Derived Beilinson–Flach elements and the arithmetic of the adjoint of a modular form

Abstract

Kings, Lei, Loeffler and Zerbes constructed in [LLZ], [KLZ1] a three-variable Euler system $\kappa (\text{g},\text{h})$ of Beilinson–Flach elements associated to a pair of Hida families $(\text{g},\text{h})$ and exploited it to obtain applications to the arithmetic of elliptic curves, extending the earlier work [BDR]. The aim of this article is to show that this Euler system also encodes arithmetic information concerning the group of units of the associated number fields. The setting becomes specially novel and intriguing when $\text{g}$ and $\text{h}$ specialize in weight 1 to $p$-stabilizations of eigenforms such that one is dual to the other. We encounter an exceptional zero phenomenon which forces the specialization of $\kappa (\text{g},\text{h})$ to vanish and we are led to study the derivative of this class. The main result we obtain is the proof of the main conjecture of [DLR4] on iterated integrals and the main conjecture of [DR1] for Beilinson–Flach elements in the adjoint setting. The main point of this paper is that the methods of [DLR1], [DLR4] and [CH], where the above conjectures are proved when the weight 1 eigenforms have CM, do not apply to our setting and new ideas are required. In the previous works, a crucial ingredient is a factorization of $p$-adic $L$-functions, which in our scenario is not available due to the lack of critical points. Instead we resort to the principle of improved Euler systems and $p$-adic $L$-functions to reduce our problems to questions which can be resolved using Galois deformation theory. We expect this approach may be adapted to prove other cases of the Elliptic Stark Conjecture and of its generalizations that appear in the literature.