STNB2018 (32nd edition)

Towards the rationality of Darmon-Dasgupta units

Presenters

The multiple analogies between units in number fields and points in elliptic curves are also present in which concerns $p$-adic conjectural constructions in the real quadratic case: Stark-Heegner points are natural substitutes of Heegner points, while Darmon-Dasgupta units play the role of elliptic units. Some years ago, Park gave some evidence towards the rationality of these units mimicking the study that Bertolini and Darmon had made for Stark-Heegner points (following a beautiful argument with reminiscences of Kronecker's limit formula). In a joint work with Victor Rotger, we plan to study the $\psi$-isotypical conjecture of Darmon and Dasgupta, in the less well understood case in which $\psi$ is not quadratic. For that, we will make use of the Euler system of Beilinson-Flach elements, and of the reciprocity laws relating it with $p$-adic $L$-functions. In this talk we will give an overview of the general theory and explain the main ingredients of our approach.