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

摘要

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.