Selmer Groups Of Abelian Varieties Research Articles

Boundedness of Iwasawa Invariants of Fine Selmer Groups and Selmer Groups

In this article we consider Selmer groups and fine Selmer groups of abelian varieties over a number field K. Following a classical approach of Monsky for Iwasawa modules from ideal class groups, we give sufficient conditions for the Iwasawa mu -invariants of the fine Selmer groups and the mu -invariants of the Selmer groups to be bounded as one runs over the mathbb {Z}_p-extensions of K. Moreover, we describe a criterion for the boundedness of Iwasawa lambda -invariants of Selmer groups and fine Selmer groups over multiple mathbb {Z}_p-extensions which generalises a criterion of Monsky from dimension 2 to arbitrary dimension.

Fine Selmer groups and ideal class groups

Let K be a number field, let A be an abelian variety defined over K and let $$K_\infty /K$$ be a uniform p-adic Lie extension. We compare several arithmetic invariants of Iwasawa modules of ideal class groups on the one side and fine Selmer groups of abelian varieties on the other side. If $$K_\infty $$ contains sufficiently many p-power torsion points of A, then we can compare the ranks and the Iwasawa $$\mu $$ -invariants of these modules over the Iwasawa algebra. In several special cases (e.g. multiple $$\mathbb {Z}_p$$ -extensions), we can also prove relations between suitably generalised Iwasawa $$\lambda $$ -invariants of the two types of Iwasawa modules. In the literature, two different kinds of generalised Iwasawa $$\lambda $$ -invariants have been introduced for ideal class groups and Selmer groups. We define analogues of both concepts for fine Selmer groups and compare the resulting invariants. In order to obtain some of our main results, we prove new asymptotic formulas for the growth of ideal class groups and fine Selmer groups in multiple $$\mathbb {Z}_p$$ -extensions.

ON FINE SELMER GROUPS AND SIGNED SELMER GROUPS OF ELLIPTIC MODULAR FORMS

AbstractLetfbe an elliptic modular form andpan odd prime that is coprime to the level off. We study the link between divisors of the characteristic ideal of thep-primary fine Selmer group offover the cyclotomic$\mathbb {Z}_p$extension of$\mathbb {Q}$and the greatest common divisor of signed Selmer groups attached tofdefined using the theory of Wach modules. One of the key ingredients of our proof is a generalisation of a result of Wingberg on the structure of fine Selmer groups of abelian varieties with supersingular reduction atpto the context of modular forms.

On non-abelian higher special elements of p-adic representations

We develop a theory of ‘non-abelian higher special elements’ in the non-commutative exterior powers of the Galois cohomology of p-adic representations. We explore their relation to the theory of organising matrices and thus to the Galois module structure of Selmer modules. In concrete applications, we relate our general theory to the formulation of refined conjectures of Birch and Swinnerton-Dyer type and to the Galois structure of Tate–Shafarevich and Selmer groups of abelian varieties.

Bounding the Iwasawa invariants of Selmer groups

AbstractWe study the growth of p-primary Selmer groups of abelian varieties with good ordinary reduction at p in ${{Z}}_p$ -extensions of a fixed number field K. Proving that in many situations the knowledge of the Selmer groups in a sufficiently large number of finite layers of a ${{Z}}_p$ -extension over K suffices for bounding the over-all growth, we relate the Iwasawa invariants of Selmer groups in different ${{Z}}_p$ -extensions of K. As applications, we bound the growth of Mordell–Weil ranks and the growth of Tate-Shafarevich groups. Finally, we derive an analogous result on the growth of fine Selmer groups.

Variation of Tamagawa numbers of semistable abelian varieties in field extensions

AbstractWe investigate the behaviour of Tamagawa numbers of semistable principally polarised abelian varieties in extensions of local fields. In particular, we give a simple formula for the change of Tamagawa numbers in totally ramified extensions and one that computes Tamagawa numbers up to rational squares in general extensions. As an application, we extend some of the existing results on thep-parity conjecture for Selmer groups of abelian varieties by allowing more general local behaviour. We also give a complete classification of the behaviour of Tamagawa numbers for semistable 2-dimensional principally polarised abelian varieties that is similar to the well-known one for elliptic curves. The appendix explains how to use this classification for Jacobians of genus 2 hyperelliptic curves given by equations of the formy2=f(x), under some simplifying hypotheses.

Growth of Selmer Groups of CM Abelian Varieties

AbstractLet p be an odd prime. We study the variation of the p-rank of the Selmer groups of Abelian varieties with complex multiplication in certain towers of number fields.

Compatibility of arithmetic and algebraic local constants (the case )

We show that arithmetic local constants attached by Mazur and Rubin to pairs of self-dual Galois representations which are congruent modulo a prime number$p>2$are compatible with the usual local constants at all primes not dividing$p$and in two special cases also at primes dividing$p$. We deduce new cases of the$p$-parity conjecture for Selmer groups of abelian varieties with real multiplication (Theorem 4.14) and elliptic curves (Theorem 5.10).

On Selmer groups of abelian varieties over ℓ-adic Lie extensions of global function fields

Let F be a global function field of characteristic p > 0 and A/F an abelian variety. Let K/F be an ℓ-adic Lie extension (ℓ ≠ p) unramified outside a finite set of primes S and such that Gal(K/F) has no elements of order ℓ. We shall prove that, under certain conditions, Sel A (K) ℓ ∨ has no nontrivial pseudo-null submodule.

Organizing Matrices for Arithmetic Complexes

We extend and refine the theory of “organizing modules” of Mazur and Rubin to construct a canonical class of matrices that encodes a range of information about natural families of complexes in arithmetic. We then describe several concrete applications of this theory including the proof of new results on the explicit structures of Galois groups, ideal class groups, and wild kernels in higher algebraic K-theory and the formulation of a range of explicit conjectures concerning both the ranks and Galois structures of Selmer groups of abelian varieties over finite (nonabelian) Galois extensions of number fields.

Selmer groups of abelian varieties in extensions of function fields

Let k be a field of characteristic q, \({\mathcal{C}}\) a smooth geometrically connected curve defined over k with function field \({K := k(\mathcal{C})}\) . Let A/K be a non-constant abelian variety defined over K of dimension d. We assume that q = 0 or > 2d + 1. Let p ≠ q be a prime number and \({\mathcal{C}' \to \mathcal{C}}\) a finite geometrically Galois and etale cover defined over k with function field \({K' := k(\mathcal{C}')}\). Let (τ′, B′) be the K′/k-trace of A/K. We give an upper bound for the \({\mathbb{Z}_p}\) -corank of the Selmer group Selp(A ×KK′), defined in terms of the p-descent map. As a consequence, we get an upper bound for the \({\mathbb{Z}}\) -rank of the Lang–Neron group A(K′)/τ′B′(k). In the case of a geometric tower of curves whose Galois group is isomorphic to \({\mathbb{Z}_p}\), we give sufficient conditions for the Lang–Neron group of A to be uniformly bounded along the tower.

A note on Selmer groups of abelian varieties over the trivializing extensions

We prove that for any abelian variety A A defined over a number field K K that is not isogenous to a product of CM elliptic curves, the pontrjagin dual of the Selmer group of the abelian variety over the trivializing extension K ( A [ p ∞ ] ) K(A[p^\infty ]) has no nonzero pseudo-null submodules.

On Primitive and Realisable Classes

Let S be a scheme, and let G be a finite, flat, commutative group scheme over S. In this paper we show that (subject to a mild technical assumption) every primitive class in Pic(G) is realisable. This gives an affirmative answer to a question of Waterhouse. We also discuss applications to locally free classgroups and to Selmer groups of Abelian varieties.