268 results on '"Cusp form"'
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2. Bounds toward Hypothesis S for cusp forms
- Author
-
Yangbo Ye
- Subjects
Cusp (singularity) ,Pure mathematics ,Algebra and Number Theory ,Mathematics::Number Theory ,Product (mathematics) ,Large sieve ,Arithmetic progression ,Holomorphic function ,Cusp form ,Fourier series ,Mathematics ,Exponential function - Abstract
Iwaniec, Luo, and Sarnak proposed Hypothesis S and its generalization which predicts non-trivial bounds for a smooth sum of the product of an arithmetic sequence { a n } and a fractional exponential function. When a n is the Fourier coefficient λ f ( n ) of a fixed holomorphic cusp form f, however, a resonance phenomenon prohibits any improvement of the bound beyond a barrier. It is believed that this resonance barrier could be overcome when the weight k of f tends to infinity. The present paper is a first step toward this goal by proving non-trivial bounds for this sum when k and the summation length X both tend to infinity. No such non-trivial bounds are previously known if the form f is allowed to move. Similar bounds are also proved for linear phases and for Maass forms. The main technology is improved large sieve inequalities over a short interval.
- Published
- 2022
3. Some results on divisor problems related to cusp forms
- Author
-
Wei Zhang
- Subjects
Combinatorics ,Cusp (singularity) ,Algebra and Number Theory ,Number theory ,Integer ,Divisor ,Holomorphic function ,Lambda ,Fourier series ,Cusp form ,Mathematics - Abstract
Let $$\lambda _{f}(n)$$ be the normalized Fourier coefficients of a holomorphic Hecke cusp form of full level. We study a generalized divisor problem with $$\lambda _{f}(n)$$ over some special sequences. More precisely, for any fixed integer $$k\ge 2$$ and $$j\in \{1,2,3,4\},$$ we are interested in the following sums $$\begin{aligned} S_{k}(x,j):=\sum _{n\le x}\lambda _{k,f}(n^{j})=\sum _{n\le x}\sum _{n=n_{1}n_{2}\cdots n_{k}}\lambda _{f}(n_{1}^{j})\lambda _{f}(n_{2}^{j})\cdots \lambda _{f}(n_{k}^{j}). \end{aligned}$$
- Published
- 2020
4. On the standard L-function for $$\mathrm{GSp}_{2n} \times \mathrm{GL}_1$$ and algebraicity of symmetric fourth L-values for $$\mathrm{GL}_2$$
- Author
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Abhishek Saha, Ameya Pitale, and Ralf Schmidt
- Subjects
Cusp (singularity) ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Holomorphic function ,Reciprocity law ,01 natural sciences ,Cusp form ,Lift (mathematics) ,Number theory ,Pullback ,0103 physical sciences ,010307 mathematical physics ,L-function ,0101 mathematics ,Mathematics - Abstract
We prove an explicit integral representation—involving the pullback of a suitable Siegel Eisenstein series—for the twisted standard L-function associated to a holomorphic vector-valued Siegel cusp form of degree n and arbitrary level. In contrast to all previously proved pullback formulas in this situation, our formula involves only scalar-valued functions despite being applicable to L-functions of vector-valued Siegel cusp forms. The key new ingredient in our method is a novel choice of local vectors at the archimedean place which allows us to exactly compute the archimedean local integral. By specializing our integral representation to the case $$n=2$$ n = 2 we are able to prove a reciprocity law—predicted by Deligne’s conjecture—for the critical values of the twisted standard L-function for vector-valued Siegel cusp forms of degree 2 and arbitrary level. This arithmetic application generalizes previously proved critical-value results for the full level case. By specializing further to the case of Siegel cusp forms obtained via the Ramakrishnan–Shahidi lift, we obtain a reciprocity law for the critical values of the symmetric fourth L-function of a classical newform.
- Published
- 2020
5. On degree 2 Siegel cusp forms and its Fourier coefficients
- Author
-
Yves Martin
- Subjects
Cusp (singularity) ,Pure mathematics ,Algebra and Number Theory ,Series (mathematics) ,Mathematics::Number Theory ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Cusp form ,Integer ,Diagonal matrix ,0101 mathematics ,Fourier series ,Laplace operator ,Eigenvalues and eigenvectors ,Mathematics - Abstract
We present a set of diagonal matrices which index enough Fourier coefficients for a complete characterization of all Siegel cusp forms of degree 2, weight k, level N and character χ, where k is an even integer ≥4, N is an odd, square-free positive integer, and χ has conductor equal to N. As an application, we show that the Koecher-Maass series of any F ∈ S k 2 twisted by the set of Maass waveforms whose eigenvalues are in the continuum spectrum of the hyperbolic Laplacian determines F. We also generalize a result due to Skogman about the non-vanishing of all theta components of a Jacobi cusp form of even weight and prime index, which may have some independent interest.
- Published
- 2020
6. On the Functional Independence of Zeta-Functions of Certain Cusp Forms
- Author
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Antanas Laurinčikas
- Subjects
Cusp (singularity) ,Pure mathematics ,General Mathematics ,010102 general mathematics ,02 engineering and technology ,Space (mathematics) ,01 natural sciences ,Cusp form ,symbols.namesake ,020303 mechanical engineering & transports ,Operator (computer programming) ,0203 mechanical engineering ,Functional independence ,symbols ,0101 mathematics ,Fourier series ,Dirichlet series ,Mathematics ,Analytic function - Abstract
The zeta-function ζ(s, F), s = σ + it of a cusp form F of weight κ in the half-plane σ > (κ + 1)/2 is defined by the Dirichlet series whose coefficients are the coefficients of the Fourier series of the form F. The compositions V(ζ(s,F)) with an operator V on the space of analytic functions are considered, and the functional independence of these compositions for certain classes of operators V is proved.
- Published
- 2020
7. Weyl bound for GL(2) in t-aspect via a simple delta method
- Author
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Keshav Aggarwal
- Subjects
Cusp (singularity) ,Delta method ,Pure mathematics ,Algebra and Number Theory ,Simple (abstract algebra) ,Mathematics::Quantum Algebra ,Mathematics::Number Theory ,Holomorphic function ,Mathematics::Representation Theory ,Cusp form ,Mathematics - Abstract
We use a simple delta method to prove the Weyl bound in t-aspect for the L-function of a holomorphic or a Hecke-Maass cusp form of arbitrary level and nebentypus. In particular, this extends the results of Meurman and Jutila for the t-aspect Weyl bound, and the recent result of Booker, Milinovich and Ng to Hecke cusp forms of arbitrary level and nebentypus.
- Published
- 2020
8. Explicit subconvexity savings for sup-norms of cusp forms on PGLn(R)
- Author
-
Nate Gillman
- Subjects
Cusp (singularity) ,Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Laplace operator ,Cusp form ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Blomer and Maga [2] recently proved that, if F is an L 2 -normalized Hecke-Maass cusp form for SL n ( Z ) , and Ω is a compact subset of PGL n ( R ) / PO n ( R ) , then we have ‖ F | Ω ‖ ∞ ≪ Ω λ F n ( n − 1 ) / 8 − δ n for some δ n > 0 , where λ F is the Laplacian eigenvalue of F. In the present paper, we prove an explicit version of their result.
- Published
- 2020
9. On the sup-norm of SL 3 Hecke–Maass cusp forms
- Author
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Kevin Nowland, Emmanuel Royer, Guillaume Ricotta, and Roman Holowinsky
- Subjects
Cusp (singularity) ,Pure mathematics ,Uniform norm ,Compact space ,Mathematics::Number Theory ,Mathematics::Spectral Theory ,Cusp form ,Eigenvalues and eigenvectors ,Mathematics - Abstract
This work contains a proof of a non-trivial explicit quantitative bound in the eigenvalue aspect for the sup-norm of a SL(3,Z) Hecke-Maass cusp form restricted to a compact set.
- Published
- 2019
10. On comparing Hecke eigenvalues of cusp forms
- Author
-
H. X. Lao
- Subjects
Cusp (singularity) ,General Mathematics ,Cuspidal representation ,010102 general mathematics ,Holomorphic function ,010103 numerical & computational mathematics ,Lambda ,01 natural sciences ,Cusp form ,Combinatorics ,Modular group ,0101 mathematics ,Fourier series ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Let f(z) be a primitive holomorphic cusp form of even integral weight k for the full modular group. Denote its nth Hecke eigenvalue or normalized Fourier coefficient by \(\lambda_{f}(n)\). Let g(z) be another distinct primitive holomorphic cusp form of even integral weight \(\ell\) for the full modular group. In this paper, we establish that the set $$\{p| \lambda_f(p^j)
- Published
- 2019
11. On the first sign change of Fourier coefficients of cusp forms at sum of two squares
- Author
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Prashant Tiwari and Manish Kumar Pandey
- Subjects
Cusp (singularity) ,Combinatorics ,Sequence ,Algebra and Number Theory ,Number theory ,Fermat's theorem on sums of two squares ,Cusp form ,Fourier series ,Upper and lower bounds ,Mathematics ,Sign (mathematics) - Abstract
In this article, we have considered the problem of first sign change of Fourier coefficients of a primitive cusp form at a sparse sequence, where elements of the sparse sequence can be written as a sum of two squares. Following the methods of Kohnen–Sengupta, we give an effective upper bound on elements of such a sparse sequence to have first sign change.
- Published
- 2021
12. Rankin--Cohen brackets on Hermitian Jacobi forms and the adjoint of some linear maps
- Author
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Singh Sujeet Kumar and S Sumukha
- Subjects
Cusp (singularity) ,Pure mathematics ,Mathematics::Number Theory ,General Mathematics ,Scalar (mathematics) ,Special values ,Mathematics::Geometric Topology ,Hermitian matrix ,Cusp form ,symbols.namesake ,Product (mathematics) ,symbols ,Mathematics::Differential Geometry ,Fourier series ,Dirichlet series ,Mathematics - Abstract
Given a fixed Hermitian Jacobi cusp form, we define a family of linear operators between spaces of Hermitian Jacobi cusp forms using Rankin--Cohen brackets. We compute the adjoint maps of such a family with respect to the Petersson scalar product. The Fourier coefficients of the Hermitian Jacobi cusp forms constructed using this method involve special values of certain Dirichlet series associated to Hermitian Jacobi cusp forms.
- Published
- 2021
13. Joint Universality of the Zeta-Functions of Cusp Forms
- Author
-
Renata Macaitienė
- Subjects
Physics ,Cusp (singularity) ,Rational number ,Mathematics::General Mathematics ,General Mathematics ,Mathematics::Number Theory ,Field (mathematics) ,joint universality ,Cusp form ,Riemann zeta function ,symbols.namesake ,Modular group ,zeta-function ,Computer Science (miscellaneous) ,symbols ,Hecke-eigen cusp form ,universality ,QA1-939 ,Algebraic number ,Engineering (miscellaneous) ,Mathematics ,Mathematical physics ,Analytic function - Abstract
Let F be the normalized Hecke-eigen cusp form for the full modular group and ζ(s,F) be the corresponding zeta-function. In the paper, the joint universality theorem on the approximation of a collection of analytic functions by shifts (ζ(s+ih1τ,F),…,ζ(s+ihrτ,F)) is proved. Here, h1,…,hr are algebraic numbers linearly independent over the field of rational numbers.
- Published
- 2021
14. On the global sup-norm of GL(3) cusp forms
- Author
-
Valentin Blomer, Gergely Harcos, and Péter Maga
- Subjects
Pointwise ,Cusp (singularity) ,Pure mathematics ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,Function (mathematics) ,Space (mathematics) ,01 natural sciences ,Cusp form ,Upper and lower bounds ,Uniform norm ,010201 computation theory & mathematics ,0101 mathematics ,Mathematics::Representation Theory ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Let φ be a spherical Hecke–Maas cusp form on the non-compact space PGL3(ℤ)PGL3(ℝ). We establish various pointwise upper bounds for φ in terms of its Laplace eigenvalue λφ. These imply, for φ arithmetically normalized and tempered at the archimedean place, the bound $$||\phi |{|_\infty } \ll_\varepsilon \lambda _\phi ^{39/40 + \varepsilon }$$ for the global sup-norm (without restriction to a compact subset). On the way, we derive a new uniform upper bound for the GL3 Jacquet–Whittaker function.
- Published
- 2019
15. Stark points on elliptic curves via Perrin-Riou's philosophy
- Author
-
Henri Darmon and Alan G. B. Lauder
- Subjects
Cusp (singularity) ,Conjecture ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Formal group ,01 natural sciences ,Cusp form ,Combinatorics ,Elliptic curve ,symbols.namesake ,Number theory ,0103 physical sciences ,Eisenstein series ,De Rham cohomology ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In the early 90’s, Perrin-Riou (Ann Inst Fourier 43(4):945–995, 1993) introduced an important refinement of the Mazur–Swinnerton-Dyer p-adic L-function of an elliptic curve E over $$\mathbb {Q}$$ , taking values in its p-adic de Rham cohomology. She then formulated a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for this p-adic L-function, in which the formal group logarithms of global points on E make an intriguing appearance. The present work extends Perrin-Riou’s construction to the setting of a Garret–Rankin triple product (f, g, h), where f is a cusp form of weight two attached to E and g and h are classical weight one cusp forms with inverse nebentype characters, corresponding to odd two-dimensional Artin representations $$\varrho _g$$ and $$\varrho _h$$ respectively. The resulting p-adic Birch and Swinnerton-Dyer conjecture involves the p-adic logarithms of global points on E defined over the field cut out by $$\varrho _g\otimes \varrho _h$$ , in the style of the regulators that arise in Darmon et al. (Forum Math 3(e8):95, 2015), and recovers Perrin-Riou’s original conjecture when g and h are Eisenstein series.
- Published
- 2021
16. Sign Changes of Fourier Coefficients of Cusp Forms of Half-Integral Weight Over Split and Inert Primes in Quadratic Number Fields
- Author
-
Zilong He and Ben Kane
- Subjects
Cusp (singularity) ,Pure mathematics ,Algebra and Number Theory ,Mathematics - Number Theory ,Mathematics::Number Theory ,Algebraic number field ,Ring of integers ,Cusp form ,Square (algebra) ,Number theory ,Discriminant ,FOS: Mathematics ,11F37, 11F30, 11N69, 11R11, 11E20 ,Number Theory (math.NT) ,Mathematics ,Sign (mathematics) - Abstract
In this paper, we investigate sign changes of Fourier coefficients of half-integral weight cusp forms. In a fixed square class $$t\mathbb {Z}^2$$ , we investigate the sign changes in the $$tp^2$$ -th coefficient as p runs through the split or inert primes over the ring of integers in a quadratic extension of the rationals. We show that infinitely many sign changes occur in both sets of primes when there exists a prime dividing the discriminant of the field which does not divide the level of the cusp form and find an explicit condition that determines whether sign changes occur when every prime dividing the discriminant also divides the level.
- Published
- 2021
17. Cusp forms as p-adic limits
- Author
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Marie Jameson and Michael Hanson
- Subjects
Cusp (singularity) ,Pure mathematics ,Algebra and Number Theory ,Mathematics - Number Theory ,Mathematics::Number Theory ,Modular form ,Holomorphic function ,Complex multiplication ,Duality (optimization) ,11F33 ,Space (mathematics) ,Cusp form ,FOS: Mathematics ,Number Theory (math.NT) ,Fourier series ,Mathematics - Abstract
Ahlgren and Samart relate three cusp forms with complex multiplication to certain weakly holomorphic modular forms using $p$-adic bounds related to their Fourier coefficients. In these three examples, their result strengthens a theorem of Guerzhoy, Kent, and Ono which pairs certain CM forms with weakly holomorphic modular forms via $p$-adic limits. Ahlgren and Samart use only the theory of modular forms and Hecke operators, whereas Guerzhoy, Kent, and Ono use the theory of harmonic Maass forms. Here we extend Ahlgren and Samart's work to all cases where the cusp form space is one-dimensional and has trivial Nebentypus. Along the way, we obtain a duality result relating two families of modular forms that arise naturally in each case., Comment: Submitted 24 February, 2021. 12 pages
- Published
- 2021
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18. On Hecke eigenvalues of cusp forms in almost all short intervals
- Author
-
Jiseong Kim
- Subjects
Cusp (singularity) ,Pure mathematics ,Algebra and Number Theory ,Mathematics - Number Theory ,Computer Science::Information Retrieval ,Modular form ,Astrophysics::Instrumentation and Methods for Astrophysics ,Holomorphic function ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,Function (mathematics) ,Cusp form ,11F30 ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,FOS: Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Computer Science::General Literature ,Number Theory (math.NT) ,ComputingMilieux_MISCELLANEOUS ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Let $\psi$ be a function such that $\psi(x) \rightarrow \infty$ as $x \rightarrow \infty.$ Let $\lambda_{f}(n)$ be the $n$-th Hecke eigenvalue of a fixed holomorphic cusp form $f$ for $SL(2,\mathbb{Z}).$ We show that for any real valued function $h(x)$ such that $(\log X)^{2-2\alpha} \ll h(X) =o(X),$ $$\sum_{n=x}^{x+h(X)} |\lambda_{f}(n)| \ll_{f} h(X)\psi(X)(\log X)^{\alpha-1}$$ for all but $O_{f}( X\psi(X)^{-2})$ many integers $x\in [X,2X-h(X)],$ in which $\alpha$ is the average value of $|\lambda_{f}(p)|$ over primes. We generalize this for $|\lambda_{f}(n)|^{2^{k}}$ for $k \in \mathbb{Z^{+}}.$, Comment: To appear in International Journal of Number Theory
- Published
- 2020
19. On the signs of Fourier coefficients of Hilbert cusp forms
- Author
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Ritwik Pal
- Subjects
Cusp (singularity) ,Algebra and Number Theory ,Degree (graph theory) ,Mathematics - Number Theory ,010102 general mathematics ,0102 computer and information sciences ,Algebraic number field ,01 natural sciences ,Cusp form ,Combinatorics ,Number theory ,010201 computation theory & mathematics ,FOS: Mathematics ,Ideal (ring theory) ,Number Theory (math.NT) ,0101 mathematics ,Mathematics::Representation Theory ,Fourier series ,Mathematics - Abstract
We prove that given any $$\epsilon > 0$$ and a primitive adelic Hilbert cusp form f of weight $$k=(k_1,k_2,\ldots ,k_n) \in (2 {\mathbb {Z}})^n$$ and full level, there exists an integral ideal $${\mathfrak {m}}$$ with $$N({\mathfrak {m}}) \ll _{\epsilon } Q_{f}^{9/20+ \epsilon } $$ such that the $${\mathfrak {m}}$$ -th Fourier coefficient of $$C_{f} ({\mathfrak {m}})$$ of f is negative. Here n is the degree of the associated number field, $$N({\mathfrak {m}})$$ is the norm of integral ideal $${\mathfrak {m}}$$ and $$Q_{f}$$ is the analytic conductor of f. In the case of arbitrary weights, we show that there is an integral ideal $${\mathfrak {m}}$$ with $$N({\mathfrak {m}}) \ll _{\epsilon } Q_{f}^{1/2 + \epsilon }$$ such that $$C_{f}({\mathfrak {m}})
- Published
- 2020
20. Omega Results for Fourier Coefficients of Half-Integral Weight and Siegel Modular Forms
- Author
-
Soumya Das
- Subjects
Cusp (singularity) ,Pure mathematics ,Conjecture ,Degree (graph theory) ,Mathematics::Number Theory ,Congruence (manifolds) ,Omega ,Cusp form ,Fourier series ,Mathematics ,Siegel modular form - Abstract
We prove an \(\Omega \)-result for the Fourier coefficients of a half-integral weight cusp form of arbitrary level, nebentypus and weights. In particular, this implies that the analogue of the Ramanujan-Petersson conjecture for such forms is essentially the best possible. As applications, we show similar \(\Omega \)-results for Fourier coefficients of Siegel cusp forms of any degree and on Hecke congruence subgroups.
- Published
- 2020
21. p-adic Asai L-functions Attached to Bianchi Cusp Forms
- Author
-
Baskar Balasubramanyam, Ravitheja Vangala, and Eknath Ghate
- Subjects
Cusp (singularity) ,Pure mathematics ,Mathematics::Number Theory ,Congruence relation ,Measure (mathematics) ,Cusp form ,Mathematics - Abstract
We establish a rationality result for the twisted Asai L-values attached to a Bianchi cusp form and construct distributions interpolating these L-values. Using the method of abstract Kummer congruences, we then outline the main steps needed to show that these distributions come from a measure.
- Published
- 2020
22. On the first sign change of Fourier coefficients of cusp forms
- Author
-
Xiaoguang He and Lilu Zhao
- Subjects
Cusp (singularity) ,Algebra and Number Theory ,010102 general mathematics ,0102 computer and information sciences ,Square-free integer ,01 natural sciences ,Cusp form ,Combinatorics ,010201 computation theory & mathematics ,0101 mathematics ,Fourier series ,Sign (mathematics) ,Mathematics ,Congruence subgroup - Abstract
Let f be a nonzero cusp form of even integral weight k ⩾ 2 on the Hecke congruence subgroup Γ 0 ( N ) with N squarefree. Suppose that the normalized Fourier coefficients λ f ( n ) of f are real. We prove that the first sign change of λ f ( n ) occurs in the range n ≪ ( k N ) 2 + e . This improves upon the earlier result of Choie and Kohnen [2] .
- Published
- 2018
23. A study on twisted Koecher–Maass series of Siegel cusp forms via an integral kernel
- Author
-
Yves Martin
- Subjects
Cusp (singularity) ,Pure mathematics ,Algebra and Number Theory ,Series (mathematics) ,Mathematics::Number Theory ,Analytic continuation ,010102 general mathematics ,0102 computer and information sciences ,Mathematics::Spectral Theory ,01 natural sciences ,Cusp form ,symbols.namesake ,Kernel (algebra) ,010201 computation theory & mathematics ,Poincaré series ,symbols ,0101 mathematics ,Dirichlet series ,Eigenvalues and eigenvectors ,Mathematics - Abstract
We find an explicit integral kernel for the twisted Koecher–Maass series of any degree two Siegel cusp form F, where the twist is realized by an arbitrary Maass waveform whose eigenvalue is in the continuum spectrum. We also obtain the analytic properties of such a kernel (functional equations and analytic continuation), as well as a series representation of it in terms of the degree two Siegel Poincare series. From these properties we deduce the analytic properties of the twisted Koecher–Maass series. Moreover, we express the later as a multiple Dirichlet series involving the Dirichlet series associated to the Fourier–Jacobi coefficients of F. Finally, we get the integral kernel of the untwisted Koecher–Maass series (first studied by Kohnen and Sengupta in any degree) as a limit case of our construction.
- Published
- 2018
24. On mean values of mollifiers and L-functions associated to primitive cusp forms
- Author
-
Nicolas Robles, Patrick Kühn, and Dirk Zeindler
- Subjects
Cusp (singularity) ,Pure mathematics ,Current (mathematics) ,Mathematics - Number Theory ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Holomorphic function ,11M36 (Primary), 11M06, 11N64 (Secondary) ,Second moment of area ,Extension (predicate logic) ,01 natural sciences ,Cusp form ,Upper and lower bounds ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Mollifier ,Mathematics - Abstract
We study the second moment of the L-function associated to a holomorphic primitive cusp form of even weight perturbed by a new family of mollifiers. This family is a natural extension of the mollifers considered by Conrey and by Bui, Conrey and Young. As an application, we improve the current lower bound on critical zeros of holomorphic primitive cusp forms., Comment: 2 Figures. Differences to version 1. There are several improvments in the presentation, including a better fitting title and abract. We also have corrected some minor mathematical errors
- Published
- 2018
25. Exact Cutting in Spaces of Cusp Forms with Characters
- Author
-
G. V. Voskresenskaya
- Subjects
Cusp (singularity) ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Modular form ,Multiplicative function ,Structure (category theory) ,02 engineering and technology ,Function (mathematics) ,01 natural sciences ,Cusp form ,symbols.namesake ,020303 mechanical engineering & transports ,Quadratic equation ,0203 mechanical engineering ,symbols ,Dedekind eta function ,0101 mathematics ,Mathematics - Abstract
Structure theorems for spaces of cusp forms with quadratic characters are presented. It is proved that such spaces of levels N ≠ 3, 17, 19 admit exact cutting if and only if the cutting function is a multiplicative η-product. The cases of the levels N = 3, 17, 19 are also studied.
- Published
- 2018
26. Non-vanishing of the first derivative of GL(3) ×GL(2) L-functions
- Author
-
Xiaofei Yan and Guohua Chen
- Subjects
Cusp (singularity) ,Pure mathematics ,Algebra and Number Theory ,Computer Science::Information Retrieval ,Mathematics::Number Theory ,010102 general mathematics ,Astrophysics::Instrumentation and Methods for Astrophysics ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,01 natural sciences ,Cusp form ,Orthogonal basis ,0103 physical sciences ,Computer Science::General Literature ,Asymptotic formula ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Let [Formula: see text] be a fixed self-dual Hecke–Maass cusp form for [Formula: see text] and [Formula: see text] be an orthogonal basis of odd Hecke–Maass cusp forms for [Formula: see text]. We prove an asymptotic formula for the average of the first derivative of the Rankin–Selberg [Formula: see text]-function of [Formula: see text] and [Formula: see text] at the center point [Formula: see text]. This implies the non-vanishing results for the first derivative of these [Formula: see text]-functions at the center point [Formula: see text].
- Published
- 2018
27. Ternary quadratic form with prime variables attached to Fourier coefficients of primitive holomorphic cusp form
- Author
-
Deyu Zhang and Yingnan Wang
- Subjects
Cusp (singularity) ,Pure mathematics ,Algebra and Number Theory ,Mathematics::Complex Variables ,Mathematics::Number Theory ,010102 general mathematics ,Mathematical analysis ,Holomorphic function ,02 engineering and technology ,01 natural sciences ,Cusp form ,Prime (order theory) ,Quadratic form ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,0101 mathematics ,Ternary operation ,Fourier series ,Mathematics - Abstract
Let λ f ( n ) be the Fourier coefficients of primitive holomorphic cusp forms for SL 2 ( Z ) . In this paper, we study the hybrid problem of ternary quadratic forms and Fourier coefficients of primitive holomorphic cusp form.
- Published
- 2017
28. Determining Hilbert modular forms by central values of Rankin–Selberg convolutions: the weight aspect
- Author
-
Alia Hamieh and Naomi Tanabe
- Subjects
Cusp (singularity) ,Algebra and Number Theory ,Mathematics - Number Theory ,Generalization ,Mathematics::Number Theory ,010102 general mathematics ,Modular form ,01 natural sciences ,Cusp form ,Combinatorics ,Number theory ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,11F41, 11F67 ,Real number ,Mathematics - Abstract
The purpose of this paper is to prove that a primitive Hilbert cusp form $\mathbf{g}$ is uniquely determined by the central values of the Rankin-Selberg $L$-functions $L(\mathbf{f}\otimes\mathbf{g}, \frac{1}{2})$, where $\mathbf{f}$ runs through all primitive Hilbert cusp forms of weight $k$ for infinitely many weight vectors $k$. This work is a generalization of a result of Ganguly, Hoffstein, and Sengupta to the setting of totally real number fields, and it is a weight aspect analogue of the authors recent work., Comment: 20 pages
- Published
- 2017
29. Short-interval averages of sums of Fourier coefficients of cusp forms
- Author
-
David Lowry-Duda, Alexander Walker, Thomas A. Hulse, and Chan Ieong Kuan
- Subjects
Cusp (singularity) ,Algebra and Number Theory ,Conjecture ,Current (mathematics) ,Mathematics - Number Theory ,Divisor ,010102 general mathematics ,Holomorphic function ,11F30 ,01 natural sciences ,Cusp form ,Combinatorics ,symbols.namesake ,0103 physical sciences ,FOS: Mathematics ,symbols ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Fourier series ,Dirichlet series ,Mathematics - Abstract
Let $f$ be a weight $k$ holomorphic cusp form of level one, and let $S_f(n)$ denote the sum of the first $n$ Fourier coefficients of $f$. In analogy with Dirichlet's divisor problem, it is conjectured that $S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{4} + \epsilon}$. Understanding and bounding $S_f(X)$ has been a very active area of research. The current best bound for individual $S_f(X)$ is $S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{3}} (\log X)^{-0.1185}$ from Wu. Chandrasekharan and Narasimhan showed that the Classical Conjecture for $S_f(X)$ holds on average over intervals of length $X$. Jutila improved this result to show that the Classical Conjecture for $S_f(X)$ holds on average over short intervals of length $X^{\frac{3}{4} + \epsilon}$. Building on the results and analytic information about $\sum \lvert S_f(n) \rvert^2 n^{-(s + k - 1)}$ from our recent work, we further improve these results to show that the Classical Conjecture for $S_f(X)$ holds on average over short intervals of length $X^{\frac{2}{3}}(\log X)^{\frac{1}{6}}$., Comment: To Appear in the Journal of Number Theory
- Published
- 2017
30. Series expansion of the period function and representations of Hecke operators
- Author
-
Subong Lim, Dohoon Choi, Wissam Raji, and Tobias Mühlenbruch
- Subjects
Cusp (singularity) ,Pure mathematics ,Algebra and Number Theory ,Mathematics::Number Theory ,010102 general mathematics ,Mathematical analysis ,Function (mathematics) ,01 natural sciences ,Cusp form ,Binomial theorem ,010101 applied mathematics ,Eigenform ,Isomorphism ,0101 mathematics ,Series expansion ,Hecke operator ,Mathematics - Abstract
The period polynomial of a cusp form of an integral weight plays an important role in the number theory. In this paper, we study the period function of a cusp form of real weight. We obtain a series expansion of the period function of a cusp form of real weight for SL ( 2 , Z ) by using the binomial expansion. Furthermore, we study two kinds of Hecke operators acting on cusp forms and period functions, respectively. With these Hecke operators we show that there is a Hecke-equivariant isomorphism between the space of cusp forms and the space of period functions. As an application, we obtain a formula for a certain L-value of a Hecke eigenform by using the series expansion of its period function.
- Published
- 2017
31. Cusp Forms Whose Fourier Coefficients Involve Dirichlet Series
- Author
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Uğur Kırmacı
- Subjects
Cusp (singularity) ,Dirichlet kernel ,symbols.namesake ,Dirichlet conditions ,Mathematical analysis ,symbols ,General Medicine ,General Dirichlet series ,Fourier series ,Cusp form ,Dirichlet series ,Mathematics - Published
- 2017
32. Spectral square moments of a resonance sum for Maass forms
- Author
-
Yangbo Ye and Nathan Salazar
- Subjects
Cusp (singularity) ,010102 general mathematics ,Mathematical analysis ,Holomorphic function ,010103 numerical & computational mathematics ,01 natural sciences ,Cusp form ,Square (algebra) ,Combinatorics ,Mathematics (miscellaneous) ,Maass cusp form ,0101 mathematics ,Oscillatory integral ,Asymptotic expansion ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Let f be a Maass cusp form for Γ0(N) with Fourier coefficients λ f (n) and Laplace eigenvalue $$\frac{1} {4} + k^2 $$ . For real α ≠ 0 and β > 0, consider the sum S X (f; α, β) = ∑ n λ f (n)e(αn β )ϕ(n/X), where ϕ is a smooth function of compact support. We prove bounds for the second spectral moment of S X (f; α, β), with the eigenvalue tending towards infinity. When the eigenvalue is sufficiently large, we obtain an average bound for this sum in terms of X. This implies that if f has its eigenvalue beyond $$X^{\tfrac{1} {2} + \varepsilon } $$ , the standard resonance main term for S X (f; $$ \pm 2\sqrt q $$ , 1/2), q ∈ ℤ+, cannot appear in general. The method is adopted from proofs of subconvexity bounds for Rankin-Selberg L-functions for GL(2) × GL(2). It contains in particular a proof of an asymptotic expansion of a well-known oscillatory integral with an enlarged range of K e ⩽ L ⩽ K 1−e . The same bounds can be proved in a similar way for holomorphic cusp forms.
- Published
- 2017
33. Remark on the paper 'On products of Fourier coefficients of cusp forms'
- Author
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Yuk-Kam Lau, Deyu Zhang, and Yingnan Wang
- Subjects
Cusp (singularity) ,Discrete group ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Holomorphic function ,02 engineering and technology ,01 natural sciences ,Cusp form ,Combinatorics ,Integer ,Product (mathematics) ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,0101 mathematics ,Fourier series ,Mathematics - Abstract
Let a(n) be the Fourier coefficient of a holomorphic cusp form on some discrete subgroup of \(SL_2({\mathbb R})\). This note is to refine a recent result of Hofmann and Kohnen on the non-positive (resp. non-negative) product of \(a(n)a(n+r)\) for a fixed positive integer r.
- Published
- 2016
34. A note on cusp forms as p-adic limits
- Author
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Scott Ahlgren and Detchat Samart
- Subjects
0301 basic medicine ,Cusp (singularity) ,Pure mathematics ,Algebra and Number Theory ,Mathematics - Number Theory ,Mathematics::Number Theory ,11F33, 11F11, 11F03 ,010102 general mathematics ,Modular form ,Mathematical analysis ,Holomorphic function ,Harmonic (mathematics) ,Mathematical proof ,01 natural sciences ,Cusp form ,03 medical and health sciences ,symbols.namesake ,030104 developmental biology ,Eisenstein series ,FOS: Mathematics ,symbols ,Number Theory (math.NT) ,0101 mathematics ,Hecke operator ,Mathematics - Abstract
Several authors have recently proved results which express cusp forms as p-adic limits of weakly holomorphic modular forms under repeated application of Atkin's U-operator. The proofs involve techniques from the theory of weak harmonic Maass forms, and in particular a result of Guerzhoy, Kent, and Ono on the p-adic coupling of mock modular forms and their shadows. Here we obtain strengthened versions of these results using techniques from the theory of holomorphic modular forms.
- Published
- 2016
35. Finite-dimensional period spaces for the spaces of cusp forms
- Author
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Subong Lim and Dohoon Choi
- Subjects
Cusp (singularity) ,Pure mathematics ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Modular form ,Space (mathematics) ,01 natural sciences ,Cusp form ,Cohomology ,Algebra ,010104 statistics & probability ,Upper half-plane ,Isomorphism ,0101 mathematics ,Mathematics ,Vector space - Abstract
Let k be a positive integer and Pk ⊂ C[X] the set of polynomials of degree less than or equal to k. There exists an isomorphism, called the Eichler–Shimura isomorphism, between the space of cusp forms of integral weight k and a first parabolic cohomology group with coefficient module Pk-2. Moreover, Pk-2 contains period functions of cusp forms of weight k. The Eichler–Shimura isomorphism was extended to the space of cusp forms of real weight with coefficient module P. Here, in contrast to the case of integral weight P is an infinite-dimensional vector space consisting of holomorphic functions on the complex upper half plane with a certain growth condition. However, period functions of cusp forms of real weight have not been described in terms of a finite-dimensional space even for the case of half-integral weight. In this paper, we construct a new isomorphism between the space of cusp forms of real weight and an Eichler–Shimura cohomology group with coefficient module P so that we obtain a finite-dimensional space of period functions containing those of cusp forms up to coboundaries. As applications, we prove analogues of the Haberland formula for the case of real weight, and we construct injective linear maps from the spaces of mixed mock modular forms to the spaces of quantum modular forms and classify Zariski closures in A1(C) of images of mixed mock modular forms under these linear maps in terms of their weights.
- Published
- 2016
36. The notion of cusp forms for a class of reductive symmetric spaces of split rank 1
- Author
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Job J. Kuit, Henrik Schlichtkrull, and Erik P. van den Ban
- Subjects
Cusp (singularity) ,Pure mathematics ,Class (set theory) ,Direct sum ,reductive symmetric space ,Mathematics::Number Theory ,Closure (topology) ,discrete series ,Rank (differential topology) ,Space (mathematics) ,Cusp form ,43A80 ,Discrete series ,Taverne ,22E46 ,cusp form ,cuspidal integral ,22E30 ,Mathematics - Abstract
We study a notion of cusp forms for the symmetric spaces $G/H$ with $G=\mathrm{SL}(n,{\mathbb{R}})$ and $H=\mathrm{S}(\mathrm{GL}(n-1,{\mathbb{R}})\times \mathrm{GL}(1,{\mathbb{R}}))$ . We classify all minimal parabolic subgroups of $G$ for which the associated cuspidal integrals are convergent and discuss the possible definitions of cusp forms. Finally, we show that the closure of the direct sum of the discrete series representations of $G/H$ coincides with the space of cusp forms.
- Published
- 2019
37. On the cusp form motives in genus 1 and level 1
- Author
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Carel Faber and Caterina Consani
- Subjects
Cusp (singularity) ,Pure mathematics ,Mathematics - Number Theory ,010102 general mathematics ,Representation (systemics) ,11F11, 11G18, 14C25, 14H10 ,01 natural sciences ,Cusp form ,Moduli space ,law.invention ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Projector ,Symmetric group ,law ,Genus (mathematics) ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
We prove that the moduli space of stable n-pointed curves of genus one and the projector associated to the alternating representation of the symmetric group on n letters define (for n>1) the Chow motive corresponding to cusp forms of weight n+1 for SL(2,Z). This provides an alternative (in level one) to the construction of Scholl., 18 pages. To appear in Moduli Spaces and Arithmetic Geometry, Advanced Studies in Pure Mathematics, 2006
- Published
- 2019
38. On the distribution of periods of holomorphic cusp forms and zeroes of period polynomials
- Author
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Asbjørn Christian Nordentoft
- Subjects
Cusp (singularity) ,0303 health sciences ,Pure mathematics ,Distribution (number theory) ,Mathematics - Number Theory ,General Mathematics ,Mathematics::Number Theory ,010102 general mathematics ,Asymptotic distribution ,01 natural sciences ,Cusp form ,03 medical and health sciences ,Equidistributed sequence ,Joint probability distribution ,FOS: Mathematics ,Kloosterman sum ,Number Theory (math.NT) ,0101 mathematics ,Twist ,030304 developmental biology ,Mathematics - Abstract
In this paper we determine the limiting distribution of the image of the Eichler--Shimura map or equivalently the limiting joint distribution of the coefficients of the period polynomials associated to a fixed cusp form. The limiting distribution is shown to be the distribution of a certain transformation of two independent random variables both of which are equidistributed on the circle $\mathbb{R}/\mathbb{Z}$, where the transformation is connected to the additive twist of the cuspidal $L$-function. Furthermore we determine the asymptotic behavior of the zeroes of the period polynomials of a fixed cusp form. We use the method of moments and the main ingredients in the proofs are additive twists of $L$-functions and bounds for both individual and sums of Kloosterman sums., Comment: 20 pages, minor changes following referee's suggestions (published in IMRN)
- Published
- 2019
- Full Text
- View/download PDF
39. Central values of additive twists of cuspidal $L$-functions
- Author
-
Asbjorn Christian Nordentoft
- Subjects
Cusp (singularity) ,Pure mathematics ,Distribution (number theory) ,Mathematics - Number Theory ,Applied Mathematics ,General Mathematics ,Holomorphic function ,Of the form ,Cusp form ,Dirichlet character ,11F67(primary), and 11M41(secondary) ,FOS: Mathematics ,Asymptotic formula ,Isomorphism ,Number Theory (math.NT) ,Mathematics - Abstract
Additive twists are important invariants associated to holomorphic cusp forms; they encode the Eichler--Shimura isomorphism and contain information about automorphic $L$-functions. In this paper we prove that central values of additive twists of the $L$-function associated to a holomorphic cusp form $f$ of even weight $k$ are asymptotically normally distributed. This generalizes (to $k\geq 4$) a recent breakthrough of Petridis and Risager concerning the arithmetic distribution of modular symbols. Furthermore we give as an application an asymptotic formula for the averages of certain 'wide' families of automorphic $L$-functions, consisting of central values of the form $L(f\otimes \chi,1/2)$ with $\chi$ a Dirichlet character., Comment: 38 pages, small changes according to the referee's comments (accepted for publication in Crelle)
- Published
- 2018
40. Resonance for Maass forms in the spectral aspect
- Author
-
Nathan Salazar
- Subjects
Cusp (singularity) ,symbols.namesake ,Pure mathematics ,Maass cusp form ,Mathematics::Number Theory ,Holomorphic function ,symbols ,Oscillatory integral ,Asymptotic expansion ,Cusp form ,Convexity ,Mathematics ,Ramanujan's sum - Abstract
Let f be a Maass cusp form for Γ0(N) with Fourier coefficients λf (n) and Laplace eigenvalue 1/4 + k. For real α 6= 0 and β > 0 consider the sum: ∑ n λf (n)e(αn )φ ( n X ) , where φ is a smooth function of compact support. We prove bounds on the second spectral moment of this sum, with the eigenvalue tending toward infinity. When the eigenvalue is sufficiently large we obtain an average bound for this sum in terms of X. The method is adopted from proofs of subconvexity bounds for Rankin-Selberg L-functions for GL(2)×GL(2). It contains in particular the Kuznetsov trace formula and an asymptotic expansion of a well-known oscillatory integral with an enlarged range of K ≤ L ≤ K1−e. The same bounds can be proved in an analogous way for holomorphic cusp forms. Furthermore, we prove similar bounds for ∑ n λf (n)λg(n)e(αn )φ ( n X ) , where g is a holomorphic cusp form. As a corollary, we obtain a subconvexity bound for the L-function L(s, f × g). This bound has the significant property of breaking convexity even with a trivial bound toward the Ramanujan Conjecture.
- Published
- 2018
41. On exponential sums involving Fourier coefficients of cusp forms over primes
- Author
-
Fei Hou
- Subjects
010101 applied mathematics ,Cusp (singularity) ,Algebra and Number Theory ,Mathematics::Number Theory ,010102 general mathematics ,Mathematical analysis ,0101 mathematics ,01 natural sciences ,Cusp form ,Fourier series ,Mathematics ,Exponential function - Abstract
In this paper, we study the exponential sums involving Fourier coefficients of cusp forms for SL ( 2 , Z ) over primes. We are able to establish some new results. In addition, similar results have been obtained for the coefficients of symmetric-square lifts.
- Published
- 2016
42. Sums of Fourier coefficients of cusp forms of level D twisted by exponential functions
- Author
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Huan Liu
- Subjects
Cusp (singularity) ,010102 general mathematics ,Mathematical analysis ,Holomorphic function ,010103 numerical & computational mathematics ,01 natural sciences ,Cusp form ,Exponential function ,Combinatorics ,Mathematics (miscellaneous) ,Beta (velocity) ,0101 mathematics ,Fourier series ,Mathematics - Abstract
Let g be a holomorphic or Maass Hecke newform of level D and nebentypus χD, and let ag(n) be its n-th Fourier coefficient. We consider the sum \({S_1} = \sum {_{X 0}{a_g}\left( n \right)e\left( {\alpha {n^\beta }} \right)\phi \left( {n/X} \right)}\) with o(x) ∈ Cc∞ (0,+∞) and prove that S2 has better upper bounds than S1 at some special α and β.
- Published
- 2016
43. Zeros of a family of approximations of Hecke L-functions associated with cusp forms
- Author
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Arindam Roy, Alexandru Zaharescu, and Junxian Li
- Subjects
Cusp (singularity) ,Algebra and Number Theory ,Approximations of π ,010102 general mathematics ,Holomorphic function ,01 natural sciences ,Cusp form ,Combinatorics ,Distribution (mathematics) ,Number theory ,Modular group ,0103 physical sciences ,Functional equation (L-function) ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
We consider a family of approximations of a Hecke L-function \(L_f(s)\) attached to a holomorphic cusp form f of positive integral weight k with respect to the full modular group. These families are of the form $$\begin{aligned} L_f(X;s):=\sum _{n\le X}\frac{a(n)}{n^s}+(-1)^{k/2}(2\pi )^{-(1-2s)}\frac{\Gamma \left( \tfrac{k+1}{2}-s\right) }{\Gamma \left( \tfrac{k-1}{2}+s\right) }\sum _{n\le X}\frac{a(n)}{n^{1-s}}, \end{aligned}$$ where \(s=\sigma +it\) is a complex variable and a(n) is a normalized Fourier coefficient of f. From an approximate functional equation, one sees that \(L_f(X;s)\) is a good approximation to \(L_f(s)\) when \(X=t/2\pi \). We obtain vertical strips where most of the zeros of \( L_f(X;s) \) lie. We study the distribution of zeros of \(L_f(X;s)\) when X is independent of t. For \(X=1\) and 2, we prove that all the complex zeros of \(L_f(X;s)\) lie on the critical line \(\sigma =1/2\). We also show that as \(T\rightarrow \infty \) and \( X=T^{o(1)} \), \(100\,\%\) of the complex zeros of \( L_f(X;s) \) up to height T lie on the critical line. Here by \(100\,\%\) we mean that the ratio between the number of simple zeros on the critical line and the total number of zeros up to height T approaches 1 as \(T\rightarrow \infty \).
- Published
- 2016
44. COMPUTATIONS OF SPACES OF PARAMODULAR FORMS OF GENERAL LEVEL
- Author
-
David S. Yuen, Cris Poor, and Jeffery Ii Breeding
- Subjects
Cusp (singularity) ,Pure mathematics ,Conjecture ,Group (mathematics) ,Mathematics::Number Theory ,General Mathematics ,Computation ,010102 general mathematics ,Zero (complex analysis) ,010103 numerical & computational mathematics ,Mathematical proof ,01 natural sciences ,Cusp form ,Algebra ,symbols.namesake ,Fourier transform ,symbols ,0101 mathematics ,Mathematics - Abstract
This article gives upper bounds on the number of Fourier- Jacobi coecients that determine a paramodular cusp form in degree two. The level N of the paramodular group is completely general throughout. Additionally, spaces of Jacobi cusp forms are spanned by using the theory of theta blocks due to Gritsenko, Skoruppa and Zagier. We combine these two techniques to rigorously compute spaces of paramodular cusp forms and to verify the Paramodular Conjecture of Brumer and Kramer in many cases of low level. The proofs rely on a detailed description of the zero dimensional cusps for the subgroup of integral elements in each paramodular group.
- Published
- 2016
45. Oscillations of Fourier coefficients of cusp forms over primes
- Author
-
Guangshi Lü and Fei Hou
- Subjects
Cusp (singularity) ,Algebra and Number Theory ,Group (mathematics) ,010102 general mathematics ,Mathematical analysis ,Holomorphic function ,01 natural sciences ,Cusp form ,010101 applied mathematics ,Combinatorics ,Maass cusp form ,0101 mathematics ,Constant (mathematics) ,Fourier series ,Mathematics - Abstract
Let f be a primitive holomorphic or Maass cusp form for the group SL ( 2 , Z ) , and a f ( n ) its nth normalized Fourier coefficient. It is proved that, for any α , β ∈ R , there exists an effective positive constant c such that ∑ n ≤ N Λ ( n ) a f ( n ) e ( α n 2 + β n ) ≪ f N exp ( − c log N ) .
- Published
- 2016
46. Subconvexity for sup-norms of cusp forms on $$\mathrm{PGL}(n)$$ PGL ( n )
- Author
-
Valentin Blomer and Péter Maga
- Subjects
Cusp (singularity) ,General Mathematics ,010102 general mathematics ,General Physics and Astronomy ,Geometry ,Diophantine approximation ,01 natural sciences ,Cusp form ,Omega ,Combinatorics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Let F be an \(L^2\)-normalized Hecke Maas cusp form for \(\Gamma _0(N) \subseteq {\mathrm{SL}}_{n}({\mathbb {Z}})\) with Laplace eigenvalue \(\lambda _F\). If \(\Omega \) is a compact subset of \(\Gamma _0(N)\backslash {\mathrm{PGL}}_n/\mathrm{PO}_{n}\), we show the bound \(\Vert F|_{\Omega }\Vert _{\infty } \ll _{ \Omega } N^{\varepsilon } \lambda _F^{n(n-1)/8 - \delta }\) for some constant \(\delta = \delta _n> 0\) depending only on n.
- Published
- 2016
47. Simultaneous sign change of Fourier-coefficients of two cusp forms
- Author
-
Sanoli Gun, Winfried Kohnen, and Purusottam Rath
- Subjects
Cusp (singularity) ,Conjugacy class ,Mathematics::Number Theory ,General Mathematics ,Mathematical analysis ,Modular form ,Algebraic number ,Automorphism ,Cusp form ,Fourier series ,Mathematics ,Sign (mathematics) - Abstract
We consider the simultaneous sign change of Fourier coefficients of two modular forms with real Fourier coefficients. In an earlier work, the second author with Sengupta proved that two cusp forms of different (integral) weights with real algebraic Fourier coefficients have infinitely many Fourier coefficients of the same as well as opposite sign, up to the action of a Galois automorphism. In the first part, we strengthen their result by doing away with the dependency on the Galois conjugacy. In fact, we extend their result to cusp forms with arbitrary real Fourier coefficients. Next we consider simultaneous sign change at prime powers of Fourier coefficients of two integral weight Hecke eigenforms which are newforms. Finally, we consider an analogous question for Fourier coefficients of two half-integral weight Hecke eigenforms.
- Published
- 2015
48. A note on small gaps between nonzero Fourier coefficients of cusp forms
- Author
-
Satadal Ganguly and Soumya Das
- Subjects
Cusp (singularity) ,Pure mathematics ,Elliptic curve ,Integer ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Interval (graph theory) ,Fourier series ,Cusp form ,Mathematics - Abstract
It is shown that there are infinitely many primitive cusp forms f of weight 2 with the property that for all X large enough, every interval (X, X + cX(1/4)), where c > 0 depends only on the form, contains an integer n such that the n-th Fourier coefficient of f is nonzero.
- Published
- 2015
49. Rankin-Cohen Brackets on Hilbert Modular forms and Special values of certain Dirichlet series
- Author
-
Moni Kumari and Brundaban Sahu
- Subjects
11F60 ,Cusp (singularity) ,Pure mathematics ,Mathematics::Number Theory ,General Mathematics ,adjoint map ,Modular form ,11F68 ,Rankin-Cohen brackets ,11F41 ,Mathematics::Geometric Topology ,Cusp form ,Ramanujan's sum ,symbols.namesake ,Genus (mathematics) ,symbols ,Dirichlet series ,Fourier series ,Hilbert modular forms ,Siegel modular form ,Mathematics - Abstract
Given a fixed Siegel cusp form of genus two, we consider a family of linear maps between the spaces of Siegel cusp forms of genus two by using the Rankin–Cohen brackets and then we compute the adjoint maps of these linear maps with respect to the Petersson scalar product. The Fourier coefficients of the Siegel cusp forms of genus two constructed using this method involve special values of certain Dirichlet series of Rankin type associated to Siegel cusp forms. This is a generalisation of the work due to Kohnen (Math Z 207:657–660, 1991) and Herrero (Ramanujan J 36:529–536, 2015) in the case of elliptic modular forms to the case of Siegel cusp forms which is also considered earlier by Lee (Complex Var Theory Appl 31:97–103, 1996) for a special case.
- Published
- 2018
50. Average behavior of Fourier coefficients of Maass cusp forms for hyperbolic $$3$$ 3 -manifolds
- Author
-
Yujiao Jiang and Guangshi Lü
- Subjects
Combinatorics ,Cusp (singularity) ,Sums of powers ,General Mathematics ,Mathematical analysis ,Lambda ,Fourier series ,Cusp form ,Mathematics - Abstract
Let \(\lambda _\phi (n)\) be the \(n\)-th Fourier coefficient of a doubly even and normalized Hecke–Maass cusp form for hyperbolic \(3\)-manifolds. In this paper, we investigate the behavior of summatory functions in the following (i) the \(j\)-th power sum of \(\lambda _\phi (n)\) $$\begin{aligned} \sum _{N(n)\le x}\lambda _\phi (n)^{j}, \end{aligned}$$ where \(j\le 8;\) (ii) the sum of \(\lambda _\phi (n)\) over the sparse sequence \({n^l}\) $$\begin{aligned} \sum _{N(n)\le x}\lambda _\phi (n^l), \end{aligned}$$ where \(l\le 4;\) (iii) the hybrid sum for \(\lambda _\phi (n)\) $$\begin{aligned} \sum _{N(n)\le x}\lambda _\phi (n^l)^{j}, \end{aligned}$$ where \(2\le l\le 4, j=2,\) or \(l=2, j=4.\)
- Published
- 2015
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