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

Details Non-abelian $p$-adic $L$-functions and Eisenstein series of unitary groups; the CM method Non-abelian $p$-adic $L$-functions and Eisenstein series of unitary groups; the CM method

2011-07-07

arxiv.org/abs/1107.1377v2

Mathematics

Number Theory

Corrections in sections 3 and 4, added references

Scientific paper

In this work we prove 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.

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