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Li coefficients for automorphic L -functions

Jeffrey C. Lagarias (2007)

Annales de l’institut Fourier

Xian-Jin Li gave a criterion for the Riemann hypothesis in terms of the positivity of a set of coefficients λ n ( n = 1 , 2 , ...

Lifting the field of norms

Laurent Berger (2014)

Journal de l’École polytechnique — Mathématiques

Let K be a finite extension of Q p . The field of norms of a p -adic Lie extension K / K is a local field of characteristic p which comes equipped with an action of Gal ( K / K ) . When can we lift this action to characteristic 0 , along with a compatible Frobenius map? In this note, we formulate precisely this question, explain its relevance to the theory of ( ϕ , Γ ) -modules, and give a condition for the existence of certain types of lifts.

Local Class Field Theory via Lubin-Tate Theory

Teruyoshi Yoshida (2008)

Annales de la faculté des sciences de Toulouse Mathématiques

We give a self-contained exposition of local class field theory, via Lubin-Tate theory and the Hasse-Arf theorem, refining the arguments of Iwasawa [9].

Local ε 0 -characters in torsion rings

Seidai Yasuda (2007)

Journal de Théorie des Nombres de Bordeaux

Let p be a rational prime and K a complete discrete valuation field with residue field k of positive characteristic p . When k is finite, generalizing the theory of Deligne [1], we construct in [10] and [11] a theory of local ε 0 -constants for representations, over a complete local ring with an algebraically closed residue field of characteristic p , of the Weil group W K of K . In this paper, we generalize the results in [10] and [11] to the case where k is an arbitrary perfect field.

Locally analytic vectors of unitary principal series of  GL 2 ( p )

Ruochuan Liu, Bingyong Xie, Yuancao Zhang (2012)

Annales scientifiques de l'École Normale Supérieure

The p -adic local Langlands correspondence for  GL 2 ( p ) attaches to any 2 -dimensional irreducible p -adic representation V of  G p an admissible unitary representation Π ( V ) of  GL 2 ( p ) . The unitary principal series of  GL 2 ( p ) are those Π ( V ) corresponding to trianguline representations. In this article, for  p > 2 , using the machinery of Colmez, we determine the space of locally analytic vectors Π ( V ) an for all non-exceptional unitary principal series Π ( V ) of  GL 2 ( p ) by proving a conjecture of Emerton.

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