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

Céline Maistret, Andrew V. Sutherland

TL;DR

This work addresses computing Euler factors $L_p(C,T)$ for genus $2$ curves $C/\,\mathbb{Q}$ at odd primes of almost good reduction, a regime where standard CAS methods struggle. It introduces the cluster picture framework to avoid constructing a regular model, reducing the problem to explicit $\mathbb{Z}$ and $\mathbb{F}_p$ computations followed by point counts on elliptic curves over $\mathbb{F}_p$ or $\mathbb{F}_{p^2}$. The authors present deterministic and Las Vegas algorithms with running time $O(\|f\|^2\log^2\|f\|/\log p + \log^5 p)$, and provide implementations in Magma and C that dramatically outperform existing Euler factor computations on large datasets. The approach enables efficient handling of primes of almost good reduction, with potential extensions to semistable cases and higher genus, significantly advancing the practical computation of $L$-polynomials and the expansion of genus $2$ curve databases.

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.

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

TL;DR

This work addresses computing Euler factors for genus curves at odd primes of almost good reduction, a regime where standard CAS methods struggle. It introduces the cluster picture framework to avoid constructing a regular model, reducing the problem to explicit and computations followed by point counts on elliptic curves over or . The authors present deterministic and Las Vegas algorithms with running time , and provide implementations in Magma and C that dramatically outperform existing Euler factor computations on large datasets. The approach enables efficient handling of primes of almost good reduction, with potential extensions to semistable cases and higher genus, significantly advancing the practical computation of -polynomials and the expansion of genus curve databases.

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.
Paper Structure (5 sections, 3 theorems, 11 equations)

This paper contains 5 sections, 3 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Background
  4. Euler factors
  5. Cluster Pictures

Key Result

Theorem 1.1

Let $C/\mathbb{Q}$ be a genus $2$ curve $y^2=f(x)=\sum_i f_ix^i\in \mathbb{Z}[x]$ with almost good reduction at an odd prime $p$. There is a deterministic algorithm that, given a nonsquare element of $\mathbb{F}_p^\times$, computes the $L$-polynomial $L_p(C,T)$ in time where $\|f\|=\max_i\log\|f_i|$. There is also a Las Vegas algorithm with the same expected running time that does not require a n

Theorems & Definitions (12)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Definition 2.4: Clusters and cluster pictures
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 2 more