
BeilinsonKato elements and the padic BSD conjecture of MazurTateTeitelbaum [pdf]
Prof. E. GroßeKlönne / Prof. T. Schmidt Date: 11th December 2013, 18th December 2013 and 22nd January 2014
Location: Humboldt University, Institute of Mathematics
Rudower Chaussee 25, 12489 BerlinAdlershof, Germany (room 2.009, time: 1113 Uhr).
Prof. Kazim Buyukboduk, Koc Univ.Istanbul
Abstract:: In order to formulate a padic Birch and Swinnerton conjecture (BSD for short) for an elliptic curve
E, Mazur, Tate and Teitelbaum (MTT) constructed a padic Lfunction attached to E. To understand its compatibility with the usual BSD, one needs to compare the order of vanishing of the padic Lfunction at s=1 to that of the HasseWeil Lfunction (where the latter is called the analytic rank of E).
When E has split multiplicative reduction mod p, MTT observed that the padic Lfunction 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 BeilinsonKato elements in the K 2 of modular curves and a GrossZagierstyle formula we prove for the padic height of the BeilinsonKato 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 pordinary. 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 $(φГ) $modules
)
11.12.2013

Prof. Kazim Buyukboduk, Koc Univ. Istanbul

Minikurs BeilisonKato elements and the padic BSD conjecture of MazurTateTeitelbaum *)
Lecture 1 Basics: Elliptic curves, BSD, padic BSD and Iwasawa theory, Kato's Euler system and appl icartions.

18.12.2013 
Prof. Kazim Buyukboduk, Koc Univ. Istanbul 
Lecture 2 Nekovar's Selmer complexes and padic heights, a (higher) padic GrossZagier formula and the conjecture of MazurTateTeitelbaum. 
22.01.2014 
Prof. Kazim Buyukboduk, Koc Univ. Istanbul 
Lecture 3  Part 1 Galois representations attached to modular forms, triangulations and the MTT conjecture for modular forms of higher weight (joint work with D. Benois). 
29.01.2014 
Prof. Kazim Buyukboduk, Koc Univ. Istanbul 
Lecture 3  Part 2 Galois representations attached to modular forms, triangulations and the MTT conjecture for modular forms of higher weight (joint work with D. Benois). 
*) Minikurs im Rahmen des Gastaufenthalts von Prof. Buyukboduk in der FG.
Audience:
We warmly invite anyone interested in this area to attend.

