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

Robert Harron; Antonio Lei

Displaying similar documents to “Iwasawa theory for symmetric powers of CM modular forms at non-ordinary primes”

$ℒ$ -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...

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...

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...

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.

A variation of Thompson's conjecture for the symmetric groups

Mahdi Abedei, Ali Iranmanesh, Farrokh Shirjian (2020)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group and let $N (G)$ denote the set of conjugacy class sizes of $G$ . Thompson’s conjecture states that if $G$ is a centerless group and $S$ is a non-abelian simple group satisfying $N (G) = N (S)$ , then $G ≅ S$ . In this paper, we investigate a variation of this conjecture for some symmetric groups under a weaker assumption. In particular, it is shown that $G ≅ Sym (p + 1)$ if and only if $| G | = (p + 1)!$ and $G$ has a special conjugacy class of size $(p + 1)! / p$ , where $p > 5$ is a prime number. Consequently, if $G$ is a centerless group with $N (G) = N (Sym (p + 1))$ , then...

Thompson’s conjecture for the alternating group of degree $2 p$ and $2 p + 1$

Azam Babai, Ali Mahmoudifar (2017)

Czechoslovak Mathematical Journal

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For a finite group $G$ denote by $N (G)$ the set of conjugacy class sizes of $G$ . In 1980s, J. G. Thompson posed the following conjecture: If $L$ is a finite nonabelian simple group, $G$ is a finite group with trivial center and $N (G) = N (L)$ , then $G ≅ L$ . We prove this conjecture for an infinite class of simple groups. Let $p$ be an odd prime. We show that every finite group $G$ with the property $Z (G) = 1$ and $N (G) = N (A_{i})$ is necessarily isomorphic to $A_{i}$ , where $i \in {2 p, 2 p + 1}$ .

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...

A new characterization of symmetric group by NSE

Azam Babai, Zeinab Akhlaghi (2017)

Czechoslovak Mathematical Journal

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Let $G$ be a group and $ω (G)$ be the set of element orders of $G$ . Let $k \in ω (G)$ and $m_{k} (G)$ be the number of elements of order $k$ in $G$ . Let nse $(G) = {m_{k} (G) : k \in ω (G)}$ . Assume $r$ is a prime number and let $G$ be a group such that nse $(G) =$ nse $(S_{r})$ , where $S_{r}$ is the symmetric group of degree $r$ . In this paper we prove that $G ≅ S_{r}$ , if $r$ divides the order of $G$ and $r^{2}$ does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.

On a system of equations with primes

Paolo Leonetti, Salvatore Tringali (2014)

Journal de Théorie des Nombres de Bordeaux

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Given an integer $n \geq 3$ , let $u_{1}, ..., u_{n}$ be pairwise coprime integers $\geq 2$ , $𝒟$ a family of nonempty proper subsets of ${1, ..., n}$ with “enough” elements, and $ε$ a function $𝒟 \to {\pm 1}$ . Does there exist at least one prime $q$ such that $q$ divides $\prod_{i \in I} u_{i} - ε (I)$ for some $I \in 𝒟$ , but it does not divide $u_{1} \dots u_{n}$ ? We answer this question in the positive when the $u_{i}$ are prime powers and $ε$ and $𝒟$ are subjected to certain restrictions. We use the result to prove that, if $ε_{0} \in {\pm 1}$ and $A$ is a set of three or more primes that contains all prime divisors of any...

Recognition of some families of finite simple groups by order and set of orders of vanishing elements

Maryam Khatami, Azam Babai (2018)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group. An element $g \in G$ is called a vanishing element if there exists an irreducible complex character $χ$ of $G$ such that $χ (g) = 0$ . Denote by $Vo (G)$ the set of orders of vanishing elements of $G$ . Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let $G$ be a finite group and $M$ a finite nonabelian simple group such that $Vo (G) = Vo (M)$ and $| G | = | M |$ . Then $G ≅ M$ . We answer in affirmative this conjecture for $M = S z (q)$ , where $q = 2^{2 n + 1}$ and either $q - 1$ , $q - \sqrt{2 q} + 1$ or $q + \sqrt{2 q} + 1$ is a prime number, and $M = F_{4} (q)$ , where...