Heegner cycles and derivatives of \(p\)-adic \(L\)-functions. (English) Zbl 1356.11040

Seveso, Marco Adamo

J. Reine Angew. Math. 686, 111-148 (2014).

Summary: Let \(f\) be an even weight \(k\geq2\) modular form on a \(p\)-adically uniformizable Shimura curve for a suitable \(\Gamma_0\)-type level structure. Let \(K/\mathbb Q\) be an imaginary quadratic field, satisfying Heegner conditions assuring that the sign appearing in the functional equation of the complex \(L\)-function of \(f/K\) is negative. We may attach to \(f\), or rather a deformation of it, a \(p\)-adic \(L\)-function of the weight variable \(\kappa\), also depending on \(K\). Our main result is a formula relating the derivative of this \(p\)-adic \(L\)-function at \(\kappa=k\) to the Abel-Jacobi images of so-called Heegner cycles.

MSC:

11G40	\(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11F80	Galois representations
11G18	Arithmetic aspects of modular and Shimura varieties
11F85	\(p\)-adic theory, local fields
11F67	Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols

Keywords:

Heegner cycle; \(p\)-adic \(L\)-function; Shimura curve; modular form; imaginary quadratic field