UC San Diego

In Part I, we provide an explicit version of the Eichler–Selberg trace formula for Shimura curves with level structure over the rationals. As an application, we provide an algorithm to compute the isogeny decomposition of the Jacobian of Shimura curves into modular abelian varieties using the method that Rouse–Sutherland–Zureick-Brown developed for classical modular curves. We also give a trace formula for definite quaternionic modular forms over the rationals.In Part II, we compile a complete list of isomorphism class representatives of curves of genus 6 over F2. We use explicit descriptions of canonical curves in each stratum of the Brill–Noether stratification of the moduli space M6, due to Mukai in the generic case. Our computed value of #M6(F2) agrees with the Lefschetz trace formula as recently computed by Bergstrom–Canning–Petersen–Schmitt. We also report progress on compiling a corresponding list in genus 7 over F2 (for which explicit descriptions of canonical curves in each stratum of the Brill–Noether stratification of the moduli space M7 are also available) and genus 5 over F3, where the censuses are complete in all except for the generic stratum in both cases.