Generalized triple product p-adic L-functions, diagonal classes and rational points on elliptic curves

Abstract

In the first part of this thesis (chapters 1 to 5, we generalize and simplify the constructions of Darmon-Rotger and Hsieh of an unbalanced triple product p-adic L-function Lpf(F,G,H) attached to a triple (F,G,H) of p-adic families of modular forms, allowing more flexibility for the choice of G and H. Assuming that G and H are families of theta series of infinite p-slope, we prove a factorization of (an improvement of) Lpf(F,G,H) in terms of two anticyclotomic p-adic L-functions. As a corollary, when F specializes in weight 2 to the newform attached to an elliptic curve E over Q with multiplicative reduction at p, we relate Heegner points on E to p-adic partial derivatives of Lpf(F,G,H) evaluated at the critical triple of weights (2,1,1). In the second part, we generalize the p-adic explicit reciprocity laws for balanced diagonal classes by Darmon-Rotger and Bertolini-Seveso-Venerucci to the case of geometric balanced triples (f,g,h) of modular eigenforms where f is ordinary at p, while g and h are supercuspidal at p. This allows to obtain a geometric interpretation of the specializations of the p-adic L-function Lpf(F,G,H) in the so-called geometric balanced region, when G and H are families of theta series of infinite p-slope.