Quaternion algebras, Heegner points and the arithmetic of Hida families

TL;DR

This paper extends Howard's results on Heegner points to quaternionic settings, constructing big Heegner points and classes, and formulates Iwasawa main conjectures for both definite and indefinite quaternion algebras.

Contribution

It introduces a unified approach to Heegner points in quaternionic settings using optimal embeddings, and develops new conjectures on Selmer groups and L-functions.

Findings

Construction of big Heegner points and classes on Shimura curves.

Formulation of two-variable Iwasawa main conjectures for quaternion algebras.

Proposed refined Greenberg conjectures on L-function vanishing.

Abstract

Given a newform f, we extend Howard's results on the variation of Heegner points in the Hida family of f to a general quaternionic setting. More precisely, we build big Heegner points and big Heegner classes in terms of compatible families of Heegner points on towers of Shimura curves. The novelty of our approach, which systematically exploits the theory of optimal embeddings, consists in treating both the case of definite quaternion algebras and the case of indefinite quaternion algebras in a uniform way. We prove results on the size of Nekov\'a\v{r}'s extended Selmer groups attached to suitable big Galois representations and we formulate two-variable Iwasawa main conjectures both in the definite case and in the indefinite case. Moreover, in the definite case we propose refined conjectures \`a la Greenberg on the vanishing at the critical points of (twists of) the L-functions of the modular forms in the Hida family of f living on the same branch as f.