Large Selmer groups over number fields

TL;DR

This paper constructs specific Galois extensions over number fields to demonstrate the unbounded growth of p-Selmer groups of elliptic curves, highlighting new possibilities in number theory and elliptic curve research.

Contribution

It proves the existence of Galois extensions with prescribed properties that lead to arbitrarily large p-Selmer groups over quadratic fields.

Findings

Existence of Galois extensions with Galois group D_{2p} containing M.

Construction of elliptic curves with large p-Selmer groups over these extensions.

Demonstration that p-Selmer groups can grow arbitrarily large in this setting.

Abstract

Let p be a prime number and M a quadratic number field, M not equal to Q(\sqrt{p}) if p is congruent to 1 modulo 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.