an:06296345
Zbl 1356.11040
Seveso, Marco Adamo
Heegner cycles and derivatives of \(p\)-adic \(L\)-functions
EN
J. Reine Angew. Math. 686, 111-148 (2014).
00332676
2014
j
11G40 11F80 11G18 11F85 11F67
Heegner cycle; \(p\)-adic \(L\)-function; Shimura curve; modular form; imaginary quadratic field
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.