All Questions
Tagged with automorphic-forms modular-forms
152 questions
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 ...
- 15.4k
51
votes
3
answers
12k
views
What is the difference between an automorphic form and a modular form?
This is more of a question about terminology than about math.
The term "automorphic form" is clearly a generalization of the term "modular form." What is not clear is exactly which generalization it ...
- 15.4k
40
votes
1
answer
2k
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 ...
- 3,799
39
votes
2
answers
4k
views
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 \...
- 497
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:\...
- 18.7k
19
votes
3
answers
1k
views
Why only half-integral weight automorphic forms?
Why is that the theory of automorphic forms concentrates on the case of half-integral weight? I read in Borel's book "Automorphic forms on $SL_2$" (Section 18.5) that by considering the finite or ...
- 955
19
votes
3
answers
2k
views
Non-vanishing of p-adic L-functions
In Non-vanishing of L-series of modular forms (easy case?) it was answered that for a cuspidal newform $f$ of weight strictly greater than 2, then $L(f,1)$ is non-zero. (Here the $L$-series is ...
- 193
17
votes
2
answers
3k
views
central/critical/special values of L-functions terminology
I have a question about the terminology for special values
of L-functions. Is the following a correct description of
standard usage:
Suppose L(s) is an L-function which satisfies a functional
...
- 743
17
votes
2
answers
4k
views
On Siegel mass formula
I have asked this question exactly here. The question is as follows:
I am interested deeply in the following problem:
Let $f$ be a (fixed) positive definite quadratic form; and let $n$ be an ...
15
votes
3
answers
1k
views
There is no lattice in PSL(2,R) which contains PSL(2,Z) properly?
How can I see that there is no lattice in $G=\mathrm{PSL}_2( \mathbb{R})$ which contains $\Gamma_1=\mathrm{PSL}_2( \mathbb{Z})$ properly, or equivalently, that $X_1 =\mathrm{PSL}_2(\mathbb{Z}) \...
- 11.2k
14
votes
3
answers
2k
views
Nonvanishing of central L-values of quadratic twists?
Let $\pi$ be a cuspidal automorphic representation of GL(2) over a number field (if you want, assume it's $\mathbb Q$ and $\pi$ comes from a holomorphic modular form).
In the case $\pi$ has trivial ...
- 6,039
13
votes
1
answer
910
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 ...
- 40.2k
13
votes
1
answer
1k
views
Reference for: CM Hilbert Modular forms arise from Hecke characters
For classical modular forms, the correspondence between the form having CM by an imaginary quadratic field $K$ and it being induced from a Hecke character on $K$ is well-known. (Ribet's paper is a ...
- 659
12
votes
2
answers
781
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 ...
- 585
12
votes
0
answers
508
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 ...
- 8,374
11
votes
2
answers
1k
views
Using Eichler-Selberg trace formula to compute dimension of modular forms?
Is it possible to use Eichler-Selberg trace formula to compute the dimension of modular forms of weight $k$ for $SL(2,\mathbb Z)$? This was computed by classical methods such as Riemann-Roch.
- 3,804
11
votes
2
answers
1k
views
modular form Fourier coefficients and associated automorphic representation
Hi,
Let $f$ be a cuspidal modular form of some weight and level $N$. Then it determines
an irreducible automorphic representation $\pi = \bigotimes'\pi_p$ of $GL_2(\mathbf Q)$.
Let $f = \sum_i a_i q^...
- 2,842
11
votes
1
answer
698
views
Are the L-functions of a normalized newform and the corresponding cuspidal representation equal?
Let $f \in S_k(\Gamma_0(N))$ be a normalized newform with Fourier expansion
$$f(z) = \sum\limits_{n=1}^{\infty} a_n e^{2\pi i z n}$$
and $a_1 = 1$. Then $f$ is an eigenfunction of all Hecke ...
- 6,180
11
votes
1
answer
490
views
conductor formula
Let $\pi_p$ be an irreducible representation of $GL_2(\mathbb{Q}_p)$. Assume $\pi_p$ is ramified,hence it will have a positive conductor. Consider $sym^3(\pi_p)$ which is a representation of $GL_4(\...
- 424
11
votes
1
answer
565
views
What can the theory of automorphic forms for $SL(n,\mathbb{Z})$ say about $SL(n,\mathbb{Z})$?
While reading "Automorphic Forms and L-functions for the Group $GL(n,R)$" by D. Goldfeld, I've got a feeling that linear groups over $\mathbb{R}$ and $\mathbb{Z}$ are considered only as technical ...
- 3,186
11
votes
1
answer
646
views
Modularity of higher genus curves
The modularity conjecture for elliptic curves over number fields is well known, and indeed, is a theorem for all elliptic curves over $\mathbb{Q}$, and at least potentially, over any CM field.
What ...
- 1,267
11
votes
0
answers
239
views
Representation of the space of lattices in $\Bbb R^n$
The space of 2D lattices in $\Bbb R^2$ can be represented with the two Eisenstein series $G_4$ and $G_6$. Each lattice uniquely maps to a point in $\Bbb C^2$ using these two invariants, and the points ...
- 4,897
11
votes
0
answers
231
views
Eisenstein series for non congruence subgoups
What is the present status of the Eisenstein series for noncongruence subgroups?
I am aware of work of A. Scholl and Rohrlich work on the subject.
Is there any specific examples that has been ...
- 248
10
votes
2
answers
404
views
Impact of the squarefreeness of the level for modular forms
I often notice papers and results that assume that the level is squarefree in the setting of modular forms, but have a hard time figuring how where this impacts or simplifies the argument. Is there in ...
- 249
9
votes
1
answer
619
views
The correct determinant exponent of the weight $k$-operator for defining Hecke operators/adelizing modular forms
For $g \in \operatorname{SL}_2(\mathbb R)$, and $\mathbb H$ the upper half plane, and $k\geq 1$ an integer, the weight $k$-operator on functions $f: \mathbb H \rightarrow \mathbb C$ is defined by
$$...
- 6,180
9
votes
2
answers
1k
views
Adelic methods for classical modular forms
Many conjectures about properties of automorphic forms on $\mathrm{GL}(2)$ can be formulated in the basic language of classical modular forms (i.e. Hecke forms that are holomorphic on $\mathbb{H}$ or ...
- 8,374
9
votes
1
answer
530
views
standard zero free region of automorphic L-function on GL(N)
Let $L(s,\pi)$ be the standard(Godement-Jacquet) $L$-function of $\pi$, where $\pi$ is a cuspidal automorphic represetation of $GL(m,A_Q)$.
What's the standard zero-free region for $L(s,\pi)$? any ...
- 3,804
9
votes
2
answers
672
views
Computing the Petersson norm of newforms of weight 2 from the symmetric square $L$-function
Let $f \in S_2(\Gamma_0(N))$ be a newform with trivial character. I want to compute the Petersson norm $\lVert f\rVert^2$ of $f$, not normalized by $1/[\operatorname{SL}_2(\mathbf{Z}):\Gamma_0(N)]$, ...
user471019
9
votes
1
answer
751
views
Spectral decomposition of product of modular functions
The eigenfunctions of the Laplacian on $SL(2,\mathbb Z)\backslash \mathbb H$ are known to be given by three types: the constant function, the real analytic Eisenstein series (which come in a ...
9
votes
1
answer
753
views
Exceptional primes
Let $f$ be a cuspidal Hecke Eigenform of weight $k \geq 2$ and let $\rho_{f, \lambda}:G_{\mathbb{Q}} \rightarrow GL_2(E_{\lambda})$ be the corresponding Galois representation with $2 \mid \lambda$ ...
- 248
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 \...
- 7,746
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 ...
- 283
9
votes
1
answer
2k
views
How did Gauss characterize the metrical relations in the uniform (4 4 4) tiling of the hyperbolic unit disk?
My purpose is to verify an historical hypothesis I have on Gauss's tesselation of the unit disk as described in John Stilwell "Mathematics and its history". Looking at the relevant pages in ...
- 2,099
9
votes
0
answers
147
views
Terminology question for the cohomology of the Hilbert modular group
Let $\Gamma$ be the Hilbert modular group of determinant one matrices with entries in the ring of integers of a real quadratic field $F$, and let $M$ be a $\Gamma$-module. Is there a standard name for ...
- 91
9
votes
0
answers
211
views
Surjectivity of reduction for Hilbert modular forms
Fix a totally real field $K$, a level $\mathfrak{n}$, a (parallel) weight $k\geq 2$ and a primitive ray class character $\chi$ modulo $\mathfrak{n}$.
Then one can form the space $S_k(\mathfrak{n},\...
- 562
9
votes
0
answers
435
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 ...
- 3,804
8
votes
2
answers
2k
views
Relation between Theta series and Eisensteinseries
In "Mackey - Unitary Group Representation in Physics, Probability and Number Theory" on page 326, George Mackey mentions a result of Ludwig Siegel, which was later generalized to semi-simple Lie ...
- 11.2k
8
votes
4
answers
1k
views
Fourier expansion at inequivalent cusps
Let $\Gamma\subset SL(2,\mathbb{R})$ be a Fuchsian group of the first kind. Let $c_1, c_2$ be inequivalent cusps of $\Gamma.$
Consider $f\in M_k(\Gamma)$ a weight $k$ holomorphic automorphic form, ...
- 83
8
votes
1
answer
658
views
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 ...
- 164
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 ...
- 424
8
votes
1
answer
888
views
Statement of classical Ramanujan-Petersson conjecture
I'm preparing for an expository talk on some topics in the representation theory of reductive p-adic groups, including tempered representations and Whittaker models, and as motivation I wanted to ...
- 815
8
votes
2
answers
739
views
Modular forms for different groups than $SL(2,\mathbb Z)$
I know some theory of "classical" modular forms, that is functions in the complex upper-half plane satisfying
$f(\frac {az+b} {cz+d})=(cz+d)^kf(z)$
I know one can study modular forms on finite-...
- 3,738
8
votes
1
answer
1k
views
Functional equation and conductor for a Rankin-Selberg convolution
Let $f$ be a Modular form/Maass form on $GL(2)$ with level $N$ and character $\eta$ and Fourier coefficients $a(n)$.
The Rankin-Selberg convolution
$$L(s,f\times\bar f)=\sum \frac{a(n)\overline{a(n)}}...
- 585
8
votes
1
answer
567
views
Symmetric power lift of modular forms
Let $f_1$ and $f_2$ be two cuspforms of weights $k_1$ and $k_2$ and nebentypus $\epsilon_1$ and $\epsilon_2$ respectively such that $f_1 \neq f_2 \otimes \chi$ for some Dirichlet character $\chi$ of ...
- 424
8
votes
1
answer
356
views
Average bounds on Rankin-Selberg coefficients for modular forms
Let $f$ be a cuspidal Hecke newform of weight $k$ and level $N$, and denote by $a_f(n)$ its $n$-th Fourier coefficient. The newform $f$ is normalized so that $a_f(1) = 1$. As a consequence of Rankin-...
- 5,424
7
votes
2
answers
2k
views
Are the $\Gamma(N)$ the only normal congruence subgroups of $\mathrm{SL}_2(\mathbb{Z})$?
The answer to the original question is no, see JSE's answer!
Are the $\Gamma(N)$ the only normal congruence subgroups of $\mathrm{PSL}_2(\mathbb{Z})$ with no finite subgroups (elliptic elements)? What ...
- 11.2k
7
votes
2
answers
787
views
Is the Sato-Tate conjecture known for Bianchi modular forms?
Originally formulated for elliptic curves, the Sato-Tate conjecture regarding the equidistribution of Frobenius trace values according to the Haar measure on a certain compact group (the Sato-Tate ...
- 2,827
7
votes
2
answers
478
views
Rankin-Selberg integral for GL(3) form with Odd Maass form on GL(2)
Let $F$ be a Hecke-Maass cusp form for $SL_3(\mathbb Z)$.
Let $u$ be a Hecke-Maass cusp form for $SL_2(\mathbb Z)$.
The following integral
$$\mathcal L(s,F\times u)=\int_{{SL}(2,\mathbb{Z})\...
- 3,804
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 ...
- 7,609
7
votes
1
answer
489
views
Complete L-function and FE of Rankin-Selberg on GL(2)?
Let $f$ be a Maass cusp form of $\Gamma_0(N)$ on the upper half plane with character $\chi$ mod $N$ and eigenvalue $1/4+\mu^2$.
What is the complete $L$-function of the Rankin-Selberg product $L(s,f\...
- 3,804