Non-abelian $p$-adic $L$-functions and Eisenstein series of unitary groups – The CM method

Bouganis, Thanasis

Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 793-891.

Dans cet article, nous démontrons divers cas particuliers de “congruences de torsion” entre les $L$-fonctions $p$-adiques abéliennes liées aux représentations automorphes de groupes unitaires définis. Ces congruences jouent un rôle central dans la théorie d’ Iwasawa non-commutative, ce qui a été mis en évidence par les résultats de Kakde, Ritter et Weiss sur la Conjecture Principale non-abélienne pour le motif de Tate. Nous nous attaquons à ces congruences pour un groupe unitaire défini général en $n$ variables, et obtenons des résultats plus explicites dans les cas $n=1$ et $n=2$. Dans ces deux cas, nous expliquons aussi leur conséquences pour certains “motifs” particuliers, comme par exemple, les courbes elliptiques munies d’une multiplication complexe. Finalement, nous discutons d’un nouveau type de congruences que nous nommons “congruences de torsion modérées”.

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”

Classification : 11R23, 11F55, 11F67, 11M36
Keywords: ($p$-adic) $L$-functions, Eisenstein Series, Unitary Groups, Congruences
Mot clés : $L$-fonctions $p$-adiques, séries d’Eisenstein, Groupes unitaires, congruences

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