Displaying similar documents to “The Heyde theorem on a-adic solenoids”

On the de Rham and p -adic realizations of the elliptic polylogarithm for CM elliptic curves

Kenichi Bannai, Shinichi Kobayashi, Takeshi Tsuji (2010)

Annales scientifiques de l'École Normale Supérieure

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In this paper, we give an explicit description of the de Rham and p -adic polylogarithms for elliptic curves using the Kronecker theta function. In particular, consider an elliptic curve E defined over an imaginary quadratic field 𝕂 with complex multiplication by the full ring of integers 𝒪 𝕂 of 𝕂 . Note that our condition implies that 𝕂 has class number one. Assume in addition that E has good reduction above a prime p 5 unramified in 𝒪 𝕂 . In this case, we prove that the specializations of the...

Iwasawa theory for symmetric powers of CM modular forms at non-ordinary primes

Robert Harron, Antonio Lei (2014)

Journal de Théorie des Nombres de Bordeaux

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Let f be a cuspidal newform with complex multiplication (CM) and let p be an odd prime at which f is non-ordinary. We construct admissible p -adic L -functions for the symmetric powers of f , thus verifying conjectures of Dabrowski and Panchishkin in this special case. We combine this with recent work of Benois to prove the trivial zero conjecture in this setting. We also construct “mixed” plus and minus p -adic L -functions and prove an analogue of Pollack’s decomposition of the admissible...

Base change for Bernstein centers of depth zero principal series blocks

Thomas J. Haines (2012)

Annales scientifiques de l'École Normale Supérieure

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Let  G be an unramified group over a p -adic field. This article introduces a base change homomorphism for Bernstein centers of depth-zero principal series blocks for  G and proves the corresponding base change fundamental lemma. This result is used in the approach to Shimura varieties with Γ 1 ( p ) -level structure initiated by M. Rapoport and the author in [15].

Lifting the field of norms

Laurent Berger (2014)

Journal de l’École polytechnique — Mathématiques

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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.

A note on p-adic valuations of Schenker sums

Piotr Miska (2015)

Colloquium Mathematicae

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A prime number p is called a Schenker prime if there exists n ∈ ℕ₊ such that p∤n and p|aₙ, where a = j = 0 n ( n ! / j ! ) n j is a so-called Schenker sum. T. Amdeberhan, D. Callan and V. Moll formulated two conjectures concerning p-adic valuations of aₙ when p is a Schenker prime. In particular, they conjectured that for each k ∈ ℕ₊ there exists a unique positive integer n k < 5 k such that v ( a m · 5 k + n k ) k for each nonnegative integer m. We prove that for every k ∈ ℕ₊ the inequality v₅(aₙ) ≥ k has exactly one solution modulo 5 k . This...

An explicit computation of p -stabilized vectors

Michitaka MIYAUCHI, Takuya YAMAUCHI (2014)

Journal de Théorie des Nombres de Bordeaux

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In this paper, we give a concrete method to compute p -stabilized vectors in the space of parahori-fixed vectors for connected reductive groups over p -adic fields. An application to the global setting is also discussed. In particular, we give an explicit p -stabilized form of a Saito-Kurokawa lift.

An alternative description of the Drinfeld p -adic half-plane

Stephen Kudla, Michael Rapoport (2014)

Annales de l’institut Fourier

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We show that the Deligne formal model of the Drinfeld p -adic half-plane relative to a local field F represents a moduli problem of polarized O F -modules with an action of the ring of integers in a quadratic extension E of F . The proof proceeds by establishing a comparison isomorphism with the Drinfeld moduli problem. This isomorphism reflects the accidental isomorphism of SL 2 ( F ) and SU ( C ) ( F ) for a two-dimensional split hermitian space C for E / F .

Dual Blobs and Plancherel Formulas

Ju-Lee Kim (2004)

Bulletin de la Société Mathématique de France

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Let k be a p -adic field. Let G be the group of k -rational points of a connected reductive group 𝖦 defined over k , and let 𝔤 be its Lie algebra. Under certain hypotheses on 𝖦 and k , wethe tempered dual G ^ of G via the Plancherel formula on 𝔤 , using some character expansions. This involves matching spectral decomposition factors of the Plancherel formulas on 𝔤 and G . As a consequence, we prove that any tempered representation contains a good minimal 𝖪 -type; we extend this result to irreducible...

-invariants and Darmon cycles attached to modular forms

Victor Rotger, Marco Adamo Seveso (2012)

Journal of the European Mathematical Society

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Let f be a modular eigenform of even weight k 2 and new at a prime p dividing exactly the level with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to f a monodromy module D f F M and an -invariant f F M . The first goal of this paper is building a suitable p -adic integration theory that allows us to construct a new monodromy module D f and -invariant f , in the spirit of Darmon. The two monodromy modules are isomorphic, and in particular the two -invariants...

On twisted group algebras of OTP representation type over the ring of p-adic integers

Leonid F. Barannyk, Dariusz Klein (2016)

Colloquium Mathematicae

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Let ̂ p be the ring of p-adic integers, U ( ̂ p ) the unit group of ̂ p and G = G p × B a finite group, where G p is a p-group and B is a p’-group. Denote by ̂ p λ G the twisted group algebra of G over ̂ p with a 2-cocycle λ Z ² ( G , U ( ̂ p ) ) . We give necessary and sufficient conditions for ̂ p λ G to be of OTP representation type, in the sense that every indecomposable ̂ p λ G -module is isomorphic to the outer tensor product V W of an indecomposable ̂ p λ G p -module V and an irreducible ̂ p λ B -module W.

Hodge-Tate and de Rham representations in the imperfect residue field case

Kazuma Morita (2010)

Annales scientifiques de l'École Normale Supérieure

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Let K be a p -adic local field with residue field k such that [ k : k p ] = p e &lt; + and V be a p -adic representation of Gal ( K ¯ / K ) . Then, by using the theory of p -adic differential modules, we show that V is a Hodge-Tate (resp. de Rham) representation of Gal ( K ¯ / K ) if and only if V is a Hodge-Tate (resp. de Rham) representation of Gal ( K pf ¯ / K pf ) where K pf / K is a certain p -adic local field with residue field the smallest perfect field k pf containing k .

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

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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 &gt; 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.

On the infinite fern of Galois representations of unitary type

Gaëtan Chenevier (2011)

Annales scientifiques de l'École Normale Supérieure

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Let E be a CM number field, p an odd prime totally split in  E , and let  X be the p -adic analytic space parameterizing the isomorphism classes of  3 -dimensional semisimple p -adic representations of  Gal ( E ¯ / E ) satisfying a selfduality condition “of type U ( 3 ) ”. We study an analogue of the infinite fern of Gouvêa-Mazur in this context and show that each irreducible component of the Zariski-closure of the modular points in  X has dimension at least 3 [ E : ] . As important steps, and in any rank, we prove that any...

Around the Littlewood conjecture in Diophantine approximation

Yann Bugeaud (2014)

Publications mathématiques de Besançon

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The Littlewood conjecture in Diophantine approximation claims that inf q 1 q · q α · q β = 0 holds for all real numbers α and β , where · denotes the distance to the nearest integer. Its p -adic analogue, formulated by de Mathan and Teulié in 2004, asserts that inf q 1 q · q α · | q | p = 0 holds for every real number α and every prime number p , where | · | p denotes the p -adic absolute value normalized by | p | p = p - 1 . We survey the known results on these conjectures and highlight recent developments. ...