Questions tagged [automorphic-forms]

Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.

In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group G to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup $\Gamma \subset G$ of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups (Wikipedia).

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The Langlands program for beginners

Assuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things must a person know before they can understand the Langlands program and its geometric analogue? What are the good books…
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Why are $L$-functions a big deal?

I've been studying modular forms this semester and we did a lot of calculations of $L$-functions, e.g. $L$-functions of Dirichlet-characters and $L$-functions of cusp-forms. But I somehow don't see, why they are considered a big deal. To me it…
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Undergraduate roadmap for Langlands program and its geometric counterpart

What are the topics which an undergraduate with knowledge of algebra, galois theory and analysis learn to understand Langlands program and its goemetric counterpart? I would also like to know what are the good books for these topics, in what order…
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Reference for automorphic forms

I would like to know some reference to learn the theory of automorphic forms. Any (good) book or online lecture notes will be fine. I am particularly interested in the arithmetic point of view (e.g. galois representations associated).
user10676
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How important are automorphic representations among admissible ones?

I'm currently studying automorphic representations on Bump's book "Automorphic forms and representations" and on Gelbart's "Automorphic forms on Adele groups". And I have some problems in understanding the general philosophy behind automorphic…
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Concrete example of non-abelian class field theory - why Langlands program *is* a non-abelian class field theory?

Abelian class field theory generalizes quadratic reciprocity laws for general number fields with abelian Galois groups, which connects class groups and Galois groups via Artin's reciprocity map. Also, quadratic reciprocity gives us some explicit…
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Residue of Rankin-Selberg L-function for non-trivial nebentypus

Let $f\in S_k(\Gamma_0(N),\chi)$ be a normalized holomorphic newform (i.e. weight $k$, level $N$, nebentypus $\chi$) and write its Fourier expansion as $$ f(z)=\sum_{n\ge 1} \lambda_f(n)n^{(k-1)/2}e^{2\pi i n z}$$ for $\Im z >0$ where…
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Do general automorphic factors arise in some canonical way?

Disclaimer: none of what I'm about to say I understand particularly well; corrections and clarifications are not only welcome but will be accepted with great gratitude. The wikipedia article on Factors of Automorphy gives the closest to an intuitive…
Vladimir Sotirov
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Fuchsian Groups of the First Kind and Lattices

I am trying to compare various definitions and theorems I have seen recently concerning Fuchsian groups. Some of these seem to contradict each other, so I was hoping to get some clarification. First, a Fuchsian group is a discrete subgroup of…
Colin Defant
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Must automorphic forms be square-integrable modulo the center?

I've recently needed to learn the basics of automorphic forms and automorphic representations. I've seen two apparently different definitions of automorphic forms, and I'm wondering which is more standard, or if they are in fact equivalent. In…
Keenan Kidwell
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Questions about the L-function for Eisenstein Series

$E_a(z,s)$ denotes the Eisenstein series expanded at the cusp $a$. For each cusp $a=\frac{u}{w}$ of $\Gamma_0(N)$, we define the Eisenstein series $$ \begin{eqnarray}E_a(z,s)&=&\sum_{\tau\in\Gamma_a\backslash\Gamma}\Im(\sigma^{-1}_a\tau…
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Where was it originally proved that $K$-finite automorphic forms are of uniform moderate growth?

I'm providing an overview of some classical results about automorphic forms in one section of a paper I'm working on, and I've realized I don't have a good reference for the following. Let $k$ denote a number field, $\mathbb{A}$ the ring of…
Dan
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An Efficient Route to Tate's Thesis

I want to learn Tate's thesis. My advisor suggested the Book "Algebraic Number Theory" By Lang. However, it seems to be a long read before I reach Tate's Thesis. I want to know what are other good references to learn Tate's Thesis. My background is…
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Deligne-Lusztig theory and cuspidal representations of $\mathrm{GL}_{2}(\mathbb{F}_{q})$

I heard that Deligne-Lusztig theory gives us geometric way to construct representations of finite algebraic groups over finite fields, such as $\mathrm{GL}_{2}(\mathbb{F}_{q})$, which arises from etale cohomology of some variety over a finite field.…
Seewoo Lee
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A convergence lemma for adelic zeta function in automorphic forms

I'm reading Godement-Jacquet's classic Zeta functions of simple algebras (1972, Springer). On page 153, the first line: " We also $\textbf{take for granted}$ the following lemma (numbered 11.3)..." The lemma: for $\Phi\in \mathcal{S}(\mathbb{A}_F)$…
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