Overconvergent modular forms

Vincent Pilloni

Displaying similar documents to “Overconvergent modular forms”

Gauss–Manin connections for $p$ -adic families of nearly overconvergent modular forms

Robert Harron, Liang Xiao (2014)

Annales de l’institut Fourier

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We interpolate the Gauss–Manin connection in $p$ -adic families of nearly overconvergent modular forms. This gives a family of Maass–Shimura type differential operators from the space of nearly overconvergent modular forms of type $r$ to the space of nearly overconvergent modular forms of type $r + 1$ with $p$ -adic weight shifted by $2$ . Our construction is purely geometric, using Andreatta–Iovita–Stevens and Pilloni’s geometric construction of eigencurves, and should thus generalize to higher rank...

Classical and overconvergent modular forms of higher level

Robert F. Coleman (1997)

Journal de théorie des nombres de Bordeaux

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We define the notion overconvergent modular forms on $Γ_{1} (N p^{n})$ where $p$ is a prime, $N$ and $n$ are positive integers and $N$ is prime to $p$ . We show that an overconvergent eigenform on $Γ_{1} (N p^{n})$ of weight $k$ whose $U_{p}$ -eigenvalue has valuation strictly less than $k - 1$ is classical.

Bounds on sup-norms of half-integral weight modular forms

Eren Mehmet Kıral (2014)

Acta Arithmetica

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Bounding sup-norms of modular forms in terms of the level has been the focus of much recent study. In this work the sup-norm of a half-integral weight cusp form is bounded in terms of the level: we prove that $| | y^{κ / 2} {f ̃ | |}_{\infty} ≪_{ε, κ} N^{1 / 2 - 1 / 18 + ε} | | y^{κ / 2} f ̃ {| |}_{L^{2}}$ for a modular form f̃ of level 4N and weight κ, a half-integer.

Dimensions of spaces of modular forms for $Γ_{H} (N)$

Jordi Quer (2010)

Acta Arithmetica

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The second moment of quadratic twists of modular L-functions

K. Soundararajan, Matthew P. Young (2010)

Journal of the European Mathematical Society

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We study the second moment of the central values of quadratic twists of a modular $L$ -function. Unconditionally, we obtain a lower bound which matches the conjectured asymptotic formula, while on GRH we prove the asymptotic formula itself.

Quadratic modular symbols on Shimura curves

Pilar Bayer, Iván Blanco-Chacón (2013)

Journal de Théorie des Nombres de Bordeaux

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We introduce the concept of modular symbol and study how these symbols are related to $p$ -adic $L$ -functions. These objects were introduced in [] in the case of modular curves. In this paper, we discuss a method to attach quadratic modular symbols and quadratic $p$ -adic $L$ -functions to more general Shimura curves.

On the slopes of the $U_{5}$ operator acting on overconvergent modular forms

L. J. P Kilford (2008)

Journal de Théorie des Nombres de Bordeaux

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We show that the slopes of the $U_{5}$ operator acting on 5-adic overconvergent modular forms of weight $k$ with primitive Dirichlet character $χ$ of conductor 25 are given by either $\{\frac{1}{4} \cdot ⌊\frac{8 i}{5}⌋ : i \in ℕ\} or \{\frac{1}{4} \cdot ⌊\frac{8 i + 4}{5}⌋ : i \in ℕ\},$ depending on $k$ and $χ$ . We also prove that the space of classical cusp forms of weight $k$ and character $χ$ has a basis of eigenforms for the Hecke operators $T_{p}$ and $U_{5}$ which is defined over $Q_{5} (\sqrt[4]{5}, \sqrt{3})$ .

On the characteristic power series of the U operator

Fernando Q. Gouvêa, Barry Mazur (1993)

Annales de l'institut Fourier

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We show that the coefficients of the characteristic power series of Atkin’s U operator acting on overconvergent $p$ -adic modular forms of weight $k$ vary $p$ -adically continuously as functions of $k$ . Are they in fact Iwasawa functions of $k$ ?

Hecke operators in half-integral weight

Soma Purkait (2014)

Journal de Théorie des Nombres de Bordeaux

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In [], Shimura introduced modular forms of half-integral weight, their Hecke algebras and their relation to integral weight modular forms via the Shimura correspondence. For modular forms of integral weight, Sturm’s bounds give generators of the Hecke algebra as a module. We also have well-known recursion formulae for the operators $T_{p^{ℓ}}$ with $p$ prime. It is the purpose of this paper to prove analogous results in the half-integral weight setting. We also give an explicit formula for how operators...

On p-adic Siegel modular forms of non-real Nebentypus

Toshiyuki Kikuta (2012)

Acta Arithmetica

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On computing Belyi maps

J. Sijsling, J. Voight (2014)

Publications mathématiques de Besançon

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We survey methods to compute three-point branched covers of the projective line, also known as Belyĭ maps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic methods, modular forms methods, and $p$ -adic methods. Along the way, we pose several questions and provide numerous examples.

Overconvergent modular symbols and $p$ -adic $L$ -functions

Robert Pollack, Glenn Stevens (2011)

Annales scientifiques de l'École Normale Supérieure

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This paper is a constructive investigation of the relationship between classical modular symbols and overconvergent $p$ -adic modular symbols. Specifically, we give a constructive proof of a (Theorem 1.1) due to the second author [19] proving existence and uniqueness of overconvergent eigenliftings of classical modular eigensymbols of . As an application we describe a polynomial-time algorithm for explicit computation of associated $p$ -adic $L$ -functions in this case. In the case of, the control...