  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.
