On the dihedral Euler characteristics of Selmer groups of Abelian varieties

Abstract The notion of the truncated Euler characteristic for Iwasawa modules is an extension of the notion of the usual Euler characteristic to the case when the homology groups are not finite. This article explores congruence relations between the truncated Euler characteristics for dual Selmer groups of elliptic curves with isomorphic residual representations, over admissible p-adic Lie extensions. Our results extend earlier congruence results from the case of elliptic curves with rank zero to the case of higher rank elliptic curves. The results provide evidence for the p-adic Birch and Swinnerton-Dyer formula without assuming the main conjecture.

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Selmer groups of abelian varieties in extensions of function fields

Mathematische Zeitschrift ◽

10.1007/s00209-008-0351-4 ◽

2008 ◽

Vol 261 (4) ◽

pp. 787-804 ◽

Cited By ~ 1

Author(s):

Amílcar Pacheco

Keyword(s):

Function Fields ◽

Abelian Varieties ◽

Selmer Groups

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Euler Characteristics and their Congruences for Multi-signed Selmer Groups

Canadian Journal of Mathematics ◽

10.4153/s0008414x21000699 ◽

2022 ◽

pp. 1-25

Author(s):

Anwesh Ray ◽

R. Sujatha

Keyword(s):

Selmer Groups ◽

Euler Characteristics

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On the Selmer groups of abelian varieties over function fields of characteristic p > 0

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004108001801 ◽

2009 ◽

Vol 146 (1) ◽

pp. 23-43 ◽

Cited By ~ 5

Author(s):

TADASHI OCHIAI ◽

FABIEN TRIHAN

Keyword(s):

Number Field ◽

Function Field ◽

Function Fields ◽

Abelian Varieties ◽

Iwasawa Theory ◽

Selmer Groups ◽

Characteristic P ◽

Field Case ◽

New Phenomena ◽

Geometric Analogue

AbstractWe study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We also propose a conjecture (Conjecture 1.6) which might be considered as a counterpart of the principal conjecture in the case over a number field.

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GENERALIZED EXPLICIT DESCENT AND ITS APPLICATION TO CURVES OF GENUS 3

Forum of Mathematics Sigma ◽

10.1017/fms.2016.1 ◽

2016 ◽

Vol 4 ◽

Cited By ~ 4

Author(s):

NILS BRUIN ◽

BJORN POONEN ◽

MICHAEL STOLL

Keyword(s):

Abelian Varieties ◽

Selmer Groups ◽

Explicit Computation ◽

Selmer Group ◽

Common Generalization ◽

Global Fields

We introduce a common generalization of essentially all known methods for explicit computation of Selmer groups, which are used to bound the ranks of abelian varieties over global fields. We also simplify and extend the proofs relating what is computed to the cohomologically defined Selmer groups. Selmer group computations have been practical for many Jacobians of curves over $\mathbb{Q}$ of genus up to 2 since the 1990s, but our approach is the first to be practical for general curves of genus 3. We show that our approach succeeds on some genus 3 examples defined by polynomials with small coefficients.

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Akashi series, characteristic elements and congruence of Galois representations

International Journal of Number Theory ◽

10.1142/s179304211650038x ◽

2016 ◽

Vol 12 (03) ◽

pp. 593-613

Author(s):

Meng Fai Lim

Keyword(s):

Galois Representations ◽

The Other ◽

Selmer Groups ◽

Euler Characteristics

In this paper, we compare the Akashi series of the Pontryagin dual of the Selmer groups of two Galois representations over a strongly admissible [Formula: see text]-adic Lie extension. Namely, we show that whenever the two Galois representations in question are congruent to each other, the Akashi series of one is a unit if and only if the Akashi series of the other is also a unit. We will also obtain a similar result for the Euler characteristics of the Selmer groups and the characteristic elements attached to the Selmer groups.

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