Generating functions for power moments of elliptic curves over $\mathbb{F}_p$

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 $\textrm{Frob}_p(A) := \{ a \, \colon -2\sqrt{p} \leq a \leq 2\sqrt{p}\, \text{ and } a \equiv p+1 \pmod{|A|}\}$, and $\widehat{Z}_p(A;t)$ is an easily computed polynomial that is determined by the first $\Big\lceil\frac{2\lfloor 2\sqrt{p}\rfloor}{|A|}\Big\rceil$ 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.