Ciaran Schembri
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View article: Rational torsion points on abelian surfaces with quaternionic multiplication
Rational torsion points on abelian surfaces with quaternionic multiplication Open
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}}…
View article: Rational torsion points on abelian surfaces with quaternionic multiplication
Rational torsion points on abelian surfaces with quaternionic multiplication Open
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 $A(\mathbb{Q})$ i…
View article: Rational points on Atkin-Lehner quotients of geometrically hyperelliptic Shimura curves
Rational points on Atkin-Lehner quotients of geometrically hyperelliptic Shimura curves Open
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 a…
View article: On rational Bianchi newforms and abelian surfaces with quaternionic multiplication
On rational Bianchi newforms and abelian surfaces with quaternionic multiplication Open
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 e…
View article: Modularity of abelian surfaces over imaginary quadratic fields
Modularity of abelian surfaces over imaginary quadratic fields Open
Modular forms for GL(2) over an imaginary quadratic field 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 associ…
View article: Examples of genuine false elliptic curves which are modular
Examples of genuine false elliptic curves which are modular Open
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 …