Year: 2025
Type: article
Abstract: We present an efficient algorithm to compute the Euler factor of a genus 2 curve $$C/\mathbb {Q}$$ 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 $$\mathbb {Z}$$ Z and $$\mathbb {F}_p$$ F p , followed by a point-counting computation on two elliptic curves over $$\mathbb {F}_p$$ F p , or a single elliptic curve over $$\mathbb {F}_{p^2}$$ F p 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.
Source: Research in Number Theory
Authors: Céline Maistret, Andrew V. Sutherland
Institutions: University of Bristol, Massachusetts Institute of Technology
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,590
OpenAlex ID: W4407767182