Coefficient bounds for level 2 cusp forms

Paul Jenkins; Kyle Pratt

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Displaying similar documents to “Coefficient bounds for level 2 cusp forms”

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

Jordi Quer (2010)

Acta Arithmetica

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On the higher mean over arithmetic progressions of Fourier coefficients of cusp forms

Yujiao Jiang, Guangshi Lü (2014)

Acta Arithmetica

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Let $λ_{f} (n)$ be the nth normalized Fourier coefficient of a holomorphic or Maass cusp form f for SL(2,ℤ). We establish the asymptotic formula for the summatory function $\sum_{\begin{matrix} n \leq x \\ n \equiv l (m o d q) \end{matrix}} {| λ_{f} (n) |}^{2 j}$ as x → ∞, where q grows with x in a definite way and j = 2,3,4.

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...

The distribution of Fourier coefficients of cusp forms over sparse sequences

Huixue Lao, Ayyadurai Sankaranarayanan (2014)

Acta Arithmetica

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Let $λ_{f} (n)$ be the nth normalized Fourier coefficient of a holomorphic Hecke eigenform $f (z) \in S_{k} (Γ)$ . We establish that $\sum_{n \leq x} λ_{f}^{2} (n^{j}) = c_{j} x + O (x^{1 - 2 / ({(j + 1)}^{2} + 1)})$ for j = 2,3,4, which improves the previous results. For j = 2, we even establish a better result.

On Baker type lower bounds for linear forms

Tapani Matala-aho (2016)

Acta Arithmetica

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A criterion is given for studying (explicit) Baker type lower bounds of linear forms in numbers $1, Θ_{1}, . . ., Θ_{m} \in ℂ *$ over the ring $ℤ$ of an imaginary quadratic field . This work deals with the simultaneous auxiliary functions case.

Koecher-Maass series of a certain half-integral weight modular form related to the Duke-Imamoḡlu-Ikeda lift

Hidenori Katsurada, Hisa-aki Kawamura (2014)

Acta Arithmetica

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Let k and n be positive even integers. For a cuspidal Hecke eigenform h in the Kohnen plus space of weight k - n/2 + 1/2 for Γ₀(4), let f be the corresponding primitive form of weight 2k-n for SL₂(ℤ) under the Shimura correspondence, and Iₙ(h) the Duke-Imamoḡlu-Ikeda lift of h to the space of cusp forms of weight k for Spₙ(ℤ). Moreover, let $ϕ_{I ₙ (h), 1}$ be the first Fourier-Jacobi coefficient of Iₙ(h), and $σ_{n - 1} (ϕ_{I ₙ (h), 1})$ be the cusp form in the generalized Kohnen plus space of weight k - 1/2 corresponding to...

A quadratic form with prime variables associated with Hecke eigenvalues of a cusp form

Guodong Hua (2022)

Czechoslovak Mathematical Journal

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Let $f$ be a normalized primitive holomorphic cusp form of even integral weight $k$ for the full modular group $SL (2, ℤ)$ , and denote its $n$ th Fourier coefficient by $λ_{f} (n)$ . We consider the hybrid problem of quadratic forms with prime variables and Hecke eigenvalues of normalized primitive holomorphic cusp forms, which generalizes the result of D. Y. Zhang, Y. N. Wang (2017).

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.

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...

On the 4-norm of an automorphic form

Valentin Blomer (2013)

Journal of the European Mathematical Society

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We prove the optimal upper bound $\sum_{f} {∥f∥}_{4}^{4} ≪ q^{ϵ}$ where $f$ runs over an orthonormal basis of Maass cusp forms of prime level $q$ and bounded spectral parameter.

Hybrid sup-norm bounds for Hecke–Maass cusp forms

Nicolas Templier (2015)

Journal of the European Mathematical Society

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Let $f$ be a Hecke–Maass cusp form of eigenvalue $λ$ and square-free level $N$ . Normalize the hyperbolic measure such that $vol (Y_{0} (N)) = 1$ and the form $f$ such that ${∥ f ∥}_{2} = 1$ . It is shown that ${∥ f ∥}_{\infty} ≪_{ϵ} λ^{\frac{5}{24} + ϵ} N^{\frac{1}{3} + ϵ}$ for all $ϵ > 0$ . This generalizes simultaneously the current best bounds in the eigenvalue and level aspects.

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.

Diagonalization in proof complexity

Jan Krajíček (2004)

Fundamenta Mathematicae

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We study diagonalization in the context of implicit proofs of [10]. We prove that at least one of the following three conjectures is true: ∙ There is a function f: 0,1* → 0,1 computable in that has circuit complexity $2^{Ω (n)}$ . ∙ ≠ co . ∙ There is no p-optimal propositional proof system. We note that a variant of the statement (either ≠ co or ∩ co contains a function $2^{Ω (n)}$ hard on average) seems to have a bearing on the existence of good proof complexity generators. In particular, we prove that...

Universal forms $\sum a_{i} {(x_{i})}^{n}$ and Waring’s problem

L. Dickson (1936)

Acta Arithmetica

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Generalized divisor problem for new forms of higher level

Krishnarjun Krishnamoorthy (2022)

Czechoslovak Mathematical Journal

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Suppose that $f$ is a primitive Hecke eigenform or a Mass cusp form for $Γ_{0} (N)$ with normalized eigenvalues $λ_{f} (n)$ and let $X > 1$ be a real number. We consider the sum $𝒮_{k} (X) : = \sum_{n < X} \sum_{n = n_{1}, n_{2}, ..., n_{k}} λ_{f} (n_{1}) λ_{f} (n_{2}) ... λ_{f} (n_{k})$ and show that $𝒮_{k} (X) ≪_{f, ϵ} X^{1 - 3 / (2 (k + 3)) + ϵ}$ for every $k \geq 1$ and $ϵ > 0$ . The same problem was considered for the case $N = 1$ , that is for the full modular group in Lü (2012) and Kanemitsu et al. (2002). We consider the problem in a more general setting and obtain bounds which are better than those obtained by the classical result of Landau (1915) for $k \geq 5$ . Since the result is valid...

Overconvergent modular forms

Vincent Pilloni (2013)

Annales de l’institut Fourier

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We give a geometric definition of overconvergent modular forms of any $p$ -adic weight. As an application, we reprove Coleman’s theory of $p$ -adic families of modular forms and reconstruct the eigencurve of Coleman and Mazur without using the Eisenstein family.

Liftings of forms to Weil bundles and the exterior derivative

Jacek Dębecki (2009)

Annales Polonici Mathematici

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In a previous paper we have given a complete description of linear liftings of p-forms on n-dimensional manifolds M to q-forms on $T^{A} M$ , where $T^{A}$ is a Weil functor, for all non-negative integers n, p and q, except the case p = n and q = 0. We now establish formulas connecting such liftings and the exterior derivative of forms. These formulas contain a boundary operator, which enables us to define a homology of the Weil algebra A. We next study the case p = n and q = 0 under the condition...

Local Indecomposability of Hilbert Modular Galois Representations

Bin Zhao (2014)

Annales de l’institut Fourier

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We prove the indecomposability of the Galois representation restricted to the $p$ -decomposition group attached to a non CM nearly $p$ -ordinary weight two Hilbert modular form over a totally real field $F$ under the assumption that either the degree of $F$ over $ℚ$ is odd or the automorphic representation attached to the Hilbert modular form is square integrable at some finite place of $F$ .