Questions tagged [automorphic-forms]
Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.
286 questions
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Convergence of Poincaré Series
I am reading Iwaniec’s Topics in Classical Automorphic Forms Chapter 3 on Poincaré Series.
The set up is for some Fuchsian group of first kind $\Gamma$, some multiplier system $\theta$ of weight $k$. ...
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Change-of-sign of the $GL_2$ Maass form Fourier coefficients $\rho_f(m)$ and $\rho_f(-m)$
Suppose that $f$ is a half-integral-weight $GL_2$ Maass form of level $q$ and nebentypus $\chi$. If $\rho_f(m)$ denotes the $m^{th}$ Fourier coefficient of $f$, then it is known that $$\rho_f(-m)=\pm\...
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Understand the definition of the idelic lift of Dirichlet characters
I am following the book of [Goldfeld, Hundley; 2011] "Automorphic Representations and L-Functions for the General Linear Group". The definition of the idelic lift of a Dirichlet character ...
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Failure of multiplicity one of Whittaker models of irreducible admissible representations restricted to $K_v$ at archimedean places
In Cogdell's notes, remark (ii) after stating theorem 4.1, says
When $v| \infty$, if we worked simply with irreducible admissible representations
of the Hecke algebra $H_v$ then the space of (...
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Eulerian Integral representations for $GL_n\times GL_m$ $L$-functions
I am familiar with $L$-functions for $\operatorname{GL}_2\times \operatorname{GL}_1$ (e.g. Bump 'Automorphic Forms and Representations'), and currently reading on $\operatorname{GL}_n\times \...
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Strong Multiplicity One - How to determine Spectral Parameters from Satake Parameters
From Strong Multiplicity One Theorem, if we know the local representation at (almost all) the finite places, the cuspidal automorphic representation is determined uniquely. So, for example, GL(2) ...
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Functional Equation of Eisenstein series using (or not?) symmetry of Laplacian (possible error in Garrett's book)
I am following the proof of the functional equation of the (non-holomorphic) Eisenstein series $E_s$ in Garrett's book "Modern Analysis of Automorphic Forms", Corollary 1.10.5. Let $f=E_{1-s}...
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Different definitions of moderate growth for automorphic forms.
Let $G=\operatorname{GL_2}$, $F$ a number field and $\mathbb A$ is its adele ring. In the definition of an automorphic form $\phi:G(\Bbb A)\to\Bbb C$, we have the notion of moderate growth. I have ...
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Notions of “weights” of Hecke characters
Let $K$ be an imaginary quadratic extension of $\mathbb{Q}$ and let $\chi: \mathbb{A}_{K}^{\times} \rightarrow \mathbb{C}^{\times}$ be a Hecke character over $K$. By class field theory, it is attached ...
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Question about a proof in Diamond and Shurman
In Diamond and Shurman's book "A First Course In Modular Forms", in the middle of a proof that if you have a congruence subgroup $\Gamma$ with $-I\notin\Gamma$, there exist non-zero ...
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the density of $C(\Gamma \backslash SL(2,\mathbb{R}),\chi)$ in $L^2(\Gamma \backslash SL(2,\mathbb{R}),\chi)$
Let $\Gamma$ is a cocompact group of SL(2,$\mathbb{R}$), which acts discontinously on SL(2,$\mathbb{R}$), then SL(2,$\mathbb{R}$)/$\Gamma$ has finite volume(of course their is an invarant measure on ...
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Reference request: Poincare Theta Series' convergence (analytic/algebraic proof)
I am trying to find a proof of the convergence of the Poincare-$\theta$ series absolutely and uniformly on compact subsets of $\mathbb{C}$.
The approach in Ford's 'Automorphic functions' is somewhat ...
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Applications of the Petersson trace formula
The Petersson trace formula (for modular forms) is simpler to establish and use than thge Kuznetsov trace formula, but often I see no applications of it. Can it be used to obtain various of the ...
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A calcultion of lifted character in local Zeta integral in Tate's thesis [closed]
I am reading Tate's thesis from Goldfeld and Hundley's book. I am currently stuck at one calculation in section 2.3 , it says if $v=p$ is ramified prime and $\omega$ is a lift of Dirichlet's character ...
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A question on properties of Mass forms.
I am studying Proposition 3.3.3 from "Automorphic Forms and L-functions for the group $GL_n(\mathbb{R})$" by D. Goldfeld, and I need a clarification regarding the behavior of Maass forms as $...
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