Heegner points and Jochnowitz congruences on Shimura curves

TL;DR

This paper establishes a congruence relation connecting derivatives and special values of L-series for elliptic curves and quaternionic forms, extending Gross-Zagier and Zhang's formulas to Shimura curves.

Contribution

It proves a Jochnowitz congruence linking Heegner points and L-series values on Shimura curves, extending prior results to quaternionic settings.

Findings

Proves a congruence between algebraic parts of L-series derivatives and values.

Extends Gross-Zagier and Zhang formulas to Shimura curves.

Establishes a relation between different types of L-series formulas.

Abstract

Given a rational elliptic curve E, a suitable imaginary quadratic field K and a quaternionic Hecke eigenform g of weight 2 obtained from E by level raising such that the sign in the functional equation for L_K(E,s) (respectively, L_K(g,1)) is -1 (respectively, +1), we prove a ``Jochnowitz congruence'' between the algebraic part of L'_K(E,1) (expressed in terms of Heegner points on Shimura curves) and the algebraic part of L_K(g,1). This establishes a relation between Zhang's formula of Gross-Zagier type for central derivatives of L-series and his formula of Gross type for special values. Our results extend to the context of Shimura curves attached to division quaternion algebras previous results of Bertolini and Darmon for Heegner points on classical modular curves.