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

by Michael Baker

Institution: University of Waterloo
Year: 2015
Keywords: elliptic curves; modular forms; Hurwitz class numbers; quadratic forms; modular curves
Posted: 02/05/2017
Record ID: 2080679
Full text PDF: http://hdl.handle.net/10012/9649


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