# Tagged Questions

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

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### $A(z)$ is a automorphic form $\Rightarrow$ $A(-1/z)z^{-k}$ is a form

I've found this: Let A be an automorphic form (holomorphic) of degree $-k$, where $k\ge 2$. Call $a_0$ the constant term in its expansion at $i\infty$, $q$ the period of the form $A(-1/z)z^{-k}$ ...
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### Class Preserving Autmorphisms [closed]

What are Camina p groups? What special properties do Camina groups of Class 2 have over those of class 3?
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### Integral formula for the local L factor of a base changed automorphic representation

Let $\Bbb A$ the ring of rational adeles and let $\pi=\bigotimes_{p\leq\infty}\pi_p$ be an automorphic (cuspidal) representation of ${\rm GL}_2(\Bbb A)$. Fix a quadratic extension $K\supset\Bbb Q$. ...
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### Does viewing an Eisenstein series as a sum over cusps explain the antagonism between Eisenstein serieses and cusp forms?

I'm trying to understand the relationship between various aspects of the concept of "Eisenstein series" (as discussed for example in Diamond & Shurman's "A First Course in Modular Forms"), in ...
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### Why is it so difficult to use geometric methods to construct weight one modular forms?

I am trying to understand why it is that geometric methods of constructing modular forms of weight one for $SL(2,\mathbb Z)$ fail. I have a relatively rudimentary understanding of it, but it would be ...
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### Complex Galois Representaions

I'm trying to understand 1 and 2 dimensional complex representations, induced representations and associated Artin L-functions for a project. I'm finding it hard to find appropriate material to help ...
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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 ...
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### The Fricke involution on arbitrary subgroups of $\operatorname{SL}_2(\textbf{Z})$

Let $\Gamma$ be a subgroup of $\operatorname{SL}_2(\textbf{Z})$. I want to understand how the Fricke involution $w_N : f \longrightarrow (\tau \longrightarrow N^{-k/2}\tau^{-k}f(-1/N\tau))$ acts on ...
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### A sufficient condition for $\eta$-quotients to be modular forms

Let $f$ be the form : $$f(\tau)=\prod_{M\mid N}{\eta(M\tau)^{a_M}} \quad (\tau \in \mathcal{H})$$ Generally, we said that $f$ is an $\eta$-quotient when $(a_M)$ is a sequence of integers. One can ...
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### On the partial zeta function

Let $F$ be a number field and $S$ be a finite set of places of $F$ including archimedian places. Let $\zeta^S(s)$ be the partial L-function, that is the meromorphic continuation of the product of ...
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### Strong Approximation for adelic quaternionic groups

Let $H$ be a definite quaternion algebra defined over $\mathbb{Q}$, unramified at $p$, and ramified at $\infty$. Denote by $D$ the multiplicative group $H^\times$ divided by its center. In this paper,...
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### phase portraits of automorphic functions (a la Indra's Pearls)

I would like to make movies of phase portraits of automorphic/kleinian functions with varying traces -- On page 375 of Indra's Pearls there is a phase portrait of an automorphic function/fonction ...
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### How to show that a representation is supercuspidal?

Let $G$ be a reductive group. If we know a representation $\pi$ of $G$ explicitly, how could we determine that $\pi$ is supercuspidal or not? Are there some references about this? Thank you very much.
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### Whittaker models and local L-functions.

If we have a Whittaker model, how to associate a local L-function to it? Are there some references? Thank you very much.
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### How to prove that $\zeta*\zeta=\zeta$?

Let $F$ be a non-archimedean local field and $\mathcal{O}_F$ the ring of integers in $F$. Let $G_F=GL_2(F)$. Let $\pi_i$, $i=1,\ldots,n$,be non-equivalent finite dimensional irreducible ...
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### Abstract Algebra: Automorphism and irreducible

I done how to prove this the question is A. Show that there is an automorphism of the ring of rational number $\mathbb{Q}(i)[x]$that has order of $4$. I suppose i can say that if $\varphi$ is an ...
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### Why a modular form is a highest weight vector of a discrete series summand of $L_2(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}))$?

It is said that a modular form is a highest weight vector of a discrete series summand of $L_2(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}))$. Why a modular form is a highest weight vector of a ...
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### Invariance of measure on upper half plane

The upper half plane has the measure $|y|^{-2}dxdy$. Show that it is invariant under the action of $SL(2, \mathbb{R})$. I don't understand what any of this means. First, I don't understand what they ...
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### Equidistribution of lattice points on spheres in dimensions $d \ge 4$ (Pommerenke's theorem)

I am looking for either an English translation of: Pommerenke, C.: Über die Gleichverteilung von Gitterpunkten auf m-dimensionalen Ellipsoiden. Acta Arith. 5, 227-257 (1959) Or a reference in ...
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### Help with an exercise in Shimura's book

Exercise 3.26(B) in Shimura's book Introduction to the Arithmetic Theory of Automorphic functions says the following. Let $\Gamma=SL_n(\mathbb{Z})$ and $G=GL_n(\mathbb{Q})$. Let $R^{(n)}_p$ denote ...
### writing $M : \Gamma_{n,0} \backslash \Gamma_n$
Let $\Gamma_n = \operatorname{Sp}_n(\Bbb Z)$ and $\Gamma_{n,0} \subset \Gamma_n$ be a subgroup. We write $$M = \begin{pmatrix}A & B\\ C & D \end{pmatrix} \in \operatorname{Sp}_n(\mathbb{Z})$$ ...