# 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 ...

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### Is there a definition of supercupidal parameter in the Local Langland correspondence?

By the recent works of Mok, and Kaletha, Shin, White, James, I know that there is a notion of tempered $L$-parameter, square integrable $L$-parameter and generic $L$-parameter of unitary groups.
...

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### Does the supercuspidal representation becomes tempered?

I am really wondering whether supercuspidal representation may become tempered representation.
If it is not true for all classical group, is it especially true for unitary group?
If it is not true ...

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### What is the spectral decomposition of $L^2(\textbf{G}(\mathbb{Q})\backslash \textbf{G}(\mathbb{A}))$ for compact quotient?

I'm trying to work out explicitly the spectral decomposition of $L^2(G(\mathbb{Q})\backslash G(\mathbb{A}))$ when $G$ is anisotropic -- it has no split tori defined over $\mathbb{Q}$. I know that this ...

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### Admissible representations of GL(3)

The finite dimensional representation of $GL(2,\mathbb{R})$ are obtained by tensoring the symmetric powers of standard representations of $GL(2,\mathbb{R})$ with the character $\chi \circ \textrm{det}$...

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### Examples of conjectural functorial transfer which has $\times GL(1)$ functional equation?

I am look for some conjectural functorial transfer $X$ which
(A)for any $GL(1)$ automorphic representation $\pi$, we have
$L(s, X\times \pi)$ is holomorphic and satisfies certain functional ...

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### On tetrahedral Artin representation

I am trying to understand the proof of existence of $\pi(\rho)$ for tetrahedral representation (Galois representation of dim 2 having image $A_4$ in $PGL_2(\mathbb{C})$) explained by Rogawski, ...

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### Definition of Admissible Representation

Let $G$ be a connected, reductive group over a number field $k$. Let $v$ be a place of $k$.
If $v$ is finite, an admissible representation of $G(k_v)$ is defined to be an abstract representation of $...

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### Reference for the proof of Langlands conjecture for $GL_n$ over function fields

Is there any reference written in English for the proof of Langlands conjecture for $GL_n$ over function fields?

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### Bound of higher rank spherical Whittaker function

I am not much familiar with the literature about upper bound of the spherical Whittaker function on higher rank real groups. Any reference or answer to the following question would be appreciated.
...

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### Galois representation and weight one Hilbert modular form

Let $f$ be a primitive weight one Hilbert modular form for the totally real field $F$ (the weight of $f$ is parallel). Assume that $f$ is $N$-new, $p$-stable and cuspidal. Let $\rho:G_F \rightarrow \...

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### How can one understand the Eisenstein series E2 in terms of automorphic representation?

The weight 2, level 1 Eisenstein series $E_2(z)$ is a non-holomorphic automorphic form. It is defined as the analytic continuation to $s = 0$ of the series
$$ E_2(z, s) = \sum_{\substack{m, n \in \...

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### Newform of a cuspidal Automorphic Representation

I was going through these notes https://www.dpmms.cam.ac.uk/~ty245/2008_AGR_Fall/2008_agr_week1.pdf . There, Theorem 9.2 states that: If $\pi ^{\infty}$ is a cuspidal automorphic representation of $\...

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### Reference Request: Definition of Induced Representation for reductive groups over a local field

Let $G$ be a connected, reductive group over a local field $F$ of characteristic zero, and $H$ a closed subgroup of $G$ which is defined over $F$. Let $\mu_H, \mu_G$ be right Haar measures on $H(F), ...

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### Whittaker functions estimates proof

I am reading the proof of the estimates of Whittaker functions from Jacquet, Hervé, Ilja Iosifovitch Piatetski-Shapiro, and Joseph Shalika. "Automorphic forms on GL (3) I." Annals of Mathematics 109.1 ...

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### Adelic Schwartz class

I am not a specialist in automorphic forms, can someone explain to me typical elements of adelic Schwartz class, $\mathcal{S}(\mathbb{A})$. Over the real numbers there are obviously elements like:
$$ ...

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### The space of Whittaker functionals is at most one-dimensional

Let $\mathbf G$ be a connected, reductive group over a local field $F$, and let $(\pi,V)$ be a smooth, irreducible, admissible representation of $G = \mathbf G(F)$. Assume there exists a Borel ...

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### Existence of a Hilbert modular form of parallel weight 6

I am new in the field of Hilbert modular forms and I could not find any Hilbert modular form of this specific weight for the 2-fold upper half-plane.
Does anybody know an example of any level?

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### What is a common name for these automorphic objects?

I am looking for a name which includes these objects:
1. automorphic forms, cusp forms and non-cusp forms
2. Rankin-Selberg convolution between automorphic forms (which is conjectured to be ...

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### Numerical Evidence for Grand Riemann Hypothesis?

Let $L(s)$ be an $L$-function coming from Hecke characters or automorphic forms (e.g. modular form on GL(2), Maass form on GL(2), and higher-rank analogues).
Is there any numerical evidence for ...

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### Reference for a formula of Kloosterman sum (in connection with Jacobi symbol) and its generalizations

The following is from wikipedia:
The lifting formulas below, however, are often as good as an explicit evaluation. If $gcd(a,p) = 1$ one also has the important transformation:
$$S(a,a;p) = \sum_{m=0}^...

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### Why are Bessel function and Kloosterman sum similar?

It is a convention to say Kloosterman sums and Bessel functions are similar.
There are papers talking about Bessel functions on $p$-adic group (associated with a representation) such as Baruch's: ...

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### The importance of relations between automorphic forms and arithmetic functions

As I understand things, one of the classical reasons to care about modular forms was their relation to interesting arithmetic functions/counting questions, i.e. on sums of squares and partitions. When ...

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### Kontorovich Lebedev transform

By the title I mean [reference: ``Spectral methods of Automorphic forms" by Iwaniec (B.41)-(B.43)] for $f\in C^\infty_c(\mathbb{R^+})$, one has
$$f(x)=\pi^{-2}\int_{-\infty}^\infty K_{it}(x)F_f(t)t\...

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### When is the local representation associated to an elliptic curve a Steinberg?

If $E$ is an elliptic curve over $\mathbb{Q}$, and $\pi$ is the automorphic representation of $\mathrm{GL}_2$ associated to $E$, then one can write $\pi = \otimes_v \pi_v$ with each $\pi_v$ an ...

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### How can this argument calculating the Haar measure on a parabolic subgroup be generalized to the non-split case?

Let $\mathbf G$ be a connected, reductive group over a local field $F$. Assume there is a maximal torus $\mathbf T$ which is split over $F$. Let $\mathbf P$ be a parabolic subgroup of $\mathbf G$ ...

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### What's the point of a Whittaker model?

Let $G$ be a quasi-split connected reductive group over a $p$-adic field $F$. Let $B$ be a Borel subgroup which is defined over $F$, with $B = TU$, $T$ defined over $F$. The choice of $T$ and $B$ ...

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### Poles and residues of degenerate Eisenstein series on GL(n)

Suppose $P \subseteq GL(n)$ is a parabolic subgroup. How do I find (or what is a reference for) the poles and residues of degenerate Eisenstein series associated to $P$?
More precisely, suppose $P_1, ...

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### Compactness of the automorphic quotient

Let $F$ be a (totally real) number field, and $E$ a (totally imaginary) quadratic extension of $F$. We consider $U$ a unitary group (with respect to a given hermitian form over $E$). The question is:
...

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### Modularity of endomorphism algebras

This question is about comparing Hecke algebras and endomorphism algebras.
Let $\mathbf{A}_f$ be the ring of finite adèles of $\mathbf{Q}$ and let $K$ be a compact open subgroup of $\mathrm{GL}_2(\...

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### On the consistency of the definition of the conductor for automorphic forms

Let $\pi$ be an irreducible admissible representation of $\mathrm{GL}_2(F)$, where $F$ is local non-archimedean. The local conudctor associated to $\pi$ can be defined in two usual manners:
By its ...

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### Selberg trace formula, quadratic L-values, and generalization

It is known that the geometric side of the Selberg trace formula on GL(2) is related to values of quadratic L-functions (due to Sarnak, Zagier, etc).
Are there any conjectures or results about its ...

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### Hilbert Modular Surface: Eigenfunctions of The Laplacian

The spectrum of the Laplacian on $L^2$ of a Hilbert modular surface decomposes into a discrete part and a continuous part $[1/4,\infty)$. The continuous part contains eigenvalues $\geq 1/4$.
I would ...

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### Clozel's unpublished manuscript

I'm looking for Clozel's unpublished manuscript
L. Clozel, Modular properties of automorphic representations I: Applications of
the Selberg trace formula (1993)
cited in Urban's Eigenvarieties ...

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### Does this particular L-series built from L-functions of prime degree define an L-function?

Throughout this question, I call 'L-function' any automorphic L-function belonging to the Selberg class.
Suppose $ (F_i)_{(i>0)} $ is a sequence of L-functions with $ F_i $ of degree $ p_i $ ...

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### Known degrees of L-functions F and G whose Rankin-Selberg convolution is an L-function

Calling '$L$-function' any automorphic $L$-function belonging to the Selberg class, what are the known $L$-functions $L(s,F)$ and $L(s,G)$ of respective degrees $d$ and $d'$ such that the Rankin-...

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### Reaching Hecke eigenvalues from a trace formula

I am interested in studying equidistribution of Hecke eigenvalues and proving statistical properties of arithmetical objects. On the road, I face the following problem: how to express sums of the form
...

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### $L^2$ norms of Whittaker vectors and zeros of Intertwining operators

For $\mu,\nu\in \mathbb{C}^2$ we denote $I(\mu,\nu)$ to be the principal series of $\mathrm{GL}_2(\mathbb{Q}_p)$ induced from $|.|^\mu\otimes |.|^\nu$. For $s=\mu-\nu$ one defines the standard ...

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### The relation of the local principal representations of $U(2)$ and $GL(2)$

Let $E/F$ be a quadratic extension of number fields and $v$ is a non-archimedean place of $F$.
Let $G=U(2)(F_v)$ be the $F_v$-points of the 2-dimension unitary group associated to $E_v/F_v$ and $B$, $...

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### Why is the L group of an even unitary group what it is?

It is fairly easy to see from the formalism of the L group that the L group of a quasisplit unitary group will be a nontrivial semidirect product of $GL_n(\mathbb{C})$ (for the appropriate value of $n$...

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### Explicit formula of base change for GL(n)

Let $E/F$ be a quadratic extension of number fields and $v$ is a place of $F$.
Let $\chi_1,\chi_2$ be the unramified characters of $F_v^{\times}$.
If $B(\chi_1,\chi_2)$ is the unramified principal ...

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### Irrelevant parabolics and inner forms of GSp(4)

In Ralf Schmidt's appendix to "Jacquet-Langlands-Shimizu correspondence for theta lifts to $\mathrm{GSp}(2)$ and its inner forms" by Narita and Okazaki , he computes the representations of $\mathrm{...

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### “Classical” description of automorphic forms on unitary groups

What I mean by classical: For the case of $GL_2$, the answer to my question would be that the automorphic forms are either Maas forms or modular forms. For $GSp(2n)$ these are the Siegel modular forms....

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### Is there a known construction of Cuspidal representations of GL(3) isomorphic to their own twist?

Take $F$ a number field, $\pi$ a cuspidal automorphic representation of $GL(3, \mathbb{A}_F).$ Suppose $\pi \cong \pi\otimes \chi.$ Comparing central characters we see that $\chi$ must be cubic.
Now ...

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### Relationship between motivic Galois groups and Langlands program [duplicate]

I would like to know if there is any relationship between the motivic Galois groups and the Langlands program.
Many thanks.

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### Relation between representations of p-adic groups and affine Hecke algebras

Let $R_n$ be the category of complex-valued smooth finite-length representations of the group $GL_n(F)$, where $F$ is a local field.
By the result of Borel, the subcategory of $R_n$ consisting of ...

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### Bernstein–Zelevinsky classification for classical groups

Bernstein and Zelevinsky classifies the irreducible complex smooth representations of a general linear group over a local field in terms of cuspidal representations. The irreducible modules are ...

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### Modularity of the Shimura-Shintani kernel

Following this paper by Bringmann, Kane, Kohnen, the kernel of the Shimura and Shintani lifts is the function $$\Omega(\tau,z) = \frac{1}{\binom{2k-2}{k-1}\pi} \sum_{D=1}^{\infty} \sum_{b^2 - 4ac = D} ...

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### Eisenstein cohomology - explicit computation and relation to Franke's trace formula

Let $G$ be a reductive group over $\mathbb{Q}_p$. Let $X_G$ be a locally symmetric space associated to the group $G$, and let $\partial X_G$ be the Borel-Serre boundary of $X_G$. The space $\partial ...

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### Explicit description of the Plancherel measure for $GL_n(\mathbb{R})$

Let $G:=\mathrm{GL}_n(\mathbb{R})$ and $f\in C_c^\infty(G)$. One can uniquely determine the Plancherel measure $d\mu_p$ on $\hat{G}$, the unitary (actually tempered) dual of $G$, by the equation
$$f(g)...

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### What is the $p$-adic Langlands conjecture for $\mathbf{GL}_1$?

In the Boston conference on Fermat's Last Theorem (Summer 1995), Barry Mazur said (around 15m into the video) about class field theory that
If you are a number-theorist and you want to cheer ...