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

TL;DR

This paper studies the arithmetic of Beilinson-Flach elements at special weights, proving conjectures related to derivatives of Euler system classes and introducing new methods beyond previous CM-based approaches.

Contribution

It proves conjectures of Darmon, Lauder, and Rotger for Beilinson-Flach elements in the adjoint setting without relying on CM assumptions, using novel deformation techniques.

Findings

Proved conjectures on iterated integrals and Beilinson-Flach elements.

Developed new methods for non-CM weight 1 eigenforms.

Connected Euler systems, $p$-adic $L$-functions, and Galois deformation theory.

Abstract

Kings, Lei, Loeffler and Zerbes constructed a three-variable Euler system $\kappa({\bf g},{\bf h})$ of Beilinson-Flach elements associated to a pair of Hida families $({\bf g},{\bf h})$ and exploited it to obtain applications to the arithmetic of elliptic curves by specializing the Euler system to points of weights $(2,1,1)$. The aim of this article is showing that this Euler system also encodes arithmetic information at points of weights $(1,1,0)$, concerning the group of units of the associated number fields. The setting becomes specially novel and intriguing when ${\bf g}$ and ${\bf h}$ specialize in weight $1$ to $p$-stabilizations of eigenforms such that one is dual of another. We encounter an exceptional zero phenomenon which forces the specialization of $\kappa({\bf g}, {\bf h})$ at $(1,1,0)$ to vanish and we are led to study the derivative of this class. The main result we obtain is the proof of a conjecture of Darmon, Lauder and Rotger on iterated integrals and another conjecture of Darmon and Rotger for Beilinson-Flach elements in the adjoint setting. The main point of this paper is that the methods of previous works, where the above conjectures are proved when the weight $1$ eigenforms have CM, do not apply to our setting and new ideas are required. Here, a factorization of $p$-adic $L$-functions 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 are appearing in the literature.