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 $$C/\mathbb {Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>/</mml:mo> <mml:mi>Q</mml:mi> </mml:mrow> </mml:math> 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 $$\mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> and $$\mathbb {F}_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> , followed by a point-counting computation on two elliptic curves over $$\mathbb {F}_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> , or a single elliptic curve over $$\mathbb {F}_{p^2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:msup> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:msub> </mml:math> . 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.