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arXiv · 1804.07225

Examples of genuine QM abelian surfaces which are modular

Ciaran Schembri
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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.

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Ciaran Schembri. 2018-04-19. Examples of genuine QM abelian surfaces which are modular. https://doi.org/10.1007/s40993-018-0150-x

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