Elliptic Curves over Finite Fields and their l-Torsion Galois Representations
Institution: | University of Waterloo |
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Department: | |
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.