Generating functions for power moments of elliptic curves over Fp

Seminal works by Birch and Ihara gave formulas for the mth power moments of the traces of Frobenius endomorphisms of elliptic curves over F-p for primes p >= 5. Recent works by Kaplan and Petrow generalized these results to the setting of elliptic curves that contain a subgroup isomorphic to a 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 mth power moments. In particular, we find that Z(p)(A; t) = (Z) over cap (p)(A; t)/Pi(a is an element of Frobp(A)) (1 - at), where Frob(p)(A) := {a : -2 root p <= a <= 2 root p and a equivalent to p + 1 (mod vertical bar A vertical bar)}, and (Z) over cap (p)(A; t) is an easily computed polynomial that is determined by the first [inverted right perpendicular 2 left perpendicular 2 root p right perpendicular inverted left perpendicular /vertical bar A vertical bar] 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. (C) 2019 Elsevier Inc. All rights reserved.