All Questions
Tagged with automorphic-forms modular-forms
152 questions
3
votes
0
answers
190
views
Voronoi formula on $\mathrm{GL}_4$ in the level aspect with ramification
$\DeclareMathOperator\GL{GL}$Let $f$ be an automorphic form on $\GL_3$ for $\Gamma_0(p)$ with $p$ being a prime (see Bump or Goldfeld's books for definitions). Recall that, in this paper-"The ...
- 41
2
votes
0
answers
164
views
Bounds of modular functions on the Ford circles
Assume a holomorphic function from a product of two upper half planes $Z: \mathbb{H}_+\times \mathbb{H}_+\rightarrow \mathbb{C}$ with an expansion of the form
$$
Z(\tau,\tau') = \sum_{(h,h')\in S} a_{...
- 21
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
8
votes
1
answer
356
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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
3
votes
1
answer
228
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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 ...
- 353
4
votes
0
answers
103
views
Automorphic forms on $\mathrm{GL}_{2}$, $\mathrm{SL}_{2}$, and $\mathrm{Mp}_{2}$ — classical counterparts
I asked exact same question on MSE but haven't got answer yet, so asking here, too. I may erase the original one once I got an answer here.
--
I'm confusing about automorphic representations of $\...
- 2,215
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
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
3
votes
0
answers
122
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Is there any notion of Poincaré series for Hermitian modular forms?
I have been studying modular forms and their generalisations for a year or so. It is a very interesting fact that the space of cusp forms $S_k$ is generated by the Poincaré series of exponential type (...
3
votes
1
answer
214
views
$p$th Fourier coefficients of newforms for ramified primes $p$
This question is about some basic(classical) results on Atkin-Lehner-Li theory of newforms. Let $f$ be a (normalized) newform of level $N$ and character $\epsilon$. Denote the $n$th Fourier ...
- 663
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:
...
- 424
2
votes
1
answer
147
views
On the square mean of Fourier coefficients of cusp forms
I have a question which may look naive for many experts here:
For any primitive holomorphic form $f$ of level $M$ ($M\in \mathbb{N}$), whether or not one has the lower bound that:
$$\sum_{X<n\le 2X}...
- 563
4
votes
1
answer
299
views
The Wilton-type bounds involving half-integral weight cusp forms
There is a basic question which puzzles me for a while, and maybe look naive for some experts here. The question is the following:
Let $f(z)=\sum_{n\ge 1} a_f(n) n^{k/2-1/4}e(nz)\in S_{k+1/2}(4N)$ be ...
- 563
3
votes
1
answer
237
views
Experiments with Voronoï summation
In order to test my understanding of the Voronoï summation formula, I tried to apply it to a simple estimation of partial sums of Fourier coefficients of cusp forms. The result I obtained cannot ...
- 545
3
votes
0
answers
96
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Discrete subgroups of $\text{Sp}_{4}(\mathbb{Q})$ parameterizing polarized Abelian surfaces plus torsion data
I want to start by considering a familiar congruence subgroup of the integral symplectic group $\text{Sp}_{4}(\mathbb{Z})$. For a positive integer $N$, let $\Gamma _{0}^{(2)}(N) \subset \text{Sp} _{4}(...
- 1,701
3
votes
1
answer
174
views
Petersson norms of quaternionic modular forms
How is the Petersson norm of a quaternionic modular form defined?
Background: In Tamiozzo, On the Bloch-Kato conjecture for Hilbert modular forms, section 3.3, it is written "We normalize $f_B$ ...
user471019
1
vote
0
answers
87
views
what is the relationship betwen $L(s,sym^mf\times sym^mg)$ symmetric L function of $f$ and $g$ and $\lambda_{f}(n^m)$, $\lambda_{g}(n^m)$?
what is the relationship betwen $L(s,sym^mf\times sym^mg)$ symmetric L function of $f$ and $g$ and $\lambda_{f}(n^m)$, $\lambda_{g}(n^m)$ ?
- 11
5
votes
1
answer
891
views
Can Taniyama-Shimura conjecture be generalized to curves of higher genus (within Langlands framework)?
The Shimura-Taniyama-Weil conjecture asserts that if E is an elliptic curve over Q, then there is an integer N ≥ 1 and a weight-two cusp form f of level N, normalized so that a1(f) = 1, such that ap(E)...
- 2,216
2
votes
1
answer
219
views
Voronoï summation for cusp forms with characters
In an attempt to solve an unrelated problem, I was led to the task of estimating/bounding from above sums of the form
$$\sum_{m=1}^\infty\lambda(m)e\left(-\frac{am}{q}\right)h(m)$$
where $\sum_{m=1}^\...
- 545
4
votes
0
answers
187
views
Understanding Shimura correspondence in context of Langlands functoriality
Recently, I started to read about automorphic forms and representations on covering groups, e.g. metaplectic groups. I set my first goal as understanding Shimura's correspondence in representation ...
- 2,215
3
votes
1
answer
93
views
Decomposition of real quasimodular forms of depth 1
Let $\widetilde{M}_k^{\leq \ell}$ be the space of weight $k$ depth $\leq \ell$ quasimodular forms, and $\widetilde{M}_{k,\mathbb R}^{\leq \ell}$ be a subspace of $\widetilde{M}_k^{\leq \ell}$ whose ...
- 663
2
votes
0
answers
98
views
Extrema of real analytic Eisenstein series and more general modular functions
The real analytic Eisenstein series defined by the Poincare sum
$$E(s,z)=\sum_{(m,n)\neq (0,0)} {y^s\over |mz+n|^{2s}}$$
for $z\in{\mathbb H}$ and ${\rm Re}(s)>1$ is a manifestly $SL(2,{\mathbb Z})$...
- 21
3
votes
0
answers
144
views
Ash–Stevens for Hilbert modular forms
In the theory of mod-$p$ modular forms, I learned a while ago about an interesting result that I think is technically due to Serre and Tate, though the proof was first published by Jochnowitz in ...
- 241
2
votes
0
answers
192
views
Modular forms and Petersson inner product via De Rham cohomology, Hodge filtration and cup products
I'm looking for an explanation on how and why you can define modular forms through De Rham cohomology via the Hodge filtration and especially how the Petersson inner product is related to the cup ...
- 804
1
vote
0
answers
151
views
Action of $T_p$ on automorphic forms, and error in Gelbart's "Automorphic forms on adele groups"?
Let $f\in\mathcal{S}_k(\Gamma_0(N),\chi)$ be a cuspidal modular form, and $\phi_f\in\mathcal{A}_0(\text{GL}_2(\mathbb{Q})\backslash\text{GL}_2(\mathbb{A}_\mathbb{Q}),\omega)$ be its corresponding ...
- 111
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 $\...
- 391
4
votes
0
answers
204
views
A question on the twisted symmetric square L-functions
Sorry to disturb. I have a puzzle which might be naive for many experts here.
Let $f$ be a Hecke newform of prime level $N$ on $\mathrm{GL}_2$, and $
\chi$ a primitive character of square-free ...
- 353
2
votes
0
answers
205
views
Two basic questions on congruence subgroups
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$I have two questions related to congruence subgroups.
Let $$\Gamma=\Gamma_0(N)=\Big\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \...
- 1,019
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://...
- 563
3
votes
0
answers
89
views
Hoffstein–Lockhart for non-congruence subgroups
Let $\Gamma$ be a non-congruence subgroup of $\operatorname{SL}(2,\mathbb{Z})$ of finite index and let $f$ be a holomorphic cuspidal modular form of weight $k$ for the group $\Gamma$. For simplicity, ...
- 589
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
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 ...
- 563
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 ...
- 563
6
votes
2
answers
392
views
A lower-bound for the square-mean of Fourier coefficients of cusp forms at primes argument
There is a basis question which puzzles me for a while. The question is the following:
Let $X\ge 2,$ and $\lambda(n)$ be the $n$-th Fourier coefficient of a $GL(2)$ newform of prime level $N>1$, ...
- 563
5
votes
1
answer
184
views
Smallest Fourier coefficient divisible by a prime
If $f$ is a cusp form of weight $k$ and level $\Gamma_1(N)$. For simplicity, suppose that the Fourier coefficients of $f$ at $\infty$ are in $\mathbb{Z}$. Let $\ell$ be a prime which does not divide ...
- 589
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
2
votes
0
answers
179
views
On Fourier coefficients of Bianchi modular forms, l-ordinary
Let $f\in S_2(\Gamma_1(N))$ be a Hecke eigenform and $\ell$ a prime number does not divide $N$. Let $a_f(\ell)$ be the $\ell$-th Fourier coefficient of $f$. Then $a_f(\ell)$ is is called $\ell$-...
- 21
0
votes
0
answers
159
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Holomorphic automorphic/cusp forms on real Lie groups
An automorphic form on a real Lie group $G$ for a discrete subgroup $\Gamma$ is a function $f:G\to\mathbb{C}$ with some properties (see Borel’s definition in Proceedings of Symposia in PURE ...
- 391
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
4
votes
1
answer
417
views
What are the known number-theoretic functions, that are related to "the number of ideals of norm $n$, that belong to the class $[c]$"?
Let $L$ be a number field, $\mathcal{O}_L$ its ring of integers, and $\mathcal{Cl(O}_L)$ its ideal class group. Let's fix an arbitrary class $[c] \in \mathcal{Cl(O}_L)$. By $r(n)=r([c], n)$, I mean ...
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
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 ...
1
vote
0
answers
134
views
Automorphic representation of weight 3 eigenforms
Let $f$ be a weight 3 eigenform with rational Fourier coefficients. As shown by Elkies and Schutt, $f$ is associated to a singular K3 surface over $\mathbb{Q}$. A construction of Shioda and Inose ...
- 167
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 ...
- 131
1
vote
0
answers
80
views
Polyharmonic Maass forms are automorphic forms on $\mathrm{SL}_2(\mathbb{R})$
Let $G=\mathrm{SL}_2(\mathbb{R})$, $K=\mathrm{SO}(2)$, and $f$ be a holomorphic modular form of weight $k$ for $\Gamma$ a Fuchsian group of the first kind. In Borel's book, 'automorphic forms on $\...
- 663
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
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
7
votes
0
answers
122
views
Theta Function Associated to Kummer Lattice
This is something which I feel must be out in the literature somewhere, but I have been unable to find anything.
If we let $\text{Km}(A)$ be the Kummer $K3$ surface associated to an abelian surface $A$...
- 1,701
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
4
votes
1
answer
211
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anti-holomorphic Hilbert modular forms as global sections
The classical definition of Hilbert modular cuspforms as given in say, Hida's "On $p$-adic Hecke algebras for $\mathrm{GL}_2$ over totally real fields", defines them as holomorphic functions ...
- 302