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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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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 Gros …
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Does weight $2$ cuspidal Bianchi modular form have infinitely many zero Fourier coefficient ...
Let $K$ be an imaginary quadratic field. Let $f \in S_2(\mathfrak{n})$ be a weight $2$ cuspidal cof level $\Gamma_0(\mathfrak{n})$ over $K$ (for definitions one can see http://www.lmfdb.org/knowledge/ …
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What automorphic forms are expected to occur in the zeta function of moduli space of curves?
Assume $g \geq 1$ and $n \geq 0$, the moduli stack ${\mathcal {M}}_{g,n}$ classifies families of smooth projective curves of genus $g$ with $n$ marked points , together with their isomorphisms. It has …
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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 cuspi …
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Restriction of irreducible representations from $G(\mathbb Q_p)$ to $[G, G](\mathbb Q_p)$
Let $\pi_p$ be a smooth irreducible representation of $G(\mathbb Q_p)$, where $G$ is a connected reductive group over $\mathbb Q_p$. Consider the restriction of $\pi_p$ to $[G, G](\mathbb Q_p)$, how d …
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Galois representations attached to discrete automorphic representations
Let $F$ be a totally real field. Let $G$ be a (split) connected reductive group over $F$. Let $\pi$ be an irreducible automorphic representation of $G$.
Recall in the work of Buzzard and Gee "The conj …
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When does a locally symmetric space have no odd degree Betti numbers?
Let $G$ be a semisimple real lie group, $K$ be a maximal compact subgroup of $G$, $\Gamma$ be a torsion-free cocompact discrete subgroup. The Betti number the locally symmetric space $X_{\Gamma}:=\Gam …
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Higher dimensional generalization of an identity between traces of Hecke operators and numbe...
In http://www.math.ubc.ca/~behrend/ladic.pdf, the author uses his generalization of Lefschetz trace formula to smooth algebraic stacks to prove an interesting identity (Proposition 6.4.11.):
$\sum_{k …
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Explicit Jacquet-Langlands correspondence for real reductive groups
Let $G$ be a connected reductive group over $\mathbb R$. Let $G'$ over $\mathbb R$ be an inner form of $G$ with ${}^LG={}^LG'$. By local Langlands correspondence over $\mathbb R$, if a $L$-packet of $ …
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