Strengthenings of Mazur's Conjecture for Higher Heegner Points

TL;DR

The article strengthens Mazur's conjecture on higher Heegner points by establishing both vertical non-torsion for large conductors $n$ and horizontal non-torsion for large primes $p$, uniformly across modular and Shimura curves. It introduces a commensurability-based framework to understand closures of diagonal subgroups in self-products, couples this with Hecke operator equidistribution and joining-measure classifications, and then transfers the results to Shimura varieties of Hodge type via Kisin's integral models and Galois–Hecke orbit relations. A key innovation is showing that the Galois orbits of Heegner points sit inside Hecke orbits and that their relative size can be quantified by $d(n)$, enabling quantitative distribution statements of supersingular reductions. This yields explicit non-torsion conclusions for traces of higher Heegner points and extends Mazur-type conjectures to a broad Shimura-curve setting, with potential implications for Iwasawa theory and the non-vanishing of Rankin–Selberg L-values. The work also suggests pathways to generalize to Shimura varieties of Hodge type over totally real fields and to broaden the ergodic-joining framework beyond Ratner-type techniques.

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.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2: Corollary \ref{["vertical Mazur's conjecture, stronger version"]}
• Theorem 1.3: Corollary \ref{["horizontal Mazur's conjecture"]}, Horizontal Mazur's conjecture
• Theorem 1.4: Corollary \ref{["generalized Mazur's conjecture"]}
• Remark 1.5
• Theorem 1.6: Theorem \ref{['S^circ']}
• Proposition 2.1
• proof
• Lemma 2.2
• proof