Session A3 - Computational Number Theory
July 12, 17:00 ~ 17:40 - Room B6
Rigorous computation of the endomorphism ring of a Jacobian
Universität Ulm, Germany - firstname.lastname@example.org
Consider a curve $X$, defined by an explicit equation, over a number field. Let $J$ be the Jacobian of $X$. The endomorphism ring $E$ of $J$ is an important arithmetic invariant of $X$; for example, the Sato-Tate group of $X$ can be recovered from the Galois module structure of $E$.
Heuristically, the ring $E$ can be determined quite quickly. In this talk, we consider the rigor of such calculations. In particular, we give an algorithm that verifies whether a putative tangent representation of an endomorphism in fact globalizes, It does not use complex approximations and is, to our knowledge, the first general-purpose algorithm over number fields that provably terminates. We then discuss how this algorithm can be combined with other techniques to rigorously compute the ring $E$.
Joint work with Edgar Costa (Dartmouth College), Nicolas Mascot (University of Warwick) and John Voight (Dartmouth College).