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...an [[orthonormal basis]] of <math>S_k(\Gamma(1))</math>, the space of cusp forms of weight <math>k>2</math> on <math>SL_2(\mathbb{Z})</math>. Then for any p * [[Henryk Iwaniec]]: ''Topics in Classical Automorphic Forms''. [[Graduate Studies in Mathematics]] 17, American Mathematics Society, P ...

...onnects [[Kloosterman sum]]s at a deep level with the spectral theory of [[automorphic form]]s. Originally this could have been stated as follows. Let ...n sums he was able to derive estimates for Fourier coefficients of modular forms in cases where [[Pierre Deligne]]'s proof of the [[Weil conjectures]] was n ...

...H''(''z'') is any [[meromorphic function]] on ''D'', then one obtains an [[automorphic function]] by averaging over &Gamma;: ...ion | last1=Kollár | first1=János | title=Shafarevich maps and automorphic forms | publisher=[[Princeton University Press]] | series=M. B. Porter Lectures | ...

| known_for = {{ubl|[[Automorphic forms]]|[[Jacquet module]]|[[Jacquet–Langlands correspondence]]|[[Arthur–Selberg ...founders of the theory of [[Automorphic forms#Automorphic representations|automorphic representations]] and their associated [[L-functions]], and his results pla ...

..._and_their_derivates.pdf "Nonvanishing theorems for L-functions of modular forms and their derivatives"]. ''[[Inventiones Mathematicae]]'', 102(1), pp.&nbsp ...riedberg, S., & Hoffstein, J. (1996). "On some applications of automorphic forms to number theory", ''Bulletin of the American Mathematical Society'', 33(2) ...

{{DISPLAYTITLE:Automorphic ''L''-function}} ...ction ''L''(''s'',π,''r'') of a complex variable ''s'', associated to an [[automorphic representation]] π of a [[reductive group]] ''G'' over a [[global field]] a ...

{{short description|Type of theorem in automorphic forms}} ...orem states that a representation of an algebraic group over the adeles is automorphic whenever the L-functions of various twists of it are well-behaved. ...

...long been a standard tool for studying analytic properties of automorphic forms and their ''L''-functions. There have been numerous results coming out the *{{Citation | last1=Good | first1=Anton | title=Cusp forms and eigenfunctions of the Laplacian| year=1984 | journal=Mathematische Anna ...

In the mathematical theory of [[automorphic representation]]s, a '''multiplicity-one theorem''' is a result about the [ ...),&nbsp;{{math|''ω''}}) denote the [[cuspidal representation|space of cusp forms with central character &omega;]] on ''G''('''A'''). This space decomposes i ...

...r form]]s is another modular form, generalizing the product of two modular forms. ...ions for [[polynomial]]s in [[derivative]]s of modular forms to be modular forms, and {{harvs|txt|last=Cohen|authorlink=Henri Cohen (number theorist)|year=1 ...

[[Category:Automorphic forms]] ...

...oral dissertation ''Arithmetic Quotients of the Complex 2-Ball and Modular Forms of Nebentypus'' was supervised by [[Ilya Piatetski-Shapiro]].<ref>{{MathGen ...he goal is to characterize the L-functions that originate from automorphic forms. For <math>GL_2</math> this was solved by [[Hervé Jacquet]] and [[Robert La ...

...ology]] instead of real cohomology, which relates the coefficients of cusp forms to eigenvalues of [[Frobenius endomorphism|Frobenius]] acting on these grou ...math> depends on ''n'' (Shimura, "Intruduction to the arithmetic theory of automorphic functions", Theorem 8.4) ...

In [[mathematics]], a '''Jacobi form''' is an [[automorphic form]] on the [[Jacobi group]], which is the [[semidirect product]] of the ...eight representations of affine [[Kac–Moody algebra]]s. Meromorphic Jacobi forms appear in the theory of [[Mock modular form]]s. ...

...includes [[Rankin–Selberg method|Rankin–Selberg]] products for cuspidal [[automorphic representation]]s of [[general linear group]]s. The method develops the the ...step refines the crude functional equation of a globally generic cuspidal automorphic representation <math>\pi = \otimes' \pi_v</math> to individual functional e ...

[[Category:Automorphic forms]] ...

...mber in the upper half-plane in which case the theta constants are modular forms, or more generally may be an element of a Siegel upper half plane in which [[Category:Automorphic forms]] ...

...cquet | first1=H. | last2=Langlands | first2=Robert P. | title=Automorphic forms on GL(2) | url=http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/JL.ht [[Category:Automorphic forms]] ...

...l integrals are an important technical tool in the theory of [[automorphic forms]], where they enter into the formulation of various [[trace formula (disamb [[Category:Automorphic forms]] ...

..., known for his research in number theory, algebraic geometry, and modular forms. ...th>_<em>p</em>{{brackets|<em>X</em>}}) attached to ordinary cusp forms," published in 1986 in [[Inventiones Mathematicae]].<ref>{{cite press relea ...