Large Selmer groups over number fields

Abstract

Let p be a prime number and M a quadratic number field, M Q( √ p) if p ≡ 1 mod 4. We will prove that for any positive integer d there exists a Galois extension F/Q with Galois group D 2p and an elliptic curve E/Q such that F contains M and the p-Selmer group of E/F has size at least p d .

On Brauer–Kuroda type relations of S-class numbers in dihedral extensions

“…Alex Bartel -Relations between class numbers in dihedral extensions as required. Next, for a subgroup H of G we can identify Hom G (G/H, Γ) with Γ H via f → f (1). We claim that under this identification, we have det (, ) 1 , Γ) which is trivial outside of G/H i is orthogonal to an element which is trivial outside of G/H k .…”

Brauer relations in finite groups

“…. , 0) such that the sum of all coefficients of these terms is not congruent to 0 modulo p. Moreover, by Corollary 7.12 (2), this relation must contain either a term in M with signature (1, . .…”

“…Thus, Brauer relations can provide non-trivial relationships between different arithmetic invariants, like the class numbers and the regulators of various intermediate fields. This point of view proved to be very fruitful to study class numbers and unit groups [10,25,34,30], related Galois module structures [9,3] and Mordell-Weil groups and other arithmetic invariants of elliptic curves and abelian varieties [16,15,2]. In a slightly different direction, a verification of the vanishing in Brauer relations of conjectural special values of L-functions can be regarded as strong evidence for the corresponding conjectures.…”

“…One concrete number-theoretic application of Brauer relations is the theory of 'regulator constants', used in the proof of the Selmer parity conjecture for elliptic curves over Q [16], questions related to Selmer growth [15,17,2], and also to analyse unit groups and higher K-groups of number fields [3,6]. The regulator constant C Θ (Γ) ∈ Q × is an invariant attached to a Z[G]module Γ and a Brauer relation Θ in G. For applications to elliptic curves the most important regulator constant is that of the trivial Z[G]-module Γ = 1, as it controls the l-Selmer rank of the curve over the ground field.…”

Index formulae for integral Galois modules

“…From now on, we closely follow [15, §4]. Suppose that we have a subset U of subgroups of G such that Θ = 1 + (|U| − 1) · G − H∈U H, 1 Here and elsewhere, "irreducible rational representation" refers to representations that are irreducible over Q, rather than absolutely irreducible representations that happen to be defined over Q. Note also, that once we substitute the above formula for the regulator constant into (2.4), (2.6), or (2.14), the torsion quotient cancels, so in any case, it is not present in the final index formulae.…”