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The equivariant main conjecture via 1-motives, and applications

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This talk presents joint work with Cristian Popescu. We introduce ``abstract l-adic 1-motives”, which are a slight generalisation of 1-motives, as used by Deligne in order to prove the Brumer-Stark conjecture for function fields. Each such 1-motive M comes with an l-adic realisation Tl M. One obtains a 1-motive M = MS,T for every G-Galois extension K/k of global fields (and suitable auxiliary sets S, T, which are familiar from the theory of L-functions). Our first main result says that Tl M is cohomologically trivial over G. We then can show that its Fitting ideal is given by an equivariant p-adic L-series, very much in the style of other and earlier Equivariant Main Conjectures.

This seems to have many applications; we will discuss two of them. The first is geometric in nature. We show that the degree zero class number of the Fermat curve xl + yl = 1 over a finite field with q elements is either 1 or at least divisible by ll-2. The underlying idea is that cohomological triviality implies congruences between L-functions, and these congruences are arithmetically meaningful. Secondly, we explain an explicit construction of Tate sequences arising from our approach. Work in progress indicates that this leads to a proof of the Rubin-Stark conjecture under certain conditions.

This talk is part of the Number Theory Seminar series.

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