Generalized Kato classes and exceptional zero conjectures

Abstract

The primary objective of this paper is the study of different instances of the elliptic Stark conjectures of Darmon, Lauder and Rotger, in a situation where the elliptic curve attached to the modular form $f$ has split multiplicative reduction at $p$ and the arithmetic phenomena are specially rich. For that purpose, we resort to the principle of improved $p$-adic $L$-functions and study their $\mathcal L$-invariants. We further interpret these results in terms of derived cohomology classes coming from the setting of diagonal cycles, showing that the same $\mathcal L$-invariant which arises in the theory of $p$-adic $L$-functions also governs the arithmetic of Euler systems. Thus, we can reduce, in the split multiplicative situation, the conjecture of Darmon, Lauder and Rotger to a more familiar statement about higher order derivatives of a triple product $p$-adic $L$-function at a point lying inside the region of classical interpolation, in the realm of the more well-known exceptional zero conjectures.