On $p$-adic Euler constants

Abhishek Bharadwaj

Displaying similar documents to “On $p$ -adic Euler constants”

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 $p \geq 5$ 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...

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.

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 $K_{\infty} / K$ is a local field of characteristic $p$ which comes equipped with an action of $Gal (K_{\infty} / K)$ . 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 $a ₙ = \sum_{j = 0}^{n} (n! / j!) n^{j}$ 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 $v ₅ (a_{m \cdot 5^{k} + n_{k}}) \geq k$ 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 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 $E / F$ .

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 $β ₁ α^{- 1} ₁ \pm β ₂ α^{- 1} ₂$ 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.

$ℒ$ -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 $k \geq 2$ 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 $ℒ$ -invariant $ℒ_{f}^{F M}$ . 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 $ℒ$ -invariant $ℒ_{f}$ , in the spirit of Darmon. The two monodromy modules are isomorphic, and in particular the two $ℒ$ -invariants...

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 $\hat{G}$ 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...

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

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 $p > 2$ , 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 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 $Gal (\bar{E} / E)$ 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 $3 [E : ℚ]$ . 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 $inf_{q \geq 1} q \cdot ∥ q α ∥ \cdot ∥ q β ∥ = 0$ holds for all real numbers $α$ and $β$ , where $∥ \cdot ∥$ denotes the distance to the nearest integer. Its $p$ -adic analogue, formulated by de Mathan and Teulié in 2004, asserts that $inf_{q \geq 1} q \cdot ∥ q α ∥ \cdot {| q |}_{p} = 0$ holds for every real number $α$ and every prime number $p$ , where ${| \cdot |}_{p}$ denotes the $p$ -adic absolute value normalized by ${| p |}_{p} = p^{- 1}$ . We survey the known results on these conjectures and highlight recent developments. ...

Optimality of the Width- $w$ Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases

Clemens Heuberger, Daniel Krenn (2013)

Journal de Théorie des Nombres de Bordeaux

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We consider digit expansions $\sum_{j = 0}^{ℓ - 1} Φ^{j} (d_{j})$ with an endomorphism $Φ$ of an Abelian group. In such a numeral system, the $w$ -NAF condition (each block of $w$ consecutive digits contains at most one nonzero) is shown to minimise the Hamming weight over all expansions with the same digit set if and only if it fulfills the subadditivity condition (the sum of every two expansions of weight $1$ admits an optimal $w$ -NAF). This result is then applied to imaginary quadratic bases, which are used for scalar...

Displaying similar documents to “On p -adic Euler constants”

Displaying similar documents to “On $p$ -adic Euler constants”