Strengthenings of Mazur's Conjecture for Higher Heegner Points

TL;DR

This paper advances Mazur's conjecture by providing quantitative results on higher Heegner points, using ergodic theory and orbit analysis to overcome previous limitations related to level structures.

Contribution

It introduces a novel approach that avoids restrictive hypotheses by analyzing Galois and Hecke orbits, leading to strengthened non-torsion results for higher Heegner points.

Findings

Proves a vertical version for large powers n.

Establishes a horizontal version for large primes p.

Reduces the problem to ergodic and equidistribution results.

Abstract

We establish quantitative strengthenings of Mazur's conjecture regarding the non-torsion property of higher Heegner points on modular and Shimura curves, confirming both a vertical version for sufficiently large powers $n$ and a horizontal version for primes $p \gg 1$. Distinct from previous strategies by Cornut and Vatsal that relied on Ratner's theorems on unipotent flows and required restrictive hypotheses on level structures, our approach circumvents these constraints by exploiting the interplay between Galois orbits and Hecke orbits. By quantifying the relative size of these orbits, we reduce the problem to ergodic results concerning the equidistribution of Hecke operators and the classification of joining measures. This method allows for the analysis of simultaneous supersingular reductions without requiring the surjectivity of reduction maps.