arXiv subjects

Ciaran Schembri

Publications and source records attributed to Ciaran Schembri.

4 recordsLinked to original sources

arXiv2023

Rational torsion points on abelian surfaces with quaternionic multiplication

Let $A$ be an abelian surface over $\mathbb{Q}$ whose geometric endomorphism ring is a maximal order in a non-split quaternion algebra. Inspired by Mazur's theorem for elliptic curves, we show that the torsion subgroup of $A(\mathbb{Q})$ is $12$-torsion and has order at most $18$. Under the additional assumption that $A$ is of $\mathrm{GL}_2$-type, we give a complete classification of the possible torsion subgroups of $A(\mathbb{Q})$.

math.NT↗

arXiv2022

Rational points on Atkin-Lehner quotients of geometrically hyperelliptic Shimura curves

Guo and Yang give defining equations for all geometrically hyperelliptic Shimura curves $X_0(D,N)$. In this paper we compute the $\mathbb{Q}$-rational points on the Atkin-Lehner quotients of these curves using a variety of techniques. We also determine which rational points are CM for many of these curves.

math.NT↗

arXiv2020

On rational Bianchi newforms and abelian surfaces with quaternionic multiplication

We study the rational Bianchi newforms (weight 2, trivial character, with rational Hecke eigenvalues) in the LMFDB that are not associated to elliptic curves, but instead to abelian surfaces with quaternionic multiplication. Two of these examples exhibit a rather special kind of behaviour: we show they arise from twisted base change of a classical newform with nebentypus character of order 4 and eight inner twists.

math.NT↗

arXiv2019

Examples of genuine QM abelian surfaces which are modular

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.

math.NT↗