Reduction of L-functions of Elliptic Curves Modulo Integers
Abstract
Let $\mathbb{F}_q$ be a finite field of size $q$, where $q$ is a power of a prime $p \geq 5$. Let $C$ be a smooth, proper, and geometrically connected curve over $\mathbb{F}_q$. Consider an elliptic curve $E$ over the function field $K$ of $C$ with nonconstant $j$-invariant. One can attach to $E$ its $L$-function $L(T,E/K)$, which is a generating function that contains information about the reduction types of $E$ at the different places of $K$. The $L$-function of $E/K$ was proven to be a polynomial in $\mathbb{Z}[T]$.
In 1985, Schoof devised an algorithm to compute the zeta function of an elliptic curve over a finite field by directly computing its numerator modulo sufficiently many primes $\ell$. By analogy with Schoof, we consider an elliptic curve $E$ over $K$ with nonconstant $j$-invariant and study the problem of directly computing the reduction modulo $\ell$ of $L(T,E/K)$. In this work, we obtain results in two directions. Firstly, given an integer $N$ different from $p$ and an elliptic curve $E$ with $K$-rational $N$-torsion, we give a formula for the reduction modulo $N$ of the $L$-function of certain quadratic twists, extending a result from Chris Hall. We also give a formula relating the $L$-functions modulo $2$ of any two quadratic twists of $E$, without any assumption on the $K$-rational $2$-torsion. Secondly, given a prime $\ell \neq p$, we give, under some relatively general conditions, formulas for the reduction of $L(T,E/K)$ modulo $\ell$.