arXiv subjects

Kestutis Cesnavicius

Publications and source records attributed to Kestutis Cesnavicius.

28 records · Page 2Linked to original sources

arXiv2017

The $\ell$-parity conjecture over the constant quadratic extension

For a prime $\ell$ and an abelian variety $A$ over a global field $K$, the $\ell$-parity conjecture predicts that, in accordance with the ideas of Birch and Swinnerton-Dyer, the $\mathbb{Z}_{\ell}$-corank of the $\ell^{\infty}$-Selmer group and the analytic rank agree modulo $2$. Assuming that $\mathrm{char} K > 0$, we prove that the $\ell$-parity conjecture holds for the base change of $A$ to the constant quadratic extension if $\ell$ is odd, coprime to $\mathrm{char} K$, and does not divide the degree of every polarization of $A$. The techniques involved in the proof include the étale cohomological interpretation of Selmer groups, the Grothendieck-Ogg-Shafarevich formula, and the study of the behavior of local root numbers in unramified extensions.

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arXiv2017

$p$-Selmer growth in extensions of degree $p$

There is a known analogy between growth questions for class groups and for Selmer groups. If $p$ is a prime, then the $p$-torsion of the ideal class group grows unboundedly in $\mathbb{Z}/p\mathbb{Z}$-extensions of a fixed number field $K$, so one expects the same for the $p$-Selmer group of a nonzero abelian variety over $K$. This Selmer group analogue is known in special cases and we prove it in general, along with a version for arbitrary global fields.

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arXiv2016

Coarse base change fails for some modular curves

For a congruence level $H \subset \mathrm{GL}_2(\hat{\mathbb{Z}})$, the formation of the modular curve $X_H$, i.e., of the coarse moduli space of the level $H$ modular stack $\mathscr{X}_H$, is known to commute with arbitrary base change in a wide range of cases. We exhibit infinitely many $H$, for instance, $H = Γ_1(4)$, for which this coarse base change property fails. In our examples failure is witnessed for base change to $\mathbb{F}_2$ and for any $\mathbb{Z}_{(2)}$-fiberwise dense open substack of $\mathscr{X}_H$. These examples fill in several open entries in a table in the book of Katz and Mazur.

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arXiv2016

The remaining cases of the Kramer-Tunnell conjecture

For an elliptic curve $E$ over a local field $K$ and a separable quadratic extension of $K$, motivated by connections to the Birch and Swinnerton-Dyer conjecture, Kramer and Tunnell have conjectured a formula for computing the local root number of the base change of $E$ to the quadratic extension in terms of a certain norm index. The formula is known in all cases except some when $K$ is of characteristic $2$, and we complete its proof by reducing the positive characteristic case to characteristic $0$. For this reduction, we exploit the principle that local fields of characteristic $p$ can be approximated by finite extensions of $\mathbb{Q}_p$--we find an elliptic curve $E'$ defined over a $p$-adic field such that all the terms in the Kramer-Tunnell formula for $E'$ are equal to those for $E$.

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arXiv2015

Poitou-Tate without restrictions on the order

The Poitou-Tate sequence relates Galois cohomology with restricted ramification of a finite Galois module $M$ over a global field to that of the dual module under the assumption that $\#M$ is a unit away from the allowed ramification set. We remove the assumption on $\#M$ by proving a generalization that allows arbitrary "ramification sets" that contain the archimedean places. We also prove that restricted products of local cohomologies that appear in the Poitou-Tate sequence may be identified with derived functor cohomology of an adele ring. In our proof of the generalized sequence we adopt this derived functor point of view and exploit properties of a natural topology carried by cohomology of the adeles.

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arXiv2015

Topology on cohomology of local fields

Arithmetic duality theorems over a local field $k$ are delicate to prove if $\mathrm{char} k > 0$. In this case, the proofs often exploit topologies carried by the cohomology groups $H^n(k, G)$ for commutative finite type $k$-group schemes $G$. These "Čech topologies", defined using Čech cohomology, are impractical due to the lack of proofs of their basic properties, such as continuity of connecting maps in long exact sequences. We propose another way to topologize $H^n(k, G)$: in the key case $n = 1$, identify $H^1(k, G)$ with the set of isomorphism classes of objects of the groupoid of $k$-points of the classifying stack $\mathbf{B} G$ and invoke Moret-Bailly's general method of topologizing $k$-points of locally of finite type $k$-algebraic stacks. Geometric arguments prove that these "classifying stack topologies" enjoy the properties expected from the Čech topologies. With this as the key input, we prove that the Čech and the classifying stack topologies actually agree. The expected properties of the Čech topologies follow, which streamlines a number of arithmetic duality proofs given elsewhere.

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arXiv2015

Selmer groups as flat cohomology groups

Given a prime number $p$, Bloch and Kato showed how the $p^\infty$-Selmer group of an abelian variety $A$ over a number field $K$ is determined by the $p$-adic Tate module. In general, the $p^m$-Selmer group $\mathrm{Sel}_{p^m} A$ need not be determined by the mod $p^m$ Galois representation $A[p^m]$; we show, however, that this is the case if $p$ is large enough. More precisely, we exhibit a finite explicit set of rational primes $Σ$ depending on $K$ and $A$, such that $\mathrm{Sel}_{p^m} A$ is determined by $A[p^m]$ for all $p \not \in Σ$. In the course of the argument we describe the flat cohomology group $H^1_{\mathrm{fppf}}(O_K, \mathcal{A}[p^m])$ of the ring of integers of $K$ with coefficients in the $p^m$-torsion $\mathcal{A}[p^m]$ of the Néron model of $A$ by local conditions for $p\not\in Σ$, compare them with the local conditions defining $\mathrm{Sel}_{p^m} A$, and prove that $\mathcal{A}[p^m]$ itself is determined by $A[p^m]$ for such $p$. Our method sharpens the known relationship between $\mathrm{Sel}_{p^m} A$ and $H^1_{\mathrm{fppf}}(O_K, \mathcal{A}[p^m])$ and continues to work for other isogenies $ϕ$ between abelian varieties over global fields provided that $\mathrm{deg} ϕ$ is constrained appropriately. To illustrate it, we exhibit resulting explicit rank predictions for the elliptic curve $11A1$ over certain families of number fields.

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arXiv2014

Local factors valued in normal domains

We give an exposition of Deligne's theory of local $ε_0$-factors over fields and discrete valuation rings under the assumption that the theory over the complex numbers is known. We then employ standard techniques from algebraic geometry to deduce the theory of local $ε_0$-factors over arbitrary normal integral schemes.

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arXiv2014

Selmer groups and class groups

Let $A$ be an abelian variety over a global field $K$ of characteristic $p \ge 0$. If $A$ has nontrivial (resp. full) $K$-rational $l$-torsion for a prime $l \neq p$, we exploit the fppf cohomological interpretation of the $l$-Selmer group $\mathrm{Sel}_l A$ to bound $\#\mathrm{Sel}_l A$ from below (resp. above) in terms of the cardinality of the $l$-torsion subgroup of the ideal class group of $K$. Applied over families of finite extensions of $K$, the bounds relate the growth of Selmer groups and class groups. For function fields, this technique proves the unboundedness of $l$-ranks of class groups of quadratic extensions of every $K$ containing a fixed finite field $\mathbb{F}_{p^n}$ (depending on $l$). For number fields, it suggests a new approach to the Iwasawa $μ= 0$ conjecture through inequalities, valid when $A(K)[l] \neq 0$, between Iwasawa invariants governing the growth of Selmer groups and class groups in a $\mathbb{Z}_l$-extension.

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arXiv2010

On a Generalization of the Flag Complex Conjecture of Charney and Davis

The Flag Complex Conjecture of Charney and Davis states that for a simplicial complex $S$ which triangulates a $(2n - 1)$-generalized homology sphere as a flag complex one has $(-1)^n \sum_{σ\in S} \left(\frac{-1}{2}\right)^{\dimσ+ 1} \ge 0$, where the sum runs over all simplices $σ$ of $S$ (including the empty simplex). Interpreting the $1$-skeleta of $σ\in S$ as graphs of Coxeter groups, we present a stronger version of this conjecture, and prove the equivalence of the latter to the Flag Complex Conjecture.

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