The Irregular Elliptic Stark Conjecture in the Cubic Unramified Case

Abstract

The entirety of this thesis is part of a joint work with Victor Rotger. It investigatesthe Irregular Elliptic Stark Conjecture, a refinement of the conjectures of Darmon–Rotger– Lauder relating p-adic iterated integrals attached to modular forms of weights (2, 1, 1) to arithmetic regulators valued in adjoint Galois representations. In the irregular setting, the ordinary overconvergent space at weight one is non-semisimple and four-dimensional, and the associated local Galois representation admits no canonical Frobenius-stable line. As a result, the conjecture requires a regulator constructed from the exterior square of the Selmer group and valued in Ad0ρg, together with a comparison map to generalized overconvergent eigenforms. We prove new cases of the conjecture in two directions. First, when g has complex multiplication by an imaginary quadratic field in which p splits, we relate the conjecture to the leading terms of triple product p-adic L-functions. Second, when g = h are induced from a cubic unramified character of an imaginary quadratic field, we show under suitable rank conditions that the modular form π_{g_α}(e_{ord}(d^{-1} f^[p] x h))is a scalar multiple of the generalized eigenform g^b_1. These results provide theoretical evidence for the Irregular Elliptic Stark Conjecture and clarify the role of generalized Hecke eigenspaces and adjoint regulators in the weight one setting.