**12**

votes

**0**answers

281 views

### No Siegel-Landau zeros for $\mathrm{GL}(n)$

The problem of non-existance of Siegel-Landau zeros seems to be uncharacteristically easier for cuspidal automorphic representations $\pi$ on $\mathrm{GL}(n)$ if $n\geq2$. We have in fact:
There ...

**12**

votes

**0**answers

1k views

### Why doesn't functoriality immediately imply the modularity theorem?

Let $E/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, the prime indexed coefficients of its $L$-function agree with those of a weight $2$ cusp eigenform $f$ with integer coefficients. ...

**11**

votes

**0**answers

171 views

### Definition of discrete spectrum and continuous and basic properties

I apologize if this is too basic for MO.
I have an embarrassing admission to make: I don't know the actual definition of the discrete/continuous spectrum of a reductive group $G/\mathbb{Q}$ (in the ...

**11**

votes

**0**answers

199 views

### Are Hecke eigenvalues on the cohomology of the Newton polygon strata automorphic?

Fix a genus $g$, a prime $p$, and a Newton polygon $\Delta$ of an abelian variety of genus $g$.
Let $\mathcal A_{g, \overline{\mathbb F}_p, \Delta}$ be the moduli stack of abelian varieties of genus ...

**10**

votes

**0**answers

189 views

### 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 ...

**9**

votes

**0**answers

267 views

### Automorphic factorization of Dedekind zeta functions

It is well known that for abelian number fields, the factorization of its Dedekind zeta function goes like this:
$$\zeta_K(s)=\zeta(s)\prod_{\chi \neq 1} L(s,\chi)$$
with the Dirichlet characters ...

**9**

votes

**0**answers

247 views

### Irreducibility of Galois representations attached to unitary groups

If $G$ is a unitary group in $n$ variables over $\mathbb Q$, attached to an hermitian form for an imaginary quadratic extension $E/\mathbb Q$ and if we suppose that the hermitian form is definite over ...

**8**

votes

**0**answers

255 views

### 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, ...

**8**

votes

**0**answers

125 views

### Algebraic construction of the modular representation of $\mathrm{SL}_2(\hat{\mathbf Z})$

The answer to this question is probably to be found in the theory of automorphic forms, but (I don't know much about it and consequently) after some tries, I did not catch it. Thus I'd be grateful if ...

**8**

votes

**0**answers

202 views

### Phillips-Sarnak conjecture in higher dimension

The Phillips-Sarnak conjecture states that for a generic Fuchsian lattice the space of Maass cusp forms is finite-dimensional. Generic here means in particular non-uniform, non-arithmetic, no special ...

**8**

votes

**0**answers

366 views

### A frustrating cohomology class on the moduli of abelian surfaces

Here's a very frustrating question that I have been stuck on for some time. I believe that my question could fit in a general framework of what happens when you restrict $L^2$-cohomology classes on a ...

**8**

votes

**0**answers

499 views

### Automorphic representations attached to abelian varieties

Let $A$ be an abelian variety defined over $\mathbb{Q}$, of dimension $d$. It is widely expected that there is an automorphic representation $\pi_A$ of $GL(2d)/\mathbb{Q}$ whose L-function agrees ...

**7**

votes

**0**answers

65 views

### Automorphic representations whose local factors are tempered almost everywhere

Let $F$ be a global field, let $\mathbf{G}$ be a reductive algebraic group over $F$, and let $\pi$ be an irreducible discrete automorphic representation of $\mathbf{G}$.
Write $\pi$ as a restricted ...

**6**

votes

**0**answers

154 views

### drinfeld shtukas over higher dimensional spaces

Everytime I encounter Drinfeld Shtukas, the definition begins with vector bundles over a curve $X$ over a finite field. My question is: why the restriction to curves? Is there any interest or results ...

**6**

votes

**0**answers

186 views

### Rankin-Selberg for Maass form GL(3)xGL(2)

Let $F$ be a Maass cusp form for $\mathrm{SL}(3,\mathbb{Z})$ (level 1 trivial character).
Let $g$ be a Maass cusp form for $\Gamma_0(N)$ with character $\chi$ mod $N$. For convenience, you may assume ...

**6**

votes

**0**answers

189 views

### Is there an integral pairing between quaternionic Hecke algebras and cusp forms?

Let $F$ be a totally real number field with integers $\mathcal{O}_F$ and $B$ a quaternion algebra over $F$ split at exactly one infinity place.Fix $n\geq 1$ and like in the special case $F=\mathbb{Q}, ...

**6**

votes

**0**answers

270 views

### Is the space of global Whittaker functions complete?

Let $f$ be a complex valued function of $GL_n(\mathbb{A})$, where $\mathbb{A}$ is the adeles of some number field. Assume $f(ug)=\psi(u)f(g)$ for any $u$ in the standard maximal unipotent subgroup ...

**5**

votes

**0**answers

218 views

### Examples of Rankin-Selberg L-functions from Eisenstein series

I've been digging for awhile to not much success, so I figure I would try here:
I am looking for some references which compute explicitly examples of Rankin-Selberg L-functions from the constant ...

**5**

votes

**0**answers

90 views

### different definitions of epsilon constants for representations of GL(2) from modular forms

I ran into this question when trying to compute the Atkin-Lehner pseudo-eigenvalue of
newforms. Let $k \geq 2$, let $\omega$ be a Dirichlet character modulo $N$ and let $f \in S_k(N,\omega)$ be a ...

**5**

votes

**0**answers

183 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 ...

**5**

votes

**0**answers

130 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 $m_\pi$ ...

**5**

votes

**0**answers

375 views

### a generalization of a formula of Shimura

Let $\phi$ be a $GL(2)$ automorphic form with Fourier coefficients $a(n)$ and $a(1)=1$.
Obviously we have $L(s,\phi)=\sum \frac{a(n)}{n^s}$.
Shimura have the following formula
$L(s, Ad\; ...

**5**

votes

**0**answers

177 views

### On Langlands Pairing and transfer factors

In the paper "On the definition of transfer factors" Langlands and Shelstad define a certain number of factors $\Delta_{I}$, $\Delta_{II}$,$\Delta_{III,1}$,$\Delta_{III,2}$, which are roots of unity.
...

**4**

votes

**0**answers

397 views

### What is the best way to learn about Modular Forms?

I am a senior Mathematics Major, and I am interesting in learning about Modular Forms. I have a layman's general sense of what they are but I was wondering if there is a lecture(I am willing to pay) ...

**4**

votes

**0**answers

88 views

### Shifted convolution problem for Coefficients of automorphic forms

The shifted convolution problem for coefficients of modular forms is well studied and many estimates were established for the shifted convolution sums of Hecke eigenvalues. So, one may ask about the ...

**4**

votes

**0**answers

109 views

### Uniqueness of cohomological holomorphic discrete series representation

In Claus Sorenson's PhD thesis, he proves a theorem about level lifting of paramodular forms whose associated automorphic representation has component $\pi_{\infty}$ that is the "cohomological ...

**4**

votes

**0**answers

73 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 ...

**4**

votes

**0**answers

252 views

### Why Whittaker functions are useful?

Whittaker functions appears in Langlands program. Recently, it is shown that some Whittaker functions can be obtained by integrating a function related to decoration over a geometric crystal in ...

**4**

votes

**0**answers

126 views

### parametrization of irreducible finite dimensional representation of Weil group

Let $F$ be a p-adic field, with p a prime denoting the residue field characteristic. Let $\mathcal{W}_F$ be the Weil group. In the local Langlands correspondence for $GL(n,F)$, it is important to know ...

**4**

votes

**0**answers

173 views

### Restriction of representations from $SO_{2n}$ to $SO_{2n-1}$ and $K$-fixed vectors

Let $F$ be a $p$-adic field, $G=SO_{2n}(F)$ the split special orthogonal group and $H=SO_{2n-1}(F)$, taken as a subgroup of $G$. Assume that we have an irreducible, admissible representation $\pi$ of ...

**4**

votes

**0**answers

270 views

### Generating function related to 2-residues of partitions

Question
Express the following power series in two commuting variables $x,y$ as an infinite product, or a short sum of infinite products:
$$
...

**4**

votes

**0**answers

545 views

### base change and Langlands' combinatorial exercise

Hi,
Is it correct that Langlands' combinatorial exercise (as he terms it in his paper
"Shimura varieties and the Selberg trace formula") is to establish base change identities between orbital ...

**4**

votes

**0**answers

267 views

### spectral decomposition for elliptic surfaces?

I'm looking for explicit formulae for the spectral decomposition of $L^2(S)$, where $S$ is an elliptic surface (of complex dimension 2).
To be precise, the elliptic surface I'm looking at is the ...

**3**

votes

**0**answers

63 views

### Noncommutivity of various lifts

Let $D$ be a definite quaternion algebra over $\mathbb{Q}$ ramified at $p$ and $\infty$ for simplicity. Then an automorphic form on $D^{x}/F^{x}$ has several different interpretations. First, through ...

**3**

votes

**0**answers

98 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 ...

**3**

votes

**0**answers

245 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) ...

**3**

votes

**0**answers

97 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 ...

**3**

votes

**0**answers

160 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 ...

**3**

votes

**0**answers

94 views

### 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 ...

**3**

votes

**0**answers

139 views

### Functoriality for triple product GL(2) x GL(2) x GL(2)

Let $f$, $g$ and $h$ be three general automorphic forms on GL(2).
Do we know that $L(s, f\times g\times h)$ comes from an automorphic form on GL(8)?

**3**

votes

**0**answers

147 views

### Are quantities involved in Generalized Ramanujan Conjecture eigenvalues of some unitary operator?

If I'm not mistaken, every automorphic L-function $L(s,\pi)$ verifies $\displaystyle{L(s,\pi)=\prod_{p}L_{p}(s,\pi_{p})}$ where ...

**3**

votes

**0**answers

124 views

### Bessel function for $GL_3(\mathfrak{R})$?

In the $GL_2(\mathfrak{R})$ case, assume that $\pi$ is an irreducible unitary representation and $W_{\pi}(g)$ is the Whittaker functional associates with $\pi$. Then there is a Bessel function ...

**3**

votes

**0**answers

133 views

### Do local L-functions/epsilon factors vary continuously with the Fell topology?

Edit due to the comment.
Consider $G=GL(2)$ over a local field $F$. The Fell topology on the unitary dual of $G(F)$ is seperable.
Given a sequence of irreducible unitary representations $(\pi_n)$ of ...

**3**

votes

**0**answers

266 views

### The operator \boxtimes and \boxplus in automorphic representations

Given two automorphic representations $\pi_1, \pi_2$ of $GL_2(\mathbb A_Q)$ and $GL_3(\mathbb A_Q)$ respectively. Let $\pi_i =\otimes_v \pi_{i, v}$.
Now, for each $v$, let $\pi_{1, v}\boxtimes ...

**3**

votes

**0**answers

728 views

### Base-Change and Automorphic-Induction for $GL_1$

Dear all, I try to understand the base-change and automorphic-induction in the theory of automorphic forms, for the simplest case: $GL_1$. Both are implied by Langlands conjectures
Base-Change
Let ...

**3**

votes

**0**answers

371 views

### Non-vanishing of twists of L functions for GL(4)

Hello,
This is a question in the spirit of
Nonvanishing of central L-values of quadratic twists?
and the application I have in mind is to p-adic L-functions a la Ash-Ginzburg.
The question is ...

**2**

votes

**0**answers

63 views

### Eigenvalues of the imaginary part of the Symplectic action on Siegel upper half plane

Let $A,B\in M_n(\mathbb{R})$ and $U=A+iB$ unitary. $R=diag(r_1,r_2,…,r_n)$ is a diagonal matrix with $r_i>0, \forall i $. I need to calculate $\det(Ae^{-R}A^T+Be^{R}B^T)$. This matrix ...

**2**

votes

**0**answers

235 views

### A problem on Hecke operators

I have a problem that is related to page 11 in http://www.cims.nyu.edu/~venkatesh/research/ml.pdf
Fix $m\in\Bbb N$.
Pick $(a,b)\in\Bbb Z_p^2$ randomly such that $m<a,b<2m$ with $a,b$ ...

**2**

votes

**0**answers

119 views

### Reference request: proofs of the theorems in the paper On the representation of the group GL(n, K) where K is a local field

In the paper On the representation of the group GL(n, K) where K is a local field by Gelfand and Kazhdan, it is said that the proofs of the theorems in the paper are published in some other papers. I ...

**2**

votes

**0**answers

104 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 ...