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Thanasis Bouganis
Non-abelian $p$-adic $L$-functions and Eisenstein series of unitary groups – The CM method
($L$-fonctions $p$-adiques non-abéliennes et série d’Eisenstein pour les groupes unitaires – La méthode CM)
Annales de l'institut Fourier, 64 no. 2 (2014), p. 793-891, doi: 10.5802/aif.2866
Article PDF | Reviews MR 3330923 | Zbl 06387293
Class. Math.: 11R23, 11F55, 11F67, 11M36
Keywords: ($p$-adic) $L$-functions, Eisenstein Series, Unitary Groups, Congruences

Résumé - Abstract

In this work we prove various cases of the so-called “torsion congruences” between abelian $p$-adic $L$-functions that are related to automorphic representations of definite unitary groups. These congruences play a central role in the non-commutative Iwasawa theory as it became clear in the works of Kakde, Ritter and Weiss on the non-abelian Main Conjecture for the Tate motive. We tackle these congruences for a general definite unitary group of $n$ variables and we obtain more explicit results in the special cases of $n=1$ and $n=2$. In both of these cases we also explain their implications for some particular “motives”, as for example elliptic curves with complex multiplication. Finally we also discuss a new kind of congruences, which we call “average torsion congruences”

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