$L$-Functions of Elliptic Curves Modulo Integers

Abstract

In 1985, Schoof devised an algorithm to compute zeta functions of elliptic curves over finite fields by directly computing the numerators of these rational functions modulo sufficiently many primes (see \cite{schoof_1985}). If $E/K$ is an elliptic curve with nonconstant $j$-invariant defined over a function field $K$ of characteristic $p \geq 5$, we know that its $L$-function $L(T,E/K)$ is a polynomial in $\mathbb{Z}[T]$ (see \cite[p.11]{katz_2002}). Inspired by Schoof, we study the reduction of $L(T,E/K)$ modulo integers. We obtain three main results. Firstly, if $E/K$ has non-trivial $K$-rational $N$-torsion for some integer $N$ coprime with $p$, we extend a formula for $L(T,E/K) \bmod N$ due to Hall (see \cite[p.133, Theorem 4]{hall_2006}) to all quadratic twists $E_f/K$ with $f \in K^\times \smallsetminus K^{\times 2}$. Secondly, without any condition on the $2$-torsion subgroup of $E(K)$, we give a formula for the quotient modulo $2$ of $L$-functions of any two quadratic twists of $E/K$. Thirdly, we use these results to compute the global root numbers of an infinite family of quadratic twists of an elliptic curve and in most cases find the exact analytic rank of each of these twists. We also illustrate that in favourable situations our second main result allows one to compute much more efficiently $L(T,E_f/K) \bmod 2$ than an algorithm of Baig and Hall (see \cite{baig_hall_2012}). Finally, we use our formulas to compute directly some degree $2$ $L$-functions.