Equidistribution of Heegner Points and Ternary Quadratic Forms

TL;DR

This paper establishes new equidistribution results for Galois orbits of Heegner points at inert primes, extending previous theorems and providing effective surjectivity for reduction maps in the context of quadratic forms and elliptic curves.

Contribution

It generalizes existing equidistribution theorems by allowing both the fundamental discriminant and conductor to grow, and introduces an effective surjectivity result for reduction maps at inert primes.

Findings

Proves equidistribution of Heegner points with growing discriminant and conductor.

Establishes effective surjectivity of reduction maps to supersingular points.

Utilizes quadratic forms and distribution relations techniques.

Abstract

We prove new equidistribution results for Galois orbits of Heegner points with respect to reduction maps at inert primes. The arguments are based on two different techniques: primitive representations of integers by quadratic forms and distribution relations for Heegner points. Our results generalize one of the equidistribution theorems established by Cornut and Vatsal in the sense that we allow both the fundamental discriminant and the conductor to grow. Moreover, for fixed fundamental discriminant and variable conductor, we deduce an effective surjectivity theorem for the reduction map from Heegner points to supersingular points at a fixed inert prime. Our results are applicable to the setting considered by Kolyvagin in the construction of the Heegner points Euler system.