Hypergeometric L-functions in average polynomial time

TL;DR

This paper presents an efficient algorithm to compute the trace of Frobenius for hypergeometric motives across all primes up to a bound, leveraging advanced trace formulas and average polynomial time methods.

Contribution

It introduces a novel algorithm that computes mod-p reductions of Frobenius traces for hypergeometric motives in quasilinear time, combining existing trace formulas with new average time techniques.

Findings

Algorithm computes traces for all primes up to X efficiently

Achieves quasilinear time complexity in X

Combines trace formulas with average polynomial time methods

Abstract

We describe an algorithm for computing, for all primes $p \leq X$, the mod-$p$ reduction of the trace of Frobenius at $p$ of a fixed hypergeometric motive in time quasilinear in $X$. This combines the Beukers--Cohen--Mellit trace formula with average polynomial time techniques of Harvey et al.