Elliptic Curves over Finite Fields and their l-Torsion Galois Representations

Abstract

Let $q$ and $\ell$ be distinct primes. Given an elliptic curve $E$ over $\mathbf{F}_q$, we study the behaviour of the 2-dimensional Galois representation of $\mathrm{Gal}(\overline{\mathbf{F}_q}/\mathbf{F}_q) \cong \widehat{\mathbf Z}$ on its $\ell$-torsion subgroup $E[\ell]$. This leads us to the problem of counting elliptic curves with prescribed $\ell$-torsion Galois representations, which we answer for small primes $\ell$ by counting rational points on suitable modular curves. The resulting exact formulas yield expressions for certain sums of Hurwitz class numbers.