The kernel of Ribet's isogeny for genus three Shimura curves

摘要

There are exactly nine reduced discriminants D of indefinite quaternion algebras over Q for which the Shimura curve X_D attached to D has genus 3. We present equations for these nine curves and, moreover, for each D we determine a subgroup c(D) of cuspidal divisors of degree zero of Jac(X_0(D))~(new) such that the abelian variety Jac(X_0(D))~(new)/c(D) is the jacobian of the curve X_D.