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

Abstract

There are exactly nine reduced discriminants $D$ of indefinite quaternion algebras over $\mathbb Q$ 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 ${\rm Jac}(X_0(D))^{\rm new}$ such that the abelian variety ${\rm Jac}(X_0(D))^{\rm new}/c(D)$ is the jacobian of the curve $X_D$.