**12**

votes

**1**answer

214 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 ...

**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 ...

**11**

votes

**3**answers

545 views

### Philosophy behind cohomological representations

For a given real reductive Lie group $G$, we have the notion of a representation being cohomological using the Lie algebra cohomology. In particular we know that the discrete series representations of ...

**4**

votes

**1**answer

182 views

### Eisenstein Series on Siegel Space

I am looking for any reference dealing explicitly with Eisenstein series on Siegel space (the simplest case of $\rm{SP}_4$ is fine). Anything would be welcome, but in particular I'm interested in the ...

**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 $a,b$ ...

**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 ...

**1**

vote

**1**answer

121 views

### Antiholomorphic cusp forms of negative weight

Let $k\geq 2$ be an even integer and let $\Gamma=\Gamma_0(N)$. Let $f\in S_k(\Gamma)$. To $f$, one may associate an antiholomorphic cusp form of weight $k$ and level $\Gamma$ by defining ...

**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) ...

**15**

votes

**2**answers

684 views

### Do we care about multiple zeta functions?

Coming from a number-theoretic background, I certainly care about $L$-functions and in particular automorphic ones. For automorphic forms on $SL_2(\mathbb{Z}) \backslash SL_2(\mathbb{R})$, ...

**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 ...

**4**

votes

**1**answer

338 views

### Birch's conjecture from Representation Theory

Birch has a conjecture about which automorphic forms on $PGL(2)$ are the lifts from nonsplit $O(3)$. Temporarily ignore global issues, and focus on the local nonarchimedian picture. The automorphic ...

**5**

votes

**1**answer

100 views

### Reference request: normalization of intertwining operators for GL(2, C)

Take $F$ a local field and $\chi_1, \chi_2$ two characters, write $M(\chi_1, \chi_2)$ for the standard intertwining integral
$$M(\chi_1. \chi_2).f(g) := \int_{F} f\left( \begin{pmatrix} 0&-1\\ ...

**6**

votes

**1**answer

359 views

### Local Langlands correspondence and Galois equivariance

The local Langlands correspondence $\text{rec}$ for $\text{GL}_{n}$ itself is not Galois equivariant (i.e. invariant under automorphisms of its field of definition) but rather its twist by ...

**36**

votes

**1**answer

1k views

### What is the status of Arthur's book?

Arthur's long-awaited book project is now published (The endoscopic classification of representations: orthogonal and symplectic groups). However, in the book he takes some things for granted:
The ...

**16**

votes

**2**answers

348 views

### what is the equivalent of the Euler constant for higher dimensional lattices

Let $\Lambda$ be a unimodular lattice in $\mathbb R^d$. Then there are constants such that
$$\sum_{\substack{\gamma\in \Lambda\\0<|\gamma|<R\\}} \frac{1}{|\gamma|^d} = c_1 \log R + c_2 + ...

**7**

votes

**1**answer

316 views

### Arithmetically equivalent number fields and Langlands Program

Two (number) fields are arithmetically equivalent if their Dedekind zeta functions are the same. It is known that any two arithmetically equivalent fields are not necessarily isomorphic; Prasad ...

**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 ...

**14**

votes

**1**answer

659 views

### Even Galois representations “mod p”

Consider an irreducible $\mathrm{mod}$ $p$ representation:
$$\rho: \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_2(\bar{\mathbb{F}}_p)$$
If $\rho$ is odd, it was conjectured by Serre in ...

**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 ...

**7**

votes

**1**answer

137 views

### Matsushima-Murakami Isomorphism for $L^2$-cohomology

Let $\mathbf{G}$ be a reductive connected linear algebraic group over a totally real global number field, say $\mathbb{Q}$. Let $\mathbb{A}=\mathbb{R}\times\mathbb{A}_f$ be the ring of rational adele.
...

**11**

votes

**1**answer

281 views

### Non-algebraic Hecke characters

Algebraic Hecke characters are ubiquitous in modern number theory. They are in 1-1 correspondence with one dimensional complex Galois representations, and in some precise sense they are the building ...

**4**

votes

**1**answer

274 views

### Fourier expansion of automorphic forms

we know that for $r \in \{1,2,3,4\},$ $\lambda_{Sym^rf}$ is an automorphic form (here $f$ is a modular form for the full modular group) and this fact is conjectured for $r\geq 5$ by Langlands and ...

**6**

votes

**1**answer

179 views

### History of spectral methods to the study of real analytic $GL_2$-Eisenstein series

I'm trying to sort out the history of spectral methods in the study of real analytic $GL_2$-Eisenstein series. From what I read so far, I would say that the subject was really kicked off by the ...

**6**

votes

**1**answer

211 views

### Simplest case of Langlands-Shahidi method

I would like to read the simplest examples of Langlands-Shahidi method carried out to prove the functional equation of $L$-function.
Could the constant term of $\mathrm{GL}(2)$-Eisenstein series be ...

**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 ...

**6**

votes

**1**answer

281 views

### Converse to Modularity II: Maass cusp forms

(This comes from this other question. You can find more details there)
The following bijection is now a theorem:
Odd irreducible 2-dim Galois repn $\longleftrightarrow$ weight 1
newforms
...

**1**

vote

**2**answers

209 views

### Generic irreducibility of parabolic induction

In J.Bernstein's notes: REPRESENTATION OF P-ADIC GROUPS, he remarked the following result(see P.88):
Let $G$ be a reductive group defined over nonarchimedean local field $F$, $P$ parabolic subgroup of ...

**1**

vote

**1**answer

171 views

### On the definition of matrix coefficient

As far as I have known, for irreducible admissible representation $\pi$ of $p$-adic group $G$, the matrix coefficient is defined as follows:
For $v\in \pi$ and $w \in \pi ^\vee$, the contragredient ...

**6**

votes

**1**answer

335 views

### Uniform definition of $S(\mathbb{R})$ and $S(\mathbb{Q}_p)$

Let $\mathcal{P}=\{\infty, 2,3,5,7,11,\ldots\}$ be the set of primes of $\mathbb{Q}$ and let $\mathbb{Q}_p$ denote the corresponding completions, so in particular $\mathbb{Q}_{\infty}=\mathbb{R}$.
Is ...

**12**

votes

**1**answer

257 views

### To what extent are modular parametrizations expected to generalize?

By the Modularity Theorem (a.k.a. the Shimura--Taniyama--Weil Conjecture), if $E$ is an elliptic curve over $\textbf{Q}$ with conductor $N$, then there exists a “modular parametrization” $\psi: X_0(N) ...

**1**

vote

**1**answer

149 views

### Question about mean square estimate for sums of Dirichlet coefficients of Symmetric Power $L$-functions

I have a question related to Coefficients of Symmetric power $L$-functions and I would be grateful if you could answer it.
Let $\lambda_{Sym^rf}(n)$ be the $n$th Dirichlet coefficient of ...

**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 ...

**4**

votes

**1**answer

234 views

### Is the twisted symmetric fifth power $L$-function holomorphic?

Let $\pi$ be a Maass cusp form for SL($2,\mathbb Z$). Let $\omega$ be a primitive Dirichlet character.
Let us consider the $L-$ function
$$L(s,Sym^5 \pi \times \omega)$$ or $L(s,Sym^6 \pi \times ...

**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 ...

**2**

votes

**1**answer

161 views

### How to construct the symmetric power function from a modular form?

I want to understand how we construct from a modular form $f$ its symmetric power function $Sym^rf.$ I read that there is a particular representation that does this but I am not familiar with this ...

**5**

votes

**1**answer

191 views

### The infinity-type of automorphic representations in the Langlands correspondence

Let $K$ be a number field, $\rho\colon \mathrm{Gal}_K\to \mathrm{GL}_n(\overline{\mathbf{Q}_p})$ a geometric (i.e.: unramified a.e., de Rham above $p$) irreducible Galois representation. One piece of ...

**3**

votes

**1**answer

168 views

### Fourier Transform of Eisenstein Series - Sum of Divisors or Ramanujan Sums?

I am stuck on this computation of the Fourier coefficients of Eisenstein series. For $\Gamma = SL(2, \mathbb{Z})$ and $\Gamma_\infty = \left\{ \left( \begin{array}{cc} 1 & m \\ 0 & 1 ...

**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 ...

**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 ...

**2**

votes

**1**answer

243 views

### A computation about Whittaker functions and Eisenstein series

I have some questions about the computation of Eisenstein series and Whittaker functions in the book. The question is on page 29, Theorem 4.3.
My questions are in the following.
(1) I think that ...

**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 ...

**1**

vote

**0**answers

77 views

### The dimension of the space of automorphic forms with multiplier system

Let $\Gamma$ be a discrete subgroup of $SL_{2}(\mathbb{Z})$ and $\vartheta$ a multiplier system of weight $k$ for $\Gamma$, by which we mean a function $\vartheta:\Gamma \rightarrow \mathbb{C}$ ...

**2**

votes

**1**answer

301 views

### Two questions about Whittaker functions

I am watching the video: Modeling p-adic Whittaker functions, Part I. I have two questions about Whittaker functions in the video.
From 33:00 to 37:00, it is said that after changing of variables, ...

**5**

votes

**2**answers

231 views

### Analytic continuation of intertwining operator

I was trying to understand the paper "Forms of GL(2) from the analytic point of view", by Gelbart and Jacquet.
On Page 226 in Remark (4.13) they mention that the kernel of the local intertwining ...

**33**

votes

**1**answer

935 views

### What can topological modular forms do for number theory?

Topological modular forms ($TMF$) have in the recent years made an impact in algebraic topology. Roughly, the spectrum $tmf$ is the (derived) global sections of the sheaf of $E_\infty$ ring spectra ...

**1**

vote

**1**answer

257 views

### How does the solenoid structure of $\mathbb{A}/\mathbb{Q}$ lift to $PGL(2, \mathbb{A})/ PGL(2, \mathbb{Q})$?

Some papers I am reading talk about an "adelic" object $PGL(2, \mathbb{Q}) \backslash PGL(2, \mathbb{A})$ . This has sparked a lot of confusion since I don't know what such a quotient could mean.
A ...

**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 ...

**2**

votes

**1**answer

125 views

### References about identities of Gauss sum

I am reading the paper. In the end of page 10, there are the following identities of Gauss sum.
\begin{align}
& h(b) h(a+b) = q^b h(b) h(a), \\
& h(b) g(a+b) = q^b h(b) g(a), \\
& g(a+b) ...

**6**

votes

**1**answer

527 views

### Multiplicity one theorem

I am reading Dorian Goldfeld's book Automorphic forms and L functions for the groups GL(n,R) ...

**13**

votes

**2**answers

430 views

### Elliptic curves and supercuspidal representations of conductor $p^2$

Let $E$ be an elliptic curve defined over $\mathbf{Q}$. Let $p \geq 5$ be a prime of additive reduction for $E$.
Let $f$ be the newform associated to $E$, and let $\pi$ be the irreducible admissible ...