Examples of genuine QM abelian surfaces which are modular

Ciaran Schembri

doi:10.1007/s40993-018-0150-x

Abstract

Let K be an imaginary quadratic field. Modular forms for GL(2) over K are known as Bianchi modular forms. Standard modularity conjectures assert that every weight 2 rational Bianchi newform has either an associated elliptic curve over K or an associated abelian surface with quaternionic multiplication over K. We give explicit evidence in the way of examples to support this conjecture in the latter case. Furthermore, the quaternionic surfaces given correspond to genuine Bianchi newforms, which answers a question posed by Cremona as to whether this phenomenon can happen.

Highlights

Let K be an imaginary quadratic field
Motivated by the conjectural connections with Bianchi modular forms, in 1992 Cremona asked whether genuine QM surfaces over imaginary quadratic fields should exist
By carrying out a detailed analysis of the associated Galois representations and applying the Faltings–Serre–Livné criterion, we prove the modularity of these QM surfaces

Summary

Introduction

A simple abelian surface over K whose algebra of K -endomorphisms is an indefinite quaternion algebra over Q is commonly known as a QM abelian surface, or just QM surface These are often referred to false elliptic curves, coined by Serre in the 1970s [1] based on the observation that such a surface is isogenous to the square of an elliptic curve modulo every prime of good reduction [30, Lemma 6]. Motivated by the conjectural connections with Bianchi modular forms, in 1992 Cremona asked whether genuine QM surfaces over imaginary quadratic fields should exist (see Question 1.3). We answer this question in the positive by providing explicit genus 2 curves whose Jacobians are genuine QM surfaces. The main result of the present article is as follows: Theorem 1.1 The Jacobians of the following genus 2 curves are QM surfaces which are modular by a genuine Bianchi newform as in Conjecture 1.2

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