Parity of genera of Shimura curves over a real quadratic field, by Marat Sadykov

Let $K={\Q}({\sqrt{m}})$ be a real quadratic field and $V_B$ be a Shimura curve corresponding to a maximal order in a quaternion algebra $B$ over $K$ such that $B$ is ramified at exactly one infinite place. We give the necessary and sufficient conditions for the genus of $V_B$ to be even in almost all cases. This result allowed Jordan, Livne, and Varshavsky to prove that Shimura curves over $K$ always have even jacobian provided that $m$ is a prime congruent to 1 modulo 4.

Marat Sadykov <msadykov@math.columbia.edu>