Numerical Evidence for a Refinement of Deligne’s Period Conjecture for Jacobians of Curves

Abstract

Let A / Q be a Jacobian variety and let F be a totally real, tamely ramified, abelian number field. Given a character ψ of F / Q , Deligne’s Period Conjecture asserts the algebraicity of the suitably normalized value L ( A , ψ , 1 ) at s = 1 of the Hasse-Weil-Artin L-function of the ψ -twist of A. We formulate a conjecture regarding the integrality properties of the family of normalized L-values ( L ( A , ψ , 1 ) ) ψ , and its relation to the Tate-Shafarevich group of A over F. We numerically investigate our conjecture through p-adic congruence relations between these values.