TL;DR
This paper extends the Gross-Zagier and Zhang theorems to include specific CM points on higher-level Shimura curves, linking heights with derivatives of automorphic L-functions, with applications to Hida theory.
Contribution
It generalizes the Gross-Zagier and Zhang theorems to higher-level Shimura curves and explores implications for Hida theory.
Findings
Extended Gross-Zagier and Zhang theorems to higher-level Shimura curves.
Established connections between CM point heights and automorphic L-function derivatives.
Provided groundwork for applications in Hida families.
Abstract
The theorems of Gross-Zagier and Zhang relate the N\'eron-Tate heights of complex multiplication points on the modular curve X_0(N) (and on Shimura curve analogues) with the central derivatives of automorphic L-functions. We extend these results to include certain CM points on Shimura curves of higher level structure. These results are motivated by applications to Hida theory which are described in the companion article "Central derivatives of L-functions in Hida families."
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