RECIPROCITY LAWS FOR BALANCED DIAGONAL CLASSES

This article constructs a 3-variable balanced diagonal class kappa(f, g, h) in the cohomology of the Galois representation associated to a self-dual triple (f, g, h) of p-adic Hida families. Its first main result (Theorem A of Section 1.1) establishes an explicit reciprocity law relating kappa(f, g, h) to the unbalanced Garrett-Rankin p-adic L-function attached to (f, g, h). The class kappa(f, g, h) arises from the p-adic interpolation of diagonal classes in the Bloch-Kato Selmer groups of the specializations of (f, g, h) at balanced triples of classical weights. As a consequence, the value of kappa(f, g, h) at a specialization (f, g, h) of (f, g, h) at an unbalanced triple of classical weights is a p-adic limit of crystalline classes. Our second main result (Theorem B of Section 1.2) shows that the obstruction to the crystallinity of an appropriate derivative of kappa(f, g, h) at (f, g, h) is encoded in the central critical value of the complex L-function of f circle times g circle times h.