Pure Maths Colloquium: Iwasawa Theory and p-adic measures.
20 January 2014 16:00 in CM221
L-functions are generalizations of the well-known Riemann zeta function. One can associate an L-function to a Dirichlet character, an elliptic curve or even a "motive". These functions, even though analytic in nature, encode important arithmetic information. Perhaps the most prominent example is the Birch and Swinnerton-Dyer Conjecture, which relates the L-function of an elliptic curve to its Mordell-Weil group.
On the other hand these L-functions have very interesting p-adic properties (for some chosen prime number p), as for example in the case of the Riemann zeta function the famous Kummer Congruences. These properties are today understood through the notion of (abelian) p-adic measures. It was Iwasawa, and later Mazur and Greenberg, formulating their Main Conjectures, that conjectured the arithmetic significance of these p-adic measures.
In this talk we will start by discussing the notion of (abelian) p-adic measures and their role in the Main Conjectures. Then we will move to a highly conjectural generalization of them to a non-abelian setting due to Coates, Fukaya, Kato, Sujatha and Venjakob, and discuss recent developments in this direction.
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