a finite flat commutative group scheme. Then
$\mathrm{H}^{1}_{\mathrm{fppf}}(X,G)$ is finite.

This theorem is proven by reduction to the finite flat simple group schemes
$\mathbf{Z}/p$,$\mu_{p}$ and $\alpha_{p}$ over an algebraically closed field
using de Jong’s alteration theorem, Raynaud-Gruson and a dévissage argument.

Using this technical result and refining our methods from [Kel16], we obtain
the following three results:

The following theorem has been proved as [Kel16, Lemma 4.28] for $p$ prime to
the characteristic of $k$; in this article, we prove it also for $p$ equal to
the characteristic of $k$:

######  Theorem.

(See Lemma 5.1) Let $\mathscr{A}/X$ be an Abelian scheme over a proper variety
$X$ over a finite field of characteristic $p$. Then
$\Sha(\mathscr{A}/X)[p^{\infty}]$ is cofinitely generated.

In [Kel16, Theorem 4.31], we proved:

######  Theorem 1.2.

Let $X/k$ be proper, $\mathscr{A}$ and $\mathscr{A}^{\prime}$ Abelian schemes
a variety $X$ over a finite field and $f:\mathscr{A}^{\prime}\to\mathscr{A}$
an étale isogeny. Let $\ell\neq\operatorname{char}{k}$ be a prime. Then
$\Sha(\mathscr{A}/X)[\ell^{\infty}]$ is finite if and only if
$\Sha(\mathscr{A}^{\prime}/X)[\ell^{\infty}]$ is finite.

In this article, we prove it also for $\ell$ equal to the characteristic of
$k$:

######  Theorem (invariance of finiteness of under isogenies, Theorem 4.1).

Let $X/k$ be a proper variety over a finite field $k$ and
$f:\mathscr{A}\to\mathscr{A}^{\prime}$ be an isogeny of Abelian schemes over
$X$. Let $p$ be an arbitrary prime. Assume $f$ étale if
$p\neq\operatorname{char}{k}$. Then $\Sha(\mathscr{A}/X)[p^{\infty}]$ is
finite if and only if $\Sha(\mathscr{A}^{\prime}/X)[p^{\infty}]$ is finite.

In [Kel16, Theorem 4.29], we proved:

######  Theorem 1.3.

Let $f:X^{\prime}\to X$ be a morphism of normal integral varieties over a
finite field which is an alteration of degree prime to $\ell$ for a prime
$\ell$ invertible on $X$, i.e., $f$ is a proper, surjective, generically étale
morphism of generical degree prime to $\ell$. If $\mathscr{A}$ is an Abelian
scheme on $X$ such that the $\ell^{\infty}$-torsion of the Tate-Shafarevich
group $\Sha(\mathscr{A}^{\prime}/X^{\prime})$ of
$\mathscr{A}^{\prime}:=f^{*}\mathscr{A}=\mathscr{A}\times_{X}X^{\prime}$ is
finite, then the $\ell^{\infty}$-torsion of the Tate-Shafarevich group
$\Sha(\mathscr{A}/X)$ is finite.

In this article, we prove it also for $\ell$ equal to the characteristic of
$k$ and remove the condition that the generical degree is prime to $\ell$ if
$\ell$ is invertible on $X$:

######  Theorem.

(invariance of finiteness of under alterations, Theorem 5.3 and Theorem 5.5)
Let $f:X^{\prime}\to X$ be a proper, surjective, generically finite morphism
of generical degree $d$ of regular, integral, separated varieties over a
finite field of characteristic $p>0$. Let $\mathscr{A}$ be an abelian scheme
on $X$ and
$\mathscr{A}^{\prime}:=f^{*}\mathscr{A}=\mathscr{A}\times_{X}X^{\prime}$. Let
$\ell$ be an arbitrary prime. Assume $(d,\ell)=1$ if $\ell=p$. If
$\Sha(\mathscr{A}^{\prime}/X^{\prime})[\ell^{\infty}]$ is finite, so is
$\Sha(\mathscr{A}/X)[\ell^{\infty}]$.

### Notation.

Canonical isomorphisms are often denoted by “$=$”. We denote Pontrjagin
duality by $(-)^{D}$ and duals of Abelian schemes and Cartier duals by
$(-)^{t}$.

For a scheme $X$, we denote the set of codimension-$1$ points by $X^{(i)}$ and
the set of closed points by $|X|$.

For an abelian group $A$, let $A_{\mathrm{tors}}$ be the torsion subgroup of
$A$, and $A_{\mathrm{n\text{-}tors}}=A/A_{\mathrm{tors}}$. For $A$ a
cofinitely generated $\ell$-primary group, let $A_{\mathrm{div}}$ be the
maximal divisible subgroup of $A$, which equals the subgroup of divisible
elements of $A$ in this case ([Kel19, Lemma 2.1.1 (iii)]), and
$A_{\mathrm{n\text{-}div}}=A/A_{\mathrm{div}}$. For an integer $n$ and an
object $A$ of an abelian category, denote the cokernel of
$A\stackrel{{\scriptstyle n}}{{\to}}A$ by $A/n$ and its kernel by $A[n]$, and
for a prime $p$ the $p$-primary subgroup
$\operatorname{\varinjlim}_{n}A[p^{n}]$ by $A[p^{\infty}]$. Write
$A[\text{non-}p]$ or $A[p^{\prime}]$ for $\varinjlim_{p\nmid n}A[n]$. For a
prime $\ell$, let the $\ell$-adic Tate module $T_{\ell}A$ be
$\varprojlim_{n}A[\ell^{n}]$ and the rationalized $\ell$-adic Tate module
$V_{\ell}A=T_{\ell}A\otimes_{\mathbf{Z}_{\ell}}\mathbf{Q}_{\ell}$. The corank
of $A[p^{\infty}]$ is the $\mathbf{Z}_{p}$-rank of $A[p^{\infty}]^{D}=T_{p}A$.

##  2\. Vanishing of étale cohomology with supports of Abelian schemes

This is a complement to the “vanishing condition” $\mathrm{H}^{i}_{Z}(X,G)=0$
from [Kel16, (4.4)], which is proven there only for Jacobians of curves, see
[Kel16, Lemma 4.10].

######  Theorem 2.1.

Let $X$ be a regular integral Noetherian separated scheme and $G/X$ be a
finite étale commutative group scheme of order invertible on $X$. Let
$Z\hookrightarrow X$ be a closed subscheme of codimension $\geq 2$. Then
$\mathrm{H}^{i}_{Z}(X,G)=0$ for $i\leq 2$ (étale cohomology with supports in
$Z$).

###### Proof.

Let $U=X\setminus Z$. One has a long exact cohomology sequence

$\displaystyle\ldots\to\mathrm{H}^{i-1}(X,G)\to\mathrm{H}^{i-1}(U,G)\to$
$\displaystyle\quad\mathrm{H}^{i}_{Z}(X,G)\to\mathrm{H}^{i}(X,G)\to\mathrm{H}^{i}(U,G)\to\ldots,$

so one has to prove that $\mathrm{H}^{i}(X,G)\to\mathrm{H}^{i}(U,G)$ is an
isomorphism for $i=0,1$ and injective for $i=2$.

For $i=0$, the claim $\mathrm{H}^{i}_{Z}(X,G)=0$ is equivalent to the
injectivity of

$\mathrm{H}^{0}(X,G)\to\mathrm{H}^{0}(U,G),$

which is clear from [Har83, p. 105, Exercise II.4.2] since $G/X$ is separated,
$X$ is reduced and $U\hookrightarrow X$ is dense.

For $i=1$ the claim $\mathrm{H}^{i}_{Z}(X,G)=0$ is equivalent to

$\mathrm{H}^{0}(X,G)\to\mathrm{H}^{0}(U,G)$

being surjective and

$\mathrm{H}^{1}(X,G)\to\mathrm{H}^{1}(U,G)$

being injective. The surjectivity of
$\mathrm{H}^{0}(X,G)\to\mathrm{H}^{0}(U,G)$ follows e. g. from

######  Theorem 2.2.

Let $S$ be a normal Noetherian base scheme, and let $u:T\dashrightarrow G$ be
an $S$-rational map from a smooth $S$-scheme $T$ to a smooth and separated
$S$-group scheme $G$. Then, if $u$ is defined in codimension $\leq 1$, it is
defined everywhere.

###### Proof.

See [BLR90, p. 109, Theorem 1]. ∎

For the injectivity of $\mathrm{H}^{1}(X,G)\to\mathrm{H}^{1}(U,G)$: If a
principal homogeneous space $P/X$ for $G/X$ is trivial over $U$, then it is
trivial over $X$: The trivialization over $U$ gives a rational map from $X$ to
the principal homogeneous space and any such map (with $X$ a regular scheme)
extends to a morphism by Theorem 2.2.

For the surjectivity of $\mathrm{H}^{1}(X,G)\to\mathrm{H}^{1}(U,G)$: This
means that any principal homogeneous space $P/U$ extends to a principal
homogeneous space $\bar{P}/X$. By [Mil80, p. 123, Corollary III.4.7], we have
$\mathrm{PHS}(G/X)\xrightarrow{\leavevmode\nobreak\ \sim\leavevmode\nobreak\
}\mathrm{H}^{1}(X_{\mathrm{fl}},G)$ (Čech cohomology) since $G/X$ is affine.
Since $G/X$ is smooth, [Mil80, p. 123, Remark III.4.8 (a)] shows that we can
take étale cohomology as well, and by [Mil80, p. 101, Corollary III.2.10], one
can take derived functor cohomology instead of Čech cohomology. Recall:

######  Theorem 2.3 (Zariski-Nagata purity).

Let $X$ be a locally Noetherian regular scheme and $U$ an open subscheme with
closed complement of codimension $\geq 2$. Then the functor $X^{\prime}\mapsto
X^{\prime}\times_{X}U$ is an equivalence of categories from étale coverings of
$X$ to étale coverings of $U$.

###### Proof.

See [Gro71, Exp. X, Corollaire 3.3]. ∎

By Theorem 2.3, one can extend $P/U$ uniquely to a $\bar{P}/X$, for which we
have to show that it represents an element of $\mathrm{H}^{1}(X,G)$, i. e.,
that it is a $G$-torsor.

So we need to show that if $P/U$ is an $G|_{U}$-torsor and $\bar{P}$ an
extension of $P$ to a finite étale covering of $X$, then $\bar{P}/X$ is also
an $G$-torsor. For this, we use the following

######  Theorem 2.4.

Let $X$ be a connected scheme, $G\to X$ a finite flat group scheme, and
$\bar{P}\to X$ a scheme over $X$ equipped with a left action
$\rho:G\times_{X}\bar{P}\to\bar{P}$. These data define a $G$-torsor over $X$
if and only if there exists a finite locally free surjective morphism $Y\to X$
such that $\bar{P}\times_{X}Y\to Y$ is isomorphic, as a $Y$-scheme with
$G\times_{X}Y$-action, to $G\times_{X}Y$ acting on itself by left
translations.

###### Proof.

See [Sza09, p. 171, Lemma 5.3.13]. ∎

That $P/U$ is an $G|_{U}$-torsor amounts to saying that there is an operation

$G|_{U}\times_{U}P\to P$

as in the previous Theorem 2.4. Since this is étale locally isomorphic to the
canonical action

$G|_{U}\times_{U}G|_{U}\stackrel{{\scriptstyle\mu}}{{\to}}G|_{U}$

which is finite étale, by faithfully flat descent the operation defines an
étale covering, so extends by Zariski-Nagata purity (Theorem 2.3) uniquely to
an étale covering $H\to X$, which by uniqueness has to be isomorphic to
$G\times_{X}\bar{P}\to\bar{P}$. Now a routine check shows the condition in
Theorem 2.4.

There is a finite étale Galois covering $X^{\prime}/X$ with Galois group $G$
such that $G\times_{X}X^{\prime}$ is isomorphic to a direct sum of $\mu_{n}$
with $n$ invertible on $X$. The Leray spectral sequence with supports
$\mathrm{H}^{p}(G,\mathrm{H}^{q}_{Z^{\prime}}(X^{\prime},G\times_{X}X^{\prime}))\Rightarrow\mathrm{H}^{p+q}_{Z}(X,G)$
from [Kel16, p. 228, Theorem 4.9], so it suffices to show the vanishing
$\mathrm{H}^{q}_{Z}(X^{\prime},G\times_{X}X^{\prime})=0$ for $q=0,1,2$. Hence
one can assume $G\cong\mu_{n}$ for $n$ invertible on $X$.

By [Mil80, Example III.2.22], one has an injection
$\operatorname{Br}(X)\hookrightarrow\operatorname{Br}(K(X))$ with $K(X)$ the
function field of $X$ and
$\operatorname{Br}(X)\to\operatorname{Br}(U)\to\operatorname{Br}(K(X))$, so
$\operatorname{Br}(X)\to\operatorname{Br}(U)$ is injective. By the hypotheses
on $X$ and since the codimension of $Z$ in $X$ is $\geq 2$, there is a
restriction isomorphism
$\operatorname{Pic}(X)\xrightarrow{\leavevmode\nobreak\
\sim\leavevmode\nobreak\ }\operatorname{Pic}(U)$ (because of the codimension
condition and [Har83, Proposition II.6.5 (b)],
$\operatorname{Cl}{X}\xrightarrow{\leavevmode\nobreak\
\sim\leavevmode\nobreak\ }\operatorname{Cl}{U}$, and because of [Har83,
Proposition II.6.16], $\operatorname{Cl}{X}\cong\operatorname{Pic}{X}$
functorial in the scheme). Hence the snake lemma applied to the diagram

$\displaystyle{0}$$\displaystyle{\operatorname{Pic}(X)/n}$$\displaystyle{\mathrm{H}^{2}(X,\mu_{n})}$$\displaystyle{\operatorname{Br}(X)[n]}$$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{\operatorname{Pic}(U)/n}$$\displaystyle{\mathrm{H}^{2}(U,\mu_{n})}$$\displaystyle{\operatorname{Br}(U)[n]}$$\displaystyle{0}$

gives that $\mathrm{H}^{2}(X,\mu_{n})\to\mathrm{H}^{2}(U,\mu_{n})$ is
injective, so $\mathrm{H}^{2}_{Z}(X,\mu_{n})=0$. ∎

######  Corollary 2.5.

Let $X$ be a regular integral Noetherian separated scheme and $\mathscr{A}/X$
be an Abelian scheme. Let $Z\hookrightarrow X$ be a closed subscheme of
codimension $\geq 2$. Then $\mathrm{H}^{i}_{Z}(X,\mathscr{A})$ is torsion for
all $i$. Furthermore, $\mathrm{H}^{0}_{Z}(X,\mathscr{A})=0$, and for $i=1,2$,
the only possible torsion is $p$-torsion for $p$ not invertible on $X$.

###### Proof.

By [Kel16, p. 224, Proposition 4.1], $\mathrm{H}^{i}(X,\mathscr{A})$ is
torsion for $i>0$. The Kummer exact sequence
$0\to\mathscr{A}[n]\to\mathscr{A}\to\mathscr{A}\to 0$ for $n$ invertible on
$X$ yields a surjection

$\mathrm{H}^{i}_{Z}(X,\mathscr{A}[n])\twoheadrightarrow\mathrm{H}^{i}_{Z}(X,\mathscr{A})[n],$

so it suffices to show that $\mathrm{H}_{Z}^{i}(X,\mathscr{A}[n])=0$ for
$i=1,2$. But this is Theorem 2.1. The triviality
$\mathrm{H}_{Z}^{0}(X,\mathscr{A})=0$ is equivalent to the injectivity of

$\mathrm{H}^{0}(X,\mathscr{A})\to\mathrm{H}^{0}(U,\mathscr{A}),$

which is clear from [Har83, p. 105, Exercise II.4.2] since $\mathscr{A}/X$ is
separated, $X$ is reduced and $U\hookrightarrow X$ is dense. ∎

With vanishing condition (4.4) in [Kel16, Theorem 4.5] satisfied for
$\mathscr{A}/X$ by Corollary 2.5, the statement there generalizes from
$\mathscr{A}$ a Jacobian to $\mathscr{A}$ a general Abelian scheme:

######  Theorem 2.6.

Let $X$ be regular, Noetherian, integral and separated and let $\mathscr{A}$
be an Abelian scheme over $X$. For $x\in X$, denote the function field of $X$
by $K$, the quotient field of the strict Henselization of $\mathscr{O}_{X,x}$
by $K_{x}^{nr}$, the inclusion of the generic point by
$j:\\{\eta\\}\hookrightarrow X$ and let
$j_{x}:\operatorname{Spec}(K_{x}^{nr})\hookrightarrow\operatorname{Spec}(\mathscr{O}_{X,x}^{sh})\hookrightarrow
X$ be the composition. Then we have

$\mathrm{H}^{1}(X,\mathscr{A})\xrightarrow{\leavevmode\nobreak\
\sim\leavevmode\nobreak\
}\ker\Big{(}\mathrm{H}^{1}(K,j^{*}\mathscr{A})\to\prod_{x\in
X}\mathrm{H}^{1}(K_{x}^{nr},j_{x}^{*}\mathscr{A})\Big{)}.$

One can replace the product over all points by the following:

(a) the closed points $x\in|X|$: One has isomorphisms

$\displaystyle\mathrm{H}^{1}(X,\mathscr{A})\xrightarrow{\leavevmode\nobreak\
\sim\leavevmode\nobreak\
}\ker\Big{(}\mathrm{H}^{1}(K,j^{*}\mathscr{A})\to\prod_{x\in|X|}\mathrm{H}^{1}(K_{x}^{nr},j_{x}^{*}\mathscr{A})\Big{)}$

and

$\displaystyle\mathrm{H}^{1}(X,\mathscr{A})\xrightarrow{\leavevmode\nobreak\
\sim\leavevmode\nobreak\
}\ker\Big{(}\mathrm{H}^{1}(K,j^{*}\mathscr{A})\to\prod_{x\in|X|}\mathrm{H}^{1}(K_{x}^{h},j_{x}^{*}\mathscr{A})\Big{)}$

with $K_{x}^{h}=\operatorname{Quot}(\mathscr{O}_{X,x}^{h})$ the quotient field
of the Henselization if $\kappa(x)$ is finite.

or (b) the codimension-$1$ points $x\in X^{(1)}$: One has an isomorphism

$\displaystyle\mathrm{H}^{1}(X,\mathscr{A})\xrightarrow{\leavevmode\nobreak\
\sim\leavevmode\nobreak\
}\ker\Big{(}\mathrm{H}^{1}(K,j^{*}\mathscr{A})\to\bigoplus_{x\in
X^{(1)}}\mathrm{H}^{1}(K_{x}^{nr},j_{x}^{*}\mathscr{A})\Big{)}$

if one disregards the $p$-torsion ($p=\operatorname{char}{k}$) and $X/k$ is
smooth projective over $k$ finitely generated. For $\dim{X}\leq 2$, this also
holds for the $p$-torsion.

For $x\in X^{(1)}$, one can also replace $K_{x}^{nr}$ and $K_{x}^{h}$ by the
quotient field of the completions $\hat{\mathscr{O}}_{X,x}^{sh}$ and
$\hat{\mathscr{O}}_{X,x}^{h}$, respectively.

##  3\. Finiteness theorems for $\mathrm{H}^{1}_{\mathrm{fppf}}$ over finite
fields

The aim of this section is to show that $\mathrm{H}^{1}_{\mathrm{fppf}}(X,G)$
is finite for $X$ a normal proper variety over a finite field of
characteristic $p$ and $G/X$ a finite flat group scheme.

The proof is by reduction to the case of a finite flat _simple_ group scheme
over an algebraically closed field, which is isomorphic to $\mathbf{Z}/\ell$
(étale-étale), $\mathbf{Z}/p$ (étale-local), $\mu_{p}$ (local-étale) or
$\alpha_{p}$ (local-local).

We use the interpretation of $\mathrm{H}^{1}_{\mathrm{fppf}}(X,G)$ as
$G$-torsors on $X$ [Mil80, Proposition III.4.7] since $G/X$ is affine. We also
exploit de Jong’s alteration theorem [de 96, Theorem 4.1].

Let us first recall some well-known facts on flat cohomology.

######  Definition 3.1.

An _isogeny_ of commutative group schemes $G,H$ of finite type over an
arbitrary base scheme $X$ is a group scheme homomorphism $f:G\to H$ such that
for all $x\in X$, the induced homomorphism $f_{x}:G_{x}\to H_{x}$ on the
fibers over $x$ is finite and surjective on identity components.

######  Remark 3.2.

See [BLR90, p. 180, Definition 4]. We will usually consider isogenies between
abelian schemes, for example the finite flat $n$-multiplication, which is
étale iff $n$ is invertible on the base scheme or the abelian schemes are
trivial.

######  Lemma 3.3.

Let $G,G^{\prime}$ be commutative group schemes over a scheme $X$ which are
smooth and of finite type over $X$ with connected fibers and $\dim G=\dim
G^{\prime}$ and let $f:G^{\prime}\to G$ be a morphism of commutative group
schemes over $X$.

If $f$ is flat (respectively, étale) then $\ker(f)$ is a flat (respectively,
étale) group scheme over $X$, $f$ is quasi-finite, surjective and defines an
epimorphism in the category of flat (respectively, étale) sheaves over $X$.

###### Proof.

See [Kel19, Lemma 2.3.3]. ∎

######  Lemma 3.4 (Kummer sequence).

Let $f:G\to G^{\prime}$ be a faithfully flat isogeny between smooth
commutative group schemes over a base scheme $X$. Then the sequence

$\displaystyle 0\to\ker(f)\to G\stackrel{{\scriptstyle f}}{{\to}}G^{\prime}\to
0$

is exact on $X_{\mathrm{fppf}}$. This applies in particular to
$G=\mathbf{G}_{m}$ and $G=\mathscr{A}$ an abelian scheme and the
$n$-multiplication morphism for arbitrary $n\neq 0$.

###### Proof.

Since $f$ is faithfully flat, in particular surjective, it is an epimorphism
of sheaves by Lemma 3.3. An isogeny of abelian schemes is faithfully flat by
[Mil86, Proposition 8.1]. ∎

######  Lemma 3.5.

Let $G/X$ be a _smooth_ commutative group scheme. Then there are comparison
isomorphisms

$\displaystyle\mathrm{H}^{i}_{\mathrm{fppf}}(X,G)=\mathrm{H}^{i}_{\mathrm{\acute{e}t}}(X,G).$

In particular, $\mathrm{H}^{i}_{\mathrm{fppf}}(X,G)$ is finite if $X$ is
proper over a finite field and $G$ is a commutative finite étale group scheme.

###### Proof.

See [Mil80, Remark III.3.11 (b)] and note that the proof given there gives a
comparison isomorphism for any topologies between the étale and the flat site.
∎

######  Lemma 3.6.

Let $X$ be a Noetherian integral scheme with function field $K(X)$ and
$U\subseteq X$ dense open. Then there is an exact sequence

$1\to\mathbf{G}_{m}(X)\to\mathbf{G}_{m}(U)\to\bigoplus_{D\in(X\setminus
U)^{(1)}}\mathbf{Z}[D]\to\operatorname{Cl}(X)\to\operatorname{Cl}(U)\to 0.$

###### Proof.

The assumptions imply that there is a commutative diagram with exact rows

$\displaystyle{1}$$\displaystyle{\mathscr{O}_{X}(X)^{\times}}$$\displaystyle{K(X)^{\times}}$$\displaystyle{\operatorname{Div}(X)}$$\displaystyle{\operatorname{Cl}(X)}$$\displaystyle{0}$$\displaystyle{1}$$\displaystyle{\mathscr{O}_{X}(U)^{\times}}$$\displaystyle{K(U)^{\times}}$$\displaystyle{\operatorname{Div}(U)}$$\displaystyle{\operatorname{Cl}(U)}$$\displaystyle{0.}$

A diagram chase yields the result. ∎

######  Corollary 3.7.

Let $X$ be a Noetherian integral regular scheme and let $U\subseteq X$ be
dense open. Then there is an exact sequence

$1\to\mathbf{G}_{m}(X)\to\mathbf{G}_{m}(U)\to\bigoplus_{D\in(X\setminus
U)^{(1)}}\mathbf{Z}[D]\to\operatorname{Pic}(X)\to\operatorname{Pic}(U)\to 0.$

###### Proof.

By the assumptions, $\operatorname{Cl}(X)=\operatorname{Pic}(X)$ and
$\operatorname{Cl}(U)=\operatorname{Pic}(U)$. ∎

######  Corollary 3.8.

Let $X/\mathbf{F}_{q}$ be an integral Noetherian regular proper variety and
let $j:U\hookrightarrow X$ be the inclusion of an open subscheme of $X$. Then
$\mathrm{H}^{1}_{\mathrm{fppf}}(U,\mu_{p^{n}})$ is finite for all $n$ and any
prime $p$.

###### Proof.

The Kummer sequence Lemma 3.4 on $U_{\mathrm{fppf}}$ together with

$\operatorname{Pic}(U)=\mathrm{H}^{1}_{\mathrm{fppf}}(U,\mathbf{G}_{m,U})$

by Lemma 3.5 yields the exact sequence

$1\to\mathbf{G}_{m}(U)/p^{n}\to\mathrm{H}^{1}_{\mathrm{fppf}}(U,\mu_{p^{n}})\to\operatorname{Pic}(U)[p^{n}]\to
0.$

Since $\mathbf{G}_{m}(X)=\Gamma(X,\mathbf{G}_{m})^{\times}$ is finite by the
coherence theorem [GD61, Thm. (3.2.1)], since $X/\mathbf{F}_{q}$ is proper and
$\mathbf{F}_{q}$ is finite, and since $\operatorname{Pic}(X)$ is finitely
generated since its sits in a short exact sequence
$0\to\operatorname{Pic}^{0}(X)\to\operatorname{Pic}(X)\to\operatorname{NS}(X)\to
0$ and $\operatorname{Pic}^{0}(X)$ is finite since it is the group of rational
points of an Abelian variety over a finite field and $\operatorname{NS}(X)$ is
always finitely generated by [BGI71, Exp. XIII, § 5], by Corollary 3.7 and the
finiteness of $(X\setminus U)^{(1)}$, this exact sequence gives the finiteness
of $\mathbf{G}_{m}(U)/p^{n}$ and of $\operatorname{Pic}(U)[p^{n}]$. ∎

The following statements and proofs in this section are an extended version of
the sketch of Theorem 3.14 given by ‘darx’ in [use].

######  Lemma 3.9.

Let $X$ be a normal integral scheme and $G/X$ be a finite flat group scheme.
If $T$ is a $G$-torsor on $X$ trivial over the generic point of $X$, then $T$
is trivial. Hence,
$\mathrm{H}^{1}_{\mathrm{fppf}}(X,G)\to\mathrm{H}^{1}_{\mathrm{fppf}}(K(X),G)$
is injective, and if $f:Y\to X$ is birational,
$f^{*}:\mathrm{H}^{1}_{\mathrm{fppf}}(X,G)\to\mathrm{H}^{1}_{\mathrm{fppf}}(Y,G)$
is injective.

###### Proof.

Since $T$ is trivial over the generic point of $X$, generically, there is a
section of $\pi:T\to X$. This extends to a rational map
$\sigma:X\dashrightarrow T$. Take the schematic closure
$i:X^{\prime}\hookrightarrow T$ of $\sigma$. The composition $\pi\circ
i:X^{\prime}\to T\to X$ is birational and finite (as a composition of a closed
immersion and a finite morphism). By [GW10, Corollary 12.88], since $X$ is
normal, $X^{\prime}\to X$ is an isomorphism. Hence $\sigma$ is a section of
$\pi$, so $T/X$ is trivial. ∎

######  Lemma 3.10.

Let $X$ be a proper variety over a finite field and $Y/X$ be a finite flat
scheme. Let $Z/X$ be proper. Then $Y(Z)$ is finite.

###### Proof.

Since $\operatorname{Mor}_{X}(Z,Y)=\operatorname{Mor}_{Z}(Z,Y\times_{X}Z)$,
one can assume $Z=X$. So we have to show that there are only finitely many
sections to $\pi:Y\to X$. Such a section corresponds to an
$\mathscr{O}_{X}$-algebra map $\pi_{*}\mathscr{O}_{Y}\to\mathscr{O}_{X}$. But
$\mathrm{H}^{0}_{\mathrm{Zar}}(X,\mathscr{H}\mathrm{om}_{X}(\pi_{*}\mathscr{O}_{Y},\mathscr{O}_{X}))$
is finite by the coherence theorem [GD61, Thm. (3.2.1)] as it is a finite
dimensional vector space over a finite field. ∎

######  Lemma 3.11.

Let $Y\to X$ be an alteration of proper integral varieties with $X$ normal,
and $G/X$ be a finite flat commutative group scheme. Then
$\ker(\mathrm{H}^{1}_{\mathrm{fppf}}(X,G)\to\mathrm{H}^{1}_{\mathrm{fppf}}(Y,G))$
is finite. Hence $\mathrm{H}^{1}_{\mathrm{fppf}}(X,G)$ is finite if
$\mathrm{H}^{1}_{\mathrm{fppf}}(Y,G)$ is.

###### Proof.

If $Y\to X$ is a blow-up, the kernel is trivial by Lemma 3.9 since a blow-up
is birational. Hence the statement holds for blow-ups.

By [RG71, Théorème 5.2.2], there is a blow-up $f:X^{\prime}\to X$ such that
$Y^{\prime}:=Y\times_{X}X^{\prime}$ is flat over $X^{\prime}$. Since a
normalization morphism of integral schemes is birational [Liu06, Proposition
4.1.22], one can assume $X^{\prime}$ normal. There is a commutative diagram

$\displaystyle{0}$$\displaystyle{\ker(\mathrm{H}^{1}_{\mathrm{fppf}}(X,G)\to\mathrm{H}^{1}_{\mathrm{fppf}}(Y,G))}$$\displaystyle{\mathrm{H}^{1}_{\mathrm{fppf}}(X,G)}$$\displaystyle{0}$$\displaystyle{\ker(\mathrm{H}^{1}_{\mathrm{fppf}}(X^{\prime},G)\to\mathrm{H}^{1}_{\mathrm{fppf}}(Y^{\prime},G))}$$\displaystyle{\mathrm{H}^{1}_{\mathrm{fppf}}(X^{\prime},G)}$$\displaystyle{f^{*}}$

By the snake lemma, since $\ker{f^{*}}$ is finite as $f$ is a blow-up,

$\ker(\mathrm{H}^{1}_{\mathrm{fppf}}(X,G)\to\mathrm{H}^{1}_{\mathrm{fppf}}(Y,G))$

is finite if we can show that

$\ker(\mathrm{H}^{1}_{\mathrm{fppf}}(X^{\prime},G)\to\mathrm{H}^{1}_{\mathrm{fppf}}(Y^{\prime},G))$

is finite. Hence, we can assume $Y\to X$ finite flat.

Let $T\to X$ be in the kernel, i. e., it is a $G$-torsor on $X$ trivial when
pulled back to $Y$. Choose a section $\sigma:Y\to T\times_{X}Y$; there are
only finitely many of them by Lemma 3.10. Two such sections differ by an
element of $G(Y)$. Since the base change $T\times_{X}(Y\times_{X}Y)\to
Y\times_{X}Y$ is a $G$-torsor, one can take the $1$-cocycle

$\tau:=d^{0}(\sigma)=\operatorname{pr}_{0}^{*}(\sigma)-\operatorname{pr}_{1}^{*}(\sigma)\in
G(Y\times_{X}Y).$

The section $\tau$ corresponds to the isomorphism class of the $G$-torsor $T$
by the descent theory for the fppf covering $\\{Y\to X\\}$: As
$\mathrm{H}^{1}_{\mathrm{fppf}}(-,G)$ can be computed by Čech cohomology and
as the class of $T$ in
$\mathrm{H}^{1}_{\mathrm{fppf}}(X,G)=\check{H}^{1}_{\mathrm{fppf}}(X,G)=\operatorname{\varinjlim}_{\mathscr{U}}\check{H}^{1}_{\mathrm{fppf}}(\mathscr{U},G)$
(the colimit taken over the coverings of $X$; the natural morphism from the
first Čech cohomology to the first derived functor cohomology is always an
isomorphism) is trivialized by the covering $\\{Y\to X\\}$, it can be
represented as the $1$-cocycle $\tau=d^{0}(\sigma)$, which is a
$1$-coboundary:

$\check{H}^{1}(\\{Y\to
X\\},G)=\frac{\ker\big{(}G(Y\times_{X}Y)\stackrel{{\scriptstyle
d^{1}}}{{\to}}G(Y\times_{X}Y\times_{X}Y)\big{)}}{\operatorname{im}\big{(}G(Y)\stackrel{{\scriptstyle
d^{0}}}{{\hookrightarrow}}G(Y\times_{X}Y)\big{)}}$

But by Lemma 3.10, $G(Y\times_{X}Y)$ is finite. ∎

######  Lemma 3.12.

Let $X$ be an integral scheme with function field $K$ and $G/X$ be a finite
flat group scheme. Let $H_{K}\hookrightarrow G_{K}$ be a finite flat group
scheme. Then there is a blow-up $\tilde{X}/X$ such that $H_{K}$ extends to a
finite flat subgroup scheme of $G\times_{X}\tilde{X}$.

###### Proof.

Let $H\hookrightarrow G$ be the schematic closure of $H_{K}\hookrightarrow G$.
The morphism $H\to G\to X$ is finite as a composition of a closed immersion
and a finite morphism. By [RG71, Théorème 5.2.2], there is a blow-up
$X^{\prime}\to X$ such that $H^{\prime}:=H\times_{X}X^{\prime}\to X^{\prime}$
is flat. Then, $H^{\prime}$ is the schematic closure of $H_{K}\hookrightarrow
G^{\prime}:=G\times_{X}X^{\prime}$. So one can assume $H/X$ finite flat.

Let $Y\to X$ be finite flat. Since the morphism is affine, locally, one has
the diagram

$\displaystyle{A}$$\displaystyle{A\otimes_{R}\operatorname{Quot}(R)}$$\displaystyle{R}$$\displaystyle{\operatorname{Quot}(R).}$

Here, the upper horizontal arrow is injective by flatness of $R\to A$. Hence
$Y$ is the schematic closure of $Y_{K}$ in $Y$.

By flatness, the schematic closure of $H_{K}\times_{K}H_{K}$ in $G\times_{X}G$
is $H\times_{X}H$. By the universal property of the schematic closure [GW10,
(10.8)], one has the factorization

$\displaystyle{H_{K}\times_{K}H_{K}}$$\displaystyle{H_{K}}$$\displaystyle{H\times_{X}H}$$\displaystyle{H}$$\displaystyle{G\times_{X}G}$$\displaystyle{G,}$$\displaystyle{\mu}$$\displaystyle{\mu}$$\displaystyle{\mu}$

for the multiplication $\mu$, and similar for the inverse and unit section. ∎

######  Lemma 3.13.

Let $X$ be a proper integral variety over a field and $G/X$ be a finite flat
commutative group scheme. After an alteration $X^{\prime}\to X$, there exists
a filtration of $G$ by finite flat group schemes with subquotients of prime
order.

###### Proof.

Over the algebraic closure of the function field of $X$, there is such an
filtration since the only simple objects in the category of finite flat group
schemes of $p$-power order are $\mu_{p}$, $\mathbf{Z}/p$ and $\alpha_{p}$.
Since everything is of finite presentation, these are defined over a finite
extension of the function field [GW10, Corollary 10.79]. Now take the
normalization in this finite extension of function fields and use Lemma 3.12.
∎

######  Theorem 3.14.

Let $X$ be a proper integral normal variety over a finite field and $G/X$ be a
finite flat commutative group scheme. Then
$\mathrm{H}^{1}_{\mathrm{fppf}}(X,G)$ is finite.

###### Proof.

By Lemma 3.13, Lemma 3.11 and the long exact cohomology sequence one can
assume $G$ of prime order $p$ (since the case of $G/X$ étale is easily dealt
with). Since then $G$ is simple by [Sha86, p. 38] and since $F\circ V=[p]=0$
by [Sha86, p. 62] and [Mum70, p. 141], either $V=0$ or $F=0$ on $G$.

If $V=0$, by [de 93, Proposition 2.2], there is a short exact sequence

$0\to G\to\mathscr{L}\to\mathscr{M}\to 0$

with vector bundles $\mathscr{L},\mathscr{M}$. By the coherence theorem [GD61,
Thm. (3.2.1)], as $X$ is proper and lives over a finite ground field, and by
comparison of Zariski and fppf cohomology [Mil80, Proposition III.3.7], the
long exact cohomology sequence shows that
$\mathrm{H}^{i}_{\mathrm{fppf}}(X,G)$ is finite.

If $F=0$, after replacing $X$ by an alteration by Lemma 3.11 as in the proof
of Lemma 3.13, one can assume that $G$ is isomorphic to $\mu_{p}$ over the
generic point. Since for $Y,Z/X$ of finite presentation such that $Y_{K}\cong
Z_{K}$, there is a non-empty open subscheme $U\hookrightarrow X$ such that
$Y_{U}\cong Z_{U}$, there is a non-empty open subscheme $U\hookrightarrow X$
such that $G_{U}\cong\mu_{p,U}$. By [de 96], there is an alteration
$f:X^{\prime}\to X$ such that $X^{\prime}$ is regular. By Corollary 3.8,
$\mathrm{H}^{1}_{\mathrm{fppf}}(f^{-1}(U),\mu_{p})$ is finite. By Lemma 3.9,
$\mathrm{H}^{1}_{\mathrm{fppf}}(X^{\prime},G\times_{X}X^{\prime})$ is finite,
so by Lemma 3.11, $\mathrm{H}^{1}_{\mathrm{fppf}}(X,G)$ is finite. ∎

##  4\. Isogeny invariance of finiteness of , the $p$-part

In this section, we extend [Kel16, p. 240, Theorem 4.31] to
$p^{\infty}$-torsion.

######  Theorem 4.1.

Let $X/k$ be a proper variety over a finite field $k$ and
$f:\mathscr{A}\to\mathscr{A}^{\prime}$ be an isogeny of Abelian schemes over
$X$. Let $p$ be an arbitrary prime. Assume $f$ étale if
$p\neq\operatorname{char}{k}$. Then $\Sha(\mathscr{A}/X)[p^{\infty}]$ is
finite if and only if $\Sha(\mathscr{A}^{\prime}/X)[p^{\infty}]$ is finite.

###### Proof.

In the case where $\ell$ is invertible on $X$ and $f$ is étale (i. e., of
degree invertible on $X$), this is [Kel16, p. 240, Theorem 4.31].

Now assume $p=\operatorname{char}{k}$. Then the short exact sequence of flat
sheaves Lemma 3.4 yields an exact sequence in cohomology

$\mathrm{H}^{1}_{\mathrm{fppf}}(X,\ker(f))\to\mathrm{H}^{1}_{\mathrm{fppf}}(X,\mathscr{A})\stackrel{{\scriptstyle
f}}{{\to}}\mathrm{H}^{1}_{\mathrm{fppf}}(X,\mathscr{A}^{\prime})$

and note that
$\mathrm{H}^{1}_{\mathrm{fppf}}(X,\mathscr{A})=\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(X,\mathscr{A})=\Sha(\mathscr{A}/X)$
by Lemma 3.5 since $\mathscr{A}/X$ is smooth, and that
$\mathrm{H}^{1}_{\mathrm{fppf}}(X,\ker(f))$ is finite by Theorem 3.14. Note
that all groups are torsion (the Tate-Shafarevich groups by [Kel16, p. 224,
Proposition 4.1]), hence the sequence stays exact after taking
$p^{\infty}$-torsion. So $\Sha(\mathscr{A}/X)[p^{\infty}]$ is finite if
$\Sha(\mathscr{A}^{\prime}/X)[p^{\infty}]$ is.

For the converse, note that by [Kel19, Proposition 2.19], there is a
polarization $\lambda:\mathscr{A}^{t}\to\mathscr{A}$. Hence, the argument
above for $\lambda$ and $\lambda^{t}$ implies that
$\Sha(\mathscr{A}^{t}/X)[p^{\infty}]$ is finite iff
$\Sha(\mathscr{A}/X)[p^{\infty}]$ is, and analogously for
$\Sha(\mathscr{A}^{\prime}/X)[p^{\infty}]$. Taking the dual Kummer sequence
$0\to\ker(f^{t})\to\mathscr{A}^{\prime t}\to\mathscr{A}^{t}\to 0$ yields an
exact sequence

$\mathrm{H}^{1}_{\mathrm{fppf}}(X,\ker(f^{t}))\to\Sha(\mathscr{A}^{\prime
t}/X)\to\Sha(\mathscr{A}^{t}/X).$

By the same argument as above, $\Sha(\mathscr{A}^{\prime t}/X)[p^{\infty}]$ is
finite if $\Sha(\mathscr{A}^{t}/X)[p^{\infty}]$ is if
$\Sha(\mathscr{A}/X)[p^{\infty}]$ is. So
$\Sha(\mathscr{A}^{\prime}/X)[p^{\infty}]$ is finite. ∎

##  5\. Descent of finiteness of , the $p$-part

In this section, we extend [Kel16, p. 238, Theorem 4.29] to
$p^{\infty}$-torsion.

######  Lemma 5.1.

Let $\mathscr{A}/X$ be an Abelian scheme over a proper variety $X$ over a
finite field of characteristic $p$. Then $\Sha(\mathscr{A}/X)[p^{\infty}]$ is
cofinitely generated.

Recall that $\Sha(\mathscr{A}/X)$ was defined as
$\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(X,\mathscr{A})$ in [Kel16, p. 225,
Definition 4.2].

###### Proof.

The long exact cohomology sequence associated to the Kummer sequence Lemma 3.4
gives us a surjection

$\mathrm{H}^{1}_{\mathrm{fppf}}(X,\mathscr{A}[p^{n}])\twoheadrightarrow\mathrm{H}^{1}_{\mathrm{fppf}}(X,\mathscr{A})[p^{n}]\to
0$

Now, since $\mathscr{A}/X$ is a smooth group scheme, Lemma 3.5 gives us an
isomorphism
$\mathrm{H}^{1}_{\mathrm{fppf}}(X,\mathscr{A})=\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(X,\mathscr{A})$,
which by definition equals $\Sha(\mathscr{A}/X)$. By Theorem 3.14,
$\mathrm{H}^{1}_{\mathrm{fppf}}(X,\mathscr{A}[p^{n}])$ is finite since
$X/\mathbf{F}_{q}$ is proper. From this, one sees that
$\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(X,\mathscr{A})[p]$ is finite. Hence
$\Sha(\mathscr{A}/X)[p^{\infty}]$ is cofinitely generated by [Kel19, Lemma
2.38]. ∎

######  Lemma 5.2 (existence of trace morphism).

Let $f:X^{\prime}\to X$ be a finite étale morphism of constant degree $d$ and
let $\mathscr{F}$ be an fppf sheaf on $X$. Then there is a trace map
$\operatorname{Tr}_{f}:f_{*}f^{*}\mathscr{F}\to\mathscr{F}$, functorial in
$\mathscr{F}$, such that $\varphi\mapsto\operatorname{Tr}_{f}\circ
f_{*}(\varphi)$ is an isomorphism
$\operatorname{Hom}_{X^{\prime}}(\mathscr{F}^{\prime},f^{*}\mathscr{F})\to\operatorname{Hom}_{X}(\pi_{*}\mathscr{F}^{\prime},\mathscr{F})$
for any fppf sheaf $\mathscr{F}^{\prime}$ on $X^{\prime}$. Thus,
$f_{*}=f_{!}$, that is, $f_{*}$ is left adjoint to $f^{*}$, and
$\operatorname{Tr}_{f}$ is the adjunction map. The composites

$\mathscr{F}\to
f_{*}f^{*}\mathscr{F}\stackrel{{\scriptstyle\operatorname{Tr}_{f}}}{{\to}}\mathscr{F}$

and

$\mathrm{H}^{r}_{\mathrm{fppf}}(X,\mathscr{F})\xrightarrow{f^{*}}\mathrm{H}^{r}_{\mathrm{fppf}}(X^{\prime},f^{*}\mathscr{F})\xrightarrow{\operatorname{can}}\mathrm{H}^{r}_{\mathrm{fppf}}(X,f_{*}f^{*}\mathscr{F})\xrightarrow{\operatorname{Tr}_{f}}\mathrm{H}^{r}_{\mathrm{fppf}}(X,\mathscr{F})$

are multiplication by $d$.

###### Proof.

On may copy the proof of [Mil80, p. 168, Lemma V.1.12] almost verbatim: Let
$\mathscr{F}$ be a fppf sheaf on $X$. Let $X^{\prime\prime}\to X$ be finite
Galois with Galois group $G$ factoring as $X^{\prime\prime}\to X^{\prime}\to
X$; $X^{\prime\prime}\to X^{\prime}$ is Galois with Galois group $H\leq G$.
For any $U/X$ flat, we have
$\Gamma(U,\mathscr{F})\hookrightarrow\Gamma(U^{\prime},\mathscr{F})\hookrightarrow\Gamma(U^{\prime\prime},\mathscr{F})$
and $\Gamma(U,\mathscr{F})\xrightarrow{\leavevmode\nobreak\
\sim\leavevmode\nobreak\ }\Gamma(U^{\prime\prime},\mathscr{F})^{G}$, where
$U^{\prime}=U\times_{X}X^{\prime}$ and
$U^{\prime\prime}=U\times_{X}X^{\prime\prime}$. For a section
$s\in\Gamma(U,f_{*}f^{*}\mathscr{F}):=\Gamma(U^{\prime},\mathscr{F})$, we
define

$\operatorname{Tr}_{f}(s):=\sum_{\sigma\in G/H}\sigma(s|_{U^{\prime\prime}});$

as this is fixed by $G$, it may be regarded as an element of
$\Gamma(U,\mathscr{F})\xrightarrow{\leavevmode\nobreak\
\sim\leavevmode\nobreak\ }\Gamma(U^{\prime\prime},\mathscr{F})^{G}$. Clearly,
$\operatorname{Tr}_{f}$ defines a morphism
$f_{*}f^{*}\mathscr{F}\to\mathscr{F}$ such that its composite with
$\mathscr{F}\to f_{*}f^{*}\mathscr{F}$ is multiplication by the degree $d$ of
$f$.

If $X^{\prime}$ is a disjoint union of $d$ copies of $X$, obviously

$\operatorname{Hom}_{X^{\prime}}(\mathscr{F}^{\prime},f^{*}\mathscr{F})\xrightarrow{\leavevmode\nobreak\
\sim\leavevmode\nobreak\
}\operatorname{Hom}_{X}(f_{*}\mathscr{F}^{\prime},\mathscr{F}),$

and one may reduce the question to this split case by passing to a finite
étale covering of $X$, for example to $X^{\prime\prime}\to X$, and using the
fact that $\operatorname{Hom}$ is a sheaf.

In

$\mathrm{H}^{r}_{\mathrm{fppf}}(X,\mathscr{F})\xrightarrow{f^{*}}\mathrm{H}^{r}_{\mathrm{fppf}}(X^{\prime},f^{*}\mathscr{F})\xrightarrow{\operatorname{can}}\mathrm{H}^{r}_{\mathrm{fppf}}(X,f_{*}f^{*}\mathscr{F})\xrightarrow{\operatorname{Tr}_{f}}\mathrm{H}^{r}_{\mathrm{fppf}}(X,\mathscr{F})$

the composite of the first two maps is induced by $\mathscr{F}\to
f_{*}f^{*}\mathscr{F}$, and the composite of all three is induced by
$(\mathscr{F}\to
f_{*}f^{*}\mathscr{F}\xrightarrow{\operatorname{Tr}_{f}}\mathscr{F})$, which
is multiplication by $d$. ∎

######  Theorem 5.3.

Let $p$ be a prime and $X$ be a scheme of characteristic $p$. Let
$f:X^{\prime}\to X$ be a proper, surjective, generically étale morphism of
generical degree prime to $p$ of regular, integral, separated varieties over a
finite field. Let $\mathscr{A}$ be an abelian scheme on $X$ and
$\mathscr{A}^{\prime}:=f^{*}\mathscr{A}=\mathscr{A}\times_{X}X^{\prime}$. If
$\Sha(\mathscr{A}^{\prime}/X^{\prime})[p^{\infty}]$ is finite, so is
$\Sha(\mathscr{A}/X)[p^{\infty}]$.

###### Proof.

The same proof as in [Kel16, Theorem 4.29] works, one only needs
$\Sha(\mathscr{A}/X)[p^{\infty}]$ to be cofinitely generated in Step 2, which
is Lemma 5.1. The trace morphism in Step 3 for fppf cohomology comes from
Lemma 5.2. Note that the proof given there does not need the regularity of
$X,X^{\prime}$ and that varieties over a field are excellent by [Liu06,
Corollary 2.40 (a)]. For the convenience of the reader, we reproduce the proof
of [Kel16, Theorem 4.29] adapted to our situation here:

Step 1:
$\mathrm{H}^{1}_{\mathrm{fppf}}(X,f_{*}\mathscr{A}^{\prime})[p^{\infty}]$ is
finite. This follows from the low terms exact sequence

$0\to\mathrm{H}^{1}_{\mathrm{fppf}}(X,f_{*}\mathscr{A}^{\prime})\to\mathrm{H}^{1}_{\mathrm{fppf}}(X^{\prime},\mathscr{A}^{\prime})$

associated to the Leray spectral sequence

$\mathrm{H}^{p}_{\mathrm{fppf}}(X,\mathrm{R}^{q}f_{*}\mathscr{A}^{\prime})\Rightarrow\mathrm{H}^{p+q}_{\mathrm{fppf}}(X^{\prime},\mathscr{A}^{\prime})$

and the finiteness of

$\mathrm{H}^{1}_{\mathrm{fppf}}(X^{\prime},\mathscr{A}^{\prime})[p^{\infty}]=\Sha(\mathscr{A}^{\prime}/X^{\prime})[p^{\infty}].$

Step 2: The theorem holds if there is a trace morphism. Since by Lemma 5.2
there is a trace morphism $f_{*}f^{*}\mathscr{A}\to\mathscr{A}$ such that the
composition with the adjunction morphism

$\mathscr{A}\to f_{*}f^{*}\mathscr{A}\to\mathscr{A}$

is multiplication by $\deg{f}\neq 0$, the finiteness of
$\mathrm{H}^{1}_{\mathrm{fppf}}(X,\mathscr{A})[p^{\infty}]$ follows from that
of $\mathrm{H}^{1}_{\mathrm{fppf}}(X,f_{*}\mathscr{A}^{\prime})[p^{\infty}]$
because both groups are cofinitely generated by Lemma 5.1.

Step 3: Proof of the theorem in the general case. Let $\eta$ be the generic
point of $X$. Define $X^{\prime}_{\eta}$ by the commutativity of the cartesian
diagram

(5.1)
$\displaystyle{X^{\prime}_{\eta}}$$\displaystyle{X^{\prime}}$$\displaystyle{\\{\eta\\}}$$\displaystyle{X.}$$\displaystyle{g^{\prime}}$$\displaystyle{f_{\eta}}$$\displaystyle{f}$$\displaystyle{g}$

Since $f$ is generically étale, we can apply Lemma 5.2 to $f_{\eta}$ in this
commutative diagram. From the commutativity of that diagram, the kernel of
$f^{*}:\mathrm{H}^{1}_{\mathrm{fppf}}(X,\mathscr{A})\to\mathrm{H}^{1}_{\mathrm{fppf}}(X^{\prime},\mathscr{A}^{\prime})$
is contained in the kernel of the composition

$\mathrm{H}^{1}_{\mathrm{fppf}}(X,\mathscr{A})\stackrel{{\scriptstyle
g^{*}}}{{\to}}\mathrm{H}^{1}_{\mathrm{fppf}}(\\{\eta\\},\mathscr{A}_{\eta})\stackrel{{\scriptstyle
f_{\eta}^{*}}}{{\to}}\mathrm{H}^{1}_{\mathrm{fppf}}(X^{\prime}_{\eta},\mathscr{A}^{\prime}_{X^{\prime}_{\eta}}),$

so it suffices to show that the first arrow $g^{*}$ is injective. But by the
Néron mapping property $\mathscr{A}\xrightarrow{\leavevmode\nobreak\
\sim\leavevmode\nobreak\ }g_{*}g^{*}\mathscr{A}$ [Kel16, Theorem 3.3] (for the
_étale_ topology!),
$\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(X,\mathscr{A})\xrightarrow{\leavevmode\nobreak\
\sim\leavevmode\nobreak\
}\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(X,g_{*}\mathscr{A}_{\eta})$. However,
the Leray spectral sequence

$\mathrm{H}^{p}_{\mathrm{\acute{e}t}}(X,\mathrm{R}^{q}g_{*}\mathscr{A}_{\eta})\Rightarrow\mathrm{H}^{p+q}_{\mathrm{\acute{e}t}}(\\{\eta\\},\mathscr{A}_{\eta})$

gives an injection

$0\to\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(X,g_{*}\mathscr{A}_{\eta})\to\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(\\{\eta\\},\mathscr{A}_{\eta}).$

But because $\mathscr{A}/X$ and $\mathscr{A}_{\eta}/\\{\eta\\}$ are smooth
commutative group schemes, their étale cohomology agrees with their flat
cohomology, see Lemma 3.5, and the comparison of topology morphisms are
functorial. ∎

######  Theorem 5.4 (Stein factorization, alteration $=$ finite $\circ$
modification).

Let $f:X^{\prime}\to X$ be a proper morphism of Noetherian schemes. Then one
can factor $f$ into $g\circ f^{\prime}$, where $f^{\prime}:X^{\prime}\to
Y:=\mathbf{Spec}_{X}f_{*}\mathscr{O}_{X^{\prime}}$ is a proper morphism with
connected fibers, and $g:Y\to X$ is a finite morphism. If $f$ is an
alteration, $f^{\prime}$ is birational and proper (a _modification_).

###### Proof.

See [GD61, Thm. 4.3.1] for the statement on the existence of the
factorization, which includes
$Y=\mathbf{Spec}_{X}f_{*}\mathscr{O}_{X^{\prime}}$.

Assume now that $f$ is an alteration. If $U\subseteq X$ is an open subscheme
such that $f|_{U}$ is finite (in particular affine), one may shrink $U$ such
that it is affine, so by finiteness of $g$, $g:g^{-1}(U)\to U$ is finite and
can be written as $\operatorname{Spec}{B}\to\operatorname{Spec}{A}$. From the
statement of Stein factorization,
$g^{-1}(U)=\mathbf{Spec}_{U}f_{*}\mathscr{O}_{f^{-1}(U)}$, but $f^{\prime}$
has geometrically connected fibers, so
$f^{\prime}_{*}\mathscr{O}_{X^{\prime}}=\mathscr{O}_{Y}$, so
$f^{\prime}|_{g^{-1}(U)}$ is an isomorphism because it is affine. ∎

We also remove the hypotheses that $f$ is generically étale and has degree
prime to $\ell$ if $\ell$ is invertible on the base scheme in [Kel16, Theorem
4.29]:

######  Theorem 5.5.

Let $f:X^{\prime}\to X$ be a proper, surjective, generically finite morphism
of regular, integral, separated varieties over a finite field. Let
$\mathscr{A}$ be an abelian scheme on $X$ and
$\mathscr{A}^{\prime}:=f^{*}\mathscr{A}=\mathscr{A}\times_{X}X^{\prime}$. Let
$\ell$ be invertible on $X$. If
$\Sha(\mathscr{A}^{\prime}/X^{\prime})[\ell^{\infty}]$ is finite, so is
$\Sha(\mathscr{A}/X)[\ell^{\infty}]$.

###### Proof.

By the Stein factorization Theorem 5.4, $f$ factors as a proper, surjective,
birational morphism followed by a finite morphism. In particular, it is a
generically étale alteration. The finite morphism factors as a finite purely
inseparable morphism followed by a finite generically étale morphism. We prove
the finiteness assertion of the theorem for all such morphisms separately:

If $f$ is generically étale, this is Theorem 5.3. If $f$ is a proper,
surjective, birational morphism, it is generically an isomorphism, i.e.,
generically étale of degree $1$.

If $f$ is a universal homeomorphism, the étale sites of $X$ and $X^{\prime}$
are equivalent by $f^{*}$ and $f_{*}$ by [AGV72, VIII.1.1]. In particular, the
étale cohomology groups
$\Sha(\mathscr{A}/X)=\mathrm{H}_{\mathrm{\acute{e}t}}^{1}(X,\mathscr{A})$ and
$\Sha(\mathscr{A}^{\prime}/X^{\prime})=\mathrm{H}_{\mathrm{\acute{e}t}}^{1}(X^{\prime},f^{*}\mathscr{A})$
are isomorphic via $f^{*}$. ∎

## References

  * [AGV72] Michael Artin, Alexandre Grothendieck, and Jean-Louis Verdier, _Séminaire de Géométrie Algébrique du Bois Marie – 1963–64 – Théorie des topos et cohomologie étale des schémas – (SGA 4) – vol. 2_ , Lecture notes in mathematics (in French) 270, Berlin; New York: Springer-Verlag. iv+418, 1972 (french).
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Calculus of functional centrality

Djemel Ziou
---
Département d’informatique
Université de Sherbrooke
Sherbrooke, Qc., Canada J1K 2R1
<EMAIL_ADDRESS>

###### Abstract

In this document, we present another perspective for the calculus of optimal
geometrical primitives and functions according to the centrality requirements.
The shortest paths expressed in spatial and temporal domains are studied. We
show the effectiveness of this formulation by providing solutions that cannot
be easily accessed by classical formulation when using the calculus of
variations.

Keywords: Centrality, functional centrality, calculus of variations,
Brachistochrone, shortest path.

##  1 Introduction

The calculus of variations is widely used in mathematics, physics, image
processing, and other areas of knowledge [2, 3, 4]. While the optimization
aims to find a stationary point of a function defined over some domain, the
calculus of variation seeks to find stationary geometrical primitives other
than the point (e.g. line, surface, volume) or a stationary function according
to a given functional. Let us concentrate on the simplest calculus of
variations in which the unknown is univariate continuously derivable curves
$u(x):[a,b]\to\mathbb{R}$ and the first-order functional to be minimized
depends upon $u(x)$, its first derivative $u^{\prime}(x)$, and $x$. This
first-order functional is specified by a geometrical or physics feature
$F(x,u(x),u^{\prime}(x))$, such as the curve length, its curvature or the time
travelled by an object. It is given by:

$J(u)=\int_{a}^{b}F(x,u(x),u^{\prime}(x))dx$ (1)

The feature $F(x,u(x),u^{\prime}(x))$ is assumed to be twice continuously
differentiable with respect to $x$ and the first and second variations with
respect to $u$ and $u^{\prime}$ are continuous with respect to $x$. In this
case, the solution to this problem can be obtained by solving the following
Euler-Lagrange ordinary differential equation (ODE):

$\frac{\partial F(x,u(x),u^{\prime}(x))}{\partial
u}-\frac{d}{dx}\frac{\partial F(x,u(x),u^{\prime}(x))}{\partial u^{\prime}}=0$
(2)

In order to formulate the condition so that the stationary curve to be a
minimum of the functional $J(u)$, we need to establish the positive
definiteness of the second variation. Straightforward computations lead to
conclude that the critical curve u(x) is a minimum if the following inequality
holds:

$\frac{\partial^{2}F(x,u(x),u^{\prime}(x))}{\partial
u^{2}}h^{2}+2\frac{\partial^{2}F(x,u(x),u^{\prime}(x))}{\partial u\partial
u^{\prime}}hh^{\prime}+\frac{\partial^{2}F(x,u(x),u^{\prime}(x))}{\partial
u^{\prime 2}}h^{\prime 2}>0$ (3)

where $h(x)$ is a non-constant variation such that $h(a)=h(b)=0$. The reader
will find more about the calculus of variations in [2] and in other
literature. Let us now consider things from another perspective. To this end,
in this document we will limit ourselves to the case where
$F(x,u(x),u^{\prime}(x))>0$. The functional in Eq. 1 is a prior knowledge
expressing some arbitrariness because it is the result of intuition not
subject to experimentation. For example, it can be the length of a plane
curve, which can be measured in an infinite number of ways because there are
an infinite number of ways to define distance. From this point of view, the
variational problem in Eq. 1 can be expressed in an infinite number of ways.
Let us consider the transformation of the feature $F(x,u(x),u^{\prime}(x))$
yielding $F^{\alpha}(x,u(x),u^{\prime}(x))$, where $\alpha\in\mathbb{R}$. For
a given $x$, $F^{\alpha}(x,u(x),u^{\prime}(x))$ is greater or lesser than
$F(x,u(x),u^{\prime}(x))$ depending upon the values of
$F(x,u(x),u^{\prime}(x))$ and $\alpha$. Similarly to Eq. 1, the first-order
functional can be written in the transformed space as:

$C_{\alpha}^{\alpha}(u)=\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))dx$
(4)

where $\delta=b-a$. The rational explanation of the exponent $\alpha$ of
$C_{\alpha}()$ can be provided by the mean value theorem. By considering $F()$
as an explicit or implicit function of $x$, then there exists $\eta\in]a,b[$
such that
$F^{\alpha}(\eta,u(\eta),u^{\prime}(\eta))=\frac{1}{\delta}\int_{a}^{b}F(x,u(x),u^{\prime}(x))^{\alpha}dx$.
Instead of using $F(\eta,u(\eta),u^{\prime}(\eta))$, we rename it $C(u)$.
Taking the $\alpha$-root allows us to write:

$C_{\alpha}(u)=(\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))dx)^{1/\alpha}$
(5)

It is worth mentioning that when $\alpha=1$, the functional in Eq. 1 is a mean
value up to the interval length. Because $C_{\alpha}(u)$ is a limit when
($n\to+\infty$) of the Hölder mean of $n$ samples, we call it the first-order
functional centrality of Hölder. In this report, we will show that the first-
order functional centrality of Hölder leads us to solutions that cannot be
obtained by the variational formulation in Eq. 1. We present some of its
properties in the next section. In Section 3, we derive the ODE solution to
the first-order functional centrality of Hölder. The shortest paths in both
spatial and temporal domains are studied in Sections 4 and 5.

##  2 Properties

We announce and prove the following properties of the first-order functional
centrality of Hölder in Eq. 5 under the assumption $F(x,u(x),u^{\prime}(x))>0$
and it is bounded.

P1 : $\lim_{\alpha\to
0}C_{\alpha}(u)=e^{\frac{\int_{a}^{b}lnF(x,u(x),u^{\prime}(x))dx}{b-a}}$.
Indeed, let us write the limit of $lnC_{\alpha}(u)$:

$\lim_{\alpha\to 0}lnC_{\alpha}(u)=\lim_{\alpha\to
0}\frac{1}{\alpha}ln\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))dx$
(6)

The limit has the form 0/0, so by using the L’Hopital’s rule and the dominated
convergence theorem, we obtain:

$\displaystyle\lim_{\alpha\to 0}lnC_{\alpha}(u)$ $\displaystyle=$
$\displaystyle\lim_{\alpha\to
0}\frac{\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))lnF(x,u(x),u^{\prime}(x))dx}{\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))dx}$
(7) $\displaystyle=$
$\displaystyle\frac{\int_{a}^{b}lnF(x,u(x),u^{\prime}(x))dx}{\delta}$ (8)

P2: The functional $C_{\alpha}(u)$ is a increasing wrt $\alpha$. Indeed, the
derivative of $C_{\alpha}(u)$ wrt $\alpha$

$\frac{dlnC_{\alpha}(u)}{d\alpha}=\frac{1}{\alpha^{2}}\frac{\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))(lnF^{\alpha}(x,u(x),u^{\prime}(x))-ln(\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))dx)dx}{\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))dx}$
(9)

Equivalently,

$\frac{dlnC_{\alpha}(u)}{d\alpha}=\frac{1}{\alpha^{2}}\frac{\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))ln\frac{F^{\alpha}(x,u(x),u^{\prime}(x))}{\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))dx}dx}{\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))dx}$
(10)

Since $lnr\geq 1-1/r$ for all positive $x$, where the equality is true only if
$r=1$. In our case, $r=F^{\alpha}(x,u(x),u^{\prime}(x))/$
$\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))dx$ cannot be one
because $F^{\alpha}(x,u(x),u^{\prime}(x))$ cannot be a constant. Hence,

$\displaystyle\frac{dlnC_{\alpha}(u)}{d\alpha}>\frac{1}{\alpha^{2}}\frac{\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))(1-\frac{\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))dx}{F^{\alpha}(x,u(x),u^{\prime}(x))})dx}{\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))dx}$
(11)

It can also be rewritten as follows:

$\displaystyle\frac{dlnC_{\alpha}(u)}{d\alpha}>\frac{1}{\alpha^{2}}\frac{\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))dx-\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))dx\frac{1}{\delta}\int_{a}^{b}\frac{F^{\alpha}(x,u(x),u^{\prime}(x))}{F^{\alpha}(x,u(x),u^{\prime}(x))}dx}{\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))dx}$
(12)

Calculating the leftmost integral allows us to write:

$\displaystyle\frac{dlnC_{\alpha}(u)}{d\alpha}>\frac{1}{\alpha^{2}}\frac{\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))dx-\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))dx}{\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u(x),u^{\prime}(x))dx}=0$
(13)

The derivative of $lnC_{\alpha}(u)$ and $C_{\alpha}(u)$ are both positives
meaning $C_{\alpha}(u)$ is increasing in $\alpha$.

P3: $C_{-\infty}(u)\leq C_{\alpha}(u)\leq C_{+\infty}(u)$. This is a corollary
of $P2$.

P4: $C_{-\infty}(u)=\inf F(x,u(x),u^{\prime}(x))$ and $C_{+\infty}(u)=\sup
F(x,u(x),u^{\prime}(x))$. Let the sequence $\alpha_{1},\cdots,\alpha_{n}$ such
that $\alpha_{1}<\cdots<\alpha_{n}$, so according to P2 and Eq. 5,
$C_{\alpha_{1}}(u)<\cdots<C_{\alpha_{n}}(u)$. Thanks to the mean value
theorem, the last inequality can be rewritten
$C_{\alpha_{1}}(u)=F(\eta_{1},u(\eta_{1}),u^{\prime}(\eta_{1}))<\cdots<C_{\alpha_{n}}(u)=F(\eta_{n},u(\eta_{n}),u^{\prime}(\eta_{n}))$.
The sequence is built such that $\lim_{n\to\infty}\alpha_{n}=\alpha$. If
$\lim_{\alpha\to+\infty}C_{\alpha}(u)=F(\eta,u(\eta),u^{\prime}(\eta))$ is
finite, then it is the upper bound because $C_{\alpha}(u)$ is increasing in
$\alpha$. The same reasoning is used for the
$\lim_{\alpha\to-\infty}C_{\alpha}(u)$.

The properties P2, P3 and P4 ensure that the $C_{\alpha}(u)()$ is non
decreasing in $\alpha$ and it is bounded, i.e. $C_{\alpha}(u)\in[\inf F(),\sup
F()]$ for a given $\alpha$. The parameter $\alpha$ implements a transform of
the features $F()$ to a new space in which the first-order functional
centrality of Hölder is defined, i.e. Eq. 5. This transform leads to
specifying the contribution of the point $x$ to Hölder’s first-order
functional centrality using the parameter $\alpha$. In other words, the
relevance of features is specified by the parameter $\alpha$, meaning that it
is a trick for feature (data) selection [1]. Taken together these properties
indicate that the functional $C_{\alpha}(u)$ is a mean operator. The discrete
case (i.e., counting measure) of this functional is the Hölder mean. By the
analogy of the discrete case, we call it the functional arithmetic mean when
$\alpha=1$, the functional harmonic mean when $\alpha=-1$, and according to P1
the functional geometric mean when $\alpha=0$.

##  3 Solutions of the First order functional centrality of Hölder

In a space of curves, we want to find a smooth, continuously derivable curve
$u(x)$ defined over $[a,b]$ minimizing $C_{\alpha}(u)$ in Eq. 5. The curve is
assumed to be known in two points, $P=(a,u(a))$ and $Q=(b,u(b))$, where the
values $a$ and $b$ are the data of the problem to be studied. In order to find
the stationary curve $u(x)$, we vary $u(x)$ by a small amount of $h(x)$ (i.e.
$u_{\epsilon}(x)=u(x)+\epsilon h(x)$ and
$u^{\prime}_{\epsilon}(x)=u^{\prime}(x)+\epsilon h^{\prime}(x)$), and
calculate the first variation wrt $\epsilon$. Note that, all variations are
”glued” at the two points $P$ and $Q$; that is the real function $h(x)$ must
verifies $h(a)=h(b)=0$. According to P1 and Eq. 5, the two cases $\alpha\neq
0$ and $\alpha=0$ are considered separately.

###  3.1 Case $\alpha\neq 0$

The first variation wrt $\epsilon$ is:

$\frac{dC_{\alpha}(u_{\epsilon})}{d\epsilon}=\frac{d}{d\epsilon}(\frac{1}{\delta}\int_{a}^{b}F(x,u_{\epsilon},u_{\epsilon}^{{}^{\prime}})^{\alpha}dx)^{1/\alpha}$
(14)

Applying derivative rules, we obtain:

$\displaystyle\frac{dC_{\alpha}(u_{\epsilon})}{d\epsilon}$
$\displaystyle=\frac{1}{\alpha}(\frac{1}{\delta}\int_{a}^{b}F(x,u_{\epsilon},u_{\epsilon}^{{}^{\prime}})^{\alpha}dx)^{1/\alpha-1}$
$\displaystyle\frac{1}{\delta}\int_{a}^{b}\alpha
F(x,u_{\epsilon},u_{\epsilon}^{{}^{\prime}})^{\alpha-1}\frac{dF(x,u_{\epsilon},u_{\epsilon}^{{}^{\prime}})}{d\epsilon}dx=0$

Let us recall that $F()>0$, and therefore
$(\int_{a}^{b}F(x,u_{\epsilon},u_{\epsilon}^{{}^{\prime}})^{\alpha}dx)^{1/\alpha-1}\neq
0$. So, the term is:

$\int_{a}^{b}F(x,u_{\epsilon},u_{\epsilon}^{{}^{\prime}})^{\alpha-1}\frac{dF(x,u_{\epsilon},u_{\epsilon}^{{}^{\prime}})}{d\epsilon}dx=0$
(15)

The total derivative wrt to $\epsilon$ is:

$\frac{d}{d\epsilon}F(x,u_{\epsilon},u_{\epsilon}^{{}^{\prime}})=\frac{\partial
F(x,u_{\epsilon},u_{\epsilon}^{{}^{\prime}})}{\partial x}\frac{\partial
x}{\partial\epsilon}+\frac{\partial
F(x,u_{\epsilon},u_{\epsilon}^{{}^{\prime}})}{\partial
u_{\epsilon}}\frac{\partial u_{\epsilon}}{\partial\epsilon}+\frac{\partial
F(x,u_{\epsilon},u_{\epsilon}^{{}^{\prime}})}{\partial
u_{\epsilon}^{{}^{\prime}}}\frac{\partial
u_{\epsilon}^{{}^{\prime}}}{\partial\epsilon}$ (16)

Because $\frac{\partial x}{\partial\epsilon}=0$, by substituting this equation
in Eq. 15 and setting $\epsilon=0$, we obtain:

$\int_{a}^{b}F(x,u,u^{{}^{\prime}})^{\alpha-1}(\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial u}h(x)+\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial u^{{}^{\prime}}}h^{{}^{\prime}}(x))dx=0$ (17)

The integration by part of the second term leads to:

$\int_{a}^{b}h(x)(F(x,u,u^{{}^{\prime}})^{\alpha-1}\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial
u}-\frac{d}{dx}F(x,u,u^{{}^{\prime}})^{\alpha-1}\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial u^{{}^{\prime}}})dx=0$ (18)

The fundamental lemma of the calculus of variations [2] ensures that the
stationary curve $u(x)$ is a solution to the ODE:

$F(x,u,u^{{}^{\prime}})^{\alpha-1}\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial
u}-\frac{d}{dx}F(x,u,u^{{}^{\prime}})^{\alpha-1}\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial u^{{}^{\prime}}}=0$ (19)

This ODE can be rewritten as:

$F(x,u,u^{{}^{\prime}})^{\alpha-2}(F(x,u,u^{{}^{\prime}})(\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial u}-\frac{d}{dx}\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial u^{{}^{\prime}}})-(\alpha-1)\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial
u^{{}^{\prime}}}\frac{d}{dx}F(x,u,u^{{}^{\prime}}))=0$ (20)

Because $F(x,u,u^{{}^{\prime}})^{\alpha-2}\neq 0$, the final ODE is written as
follows:

$F(x,u,u^{{}^{\prime}})(\frac{\partial F(x,u,u^{{}^{\prime}})}{\partial
u}-\frac{d}{dx}\frac{\partial F(x,u,u^{{}^{\prime}})}{\partial
u^{{}^{\prime}}})-(\alpha-1)\frac{\partial F(x,u,u^{{}^{\prime}})}{\partial
u^{{}^{\prime}}}\frac{d}{dx}F(x,u,u^{{}^{\prime}})=0$ (21)

Note that if $\alpha=1$, we have the well-known Euler-Lagrange ordinary
differential equation (ODE). The presence of the term $\frac{\partial
F}{\partial u^{{}^{\prime}}}\frac{d}{dx}F(x,u,u^{{}^{\prime}})$ when
$\alpha\neq 1$ could leads to other solutions. To assert that the stationary
curve $u(x)$ solution to Eq. 21 is a minimum, we need to test the positiveness
of the second variation wrt $\epsilon$. Indeed, the curve $u(x)$ is a minimum
if $C_{\alpha}(u)<C_{\alpha}(u+\epsilon h)$. By using Maclaurin expansion wrt
to $\epsilon$, we write:

$f(\epsilon)=C_{\alpha}(u+\epsilon h)=C_{\alpha}(u)+\epsilon
f^{\prime}(0)+0.5\epsilon^{2}f^{\prime\prime}(0)+O_{3}(u)$ (22)

If $u(x)$ is a stationary curve then the first variation $f^{\prime}(0)=0$.
Consequently, the functional $C_{\alpha}(u+\epsilon h)>C_{\alpha}(u)$ if
$f^{\prime\prime}(0)$ is positive. At $\epsilon=0$, the first variation can be
rewritten as:

$\frac{dC_{\alpha}(u_{\epsilon})}{d\epsilon}|_{\epsilon=0}=f^{\prime}(0)=\frac{1}{\alpha}(\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u,u^{{}^{\prime}})dx)^{\frac{1}{\alpha}-1}(\frac{1}{\delta}\int_{a}^{b}\frac{dF^{\alpha}(x,u_{\epsilon},u_{\epsilon}^{{}^{\prime}})}{d\epsilon}|_{\epsilon=0}dx)$
(23)

The second variation is:

$\frac{d^{2}C_{\alpha}(u_{\epsilon})}{d\epsilon^{2}}|_{\epsilon=0}=f^{\prime\prime}(0)=\begin{array}[]{ll}&\frac{1}{\alpha}\frac{1}{\delta^{2}}(\frac{1}{\delta}\int_{a}^{b}F^{\alpha}(x,u,u^{{}^{\prime}})dx)^{\frac{1}{\alpha}-2}(\int_{a}^{b}F^{\alpha}(x,u,u^{{}^{\prime}})dx\int_{a}^{b}\frac{d^{2}F^{\alpha}(x,u_{\epsilon},u_{\epsilon}^{{}^{\prime}})}{d\epsilon^{2}}|_{\epsilon=0}dx+\\\
&~{}~{}~{}~{}(\frac{1}{\alpha}-1)(\int_{a}^{b}\frac{dF^{\alpha}(x,u_{\epsilon},u_{\epsilon}^{{}^{\prime}})}{d\epsilon}|_{\epsilon=0}dx)^{2})\end{array}$
(24)

The stationary curve is a minimum if
$\frac{d^{2}C_{\alpha}(u_{\epsilon})}{d\epsilon^{2}}|_{\epsilon=0}>0$. Knowing
that both $\frac{1}{\delta}$ and
$\int_{a}^{b}F^{\alpha}(x,u,u^{{}^{\prime}})dx$ are positive, and
$\int_{a}^{b}\frac{dF^{\alpha}(x,u_{\epsilon},u_{\epsilon}^{{}^{\prime}})}{d\epsilon}|_{\epsilon=0}dx=0$,
it follows that the stationary curve is a minimum if:

$\frac{1}{\alpha}\int_{a}^{b}\frac{d^{2}F^{\alpha}(x,u_{\epsilon},u_{\epsilon}^{{}^{\prime}})}{d\epsilon^{2}}|_{\epsilon=0}dx>0$
(25)

The inequality is fulfilled when $\alpha$ and
$\int_{a}^{b}\frac{d^{2}F^{\alpha}(x,u_{\epsilon},u_{\epsilon}^{{}^{\prime}})}{d\epsilon^{2}}|_{\epsilon=0}dx$
have the same sign. Two important cases that should be studied are the
explicit independence of $F(x,u,u^{{}^{\prime}})$ on $u(x)$ and $x$. For the
former case, substituting $\frac{\partial F(x,u,u^{{}^{\prime}})}{\partial
u}=0$ in Eq. 21 gives $F(x,u,u^{{}^{\prime}})\frac{d}{dx}\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial u^{{}^{\prime}}}+(\alpha-1)\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial
u^{{}^{\prime}}}\frac{d}{dx}F(x,u,u^{{}^{\prime}})=0$. An interpretation in
the general case seems to be not easy to draw. However, when $\alpha=1$, then
$\frac{d}{dx}\frac{\partial F(x,u,u^{{}^{\prime}})}{\partial
u^{{}^{\prime}}}=0$ and therefore $\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial u^{{}^{\prime}}}$ is a constant. This outcome
is known as the ignorable coordinates in the calculus of variations. When
$\alpha=2$, Eq. 21 is rewritten as
$\frac{d}{dx}F(x,u,u^{{}^{\prime}})\frac{\partial F}{\partial
u^{{}^{\prime}}}=0$, then $F(x,u,u^{{}^{\prime}})\frac{\partial F}{\partial
u^{{}^{\prime}}}$ is a constant. For the last $\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial x}=0$, straightforward manipulations leads to
the ODE $F(x,u,u^{{}^{\prime}})\frac{d}{dx}(u^{\prime}\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial
u^{{}^{\prime}}}-F(x,u,u^{{}^{\prime}}))+(\alpha-1)u^{\prime}\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial
u^{{}^{\prime}}}\frac{d}{dx}F(x,u,u^{{}^{\prime}})=0$. When $\alpha=1$, we
obtain the well-known Beltrami identity; i.e. $u^{\prime}\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial u^{{}^{\prime}}}-F(x,u,u^{{}^{\prime}})$ is a
constant. When $\alpha=2$, we have
$\frac{d}{dx}F(x,u,u^{{}^{\prime}})u^{\prime}\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial
u^{{}^{\prime}}}=F(x,u,u^{{}^{\prime}})\frac{d}{dx}F(x,u,u^{{}^{\prime}})$ and
hence $u^{\prime}\frac{\partial F(x,u,u^{{}^{\prime}})}{\partial
u^{{}^{\prime}}}-\frac{1}{2}F(x,u,u^{{}^{\prime}})$ is a constant. The case
$\alpha=2$ generalizes the Beltrami identity. The study of other values of
$\alpha$ could lead to other outcomes.

###  3.2 Case $\alpha=0$

We will now give the first and second variations in the case where $\alpha=0$.
The first order variation is:

$\displaystyle\frac{dC_{\alpha}(u)}{d\epsilon}|_{\epsilon=0}=C_{\alpha}(u)\int_{a}^{b}(\frac{\partial
lnF(x,u,u^{{}^{\prime}})}{\partial u}h+\frac{\partial
lnF(x,u,u^{{}^{\prime}})}{\partial u^{\prime}}h^{\prime})dx=0$ (26)

The corresponding ODE is:

$\frac{\partial lnF(x,u,u^{{}^{\prime}})}{\partial
u}-\frac{d}{dx}\frac{\partial lnF(x,u,u^{{}^{\prime}})}{\partial
u^{{}^{\prime}}}=0$ (27)

If $u(x)$ is an ignorable coordinate, then $\frac{\partial
lnF(x,u,u^{{}^{\prime}})}{\partial u^{{}^{\prime}}}$ is a constant. This is
another outcome generalizing the Beltrami identity. When
$F(x,u,u^{{}^{\prime}})$ is explicitly independent upon $x$, then we deduce
that $lnF(x,u,u^{{}^{\prime}})-u^{\prime}\frac{\partial
lnF(x,u,u^{{}^{\prime}})}{\partial u^{{}^{\prime}}}$ is a constant.
Straightforward manipulations lead to writing the second variation:

$\displaystyle\frac{d^{2}C_{\alpha}(u)}{d\epsilon^{2}}|_{\epsilon=0}$
$\displaystyle=C_{\alpha}(u)((\int_{a}^{b}(\frac{\partial
lnF(x,u,u^{{}^{\prime}})}{\partial u}h+\frac{\partial
lnF(x,u,u^{{}^{\prime}})}{\partial
u^{\prime}}h^{\prime})dx)^{2}+\int_{a}^{b}(\frac{1}{F(x,u,u^{{}^{\prime}})}(\frac{\partial^{2}F(x,u,u^{{}^{\prime}})}{\partial
u^{2}}h^{2}+$ (28) $\displaystyle
2\frac{\partial^{2}F(x,u,u^{{}^{\prime}})}{\partial u\partial
u^{\prime}}hh^{\prime}+\frac{\partial^{2}F(x,u,u^{{}^{\prime}})}{\partial
u^{\prime 2}}h^{\prime 2})-\frac{1}{F^{2}(x,u,u^{{}^{\prime}})}(\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial u}h+\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial u^{\prime}}h^{\prime})^{2})dx)$

The second variation at the stationary curve is obtained by substituting Eq.
26 in Eq. 28:

$\displaystyle\frac{d^{2}C_{\alpha}(u)}{d\epsilon^{2}}|_{\epsilon=0}=$
$\displaystyle
C_{\alpha}(u)\int_{a}^{b}(\frac{1}{F(x,u,u^{{}^{\prime}})}(\frac{\partial^{2}F(x,u,u^{{}^{\prime}})}{\partial
u^{2}}h^{2}+2\frac{\partial^{2}F(x,u,u^{{}^{\prime}})}{\partial u\partial
u^{\prime}}hh^{\prime}+\frac{\partial^{2}F(x,u,u^{{}^{\prime}})}{\partial
u^{\prime 2}}h^{\prime 2})$ (29)
$\displaystyle-\frac{1}{F^{2}(x,u,u^{{}^{\prime}})}(\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial u}h+\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial u^{\prime}}h^{\prime})^{2})dx$

Knowing that $C_{\alpha}(u)>0$, the second variation test leads to concluding
that the stationary curve is a minimum if:

$\displaystyle\int_{a}^{b}$
$\displaystyle\frac{1}{F(x,u,u^{{}^{\prime}})}(\frac{\partial^{2}F(x,u,u^{{}^{\prime}})}{\partial
u^{2}}h^{2}+2\frac{\partial^{2}F(x,u,u^{{}^{\prime}})}{\partial u\partial
u^{\prime}}hh^{\prime}+\frac{\partial^{2}F(x,u,u^{{}^{\prime}})}{\partial
u^{\prime 2}}h^{\prime 2})dx>$ (30)
$\displaystyle\int_{a}^{b}\frac{1}{F^{2}(x,u,u^{{}^{\prime}})}(\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial u}h+\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial u^{\prime}}h^{\prime})^{2})dx$

The RHS is non-negative, so we can write the second variation test as follows:

$\displaystyle\int_{a}^{b}\frac{1}{F(x,u,u^{{}^{\prime}})}(\frac{\partial^{2}F(x,u,u^{{}^{\prime}})}{\partial
u^{2}}h^{2}+2\frac{\partial^{2}F(x,u,u^{{}^{\prime}})}{\partial u\partial
u^{\prime}}hh^{\prime}+\frac{\partial^{2}F(x,u,u^{{}^{\prime}})}{\partial
u^{\prime 2}}h^{\prime 2})>0~{}~{}\mbox{if}~{}~{}\alpha=0$ (31)

##  4 Shortest path problem

We would like to find the curve of the shortest path between the two points
$P$ and $Q$ in the spatial domain. We can approximate the length of a plane
curve defined on $[a,b]$ by subdividing it into infinitesimal linear pieces,
measuring the length of each, and adding up all the lengths. The infinitesimal
piece length between $(x,u(x))$ and $(x+dx,u(x+dx))$ is the length of the
hypotenuse of the right triangle having sides $dx$ and $du(x)$, i.e.
$\sqrt{dx^{2}+(u(x+dx)-u(x))^{2}}=dx\sqrt{1+\frac{(u(x+dx)-u(x))^{2}}{dx^{2}}}$.
When $dx$ is too small, the infinitesimal length is $\sqrt{1+u(x)^{\prime
2}}dx$. Let us define the feature $F(x,u,u^{\prime})=\sqrt{1+u(x)^{\prime
2}}$. It should be noted that $u(x)$ and $x$ are ignorable coordinates, i.e.
$\frac{\partial F(x,u,u^{{}^{\prime}})}{\partial u}=0$ and $\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial x}=0$. The other terms involved in Eq. 21 are
$\frac{\partial F(x,u,u^{{}^{\prime}})}{\partial
u^{{}^{\prime}}}=\frac{u^{\prime}}{F}$,
$\frac{dF(x,u,u^{{}^{\prime}})}{dx}=\frac{u^{\prime}u^{\prime\prime}}{F}$, and
$\frac{d}{dx}\frac{\partial F(x,u,u^{{}^{\prime}})}{\partial
u^{{}^{\prime}}}=\frac{F(x,u,u^{{}^{\prime}})u^{\prime\prime}-\frac{u^{\prime
2}u^{\prime\prime}}{F(x,u,u^{{}^{\prime}})}}{F^{2}(x,u,u^{{}^{\prime}})}$. The
ODE in Eq. 21 is rewritten as:

$-F(x,u,u^{{}^{\prime}})\frac{F(x,u,u^{{}^{\prime}})u^{\prime\prime}-\frac{u^{\prime
2}u^{\prime\prime}}{F(x,u,u^{{}^{\prime}})}}{F^{2}(x,u,u^{{}^{\prime}})}-(\alpha-1)\frac{u^{\prime}}{F(x,u,u^{{}^{\prime}})}\frac{u^{\prime}u^{\prime\prime}}{F(x,u,u^{{}^{\prime}})}=0$
(32)
$-F(x,u,u^{{}^{\prime}})(F(x,u,u^{{}^{\prime}})u^{\prime\prime}-\frac{u^{\prime
2}u^{\prime\prime}}{F(x,u,u^{{}^{\prime}})})-(\alpha-1)u^{\prime
2}u^{\prime\prime}=0$ (33) $u^{\prime\prime}(x)(1+(\alpha-1)u^{\prime
2}(x))=0$ (34)

There are two cases $1+(\alpha-1)u^{\prime 2}=0$ and $u^{\prime\prime}=0$. The
solutions to these ODE are provided in Table 1 as function of $\alpha$. The
solution to the ODE $u^{\prime\prime}=0$ is a straight line in real space
completely defined by the two points, P and Q through which it passes, i.e
$y=sx+d$. Although the straight line does not depend on $\alpha$, the second
variation test does. It is a minimum except for the cases when the slope
$s\notin]-1,1[$ and $\alpha<1$. When $\alpha<0$ or $\alpha\in]0,1[$, the
slopes of the two straight lines solutions to the ODE $1+(\alpha-1)u^{\prime
2}=0$ are real and $\alpha$ dependant. However, these solutions are minima
when $\alpha<0$ and maxima when $\alpha\in]0,1[$. Let us make explicit the
case of the functional harmonic mean $\alpha=-1$ to indicate that this
property remains valid. For the functional geometric mean $\alpha=0$, the
solutions to the ODE are two straight lines that do not depend on $\alpha$ and
whose second variation test is inconclusive. The functional
$F(x,u,u^{\prime})$ is not real when $\alpha\in]1,2[$. For the case when
$\alpha\geq 2$, there are two lines in the complex plane which are solutions
to the ODE and which are minima. To illustrate, figure 1 presents a bundle of
lines passing through $P=(1,2)$ in the case where $\alpha\in]0,1[$. The slopes
of the lines crossing the $y$-axis at position greater than (resp. less than)
two are $-1/\sqrt{1-\alpha}$ (resp. $1/\sqrt{1-\alpha}$) for
$\alpha=0.1,0.3,0.5,0.9,0.95$. Finally, it should be noted that the functional
arithmetic mean $\alpha=1$ leads to only one ODE, $u^{\prime\prime}=0$. It is
the case that is studied in the calculus of variations. All the other cases
are uncommon.

Figure 1: Real solution to the functional centrality problem. The lines crossing the y-axis at position greater than (resp. less than) 2 have a negative (resp. positive) slope.  $\alpha$ | ODE | $F(x,u,u^{\prime})$ | 2nd variation test | $u(x)$ | space
---|---|---|---|---|---
$>2$ | $1+(\alpha-1)u^{\prime 2}(x)=0$ | $\sqrt{1+\frac{1}{\alpha-1}}$ | $\frac{1}{\delta}\frac{(\alpha-1)^{3}}{\alpha^{2}(\alpha-2)}(\frac{\alpha(\alpha-2)}{(\alpha-1)^{2}})^{\alpha/2}\int_{a}^{b}h^{\prime 2}(x)dx>0$ | $\frac{\mp ix}{\sqrt{\alpha-1}}+d$ | Complex
|  |  | minimum |  |
| $u^{\prime\prime}(x)=0$ | $\sqrt{1+s^{2}}$ | $\frac{1}{\delta}(s^{2}+1)^{\alpha/2-2}((\alpha-1)s^{2}+1)\int_{a}^{b}h^{\prime 2}(x)dx>0$ | $sx+d$ | Real
|  |  | minimum |  |
$2$ | $1+u^{\prime 2}(x)=0$ | $0$ | $\frac{1}{\delta}\int_{a}^{b}h^{\prime 2}(x)dx>0$ | $\mp ix+d$ | Complex
|  |  | minimum |  |
| $u^{\prime\prime}(x)=0$ | $\sqrt{1+s^{2}}$ | $\frac{1}{\delta\sqrt{s^{2}+1}}\int_{a}^{b}h^{\prime 2}(x)dx>0$ | $sx+d$ | Real
|  |  | minimum |  |
$]1,2[$ | $1+(\alpha-1)u^{\prime 2}(x)=0$ | Undefined | Undefined | $\frac{\mp ix}{\sqrt{\alpha-1}}+d$ | Complex
| $u^{\prime\prime}(x)=0$ | $\sqrt{1+s^{2}}$ | $\frac{1}{\delta}(s^{2}+1)^{\alpha/2-2}((\alpha-1)s^{2}+1)\int_{a}^{b}h^{\prime 2}(x)dx>0$ | $sx+d$ | Real
|  |  | minimum |  |
1 | $u^{\prime\prime}(x)=0$ | $\sqrt{1+s^{2}}$ | $\frac{1}{\delta(1+s^{2})^{3/2}}\int h^{\prime 2}(x)dx>0$ | $sx+d$ | Real
|  |  | minimum |  |
$]0,1[$ | $1+(\alpha-1)u^{\prime 2}(x)=0$ | $\sqrt{1+\frac{1}{1-\alpha}}$ | $\frac{\alpha}{\delta}(\frac{(\alpha-1)^{2}+1}{(\alpha-1)^{2}})^{\alpha/2}\frac{(\alpha-1)^{3}}{((\alpha-1)^{2}+1)^{2}}\int_{a}^{b}h^{\prime 2}(x)dx<0$ | $\frac{\mp x}{\sqrt{1-\alpha}}+d$ | Real
|  |  | maximum |  |
| $u^{\prime\prime}(x)=0$ | $\sqrt{1+s^{2}}$ | $\frac{1}{\delta}(s^{2}+1)^{\alpha/2-2}((\alpha-1)s^{2}+1)\int_{a}^{b}h^{\prime 2}(x)dx>0$ | $sx+d$ | Real
|  |  | minimum if $s^{2}<1$ |  |
0 | $u^{\prime\prime}(x)=0$ | $\sqrt{1+s^{2}}$ | $\frac{1}{\delta}\frac{1-s^{2}}{(1+s^{2})^{2}}\int h^{\prime 2}(x)dx>0$ | $sx+d$ | Real
|  |  | minimum if $s\in]-1,1[$ |  |
|  |  | maximum if $s\in]-\infty,-1[\cup]1,+\infty[$ |  |
| $u^{\prime}(x)=\mp 1$ | $\sqrt{2}$ | 0 | $\mp x+d$ | Real
|  |  | inconclusive |  |
$<0$ | $1+(\alpha-1)u^{\prime 2}(x)=0$ | $\sqrt{1+\frac{1}{1-\alpha}}$ | $\frac{\alpha}{\delta}(\frac{(\alpha-1)^{2}+1}{(\alpha-1)^{2}})^{\alpha/2}\frac{(\alpha-1)^{3}}{((\alpha-1)^{2}+1)^{2}}\int_{a}^{b}h^{\prime 2}(x)dx>0$ | $\frac{\mp x}{\sqrt{1-\alpha}}+d$ | Real
|  |  | minimum |  |
| $u^{\prime\prime}(x)=0$ | $\sqrt{1+s^{2}}$ | $\frac{1}{\delta}(s^{2}+1)^{\alpha/2-2}((\alpha-1)s^{2}+1)\int_{a}^{b}h^{\prime 2}(x)dx>0$ | $sx+d$ | Real
|  |  | minimum if $s\in]-1,1[$ |  |
Table 1: Solutions to the ODE $u^{\prime\prime}(1+(\alpha-1)u^{\prime 2})=0$
expressed as function of $\alpha$.

##  5 Fastest path problem

The Brachistochrone is an old problem known in the calculus of variations [2].
It describes a curve that carries a particle under gravity from one height to
another in minimal time. Other variations of this problem have included the
effects of friction, the motion of a disc on a hemisphere, the motion of a
cyclist in a velodrome, and even the quantum Brachistochrone problem [3].

Suppose that the particle is released from $P$ at $t=0$, and then follows a
curve $u(x)$ which reaches $Q$, so that $u(b)-u(a)$ is the height lost, and
$b-a$ is the horizontal distance traversed. The associated feature is
$F(x,u,u^{{}^{\prime}})=\sqrt{(1+u^{\prime 2}(x))/u(x)}$. It should be noted
that $\frac{\partial F(x,u,u^{{}^{\prime}})}{\partial x}=0$. The other
derivatives in Eq. 21 are $\frac{\partial F(x,u,u^{{}^{\prime}})}{\partial
u}=-\frac{\sqrt{1+u^{\prime 2}}}{2u^{3/2}}$, $\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial
u^{\prime}}=\frac{u^{\prime}}{\sqrt{u(1+u^{\prime 2})}}$,
$\frac{d}{dx}\frac{\partial F(x,u,u^{{}^{\prime}})}{\partial
u^{\prime}}=\frac{1}{\sqrt{u(1+u^{\prime
2})}}(\frac{u^{\prime\prime}}{1+u^{\prime 2}}-\frac{u^{\prime 2}}{2u})$, and
$\frac{dF(x,u,u^{{}^{\prime}})}{dx}=\frac{u^{\prime}u^{\prime\prime}}{\sqrt{u(1+u^{\prime
2})}}-\frac{u^{\prime}\sqrt{1+u^{\prime 2}}}{2u^{3/2}}$,
$F(x,u,u^{{}^{\prime}})(\frac{\partial F(x,u,u^{{}^{\prime}})}{\partial
u}-\frac{d}{dx}\frac{\partial F(x,u,u^{{}^{\prime}})}{\partial
u^{\prime}})=-\frac{1+u^{\prime
2}}{2u^{2}}-\frac{1}{u}(\frac{u^{\prime\prime}}{1+u^{\prime
2}}-\frac{u^{\prime
2}}{2u})=-\frac{1}{2u^{2}}-\frac{u^{\prime\prime}}{u(1+u^{\prime 2})}$,
$(\alpha-1)\frac{\partial F(x,u,u^{{}^{\prime}})}{\partial
u^{\prime}}\frac{dF(x,u,u^{{}^{\prime}})}{dx}=(\alpha-1)\frac{u^{\prime
2}}{u}(\frac{u^{\prime\prime}}{1+u^{\prime 2}}-\frac{1}{2u})$. The Euler-
Lagrange ODE in Eq. 21 becomes:

$-\frac{1}{2u}-\frac{u^{\prime\prime}}{1+u^{\prime 2}}-(\alpha-1)u^{\prime
2}(\frac{u^{\prime\prime}}{1+u^{\prime 2}}-\frac{1}{2u})=0$ (35)

The solution in the calculus of variations (i.e. $\alpha=1$) is the cycloid
with the parametric equations $x=r(\theta-sin\theta)$ and
$u(x)=r(1-cos\theta)$. It can be displayed by the path traced out by a point
placed on a rolling wheel of radius $r$ when it rotates by an angle $\theta$.
The general solution to the ODE when $\alpha\neq 1$ is challenging. For
example, it is easy to check that $u(x)=\frac{1}{\sqrt{\alpha-1}}x$ is a
stationary straight line. According to the second variation test in Eq. 25, it
is a minimum when $\alpha=2$ and $x>0$, in this case the second variation
$\frac{1}{\delta}\int_{a}^{b}((xh^{\prime}(x)-h(x))^{2}+h(x))/x^{3}dx>0$. We
solve the ODE numerically, using the midpoint method of Maple for three values
of $\alpha$ and two different initial conditions (see Fig. 2). Note that the
solution resembles the Trochoid at different distances from the center of a
circle of some radius.

a b

Figure 2: Numerical solution to Brachistochrone with the initial condition: a)
u(0)=0 and u(5)=2, b) u(0)=1 and u(5)=1, $\alpha=1$ (Red), $\alpha=2$ (Green),
c) $\alpha=8$ (Blue).

We consider another problem related to the Snell’s law governing the
refraction of light in its passage from one medium to another, provided that
the observed refractive index of the medium is identified with the inverse of
speed $c^{-1}(x)$. Pierre de Fermat observed that Snell’s law stems from the
principle that light travels the path that takes the least amount of time.
Specifically, the functional $F(x,u,u^{{}^{\prime}})$ is the time taken to
cover the path $u(x)$ for $x\in[a,b]$, i.e
$F(x,u,u^{\prime})=\frac{\sqrt{1+u^{\prime 2}}}{c(x)}$ and $c(x)>0$. The terms
involved in Eq. 21 are $\frac{\partial F(x,u,u^{{}^{\prime}})}{\partial u}=0$,
$\frac{\partial F(x,u,u^{{}^{\prime}})}{\partial
u^{{}^{\prime}}}=\frac{u^{\prime}}{c^{2}(x)F(x,u,u^{{}^{\prime}})}$,
$\frac{dF(x,u,u^{{}^{\prime}})}{dx}=\frac{u^{\prime}u^{\prime\prime}}{c^{2}(x)F(x,u,u^{{}^{\prime}})}-\frac{c^{\prime}(x)}{c(x)}F(x,u,u^{{}^{\prime}})$,
and $\frac{d}{dx}\frac{\partial F(x,u,u^{{}^{\prime}})}{\partial
u^{{}^{\prime}}}=\frac{-u^{\prime}(x)c^{\prime}(x)}{c^{3}(x)F(x,u,u^{{}^{\prime}})}+\frac{u^{\prime\prime}(x)}{c^{4}(x)F^{3}(x,u,u^{{}^{\prime}})}$.
Eq. 21 is rewritten as:

$F(x,u,u^{{}^{\prime}})(\frac{-u^{\prime}(x)c^{\prime}(x)}{c^{3}(x)F(x,u,u^{{}^{\prime}})}+\frac{u^{\prime\prime}(x)}{c^{4}(x)F^{3}(x,u,u^{{}^{\prime}})})-(\alpha-1)\frac{u^{\prime}}{c^{2}(x)F(x,u,u^{{}^{\prime}})}(\frac{u^{\prime}u^{\prime\prime}}{c^{2}(x)F(x,u,u^{{}^{\prime}})}-\frac{c^{\prime}(x)}{c(x)}F(x,u,u^{{}^{\prime}}))=0$
(36)

Eq. 36 can be rewritten as $\frac{\partial F(x,u,u^{{}^{\prime}})}{\partial
u^{{}^{\prime}}}=k$, where $k$ is a constant when $\alpha=1$ because $u(x)$ is
an ignorable coordinate. It follows that the derivative of the stationary
curve is $u^{\prime}(x)=\mp kc(x)/\sqrt{1-k^{2}c^{2}(x)}$. For example,
consider $c(x)=x$ and $k\neq 0$, then $u(x)=\mp\sqrt{1-k^{2}x^{2}}/k+p$, which
is the circle $(u(x)-p)^{2}+x^{2}=k^{-2}$. This is well-known solution in the
calculus of variations. When $\alpha=0$, Eq. 27 is rewritten as
$u^{\prime\prime}(x)(1-u^{\prime 2}(x))=0$. The case $u^{\prime 2}(x)=1$ is a
particular case of the $u^{\prime\prime}(x)=0$, the stationary curve $u(x)$
solution to the last ODE is independent of the speed; $u(x)=sx+d$. The second
variation in Eq.31, $\frac{\partial^{2}F}{\partial
u^{2}}=\frac{\partial^{2}F}{\partial u\partial u^{\prime}}=\frac{\partial
F}{\partial u}=0$, is $(1-s^{2})\int_{a}^{b}h^{\prime
2}(x)dx/\delta(s^{2}+1)^{2}$. The second variation is positive when $s^{2}<1$
because $h(x)$ cannot be a constant, and therefore the stationary curve is a
minimum in this case. In other words, only straight lines with slopes in
$]-1,1[$ are minimums. When $\alpha=2$, we can rewrite the ODE in Eq. 36 as
$\frac{d}{dx}F(x,u,u^{{}^{\prime}})\frac{\partial
F(x,u,u^{{}^{\prime}})}{\partial u^{\prime}}=0$; that is
$F(x,u,u^{{}^{\prime}})\frac{\partial F(x,u,u^{{}^{\prime}})}{\partial
u^{\prime}}=k$ and therefore $u^{\prime}(x)=kc^{2}(x)$. Let us set $c(x)=x$,
then $u(x)=kx^{3}/3+p$. It is a minimum because the second variation test in
Eq. 25 leads to $2\int_{a}^{b}\frac{h^{\prime 2}(x)}{x^{2}}/\delta dx>0$. If
we consider that there are two mediums, having different refractive indexes
whose variation is smooth along the x-axis. The speed function
$c(x)=\sqrt{e^{\beta(x-x_{0})}/(1+e^{\beta(x-x_{0})})}$, where $\beta$ and
$x_{0}$ are parameters describes two mediums. By considering this speed
function, the stationary curve $u(x)=kln(1+e^{\beta(x-x_{0})})/\beta+p$ is a
minimum because the second variation test in Eq. 25 is rewritten as
$2\int_{a}^{b}h^{\prime
2}(x)\sqrt{\frac{1+e^{\beta(x-x_{0})}}{e^{\beta(x-x_{0})}}}dx>0$, where $w$ is
a positive function of $a$, $b$, and $k^{2}$. This curve $u(x)$ is plotted in
Fig. 3.

a b

Figure 3: Fast path, $k=1$, and $x_{0}=5$. a) The speed is a logistic function
centered on $x_{0}$ and having the slope $\beta=2$. b) The curves u(x) when
the slopes are $\beta=0.2$ (Red), $\beta=2$ (blue).

##  6 Conclusion

In this document, the calculus of variations is revisited from a new angle.
Instead of the usual variational formulation, functional centrality based on
Hölder’s mean is used. For a given problem, the solution is a family of
curves, where each one corresponds to a certain data selection criterion
induced by the centrality measure used. Some of these solutions cannot be
obtained with the usual variational formulation. To illustrate, the fastest
and shortest paths are studied.

## References

  * [1] D. Ziou. Pythagorean Centrality for Data Selection. arXiv preprint arXiv:2301.10010, 2023.
  * [2] R. Courant and D. Hilbert. Methods of Mathematical Physics. New York Interscience Publishers, 1953.
  * [3] G.P. Benhama, C. Cohen, E. Brunet, and C. Clanet. Brachistochrone on a velodrome. Proc. of the Royal Society A, 2020.
  * [4] F. Kerouh, D. Ziou, and K.N. Lahmar. Content-based computational chromatic adaptation. Pattern Anal. Applic. 21, pp. 1109–1120, 2018.