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SUMMARY:The equivariant main conjecture via 1-motives\, and applications -
  Cornelius Greither (Munich)
DTSTART:20111129T143000Z
DTEND:20111129T153000Z
UID:TALK33024@talks.cam.ac.uk
CONTACT:Tom Fisher
DESCRIPTION:This talk presents joint work with\nCristian Popescu. We intro
 duce ``abstract _l_-adic 1-motives"\, which are a slight\ngeneralisation o
 f 1-motives\, as used by Deligne in order to prove\nthe Brumer-Stark conje
 cture for function fields. Each such\n1-motive _M_ comes with an _l_-adic 
 realisation\n_T<sub>l</sub> M_.  One obtains a 1-motive \n_M = M<sub>S\,T<
 /sub>_\nfor every _G_-Galois extension _K/k_ of global fields (and suitabl
 e auxiliary\nsets _S_\, _T_\, which are familiar from the theory of _L_-fu
 nctions).\nOur first main result says that _T<sub>l</sub> M_ is cohomologi
 cally\ntrivial over _G_.\nWe then can show that its Fitting ideal is given
  by an\nequivariant _p_-adic _L_-series\, very much in the style of other\
 nand earlier Equivariant Main Conjectures. \n\nThis seems to have many app
 lications\; we will discuss two of them.\nThe first is geometric in nature
 . We show that the degree zero class number\nof the Fermat curve _x^l^ + y
 ^l^ = 1_ over a finite field\nwith _q_ elements is either 1 or at least di
 visible\nby _l^l-2^_. The underlying idea is that cohomological\ntrivialit
 y implies congruences between _L_-functions\, and these\ncongruences are a
 rithmetically meaningful. Secondly\, we explain \nan explicit construction
  of Tate sequences arising from our\napproach. Work in progress indicates 
 that this leads to a proof\nof the Rubin-Stark conjecture under certain co
 nditions.
LOCATION:MR13
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