Computing an L-function modulo a prime

Abstract: Let $E$ be an elliptic curve with non-constant $j$-invariant over a function field $K$ with constant field of size an odd prime power $q$. Its $L$-function $L(T,E/K)$ belongs to $1 + T\mathbb{Z}[T]$. Inspired by the algorithms of Schoof and Pila for computing zeta functions of curves over finite fields, we propose an approach to compute $L(T,E/K)$. The idea is to compute, for sufficiently many primes $\ell$ coprime with $q$, the reduction $L(T,E/K) \bmod{\ell}$. The $L$-function is then recovered via the Chinese remainder theorem. When $E(K)$ has a subgroup of order $N \geq 2$ coprime with $q$, Chris Hall showed how to explicitly calculate $L(T,E/K) \bmod{N}$. We present novel theorems going beyond Hall's.

number theory

Audience: researchers in the topic

University of Arizona Algebra and Number Theory Seminar