p-Adic distribution of CM points and Hecke orbits. II: Linnik equidistribution on the supersingular locus

TL;DR

This paper investigates the distribution of CM points on the p-adic moduli space of elliptic curves, revealing infinitely many measures and connecting the problem to a p-adic Linnik distribution, with implications for Hecke orbits.

Contribution

It identifies all measures describing the asymptotic distribution of CM points in the p-adic setting and links the problem to a p-adic Linnik distribution, extending classical equidistribution results.

Findings

Classifies all asymptotic measures for CM points in the p-adic context.

Establishes a p-adic Linnik problem and solves it using modular form bounds.

Determines accumulation measures for arbitrary Hecke orbits.

Abstract

For a prime number $p$, we study the asymptotic distribution of CM points on the moduli space of elliptic curves over $\mathbb{C}_p$. In stark contrast to the complex case, in the $p$-adic setting there are infinitely many different measures describing the asymptotic distribution of CM points. In this paper we identify all of these measures. A key insight is to translate this problem into a $p$-adic version of Linnik's classical problem on the asymptotic distribution of integer points on spheres. To do this translation, we use the close relationship between the deformation theories of elliptic curves and formal modules and then apply results of Gross and Hopkins. We solve this $p$-adic Linnik problem using a deviation estimate extracted from the bounds for the Fourier coefficients of cuspidal modular forms of Deligne, Iwaniec and Duke. We also identify all accumulation measures of an arbitrary Hecke orbit.