The Irregular Elliptic Stark Conjecture for CM Theta series

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 $\rho_g,\rho_h : \text{Gal}(H/{\mathbb{Q}}) \rightarrow GL_2(L)$ such that the selmer group $\text{Hom}_{G_{\mathbb{Q}}}(\rho_g \otimes \rho_h , E(H))$ is two-dimensional over $L$. When the eigencurve is etale at g (regular case), The conjectures construct a certain regulator from "rational points" in $\text{Hom}_{G_{\mathbb{Q}}}(\rho_g \otimes \rho_h , E(H))$ and relate to a certain $p$-adic iterated integral $\pi_g(e_{\text{ord}}(d^{-1}f^{[p]} \times h))$. 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 hecke characters of an imaginary quadratic field $K$.