Computing points on modular curves over finite fields

TL;DR

This paper introduces a probabilistic algorithm under GRH to efficiently compute the number of points on modular curves over finite fields, enabling calculations for very large primes.

Contribution

The paper develops a novel probabilistic method for counting points on modular curves over finite fields, improving computational feasibility for large primes.

Findings

Algorithm operates in sub-exponential time under GRH.

Able to compute point counts for extremely large primes, e.g., 10^1000+1357.

Demonstrates practical application for modular curve $X_1(17)$.

Abstract

In this paper, we present a probabilistic algorithm to compute the number of $\mathbb{F}_p$-points of modular curve $X_1(n)$. Under the Generalized Riemann Hypothesis(GRH), the algorithm takes $\textrm{O}(n^{56+\delta+\epsilon}\log^{9+\epsilon} p)$ bit operations, where $\delta$ is an absolute constant and $\epsilon$ is any positive real number. As an application, we can compute $#X_1(17)(\mathbb{F}_p)\textrm{mod} 17$ for huge primes $p$. For example, we have $#X_1(17)(\mathbb{F}_{10^{1000}+1357})\textrm{mod} 17=3$.