# Algebra/Number Theory Seminar

## Computing an L-function modulo a prime

Félix Baril Boudreau, U Western Ontario, Canada

Thursday, October 14th, 2021
2:00 PM - 3:00 PM
Zoom id 566 385 6457 (no password)

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 $$\ell$$ invertible in $$K$$, 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.

Event contact: dinesh dot thakur at rochester dot edu