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7 votes
0 answers
103 views

Langlands correspondence of coverings of $\mathrm{SL}_2(\mathbb R)$ and modular forms with fractional weights

$\DeclareMathOperator\SL{SL}$Let $G \to \SL_2(\mathbb R)$ be a finite covering of degree $d \geq 2$. Then $G$ is a connected Lie group with semisimple Lie algebra $\mathfrak{g}=\mathfrak{sl}_2$ and ...
1 vote
0 answers
111 views

Whether or not the root number of GL$_3\times$GL$_2$ $L$-function $L(s, F \otimes g)$ contains the coefficients $\lambda_g(n)$ of $g$?

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $p$ and $q$ be two distinct primes. Let $$\Gamma_0(p)= \left\{ g\in \GL_3(\mathbb{Z}):g \equiv \left(\begin{matrix} \ast &\ast&\...
8 votes
1 answer
567 views

Symmetric power lift of modular forms

Let $f_1$ and $f_2$ be two cuspforms of weights $k_1$ and $k_2$ and nebentypus $\epsilon_1$ and $\epsilon_2$ respectively such that $f_1 \neq f_2 \otimes \chi$ for some Dirichlet character $\chi$ of ...
51 votes
7 answers
11k views

How is representation theory used in modular/automorphic forms?

There is certainly an abundance of advanced books on Galois representations and automorphic forms. What I'm wondering is more simple: What is the basic connection between modular forms and ...
3 votes
1 answer
228 views

On the local factor of Rankin-Selberg L-functions

I have a puzzle on the local factors of Rankin-Selberg $L$-functions. Consider two newforms on $\text{GL}_2$. Let $f$ be a newform of square-free level $N$, and $g$ a newform of trivial level. As ...
9 votes
1 answer
680 views

Roadmap to Carayol-Deligne-Langlands

Having begun self-study of Fermat's Last Theorem a few years ago, I have only recently begun to understand and appreciate the theorem of Carayol-Deligne-Langlands on local-global compatibility for ...
3 votes
1 answer
131 views

Depth of the filtration of higher ramification groups in the ramified case in Serre's modularity conjecture

I am studying Serre's paper "Sur les représentations modulaires de degré 2 de $\mathrm{Gal(\overline{\mathbb Q}/\mathbb Q)}$" and I have some questions about Serre's definition of "peu ...
28 votes
1 answer
3k views

Are there Maass forms where the expected Galois representation is $\ell$-adic?

Recall that by theorems of Deligne and Deligne--Serre, there is the following dichotomy: Modular forms on the upper half plane of level $N$ and weight $k\geq 2$ correspond to representations $\rho:\...
5 votes
2 answers
1k views

Are umbral moonshine and umbral calculus connected?

In a 2013 article, Cheng, Duncan and Harvey introduce the concept of umbral moonshine as a generalization of monstrous moonshine. The terminology they use, starting with the title, is common in umbral ...
6 votes
1 answer
574 views

Automorphic representation of GL(1)

These might be very silly questions, but somehow I am not able to understand it or I might have misunderstood something. I am reading automorphic forms from this book. What I have understood till now: ...
8 votes
2 answers
744 views

A question related to Hilbert modular form

This is a question related to Hilbert modular forms. Let $\mathbb{K}=\mathbb{Q}(\sqrt D)$ be an imaginary quadratic field with discriminant $D<0$ and $\zeta (\text{mod } m)$ a Hecke character such ...
1 vote
0 answers
103 views

Global irreducible admissible representations analogue

Let ${A}_{\mathbb{Q}}$ denote the adele over the rational numbers. Then it is known that cuspidal modular forms of level $N$ correspond to some unitary automorphic representation of $\operatorname{GL}...
5 votes
2 answers
237 views

Irreducibility of the $n$th symetric power of the reduction of the Galois representation of a non-CM newform

In "On $\ell$-adic representations attached to modular forms II", Ribet proved that the $\ell$-adic representation $\rho_{f,\ell}$ attached to a non-CM newform form $f$ satisfies $${\rm SL}...
2 votes
0 answers
128 views

Kac-Peterson modular forms and shifted theta functions

Let $\Lambda$ be the root lattice corresponding to an ADE root system $R$ of rank $n$. With the ADE assumption, the weight lattice is simply the dual lattice $\Lambda^{\vee}$. Given any weight vector $...
4 votes
0 answers
155 views

How to obtain the harmonic theta series via the global theta correspondence explicitly?

I am interested in the following kind of modular forms: let $(L,q)$ be an even unimodular lattice inside $V:=L\otimes \mathbb{Q}$, and $P$ is a harmonic homogeneous polynomial of degree $d$ on $\...
1 vote
0 answers
109 views

Whether or not the Maass form for $\Gamma _0(N)$ on $GL(3)$ covers the classical symmetric lift of a newform on $GL(2)$?

I have a blur which needs some help from the experts here, and may look naive for some experts. Recently I read Zhou's paper "The Voronoi formula on $GL{(3)}$ with ramification" (https://...
4 votes
0 answers
135 views

Values at 1 of symmetric power L-functions of Maass cusp forms

I have a blur that whether one has $L(1,\text{sym}^2f)\ll \log^A q$ for some $A>0$? Here $f$ is assumed to be a Maass cusp form of square-free level $q$. If any experts here know something about ...
1 vote
1 answer
323 views

How to relate Rankin triple L-function to its Dirichlet series

I have a very tricky question which may look naive to many experts here. Let $f$ be a newform of level prime $P$, and $g,h$ two newforms of level 1, respectively. These three forms $f,g,h$ are all of ...
4 votes
0 answers
204 views

$\DeclareMathOperator\sym{sym}$Does $L(s, \sym^2 f \times \sym^2 g)$ have a pole at $s=1$?

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\sym{sym}$I encountered a question on the poles of $\GL_3\times \GL_3$ $L$-function, which needs the knowledge of the experts ...
7 votes
1 answer
582 views

Hurwitz numbers and $t$-cores

For integers $k \geq 0$ and $d \geq 1$ let $H(k,d)$ be the Hurwitz number which, for the purposes of this posting, will be defined by: \begin{equation} H(k,d) \, := \ d! \, \sum_{\lambda \, \vdash d}...
3 votes
0 answers
95 views

Principal series representation of $SL(2,\mathbb{R})$ restricted to principal congruence subgroup

Given a principal congruence subgroup $\Gamma(N)$ of $SL(2,\mathbb{R})$, since $\Gamma(N)$ is free, consider a probability distribution $\mu_1$ of a simple random walk on $\Gamma(N)$ and consider its ...
6 votes
0 answers
190 views

Eigenvalues of spherical function on $\mathrm{SL}(2,\mathbb{R})$

Lie algebraically, the eigenvalue of the spherical function \begin{align*} \phi_{\lambda}(g)=\int_{K} e^{(i \lambda+\rho)(A(k g))} \mathrm{d} k \quad (g \in G,\,\lambda\in\mathfrak{a}^*) \end{align*} ...
9 votes
1 answer
787 views

The cohomology of modular curves as a module over the Galois group

Consider the modular curve $\pi: X(N) \to X(1)$ where this map has Galois group $G = PSL_2(\mathbb Z/N\mathbb Z)$. In particular, $G$ acts on the singular cohomology $H^1(X(N),\mathbb Z)\otimes \...
4 votes
1 answer
517 views

Surjectivity in Deligne-Serre

Let $f$ be a newform of weight $k$ and level $N$ with integer coefficients. Deligne-Serre theorem theorem says there exist a nice associated representation $\rho_{f}^{(\ell)}:\text{Gal}(\overline{\...
3 votes
1 answer
277 views

Level vs. conductor of a supercuspidal representation

What is the relation between level and conductor of a supercuspidal representation of $\operatorname{GL}_2(\mathbb{Q}_p)$ for some prime $p$? Proposition 3.4 in Loeffler and Weinstein - On the ...
2 votes
0 answers
107 views

Paramodular forms with level and Iwahori subgroups?

Given an integer $N>0$, not necessarily prime, we have the paramodular group $K(N) \subset \text{Sp}_{4}(\mathbb{Q})$, which consists of matrices of the form $$\begin{bmatrix} * & *N & * &...
1 vote
1 answer
274 views

Analogous theorem for Hilbert modular forms

I have studied modular forms and saw a correspondence like a newform correspond to a automorphic representation of $\mathrm{GL}_n(\mathbb{A_Q})$. Does any similar result holds for Hilbert modular ...
3 votes
0 answers
147 views

Classical Hecke operators and Hecke algebra of type $A_1$

What's the relation between the classical Hecke operators (as defined in J. P. Serre's A course in arithmetic chapter 6) and the Hecke algebra of type $A_1$, i.e. the algebra generated by the vertices ...
6 votes
2 answers
1k views

About different cohomology theories used to study Shimura varieties

The classical theory of Eichler-Shimura realizes the space of cusp forms of certain weights and levels in the parabolic cohomology of modular curves, which is the image of the cohomology with compact ...
1 vote
0 answers
138 views

Hilbert modular form as a representation of Hecke algebra

I am reading some notes by Snowden and I don't understand a sentence. Clearly, if we have an appropriate $R = T$ theorem then we get a modularity lifting theorem, as $\rho$ defines a homomorphism ...
3 votes
0 answers
166 views

Automorphy Factor from Vector Bundles on Compact Dual

So I'm coming from an algebraic geometry perspective and I'm trying to carefully piece together the story of interpreting automorphic forms as sections of vector bundles on Shimura varieties. I think ...
6 votes
1 answer
371 views

Classification of quasi-lisse vertex algebras

Quasi-lisse vertex algebras were introduced by Arakawa and Kawasetsu in Quasi-lisse vertex algebras and modular linear differential equations . They satisfy the property that the normalized character ...
5 votes
1 answer
461 views

To what extent does the $(\mathfrak{g},K_{\infty})$ module determines the automorphic representation?

In this question, we will follow the notations and definitions from the book "Automorphic Representations and L-Functions for the General Linear Group" by Goldfeld and Hundley. For simplicity, let's ...
6 votes
2 answers
846 views

Serre's conjecture for mod-p^n representations?

I think this may be a silly question, but here goes. Let $\rho:\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to \mathrm{GL}_2(\overline{\mathbf{F}_p})$ be a representation; say $\rho$ is of S-type ...
2 votes
0 answers
124 views

Modular transformation of affine characters of non-simply connected groups$.$

Consider an (untwisted) affine algebra corresponding to a compact and simply-connected Lie group $G$. Under a modular transformation, its characters transform as (cf. 9612078) $$ \chi_\mu\to\sum_{\nu\...
9 votes
2 answers
1k views

Hecke algebra of GL(2,F)

I was studying about the Hecke algebra from Bernstein's notes on p-adic representation theory and various other sources. First a disclaimer: everything below is fairly new to me so please feel free to ...
5 votes
0 answers
158 views

Maass-Saito-Kurokawa Lift of Weak Jacobi Forms

Given a holomorphic Jacobi form $\varphi_{k,1} \in \mathbb{J}_{k,1}$ of weight $k$ and index 1, we know we can use the Hecke operators $V_{m}$ to lift $\varphi_{k,1}$ to a Siegel modular form $$\...
6 votes
1 answer
678 views

Root number of the Rankin-Selberg convolution of two newforms

Let $p$ and $q$ be two distinct primes. Let $f\in \mathcal{S}_k^{\ast}(pq,\psi)$ be a holomorphic newform of level $pq$, nebentypus $\psi$, and weight $k$, where $\psi = \chi_p \chi_{0(q)}$, with $\...
1 vote
1 answer
119 views

eigen forms of integer weights and multiplicative functions

Let $f=\sum a(n)q^n$ a modular cuspidal form of integer weight on some congruece subgroup. Suppose that $a(1)=1$ and that $f$ is proper to Hecke operators. Its is known that the coefficients $a(n)$ ...
3 votes
0 answers
94 views

Representations of $\mathbb{H}^{\times}$ and $\mathbb{H}^{\times}/\mathbb{R}^{\times}$

In an attempt to recapture Eichler's theta correspondence I have hit a stumbling block. Let $D$ be a quaternion algebra over $\mathbb{Q}$, ramified at $p,\infty$. Also let $V_j = \text{Symm}^j(\...
0 votes
1 answer
387 views

On a claim of Zagier on extending a map to cocycle

Zagier, in his paper 'Some Surprising Consequences of the Cohomology of SL$_2(\bf{ Z})$' (link, p. 6), studies the action of $\Gamma=PSL_2(\bf Z)$ on a vector space $V$, denoting the action by $v\ |\ \...
4 votes
3 answers
568 views

Functions on hyperbolic space and modular curves

The decomposition of $L^{2}\left(S^{2}\right)$ under $SO\left(3,\mathbb{R}\right)$ is well-known. Focus now on the hyperbolic plane $H$ presented as the quotient $SL\left(2,\mathbb{R}\right)/SO\left(...
2 votes
0 answers
212 views

Conceptual reason behind Shimura lifts

Shimura lifts are correspondence between integer weight and half-integral weight automorphic forms. Half integral weight things are associated to representation of a double cover of $G =SL_2(\mathbb{R}...
7 votes
2 answers
1k 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 ...
16 votes
2 answers
1k views

Etymology of cuspidal representations

In the literature on representation theory of $GL_2(\Bbb F_p)$ and $GL_2(\Bbb Q_p)$, the irreducible representations with trivial Jacquet module are often called "cuspidal" or "supercuspidal". Why are ...
7 votes
2 answers
667 views

Representation-theoretic operations on modular forms

Let $A$ and $B$ be Hecke eigenforms of some weight $k$ and level $N$. We know that there are irreducible representations $\rho_a$, $\rho_b$ of the absolute Galois group of $\mathbb{Q}$ whose trace of ...
2 votes
1 answer
212 views

Modular Forms w.r.t. different representations linearly independent?

Let $V$ be a finite dimensional vector space over $\mathbb{C}$. Let $G = \text{SL}_2(\mathbb{Z})$, say, and $\rho : G \to \text{GL}_\mathbb{C}(V)$ a representation. Let us take a fixed basis $b_1, ...,...
23 votes
3 answers
4k views

Subgroups of GL(2,q)

I have been wondering about something for a while now, and the simplest incarnation of it is the following question: Find a finite group that is not a subgroup of any $GL_2(q)$. Here, $GL_2(q)$ is ...
22 votes
3 answers
2k views

Is SL(2,C)/SL(2,Z) a quasi-projective variety?

Consider the complex 3-fold $SL(2,\mathbb C)/SL(2,\mathbb Z)$ (just for clarity: note that $SL(2,\mathbb Z)$ acts without stabilizers, so this is a complex manifold, not a complex orbifold). Is $SL(...
9 votes
0 answers
520 views

Why is this vector space related to modular forms?

In the course of doing some calculations on a project I am working on, I came across the following presentation of a vector space. It is generated by homogenous polynomials of even degree $n$ over a ...