Corpus ID: 18030832

# Selmer groups as flat cohomology groups

@article{Cesnavicius2013SelmerGA,
title={Selmer groups as flat cohomology groups},
author={Kestutis Cesnavicius},
journal={arXiv: Number Theory},
year={2013}
}
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 $\Sigma$ depending on $K$ and $A$, such that $\mathrm{Sel}_{p^m… Expand Selmer groups and class groups Abstract Let$A$be an abelian variety over a global field$K$of characteristic$p\geqslant 0$. If$A$has nontrivial (respectively full)$K$-rational$l$-torsion for a prime$l\neq p$, we exploitExpand On the$\Lambda$-cotorsion subgroup of the Selmer group Let$E$be an elliptic curve defined over a number field$K$with supersingular reduction at all primes of$K$above$p$. If$K_{\infty}/K$is a$\mathbb{Z}_p$-extension such thatExpand p-Selmer growth in extensions of degree p This Selmer group analogue is known in special cases and it is proved in general, along with a version for arbitrary global fields. Expand The geometric average size of Selmer groups over function fields We show, in the large$q$limit, that the average size of$n$-Selmer groups of elliptic curves of bounded height over$\mathbb F_q(t)$is the sum of the divisors of$n$. As a corollary, again in theExpand On Tate-Shafarevich groups of 1-motives over Galois extensions Let K/F be a finite Galois extension of global fields with Galois group G and let M be a 1-motive over F. We discuss the kernel and cokernel of the restriction map Sha^{i}(F,M) --> Sha^{i}(K,M)^{G}Expand Ranks of abelian varieties in cyclotomic twist families • Mathematics • 2021 Let A be an abelian variety over a number field F , and suppose that Z[ζn] embeds in EndF̄ A, for some root of unity ζn of order n = 3 . Assuming that the Galois action on the finite group A[1− ζn]Expand The ℓ-parity conjecture over the constant quadratic extension Abstract For a prime ℓ and an abelian variety A over a global field K, the ℓ-parity conjecture predicts that, in accordance with the ideas of Birch and Swinnerton–Dyer, the ℤℓ-corank of the ℓ∞-SelmerExpand ON THE AVERAGE OF p-SELMER RANK IN QUADRATIC TWIST FAMILIES OF ELLIPTIC CURVES OVER FUNCTION FIELD We show that if the quadratic twist family of a given elliptic curve over Fq[t] with Char(Fq) ≥ 5 has an element whose Néron model has a multiplicative reduction away from ∞, then the averageExpand Chevalley–Weil theorem and subgroups of class groups • Mathematics • 2016 We prove, under some mild hypothesis, that an ´etale cover of curves defined over a number field has infinitely many specializations into an everywhere unramified extension of number fields. ThisExpand #### References SHOWING 1-10 OF 92 REFERENCES Selmer groups and class groups Abstract Let$A$be an abelian variety over a global field$K$of characteristic$p\geqslant 0$. If$A$has nontrivial (respectively full)$K$-rational$l$-torsion for a prime$l\neq p\$, we exploitExpand
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