Displaying similar documents to “ C ∞ -antiderivatives of p -adic C ∞ -functions”

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 ₙ = ∑ 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 ₅ ( 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...

Locally analytic vectors of unitary principal series of  GL 2 ( ℚ 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 ( ℚ p ) attaches to any 2 -dimensional irreducible p -adic representation V of  G ℚ p an admissible unitary representation Π ( V ) of  GL 2 ( ℚ p ) . The unitary principal series of  GL 2 ( ℚ 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 ( ℚ p ) by proving a conjecture of Emerton.

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...

The Heyde theorem on a-adic solenoids

Margaryta Myronyuk (2013)

Colloquium Mathematicae

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We prove the following analogue of the Heyde theorem for a-adic solenoids. Let ξ₁, ξ₂ be independent random variables with values in an a-adic solenoid Σ a and with distributions μ₁, μ₂. Let α j , β j be topological automorphisms of Σ a such that β ₁ α - 1 ₁ ± β ₂ α - 1 ₂ are topological automorphisms of Σ a too. Assuming that the conditional distribution of the linear form L₂ = β₁ξ₁ + β₂ξ₂ given L₁ = α₁ξ₁ + α₂ξ₂ is symmetric, we describe the possible distributions μ₁, μ₂.

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].

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.

Bruhat-Tits theory from Berkovich’s point of view. I. Realizations and compactifications of buildings

Bertrand Rémy, Amaury Thuillier, Annette Werner (2010)

Annales scientifiques de l'École Normale Supérieure

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We investigate Bruhat-Tits buildings and their compactifications by means of Berkovich analytic geometry over complete non-Archimedean fields. For every reductive group G over a suitable non-Archimedean field k we define a map from the Bruhat-Tits building ℬ ( G , k ) to the Berkovich analytic space G an associated with G . Composing this map with the projection of G an to its flag varieties, we define a family of compactifications of ℬ ( G , k ) . This generalizes results by Berkovich in the case of split groups. Moreover,...

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 .

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 .