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
TLDR
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
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
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
Torsor Dual Pairs
...
1
2
...

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
THE FLAT COHOMOLOGY OF GROUP SCHEMES OF RANK b.
The intent of this paper is to determine the first flat cohomology groups of certain finite fiat group schemes which are defined over the spectrum of the ring of integers in a local number field. WeExpand
On the Birch-Swinnerton-Dyer quotients modulo squares
Let A be an abelian variety over a number field K. An identity between the L-functions L(A/K i , s) for extensions K i of K induces a conjectural relation between the Birch-Swinnerton-Dyer quotients.Expand
How to do a p-descent on an elliptic curve
In this paper, we describe an algorithm that reduces the computation of the (full) p-Selmer group of an elliptic curve E over a number field to standard number field computations such as determiningExpand
Finding Large Selmer Rank via an Arithmetic Theory of Local Constants
We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose K/k is a quadratic extension of number fields, E is an elliptic curve defined over k, andExpand
On the Equation xp + yq = zr: A Survey
AbstractLet p, q and r be three integers ≥ 2. During the last ten years many new ideas have emerged for the study of the Diophantine equation x + y = z. This study is divided into three partsExpand
Endomorphisms of abelian varieties over finite fields
Almost all of the general facts about abelian varieties which we use without comment or refer to as "well known" are due to WEIL, and the references for them are [12] and [3]. Let k be a field, k itsExpand
Visualizing Elements in the Shafarevich—Tate Group
TLDR
A number of ways of “visualizing” the elements of the Shafarevich–Tate group of an elliptic curve E over a number field K are reviewed, finding the answer to the question to be yes in the vast majority of cases. Expand
L-Functions and Tamagawa Numbers of Motives
The notion of a motif was first defined and studied by A. Grothendieck, and this paper is an attempt to understand some of the implications of his ideas for arithmetic. We will formulate a conjectureExpand
Class groups and Selmer groups
It is often the case that a Selmer group of an abelian variety and a group related to an ideal class group can both be naturally embedded into the same cohomology group. One hopes to compute one fromExpand
...
1
2
3
4
5
...