Minikurs: Beilinson-Kato elements and the p-adic BSD conjecture of Mazur-Tate-Teitelbaum.

In order to formulate a p-adic Birch and Swinnerton conjecture (BSD for short) for an elliptic curve E, Mazur, Tate and Teitelbaum (MTT) constructed a p-adic L-function attached to E. To understand its compatibility with the usual BSD, one needs to compare the order of vanishing of the p-adic L-function at s=1 to that of the Hasse-Weil L-function (where the latter is called the analytic rank of E). When E has split multiplicative reduction mod p, MTT observed that the p-adic L-function always vanishes at s=1 and they conjectured that its order of zero is exactly one more than the analytic rank of E. In 1992, Greenberg and Stevens proved this conjecture when the analytic rank is zero.

In the first two lectures of this talk, I will explain a proof of the MTT conjecture when the analytic rank is one. The main ingredients for the proof are the Beilinson-Kato elements in the K₂ of modular curves and a Gross-Zagier-style formula we prove for the p-adic height of the Beilinson-Kato elements. In the last part of the talk, I will discuss an extension (in a joint work with D. Benois) of this result to the case of a modular form f of weight greater than 2. The main difficulty in this case lies in the fact the Galois representation V attached to f by Deligne, in the presence of "extra zeros", is no longer p-ordinary. This difficulty is circumvented relying on the fact that the (local Galois representation) V admits a triangulation over the Robba ring (thence it is *ordinary* in the level of the associated ($\varphi$ - Γ)-modules).