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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Can any pair of associate parabolics be related by opposite parabolics?
Let $G$ be a reductive group, say over an algebraically closed field of characteristic zero.
We have the following definitions for a pair of parabolic subgroups $P_1$ and $P_2$ with Levi quotients $...
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On the notion of cuspidality
Let $k/\mathbb{Q}$ be a number field and $\mathbb{A}$ its ring of adèles. As usual $\mathbb{A} = \mathbb{A_f} \times \mathbb{A_{\infty}}$.
The standard definition of an automorphic representation $(\...
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Shintani's unpublished paper on automorphic forms
I'm trying to find Shintani's preprint:
Shintani T., On automorphic forms on unitary groups of order 3, unpublished, 1979.
It seems to be impossible to find, even though several authors quote it. I ...
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Several L-functions but one Galois representation: How to choose
Let $\mathbf{G}$ be a reductive group which enjoys all the nice properties a reducive group can dream of. Fix $(\mathbf{G},X)$ a Shimura datum associated with it and assume that if $K\leq\mathbf{G} $ ...
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Experiments with Voronoï summation
In order to test my understanding of the Voronoï summation formula, I tried to apply it to a simple estimation of partial sums of Fourier coefficients of cusp forms. The result I obtained cannot ...
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Automorphic classification of different types of abelian surfaces
For elliptic curves over $\mathbb{Q}$ the Mumford-Tate group is either $\mathrm{GL}_2$ or $\mathrm{Res}_\mathbb{Q}^F (\mathbb{G}_m)$ if it has CM with the imaginary quadratic field $F$. In this case ...
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Discrete subgroups of $\text{Sp}_{4}(\mathbb{Q})$ parameterizing polarized Abelian surfaces plus torsion data
I want to start by considering a familiar congruence subgroup of the integral symplectic group $\text{Sp}_{4}(\mathbb{Z})$. For a positive integer $N$, let $\Gamma _{0}^{(2)}(N) \subset \text{Sp} _{4}(...
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Question on the relation of global theta lifting and local theta lift
Let $F$ be a number field and $G$ (resp. $H$) an odd orthogonal (resp. metaplectic group) over $F$.
Let $v$ be a finite place of $F$ and $\sigma_v$ a supercuspidal representation of $G_v(F_v)$. Let $\...
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Correspondence between motives and automorphic representations
What I know:
I understand motives via its realization; in Coates' and Perrin-Riou's paper On $p$-adic L-functions Attached to Motives over $\mathbb{Q}$ (see http://doi.org/10.2969/aspm/01710023), the ...
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Zeroes of certain $L$-functions on the critical line and GGP conjectures
Global Gan-Gross-Prasad conjecture (on various groups) says that nonvanishing of certain automorphic $L$-function $L(s, \pi)$ (of cuspidal representation $\pi$ of some reductive group $G$) at $s = 1/2$...
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Whittaker function oscillation on diagonal near 0
In Proposition 1 of Blomer - Applications of the Kuznetsov formula on GL(3), bounds are given for the Whittaker function
$$W_{\nu_1, \nu_2}(y_1, y_2) = \mathcal W_{\nu_1, \nu_2}\begin{pmatrix} y_1y_2 &...
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The error term for the second moment of Fourier coefficients of cusp forms with the level explicitly determined
There is a basis question which puzzles me for a while. The question is the following:
Let $p$ be a prime and $X\ge 2$. Let $f$ be a $GL_2$-newform of level $p$ and non-trivial
nebentypus, with the $n$...
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Residue of a local $\gamma$-factor and its relation with adjoint $\gamma$-factor
I met the following relation (if my understanding is correct) of local $\gamma$-factors when I was reading Hiraga-Ichino-Ikeda's paper "Formal Degrees and Adjoint $\gamma$-Factors": Let $F$ ...
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A list of $R=T$ theorems for $\mathbf{GSp}_4$
I know of only two cases of theorem $R=T$ where $T$ is a Hecke algebra acting on an automorphic forms (or representations) space of $\mathbf{GSp}_4$, the first one was proved by A.Genestier and J....
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What is the conductor of an automorphic representation for $\Gamma_0(q)$ in $GSp(4)$?
Let $\pi$ be a generic cuspidal automorphic representation on $GSp(4)$, with level $\Gamma_0(q)$ (the group of symplectic matrices with lower left block divisible by $q$), i.e.
$$\Gamma_0(q) = \left\{ ...
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Adelic functions of moderate growth
Let $f:GL_2(\mathbb A_\mathbb Q)\to\mathbb C$.
For $g\in GL_2(\mathbb A)$ and a place $v$ of $\mathbb Q$, define $\|g\|_v=\max_{1\le i,j\le 2}(|g_{ij}|_v,|(g^{-1})_{ij}|_v).$
I have seen several ...
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Petersson norms of quaternionic modular forms
How is the Petersson norm of a quaternionic modular form defined?
Background: In Tamiozzo, On the Bloch-Kato conjecture for Hilbert modular forms, section 3.3, it is written "We normalize $f_B$ ...
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What is the sum operation on torsors induced by Weil uniformization?
Let $k$ be an algebraically closed field, $G$ a reductive group, and $C$ a curve. The algebraic version of the Weil uniformization theorem (see e.g. arXiv:1511.06271v2) says that groupoid of $G$-...
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Modular forms on central division algebra of degree $\ge 3$
I just learned from some online sources including Buzzard's note and Emerton's answer on this MO question about quaternionic modular forms and explicit version of Jacquet-Langlands correspondence in ...
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what is the relationship betwen $L(s,sym^mf\times sym^mg)$ symmetric L function of $f$ and $g$ and $\lambda_{f}(n^m)$, $\lambda_{g}(n^m)$?
what is the relationship betwen $L(s,sym^mf\times sym^mg)$ symmetric L function of $f$ and $g$ and $\lambda_{f}(n^m)$, $\lambda_{g}(n^m)$ ?
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Divergence of integrals in the trace formula
I am trying to understand the following situation for $G=GL(2)$, when going from the compact trace formula to the non-compact case.
The integral over $G(\mathbb{A})^1_\gamma \backslash G(\mathbb{A})^1$...
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Weil's "Sur la formule de Siegel dans la théorie des groupes classiques"
I am struggling to understand the following passage on Weil's text "Sur la formule de Siegel dans la théorie des groupes classiques" on page 16.
If I understood correctly, in the second ...
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Do people prefer working on $\mathrm{GSp}$ and $\mathrm{GU}$ rather than $\mathrm{Sp}$ and $\mathrm{U}$, and why?
I am a new learner of Iwasawa theory and currently reading the famous paper by Skinner-Urban in 2014, and the following-up works by many other people. When reading these papers, I found that some ...
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Spectral decomposition of $\Gamma\backslash X$
Let $X$ be a reasonable manifold of non-positive curvature (could be $\mathbb{H}^n$, symmetric or locally symmetric space, homogeneous Hadamard manifold etc.), and let $\Gamma$ be a reasonable group ...
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Does the Weil representation depend only on the discriminant group?
Forgive me for asking what is undoubtedly an elementary question.
The Weil representation (defined below) of the metaplectic group $\operatorname{Mp}_2(\mathbb{Z})$ can be defined in terms of the ...
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Theta lifting over function fields
Let $F$ be a number field and $\mathbb{A}$ its adele ring.
For a dual reductive pair $G$ and $H$, let $\pi$ be a cuspidal irreducible representation of $G(\mathbb{A})$. Let $\Theta(\pi)$ be the global ...
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Waldspurger's formula and toric periods — classical and adelic versions
As far as I know, there are two versions of Waldspurger's formula (classical and adelic), which can be vaguely stated as follows
(Classical version) Let $f$ be a half-integral weight modular form of ...
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Basic results concerning the intertwining operator in the $\mathrm{SL}_2$ case
I am reading [Ikeda, Tamotsu, On the location of poles of the triple L-functions]. On page 194, the author recalled some known results concerning $\operatorname{SL}_2$. I would like to know any ...
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Meaning of the meromorphic continuation of intertwining operators
I am trying to make sure the meaning of the meromorphic continuation of the intertwining operators.
Assume we deal with a non-Archimedean field $F$ and just consider $G= SL_2$, for simplicity. We fix ...
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$L^2$-spectrum versus automorphic discrete spectrum
Let $G$ be a classical group defined over a number field $F$.
In his monumental book, (https://www.ams.org/books/coll/061/coll061-endmatter.pdf) Arthur described a spectral decomposition of $L_{disc}^...
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Local component of cuspidal automorphic representation
Let $F$ be a number field and $\mathbb{A}$ its adele ring. $G$ be a classical group and $
\pi$ be a unitary cuspidal automorphic representation of $G(\mathbb{A})$.
Then I am wondering whether there is ...
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A detail in I. Piatetski-Shapiro and S. Rallis's "Doubling paper": computing the integral on negligible orbits
I'm currently reading the paper "L-functions for the classical groups" by I. Piatetski-Shapiro and S. Rallis, where they introduced the doubling method over classical groups.
I'm confused at ...
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Which automorphic L-functions have an integral representation?
Is there a list of which automorphic L-functions are known to have an integral representation?
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On decomposition of the space of automorphic forms (via central characters)
Let $F$ be a number field and $\mathbb{A}$ be its adele. For simplicity, we assume $G$ is a connected reductive group.
Given a unitary central character $\chi: Z_{G}(F)\backslash Z_{G}(\mathbb{A}) \to ...
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Understand the $p$-adic local Langlands correspondence with examples
Let $\rho:G_{\mathbb{Q}}\rightarrow \mathbf{Gl}_{n}(\mathbb{Q}_{p})$. I would like to understand in depth why the local Langlands correspondence for $\rho_{|\mathbb{Q}_{p}}$ must consider $p$-adic ...
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Can Taniyama-Shimura conjecture be generalized to curves of higher genus (within Langlands framework)?
The Shimura-Taniyama-Weil conjecture asserts that if E is an elliptic curve over Q, then there is an integer N ≥ 1 and a weight-two cusp form f of level N, normalized so that a1(f) = 1, such that ap(E)...
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Basic question on the Langlands conjectures for $GL_n$ over global field of positive characteristic
My field is far from the Langlands conjectures. I am just trying to understand some basic ideas.
At the moment I am interested in a global field $K$ of positive characteristic and the group $G=GL_n$. ...
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Spectral decomposition of the automorphic space for a unipotent group
Let $k$ be a global field of positive characteristic, $\mathbb{A}$ its adele ring. Let $U$ be a unipotent algebraic group over $k$, of dimension sufficiently small relative to ${\rm char} (k)$. Is ...
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Explicit expression of automorphic representations as automorphic forms
Let‘s take $G=GL_n$ over a number field $F$ for example.
It's already known that every irreducible automorphic representation $\pi$ is a irreducible component of a induced representation $I(G,P;\...
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About the structure of smooth automorphic forms
Recently I read Prof. Cogdell's notes: Lectures on L-functions, Converse Theorems, and Functoriality for $GL_n$. (Co)
In chap.2.3, the conception of smooth automorphic forms is introdued. Explicitly, ...
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Voronoï summation for cusp forms with characters
In an attempt to solve an unrelated problem, I was led to the task of estimating/bounding from above sums of the form
$$\sum_{m=1}^\infty\lambda(m)e\left(-\frac{am}{q}\right)h(m)$$
where $\sum_{m=1}^\...
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Is there a generic representation for non-quasi split $p$-adic group?
It seems that generic representation only occurs for quasi-split groups.
For non-quasi split groups, is it expected that generic representation doesn’t exist?
Thank you in advance!
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Comparing Selberg and Eichler-Selberg trace formulas
The trace formula of Selberg gives an equality between trace of Hecke operators (a spectral sum) on spaces of Maass forms and sums over closed geodesics mostly. The Eichler-Selberg trace formula, ...
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Automorphic representations on non-cyclic covering groups
The theory of theta functions can be interpreted as automorphic representations on metaplectic groups (2-fold covering groups of $\mathrm{Sp}_{2}$, or $\mathrm{GL}_2$), and there's also a notion of $n$...
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Understanding Shimura correspondence in context of Langlands functoriality
Recently, I started to read about automorphic forms and representations on covering groups, e.g. metaplectic groups. I set my first goal as understanding Shimura's correspondence in representation ...
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$l$-adic sheaf associated to an algebraic representation of $\mathrm{GSp}_{4}(\mathbb{Q})$
Let $Y (N) $ be the moduli scheme of dimension two principally polarized Abelian schemes with level $N$. It is claimed in "G.Laumon - Fonctions zeta des variétés de Siegel" (Lemma 4.1) that ...
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Naive construction of modular forms of rational weights
There are some half-integral weight modular forms like Jacobi's theta functions and it can be interpreted as automorphic representations on metaplectic groups (double cover of symplectic groups). I ...
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Local Rankin-Selberg Zeta-function and Coates' p-adic L-Functions
$\DeclareMathOperator\Kern{Kern}\DeclareMathOperator\diag{diag}$
Let $F$ be a non-archimedean local field, $\mathcal{O}$ its ring of integers, $\mathfrak{p}$ its maximal ideal and $\pi$ a uniformizer.
...
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Decomposition of real quasimodular forms of depth 1
Let $\widetilde{M}_k^{\leq \ell}$ be the space of weight $k$ depth $\leq \ell$ quasimodular forms, and $\widetilde{M}_{k,\mathbb R}^{\leq \ell}$ be a subspace of $\widetilde{M}_k^{\leq \ell}$ whose ...
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Extrema of real analytic Eisenstein series and more general modular functions
The real analytic Eisenstein series defined by the Poincare sum
$$E(s,z)=\sum_{(m,n)\neq (0,0)} {y^s\over |mz+n|^{2s}}$$
for $z\in{\mathbb H}$ and ${\rm Re}(s)>1$ is a manifestly $SL(2,{\mathbb Z})$...
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