On Serre's Conjecture over Imaginary Quadratic Fields

TL;DR

This paper investigates an analog of Serre's conjecture over imaginary quadratic fields, providing computational evidence supporting the weight recipe and establishing the validity of a modular symbols method for arbitrary weights.

Contribution

It offers computational support for the weight recipe over imaginary quadratic fields and proves the modular symbols method's applicability in this setting.

Findings

Computational evidence supports the weight recipe of Buzzard, Diamond, and Jarvis.

Proves modular symbols method works for arbitrary weights over imaginary quadratic fields.

Validates the conjectural framework in the context of imaginary quadratic fields.

Abstract

We study an analog of Serre's conjecture over imaginary quadratic fields. In particular, we ask whether the weight recipe of Buzzard, Diamond and Jarvis will hold in this setting. Using a program written by the author, we provide computational evidence that this is in fact the case. In order to justify the method used in the program, we prove that a modular symbols method will work for arbitrary weights over imaginary quadratic fields.