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- W2809833019 abstract "Seminal works by Birch and Ihara gave formulas for the $m$th power moments of the traces of Frobenius endomorphisms of elliptic curves over $mathbb{F}_{p}$ for primes $p geq 5$. Recent works by Kaplan and Petrow generalized these results to the setting of elliptic curves that contain a subgroup isomorphic to a fixed finite abelian group $A$. We revisit these formulas and determine a simple expression for the zeta function $Z_p(A; t)$, the generating function for these $m$th power moments. In particular, we find that [ Z_p(A;t) = frac{widehat{Z}_p(A; t)}{displaystyle prod_{a in textrm{Frob}_p(A)}(1 - at)},] where $textrm{Frob}_p(A) := { a , colon -2sqrt{p} leq a leq 2sqrt{p}, text{ and } a equiv p+1 pmod{|A|}}$, and $widehat{Z}_p(A;t)$ is an easily computed polynomial that is determined by the first $Biglceilfrac{2lfloor 2sqrt{p}rfloor}{|A|}Bigrceil$ power moments. These rational zeta functions have two natural applications. We find rational generating functions in weight aspect for traces of Hecke operators on $S_k(Gamma)$ for various congruence subgroups $Gamma$. We also prove congruence relations for power moments by making use of known congruences for traces of Hecke operators." @default.
- W2809833019 created "2018-07-10" @default.
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- W2809833019 date "2018-07-02" @default.
- W2809833019 modified "2023-09-27" @default.
- W2809833019 title "Generating functions for power moments of elliptic curves over $mathbb{F}_p$" @default.
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