Algebra and Number Theory Seminar: Rohit Kumar

Title: The Irregular Elliptic Stark Conjecture

Abstract:  In a joint work with Victor Rotger, we study the irregular elliptic Stark conjecture of Darmon, Rotger, and Lauder. The fixed data consists of a rational elliptic curve $E_f$ and 2-dimensional artin representations $\rho_g, \rho_h : Gal(H/\mathbb{Q}) \to GL_2(L)$ such that the selmer group $Hom_{G_{\mathbb{Q}}}(E(H), \rho_g \otimes \rho_h)$ is two-dimensional over L. When the eigencurve is etale at g (regular case), The conjectures construct a certain regulator from ``rational points'' in $Hom_{G_{\mathbb{Q}}}(E(H), \rho_g \otimes \rho_h)$ and relate to a certain p-adic iterated integral $\pi_g(e_{ord}(d^{-1} f^{[p]} \times h))$. We investigate the case when the eigencurve is not smooth at g (irregular case). In this case, first we connect this p-adic iterated integral to p-adic triple product L-functions. Then, we use this to prove instances of this conjecture when g = h are induced from cubic unramified characters of an imaginary quadratic field K.