Big Heegner points and special values of $L$-series

TL;DR

This paper relates higher weight specializations of big Heegner points on Shimura curves to higher weight analogues of theta elements, deriving some conjectures from recent results and advancing the understanding of special values of L-series.

Contribution

It establishes a connection between big Heegner points and theta elements in higher weights, extending Howard's work to a broader setting and deriving conjectures from recent results.

Findings

Relation between big Heegner points and theta elements in higher weights

Deduction of conjectures from Chida-Hsieh's recent results

Extension of Howard's construction to Shimura curves

Abstract

In \cite{LV}, Howard's construction of big Heegner points on modular curves was extended to general Shimura curves over the rationals. In this paper, we relate the higher weight specializations of the big Heegner points of \emph{loc.cit.} in the definite setting to certain higher weight analogues of the Bertolini-Darmon theta elements. As a consequence of this relation, some of the conjectures formulated in \cite{LV} are deduced from recent results of Chida-Hsieh.