Primes of superspecial reduction for QM abelian surfaces. (English) Zbl 1220.11076
Summary: We show that any abelian surface with multiplication by the quaternion \(\mathbb Q\)-algebra of discriminant 6, with field of moduli \(\mathbb Q\) and which is a Jacobian in characteristic 2 and 3, has infinitely many primes of superspecial reduction. This is done by examining complex multiplication points in characteristic 0 and \(p\) and the values of a certain \(j\)-function on the associated moduli space at these points.
MSC:
11G18 | Arithmetic aspects of modular and Shimura varieties |
11G15 | Complex multiplication and moduli of abelian varieties |
14G35 | Modular and Shimura varieties |
11G25 | Varieties over finite and local fields |