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

elliptic curves;modular forms;Hurwitz class numbers;quadratic forms;modular curves;Pure Mathematics

Baker, Michael

University of Waterloo

关键词: elliptic curves;    modular forms;    Hurwitz class numbers;    quadratic forms;    modular curves;    Pure Mathematics;   

Others  :  https://uwspace.uwaterloo.ca/bitstream/10012/9649/3/Baker_Michael.pdf

瑞士|英语

来源: UWSPACE Waterloo Institutional Repository

【 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.