Elliptic curves over a finite field with a specified subgroup and the trace formula
Tadahiro Katsuoka
TL;DR
This work extends the Ihara–Birch and Kaplan–Petrow framework by computing moments of traces of elliptic curves over a finite field with a subgroup isomorphic to a group $A$ when the order of $A$ is divisible by the characteristic $p$. The authors develop and apply an Eichler–Selberg trace formula for congruence subgroups $\Gamma(p^rN,M)$, express the elliptic term in terms of Hurwitz–Kronecker class numbers via $H_{n_1,n_2}(t,q,d)$, and connect these with moments of traces through the Chebyshev polynomials $U_{k-2}$. The main result provides an explicit formula for $\mathbb{E}_q(U_{k-2}(t_E,q)\Phi_A)$ in terms of traces $T_{p^r n_1,n_2\nu}(q,1)$ when $q\equiv1 \pmod{n_2}$, and shows vanishing otherwise; the case with $p|t$ is ruled out by pairing considerations. Overall, the paper bridges arithmetic of imaginary quadratic orders, modular-trace theory, and elliptic-curve statistics under mandated finite-subgroup constraints, extending previous coprime-to-$p$ results to $p$-divisible settings with explicit, usable formulas.
Abstract
Ihara and Birch obtained a formula expressing the sum of powers of the traces of elliptic curves over a fixed finite field of characteristic $p$ in terms of the traces of Hecke operators for $\mathrm{SL}_2(\mathbb{Z})$. Generalizing the theorems of Ihara and Birch, for a finite abelian group $A$ whose order is coprime to $p$, Kaplan and Petrow gave a formula for statistical description of powers of the traces of elliptic curves which contain subgroups isomorphic to $A$. In this paper, we generalize the theorems of Ihara, Birch, and Kaplan--Petrow to the case where the order of $A$ is divisible by $p$.