PhD Public Lecture (Math) - Felix Baril Boudreau

Let F_q be a finite field of size q where q is a power of a prime p ≥ 5. Let C be a smooth, proper, and geometrically connected curve over 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 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 ℓ. 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 of L(T,E/K) modulo ℓ. 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 of Chris Hall. We also give a formula relating the L-functions modulo 2 of any two quadratic twists of E, without any assumptions on the K-rational 2-torsion. Secondly, given a prime ℓ ≠ p, we give, under some relatively general conditions, formulas for the reduction of L(T,E/K) modulo ℓ. The formulas in this work are amenable to computation by algorithms that are more efficient than naive point-counting methods.