Interpolation of generalized Heegner classes along quaternionic Coleman families
Eduardo Rocha Walchek
TL;DR
This paper develops a p-adic interpolation framework for generalized Heegner cycles in the indefinite quaternionic setting, constructing big generalized Heegner classes along Coleman families of quaternionic modular forms. By combining Shimura-curve geometry, Lieberman’s trick, Ancona’s motivic framework, and Loeffler–Zerbes overconvergent projector, it builds big Galois representations and proves a precise specialization formula linking big classes to classical Heegner data via the U_p eigenvalue a_p(\mathcal{F}_k). The work extends the Birchtoli–Darmon–Prasanna paradigm to finite slope families and establishes Euler-system–style norm relations across p-power levels, with a vision toward an explicit reciprocity law for families of L-functions. The constructions yield big generalized Heegner classes defined over K, providing a robust tool for studying Bloch–Kato phenomena in the quaternionic automorphic setting and their p-adic L-function interpolations. The methods have potential implications for understanding the arithmetic of quaternionic modular forms through-p-adic analytic families and their associated Euler systems.
Abstract
We construct big generalized Heegner classes by interpolating $p$-adically the generalized Heegner classes associated to quaternionic modular forms along a Coleman (finite slope) family, following the approach introduced by Jetchev--Loeffler--Zerbes.