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.
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Cite this version of the work
Michael Baker
(2015).
Elliptic Curves over Finite Fields and their l-Torsion Galois Representations. UWSpace.
http://hdl.handle.net/10012/9649
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