The automorphic-forms tag has no wiki summary.

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### Geometric interpretation of Cusps for general groups?

Let $\mathrm{G}$ be a reductive group over a number field $F$, but for simplicity we can think about $\mathrm{G}=\mathrm{GL_n}$ for $n>2$ and $F =\mathbb{Q}$.
Then for an automorphic form,
...

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205 views

### Is there a non-tempered representation of U(2)?

I am wondering why the first well known example of irreducible admissible representation of $p$-adic group $U(n)$ should be $U(3)$. Because, Gelbart and Rogawski suggested the non-tempered ...

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86 views

### About algebraic groups defined over Q [migrated]

I'm studying automorphic forms and there's something I don't understand, when we talk about a connected reductive algebraic group $G$ defined over $\mathbb{Q}$, connected means connected as an ...

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62 views

### On continuous part of the L^2 spectrum

Suppose $G$ is a real reductive Lie group and $\Gamma$ is a lattice in $G$ (of finite co-volume). I am reading Langlands's paper " On the functional equation satisfied by the Eisenstein series". I ...

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133 views

### Euler Subgroups and Automorphic L-functions

Recently, I have read about the Whittaker expansion for $\mathrm{GL}_n$ and was struck by the utility of the mirabolic subgroup, $\mathrm{P}_n\subset \mathrm{GL}_n$ of matrices with bottom row $(0\; 0 ...

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51 views

### Global theta series on GL

Let $E/F$ be a quadratic extension of number field and let $V$ be a Hermitian space over $E$. Then we have Weil representations for the dual pair $U(n,n)\times U(V)$, and we can consider the theta ...

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160 views

### Rankin-Selberg convolution and product of degrees

As I'm kinda obsessed with the Selberg class and because of the general converse conjecture, I'm still trying to get a rough idea of what automorphic representations and their L-functions as well as ...

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67 views

### About Theorem $3.1.3$ in Kubota's book: Elementary theory of eisenstein series

My question is about the proof of Theorem $3.1.3$ given in kubota's book, which shows how the function $\varphi(s)$ appearing in the Fourier expansion of eisenstein series can be continued ...

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153 views

### Is the restriction of a representation semisimple?

Let $F$ be local field of characteristic zero and $\pi$ be a irreducible admissible representation of $GL_n(F)$.
Let us consider its restriction to $GL_{n-1}(F)$. Then I want to know whether ...

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71 views

### faithful action of Hecke algebra

Let $G$ be a connected reductive group split over a number field $F$, $\mathbb{A}$ the adeles.
Let $v$ be a finite place and $\mathcal{H}_{v}$ the spherical hecke algebra at palce $v$.
...

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91 views

### Newvectors in tensor product representations

Let $\pi_p$ and $\pi_p^\prime$ two smooth admissible irreducible complex representations of ${\rm GL}_2(F)$ where $F$ is a non archimedean local field of residual characteristic $p$ of central ...

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209 views

### On the local root number(or local $\epsilon$-factor)

I want to ask some question related to the local root number.
Let $E/F$ be a quadratic extension of p-adic local fields and $\psi:E \to \mathbb{C}$ is an additive character of $E$.
Let $\phi:WD(E) ...

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92 views

### “simulteneous eigenvectors” under the full set of weighted Laplacians on a $g$-fold product of the Poincare half plane

This question is closely related to the following MO question Characterizing the real analytic Eisenstein series
Let $\mathfrak{h}=\{z=x+iy\in\mathbf{C}\}$ be the Poincare upper half plane endowed ...

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1k views

### Underlying idea for (automorphic) L-function?

Edit: So with a few more months of math under my belt, I recognize some of the issues with this question. I still hope for an answer, so let me say a few things.
Within the Langlands philosophy, ...

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110 views

### Functoriality for non-split orthogonal groups

I am trying to understand the functoriality conjectures of Langlands. We know that the functoriality conjectures imply that automorphic $L$-functions of a connected reductive group are equal to ...

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86 views

### Oscillatory integral moments of $L(\frac{1}{2} + it, f \times f)$

Understanding moments and subconvexity bounds for $L$-functions is a big topic with a lot of activity. I'm currently looking at a related problem, bounding
$$
\int_0^T L\left(\tfrac{1}{2} + it, f ...

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163 views

### Voronoi formula and twists by additive characters

I was wondering if there are any references for the error term in the problem
$$\sum_{n\leq x} r(n) \exp(2\pi i\frac{a}{q}n)$$
where $r(n)$ is the number of representations of $n$ as a sum of two ...

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260 views

### Lindelof Hypothesis implying Selberg Eigenvalue Conjecture?

(General) Lindelof Hypothesis which says for any $L$-function we have $$L(1/2+it)\ll Q(t)^{\epsilon}$$ for any $\epsilon>0$ where $Q(t)$ is the conductor of $L(s)$ at $t$.
For a Maass form $\phi$ ...

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**1**answer

166 views

### Asymptotic behaviour of $K$-Bessel function in transition range

It is known that the famous mistake of Iwaniec-Sarnak in their paper of $L^\infty$ norm of eigenfucntion of non-cocompact arithmetic surfaces in lemma (A1) is because of they did not consider the bump ...

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172 views

### About Section 4.2 in 'introduction to the spectral theory of automorphic forms' by Iwaniec

In Section 4.2 in 'introduction to the spectral theory of automorphic forms' by Iwaniec, the author said that the kernel of the invariant integral operator
$$
(Lf)(z)=\int_{\mathbb{H}}k(z,w)f(w)d\mu w
...

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171 views

### The sixth power integral moment of automorphic L-function attached to Maass Forms

It is known that the sixth power integral moment of automorphic L-function attached to Cusp Forms has been proved by M. Jutila, that is $\int_{0}^{T}|L(1/2+it,f)|^{6}dt \ll T^{2+\varepsilon}$.
And ...

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460 views

### Characterizing the real analytic Eisenstein series

Consider the classical real analytic Eisenstein series
$$
E(z,s)=\left(\pi^{-s}\Gamma(s)\frac{1}{2}\right)\sum_{(m,n)\neq(0,0)}\frac{y^s}{|mz+n|^{2s}},
$$
where $z=x+iy$. We think of $E(z,s)$ as a ...

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109 views

### On the Saito Kurokawa representation

I know Saito-Kurokawa(SK) representation is the famous non-tempered representation of $SO(5)$. But since the tempered or non-tempered terms are concerned with local phenomenon, I am wondering that ...

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123 views

### Lower bound of first moment of $L$-function on $\mathrm{GL}(3)$

Let $\pi$ be an automorphic form on $\mathrm{GL}(3,\mathbb{A}_{\mathbb{Q}})$.
Do we know any case that
\[\int_0^{T} \left|L(\frac{1}{2} + it, \pi)\right| dt \gg T\]
holds unconditionally?
I know the ...

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### Weyl law for Maass forms with nontrivial character

The classical Weyl law for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$ counts the number of Maass cusp forms on $\Gamma \backslash \mathbb{H}$ with Laplace eigenvalue less than $T$. This is originally due to ...

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166 views

### on the Rankin-Selberg L-function

Let $n,m$ be two different positive integers.
I heard that for cuspidal tempered automorphic representations $\pi_{n}$ and $\pi_m$ of $GL_n$ and $GL_m$, the Rankin-Selberg L-function $L(s,\pi_n ...

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128 views

### On the reductive group [closed]

I know that the automorphic representation can be defined only for reductive algebraic group.
What property of algebriac group makes it hinder to define for all algebraic group and what nice property ...

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186 views

### Lower bound of Hecke eigenvalues of Maass form

If $f$ is a Maass form and $p$-Hecke eigenvalue (i.e. Hecke eigenvalue of usual Hecke operator $T_p$) of $f$ is $\lambda_f(p)$, do we know anything about lower bound of the sum$$S(x) = \sum_{x\le p\le ...

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369 views

### Holomorphic cusp forms and cohomology of GL(2,Z)

Let $V_{k}$ denote the complex representation of $\mathrm{GL}(2)$ given by $\mathrm{Sym}^k(V)$, where $V$ is the defining 2-dimensional representation. Assume that $k$ is even. I would like to compute ...

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### limit of regular hyperbolic integrals is a unipotent integral (GL2 calculation)

In developing a simple trace formula for $G$=GL$_2$ over a number field $F$ one encounters the following identity of local integrals (for example, in Gelbart-Jacquet, 1979):
$$\int_{Z_v N_v\backslash ...

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51 views

### Definition of Degenerate hyper-Kloosterman Sum

If $(a,q)=1$, we all know the usual hyper-Kloosterman sum
$$Kl_n(a,q)= \sum_{x_1...x_n\equiv a \mod q} e(\frac{x_1+\dots+x_n}{q}).$$
I wonder what would be the correct definition when $(a,q)\neq 1$.
...

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139 views

### multiplicity of automorphic representation of unitary similitude group

Let $G$ be a unitary similitude group over $\mathbb{Q}$ (as in the book of Harris-Taylor), $\pi$ an irreducible automorphic representation of $G(\mathbb{A})$. I'm looking for some results on its ...

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127 views

### On the pole of local L-function

Let $F$ be a number field and $v$ a finite place of $F$.
Let $\chi_v$ be a unramified unitary character of $F_v$.
Then we define local L-function $L_v(s,\chi_v):=\frac{1}{1-\chi_v(\omega)q^{-s}}$ ...

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117 views

### Asymptotic expansion of an integral, related to Maass forms

I am trying to compute the asymptotic expansion of the integral
$I(t) = \int_{C} e^{\sqrt{1+u}(\frac{1}{t}+\frac{t^2}{\sqrt{u}})}\frac{u^\eta}{\sqrt{1+u}}du$
as $t$ is real and $t\rightarrow +\infty$, ...

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### nonvanishing of global theta lifting from U(1) to U(1?)

I understanding nonvanishing of theta lifting, either global or local, is a difficult and open problem. But I wanna know if there is an answer for the following simplest case.
Let $E/F$ be a ...

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118 views

### Multiplicity of automorphic representation

Let $\pi$ be an automorphic subrepresentation of a reductive group $G$. Here by this, I mean an irreducible representation realized in a subspace of the space of automorphic forms on $G$.
Let $m_\pi$ ...

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### Modular form, number of divisors [duplicate]

The Fourier expansion of Eisenstein series $E_k$ $(k \ge 4)$, which are modular forms, as well as the quasimodular $E_2$, involves powers-of-divisors $\sigma_{k-1}(n) = \sum_{d|n} d^{k-1}$.
Is there ...

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293 views

### Hecke-module structure implicit in definition of automorphic forms in Borel-Jacquet's Corvallis article

Let $G$ be a connected reductive group over a number field $F$, $G_\infty=\prod_{v\mid\infty} G(F_v)$, $\mathbf{A}$ the adèles of $F$, $\mathbf{A}_f$ the finite adèles of $F$. Fix a maximal compact ...

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### Base change of discrete series

Let $\pi_f$ be an automorphic representation of $GL_2(A_Q)$ (attached to a modular form $f$), and suppose we want to look at its base change lift to say a quadratic imaginary field. Which are the ...

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395 views

### Which L-functions are not “Langlands-Shahidi L-functions”?

The Langlands-Shahidi method, among other things, obtains certain L-functions from the constant term of Eisenstein series attached to so-called $(G,M)$ pairs, where $G$ is a reductive group, $M$ a ...

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### Symmetric Fifth Power Lift of GL(2) Automorphic Form

Let $\pi$ be an automorphic representation of $GL(2)/\mathbb{Q}$. For simplicity, you can take it to be a Maass form for $SL(2,\mathbb Z)$. Kim, Shahidi, Gelbart-Jacquet prove that
$$L(s, \pi, ...

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### Self-dual automorphic forms on $GL(4)$

As is known among experts, all self-dual automorphic forms on $GL(3)$ come from symmetric square lifts from $GL(2)$. You can find this in Ramakrishnan ...

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206 views

### Newform and Galois representation (Shimura-Deligne Reciprocity Law)

Shimura-Deligne Reciprocity Law implies that to a newform $f \colon =f(z) \in S^{\mathrm{new}}_k(\Gamma_0(N))$ of weight $k$, one can associate Galois representation $\rho_{f,\lambda} \colon ...

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### Godement-Jacquet and L-functions

Let $M_{r}(F)$ be the matrices with coefficients in a local nonarchimedean field $F$ and $q$ the cardinal of the residue field.
We have a Fourier tansform on $M_{r}(F)$ with kernel ...

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### Order of individual Fourier coefficient of a Maass form

Let $D$ be a definite quaternion division algebra over $\mathbb{Q}$ and $\mathcal{O}$ be an Eichler order of $D$. Let $F$ be a Maass form in $L^2(PGL_2(\mathcal{O})\backslash ...

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### looking for reference on dihedral, tetrahedral, or octahedral forms

I am looking for a reference on dihedral, tetrahedral, or octahedral forms. As far as I read, they are some cuspidal automorphic forms on $GL(2)$ induced from $GL(1)$. Dihedral is from $GL(1)/K$ to ...

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264 views

### Is the adjoint L-function on GL(m) holomorphic?

Let $\pi$ be an automorphic representation on GL($m$)/$\mathbb{Q}$.
Define $$L(s,\pi,Ad):=\frac{L(s,\pi\times\overline{\pi})}{\zeta(s)}.$$ This is an L-function with euler products of degree $m^2-1$.
...

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225 views

### absolute convergence of Rankin-Selberg series

Let $\pi$ and $\pi'$ be two general automorphic representation on $GL(n)$ and $GL(n')$ over $\mathbb{Q}$.
I heard that the rankin-selberg convolution L-function $L(s,\pi\times\pi')$ is absolutely ...

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### Eisenstein series over a definite division algebra

Let $D$ be the definite quaternion division algebra over $\mathbb{Q}$. $\mathcal{O}$ is a maximal order inside $D$, let's fix $\mathcal{O}$ to be the Hurwitz quaternion. Let ...

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392 views

### Langlands' original observation about Ramanujan conjecture

Obviously functoriality of arbitrary high symmetric power lifts of automorphic forms on GL(2) will lead to the Ramanujan conjecture. But I guess that is too strong for Ramanujan. I came across some ...