Examples of genuine QM abelian surfaces which are modular

TL;DR

This paper provides explicit examples of genuine quaternionic abelian surfaces over imaginary quadratic fields, supporting the conjecture that such surfaces correspond to Bianchi modular forms and answering a question about their existence.

Contribution

It offers the first explicit examples of genuine quaternionic abelian surfaces associated with Bianchi modular forms, confirming a key conjecture in the field.

Findings

Explicit examples of genuine quaternionic abelian surfaces are constructed.

These surfaces correspond to genuine Bianchi newforms, confirming their existence.

Supports the modularity conjecture relating Bianchi forms to abelian surfaces.

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 J. Cremona as to whether this phenomenon can happen.