# Questions tagged [automorphic-forms]

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.

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### Question on the proper sub-representation of induced representation

$\DeclareMathOperator\Ind{Ind}$Let $G$ be a reductive group over a $p$-adic local field $F$, and $P=MN$ a parabolic subgroup. Let $\sigma$ be an irreducible representation of $M(F)$ and consider its ...
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### The Fontaine Mazur conjecture for $\text{GL}_1$ over $\mathbf{Q}$

The Fontaine-Mazur cojectures says that if $\chi : \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \mathbf{Q}_p^\times$ is a character which is unramified almost everywhere and de Rham at $p$ then it ...
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### Level vs. conductor of a supercuspidal representation

What is the relation between level and conductor of a supercuspidal representation of $\operatorname{GL}_2(\mathbb{Q}_p)$ for some prime $p$? Proposition 3.4 in Loeffler and Weinstein - On the ...
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### Intertwining operator is not an isomorphism?

Let $F$ be a number field and $G$ a symplectic group over $F$. Let $P=MN$ is a maximal parabolic subgroup of $G$ and $W_M=N_G(M)/M$. Since $P$ is maximal, $W_M \simeq S_2$. Let $w$ be a non-trivial ...
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### A question related to newform and irreducible cuspidal representation of $\operatorname{GL}_n$

I was reading adelization of classical automorphic forms and learnt that each cusp form corresponds to an automorphic representation of $\operatorname{GL}_n(\mathbb{A}_\mathbb{Q})$. I understood the ...
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### Conductor of Principal series representation

Let $\mathbb{F}$ be a local field and let $\pi$ be a principal series representation of $GL_2(\mathbb{F})$ that is $\pi=Ind_B^{GL_2}(\chi_1\otimes\chi_2)$ for two characters $\chi_1$ and $\chi_2$ of ...
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### Is weakly holomorphic modular form finitely generated as module of modular function?

Let's use $M_k^{!}(\Gamma_{0}(N))$ to denote weakly holomorphic modular form with weight k, level $\Gamma_0(N)$(in particular, k might be negative). Then obviously $M_0^{!}(\Gamma_{0}(N))$ acts on it ...
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### Does the symmetric square L-function vanish at one?

Take a cuspidal automorphic representation $\pi$ of $GL(3)$ over a number field. My question is quite straightforward and can be related to this one : Can $L(1, \pi, \mathrm{sym}^2)$ be zero? If ...
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### Modular form attached to a hecke character of cubic/quartic extension of rational numbers

We know that for each Hecke character of a quadratic extension of $\mathbb{Q}$, we can define a modular form (in fact a cusp form). We can find this construction in the book Topics in Classical ...
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### Proving automorphy of the Galois representations of number fields without considering the residual representation

All the papers proving automorphy of the representations of Galois groups of number fields that I have come across seem to first reduce the representation modulo a prime, prove the automorphy of the ...
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### Why is Langlands functoriality usually related with period integral in a third group?

In the introduction of "PERIODS OF AUTOMORPHIC FORMS "by HERVE JACQUET, EREZ LAPID, and JONATHAN ROGAWSKI, they said "In many cases, it should be possible to characterize the $H$-distinguished ...
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### Double quotients and the isomorphism theorems

I am working on some first aspects of modular forms and automorphic representations, and would like to understand better the formal dictionary between both. It seems to be essentially based on the ...
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### Fundamental lemma and transfer of characteristic functions of congruent subgroups

Let $E/F$ be an unramified quadratic extension of $p$-adic fields. The Jacquet-Rallis fundamental lemma states that $1_{S_n(O_F)}$ and $(1_{U_n(O_F)}, 0)$ are transfer of each other, see ”On the ...
Let $\pi$ be an automorphic representation of $GSp(4)$. Provided a representation $r$ of the Langlands dual group of $GSp(4)$ (namely, the standard or the spinor one), it is possible to define a ...