# Questions tagged [automorphic-forms]

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

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### Definition unramified local representation

We can define unramified representation of $GL(2,\mathbb Q_p$) as a representation $(\pi , V)$ such that $V^K\neq \{0\}$ for $K$ the maximal compact subgroup i.e. $GL(2,\mathbb Z_p)$. However, I do ...
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### Definition of Bessel/Fourier-Jacobi/Whittaker models

I've seen the notion of the models in the title a lot in the context of automorphic forms and representations, but I wonder if there's any nice reference for the definition of them for general ...
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### Reference for classical Ramanujan conjecture for number fields

The classical Ramanujan conjecture is given in terms of bounds for the Hecke eigenvalues of normalized eigenforms. The modern formulation of Ramanujan conjecture is given in terms of Satake parameters ...
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### Fricke involution’s effect on character

I’m using the definition $W_Nf(\tau)=i^kN^{-k/2}\tau^{-k}f(-1/N\tau)$. Now suppose $f\in M_k(\Gamma_1(N),\chi)$, show $W_Nf\in M_k(\Gamma_1(N),\chi^{-1})$. I know this is pure calculation but I’m ...
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### Why are Rankin-Selberg convolutions different when $n=m$?

Let $\pi$ an automorphic cuspidal representation on $GL_n(\mathbb{A})$ (and similarly $\pi'$ on $GL_m$). For $m=n-1$, it is standard to introduce the Rankin-Selberg L-function as (the gcd of the ...
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### The notions of $K$-finite vectors in admissible representations

Let $G$ be a real Lie group and $K$ be a maximal compact subgroup. Let $(\pi, V)$ be a Hilbert space continuous representation of $G$. It can be restricted to $K$ as a representation of $K$. For each ...
1 vote
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### About holomorphic functions on the upper half plane with respect to SO(2).

Let $\mathbb{H}$ be the upper half plane. For $g=\left(\begin{array}{cc}a&b\\c&d\end{array}\right) \in SL_2(\mathbb{R})$ and $z \in \mathbb{H}$, set $J(g,z)=cz+d$. Let $SO(2)$ be the special ...
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### *Why* the converse theorem (conjecture) should be true?

Currently, the following $\mathrm{GL}(n)$ converse theorem due to Cogdell and Piatetski-Shapiro is known: For an admissible irreducible representation $\pi$ of $\mathrm{GL}(n, \mathbb{A})$, $\pi$ is ...