Generalized Kato classes and exceptional zero conjectures

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.