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Results tagged with automorphic-forms
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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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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 p …
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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 t …
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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 comput …
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Behaviour of a certain $L$ function at $s=1$
I was going through this paper. Corollary 7.3.4 says the $L$-function $L(s,\pi, \rm{sym}^4)$ is holomorphic except possibly at $s=0,1$ and gives a necessary and sufficient condition for it to have a p …
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conductor formula
Let $\pi_p$ be an irreducible representation of $GL_2(\mathbb{Q}_p)$. Assume $\pi_p$ is ramified,hence it will have a positive conductor. Consider $sym^3(\pi_p)$ which is a representation of $GL_4(\ma …
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A question related to Hilbert modular form
This is a question related to Hilbert modular forms.
Let $\mathbb{K}=\mathbb{Q}(\sqrt D)$ be an imaginary quadratic field with discriminant $D<0$ and $\zeta (\text{mod } m)$ a Hecke character such tha …
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Analogous theorem for Hilbert modular forms
I have studied modular forms and saw a correspondence like a newform correspond to a automorphic representation of $\mathrm{GL}_n(\mathbb{A_Q})$. Does any similar result holds for Hilbert modular for …
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Automorphic representation of GL(1)
These might be very silly questions, but somehow I am not able to understand it or I might have misunderstood something.
I am reading automorphic forms from this book.
What I have understood till now: …
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Symmetric power lift of modular forms
Let $f_1$ and $f_2$ be two cuspforms of weights $k_1$ and $k_2$ and nebentypus $\epsilon_1$ and $\epsilon_2$ respectively such that $f_1 \neq f_2 \otimes \chi$ for some Dirichlet character $\chi$ of o …
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