Computing Euler factors of genus 2 curves at odd primes of almost good reduction

Abstract

We present an efficient algorithm to compute the Euler factor of a genus 2 curve C/Q at an odd prime p that is of bad reduction for C but of good reduction for the Jacobian of C (a prime of ``almost good'' reduction). Our approach is based on the theory of cluster pictures introduced by Dokchitser, Dokchitser, Maistret, and Morgan, which allows us to reduce the problem to a short, explicit computation over Z and Fp, followed by a point-counting computation on two elliptic curves over Fp, or a single elliptic curve over Fp^2. A key feature of our approach is that we avoid the need to compute a regular model for C. This allows us to efficiently compute many examples that are infeasible to handle using the algorithms currently available in computer algebra systems such as Magma and Pari/GP.