Resumen de Stark-Heegner points and p-adic L-functions

Daniele Casazza

• Let K|Q be a number field and let Z(K,s) be its associated complex L-function. The analytic class number formula relates special values of Z(K,s) with algebraic invariants of the field K itself. It admits a Galois equivariant refinement known as Stark conjectures.

  We have a very similar picture in the case of elliptic curves. Let E/Q be an elliptic curve and let L(E/Q,s) be its associated complex L-function. The conjecture of Birch and Swinnerton-Dyer relates the behaviour of L(E/Q,s) at s=1 to the structure of rational solutions of the equation defined by E. The equivariant Birch and Swinnerton-Dyer conjecture is obtained including in the picture the action of Galois groups.

  The elliptic Stark conjecture formulated by H. Darmon, A. Lauder and V. Rotger purposes a p-adic analogue of the equivariant Birch and Swinnerton-Dyer conjecture, under several assumption. In their paper, the authors formulate the conjecture and prove it in some cases of good reduction of E at p using Garrett-Hida method and performing a factorization of p-adic L-functions. In this dissertation we focus on the elliptic Stark conjecture and we show how it is possible to extend the result of Darmon, Lauder and Rotger.

  In the case of good reduction of E at p we can slightly extend the result using Hida-Rankin method. This method also gives us a better control of the constants appearing in the result, thus yielding an explicit formula which contains invariants associated with the elliptic curve. To achieve the proof we mimic the main result of Darmon, Lauder and Rotger in our setting and we make use of a p-adic Gross-Zagier formula which relates special values of the Bertolini-Darmon-Prasanna p-adic L-function to Heegner points.

  In a second moment we extend both our result and Darmon-Lauder-Rotger result to the case of multiplicative reduction of E at p. In this setting we cannot use Bertolini-Darmon-Prasanna p-adic L-function due to some technical reasons. To avoid the problem we consider Castella's two variables p-adic L-function. We use both Garrett-Hida method and Hida-Rankin method. In the two cases we obtain formulae which are similar to those of the good reduction setting.