All Questions
Tagged with analytic-number-theory modular-forms
140 questions
1
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111
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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&\...
- 41
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
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votes
2
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223
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What is the definition of Tr in the context of Hilbert modular forms?
I am currently reading Garrett's book "Holomorphic Hilbert Modular Forms". But I meet trouble at the starting line. Let $F = \mathbb{Q}(\sqrt D)$ be a real quadratic field, $u= a + b\sqrt{D}\...
- 185
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-...
CommunityBot
- 1
3
votes
1
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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 ...
- 148k
2
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1
answer
147
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Finiteness and bounds for elliptic curves realizing a given galois representation
Let $\rho: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{Z}_p)$ be a continuous, irreducible Galois representation. Consider the set $\mathcal{L}_\rho$ of all elliptic curves $...
- 37k
5
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126
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Using Lang–Trotter to get bounds on averages of Fourier coefficients
Let $E$ be an elliptic curve over $\mathbf{Q}$ and let $(a_n)$ be the sequence of Fourier coefficients for the weight two newform attached to $E$. The coefficients $a_p$ are the Frobenius traces given ...
- 12.9k
3
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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 (...
- 4,713
2
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1
answer
229
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Generating function over primes in an arithmetic progression
Given a newform $\sum_{n=1}^{\infty}a(n)q^n$. Is the generating function
$$
\sum_{p\equiv a\pmod{m}}a(p)q^p
$$
over the primes $p\equiv a\pmod{m}$ still a modular form? Any help is highly appreciated! ...
- 105k
4
votes
1
answer
229
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Abscissa of convergence of the $\tau$ Dirichlet series
Define the $\tau$ Dirichlet series $L$ by
$$L(s)=\sum_{n=1}^\infty \frac{\tau (n)}{n^s}$$
where $\tau$ is defined by
$$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty \tau (n)q^n$$
where $|q|\lt 1$....
- 105k
2
votes
1
answer
147
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On the square mean of Fourier coefficients of cusp forms
I have a question which may look naive for many experts here:
For any primitive holomorphic form $f$ of level $M$ ($M\in \mathbb{N}$), whether or not one has the lower bound that:
$$\sum_{X<n\le 2X}...
- 5,089
4
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1
answer
192
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Identity related to Ramanujan's congruences
A very simple question: how do you prove the following identity:
$$\sum_{k=0}^\infty p_{5k+4}x^k=5\frac{\phi(x^5)^5}{\phi(x)^6},$$
where
$$\phi(x)=\prod_{n=1}^\infty 1-x^n,$$
and $p_n$ is the ...
- 75
1
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0
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167
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On two formulas involving the $k$-fold divisor function $d_k$ and the function $r_k$
I have a puzzle which needs some help form the experts here. Let $d(n)$ be the divisor function, and $d_k(n)$ the $k$-fold divisor function.
I) It is known that, for any positive integer $h$,
$$d(n+h)...
- 10.4k
2
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0
answers
286
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Is Sturm's theorem able to do these?
$\newcommand{\Ord}{\operatorname{Ord}}$Let $p$ be a positive integer and $F(q)=\sum A(m)q^m$ be a formal power series with integer coefficients. Then $\Ord_p(F(q))$ is defined by
$$\Ord_p(F(q)):=\min\{...
- 6,367
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 ...
- 6,367
12
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3
answers
2k
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Kronecker's Jugendtraum for real quadratic fields?
Kronecker's Jugendtraum (or Hilbert's 12'th problem) is to find abelian extensions of arbitrary number fields by adjoining `special' values of transcendental functions. The Kronecker-Weber theorem was ...
- 18.4k
3
votes
1
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237
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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 ...
- 1,692
1
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0
answers
87
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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
7
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1
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264
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Upper bound of the analytic rank of the modular Jacobian varieties $J_1(N)$
Does there exist an upper bound of the analytic rank of the modular Jacobian varieties $J_1(N)$?
(Or more generally of $J_\Gamma$ for a congruence subgroup $\Gamma_0 \subseteq \Gamma \subseteq \...
- 37k
2
votes
1
answer
219
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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}^\...
- 1,692
4
votes
0
answers
509
views
Ramanujan's conjecture on modular forms and Riemann hypothesis
I just watched Kannan Soundararajan's talk on the distributions of valus of zeta and $L$-functions at virtual ICM 2022. In his talk, he introduced a theorem on Ramanujan's ternary form $\phi_{1}: x^{2}...
- 8,374
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2
answers
744
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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 ...
- 23k
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
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137
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Entropy of a sequence
I am reading the paper Sign Changes in Hecke Eigenvalues by Matomaki and Radziwill, and in one place they mention the following,
It would be interesting to rule out the possibility of $\lambda_f(n)$ ...
- 589
1
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0
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103
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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}...
- 145
7
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1
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435
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Fricke involution and Atkin operator
Let $f\in S_k(\Gamma_0(N))$ be a cusp form for $N>1$. Consider the following operators acting on $f$ via the natural action of $GL_2^{+}(\mathbb{R})$ :
$$ W_N=\begin{pmatrix}
0 & -1\\
N & 0
...
- 20.8k
4
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0
answers
204
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A question on the twisted symmetric square L-functions
Sorry to disturb. I have a puzzle which might be naive for many experts here.
Let $f$ be a Hecke newform of prime level $N$ on $\mathrm{GL}_2$, and $
\chi$ a primitive character of square-free ...
- 8,374
2
votes
0
answers
245
views
Ambiguity about the exact definition of coefficients of modular forms
You can see the parts after my questions in the boxes. I received the answer to my first question in the comments.
I am confused about the definition of $a_n$ and $b_n$ in Part II below. I know the ...
- 2,784
1
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0
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109
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Whether or not the Maass form for $\Gamma _0(N)$ on $GL(3)$ covers the classical symmetric lift of a newform on $GL(2)$?
I have a blur which needs some help from the experts here, and may look naive for some experts. Recently I read Zhou's paper "The Voronoi formula on $GL{(3)}$ with ramification" (https://...
- 563
6
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1
answer
943
views
Voronoi formula for the symmetric $L$-function with level $N $
Sorry to disturb. Does any experts here know something upon the Voronoi type for the symmetric $L$-functions$$\sum_{n\le X} A_F(1,n)e\left ( \frac{an}{c}\right)=?$$
Here $F$ is a symmetric-lift of a $...
- 571
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votes
1
answer
163
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Lower bound related to derivative of $j$-invariant
Recall the $j$-invariant function, namely,
$$
j(\tau)=\frac{1}{q}+\sum_{k\geq 0}c_kq^k,
$$
where $q=e^{2\pi i \tau}$ and the coefficients $(c_k)_k$ are in the OEIS sequence A000521.
By using some ...
- 3,403
3
votes
3
answers
483
views
Growth of the coefficients of the inversion of the $j$-invariant function
We have the $j$-invariant defined as
I have that
$$
j(\tau)=\frac{1}{q}+\sum_{k\geq 0}c_kq^k,
$$
where $q=e^{-2\pi t}$ ($\tau=it$) and $c_k\sim e^{4\pi\sqrt{k}}/(k^{3/4}\sqrt{2})$.
The inversion ...
- 3,403
6
votes
0
answers
456
views
Conditions under which an $\eta$-quotient becomes a **weak** modular form (reference request for theorems similar to Ligozat's theorem)
For any $z \in \mathcal{H}$, let $q = e^{2\pi iz}$; and the eta function is defined as
${\displaystyle \eta (q)
=q^{\frac {1}{24}}\prod _{n=1}^{\infty }\left(1-q^{n}\right).}$
By an $\eta$-quotient ...
2
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1
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159
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On some claims on cyclic modules over Hecke algebra used in Serre's "Quelques applications du théorème de densité de Chebotarev"
I have been reading section 7 of Serre's "Quelques applications du théorème de densité de Chebotarev" (http://www.numdam.org/item/PMIHES_1981__54__123_0/), and in particular have been trying ...
- 105k
2
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0
answers
270
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Generalized Siegel Weil formula
I am studying the following Poincare-like series,
\begin{equation}
F_k(\tau,\bar{\tau})=\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma}\sqrt{\text{Im}\gamma\tau}(q_{\gamma}\bar{q}_{\gamma})^k,
\end{...
- 103
11
votes
1
answer
699
views
"strange" diophantine and parity of the partition function
Let $\{x_i\}:=\{x_1=5, x_2=13, x_3=29, x_4=37, x_5=45, \dots \}$
be the sequence of those positive integers of the form
$$
p^{4\alpha+1}n^2$$
in increasing order where $p\equiv 5\pmod 8$ is prime ...
- 43.2k
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
5
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1
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358
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Coefficients of modular forms and the Sato-Tate distribution
Let $a(n)$ be the $n$th Fourier coefficient of a normalized Hecke eigenform $f(z)=\sum_{n=1}^{\infty}a(n)q^n$ of weight $k$ with respect to the full modular group, where $q=e^{i2\pi z}$.
A new paper [...
- 51
6
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2
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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
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 ...
- 148k
4
votes
0
answers
204
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$\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 ...
- 12.9k
9
votes
0
answers
208
views
Unexpected patterns on the graph of an L-function on the critical line
Let $L(s)$ be the $L$-function associated to the (only) classical modular form of weight $26$ and level $1$.
The completed L-function $\Lambda(s)=2(2\pi)^{-s}\Gamma(s) L(s)$ is symmetric with respect ...
- 9,501
10
votes
2
answers
1k
views
Modular forms with finitely many or very few non-zero Fourier coefficients
I have an elementary question on modular forms, but which I don't know how to solve.
a) Is there a congruence subgroup $\Gamma \leq \mathrm{SL}_2(\Bbb Z)$, an integer $k \in \Bbb Z$ and a non-...
- 28.2k
2
votes
0
answers
86
views
Ordinary primes for a weak form corresponding to a CM newform
Setup: Let $f$ be a harmonic Maass form of weight $2-k$ ($k \in \mathbb{N}$), level $N$, and character $\chi$. Letting $q := e^{2\pi i z}$ and considering the Fourier expansion of any harmonic Maass ...
- 26
3
votes
0
answers
186
views
Maximum value of newform from Galois representation
One can attach $\ell$-adic Galois representations to holomorphic cuspidal newforms of weight $2$ on the upper half-plane.
If a newform is $L^2$-normalized, can one extract its maximum value from the ...
- 105k
1
vote
0
answers
132
views
Stabilizers of points in the upper half-plane
Suppose that $\Gamma$ is a group acting discontinuously on $\mathcal{H} = \{ z \in \mathbb{C} : \operatorname{Im}(z) > 1\}$. In order to keep things simple, suppose that $\Gamma \subseteq \...
- 163
1
vote
0
answers
129
views
Siegel's formula for generalized theta series with characteristics?
Siegel's formula(Siegel-Weil) directly relates the weighted sum of theta functions to Eisenstein series. (Or equivalently, the weighted sum of the cusp form is zero). I wonder if there is a ...
- 11
2
votes
1
answer
163
views
Theta series analogues for higher degree forms
It is simple to see that the following series converges absolutely and uniformly on $\mathcal{H}$ for all k positive:
$F_{2k}(z) = \sum_{n \in \mathbb{Z}} q^{n^{2k}}$
And this series being a ...
- 1,243
4
votes
0
answers
288
views
Lower bound on symmetric square L-function
In a paper of Soundararajan, equation (16c) states that for any Hecke eigenform $f$ of weight $k$, the symmetric square L-function at $1$ satisfies the bound
$$(\log k)^{-2}\ll L(1, \mathrm{sym}^2f)\...
- 1,890
4
votes
1
answer
282
views
Mean square estimate for the Kloosterman sums
For $m,n\in \mathbb{N}$, denote the Kloosterman sum
$$S(m,n;c)=\sum_{a\bmod c}e\left( \frac{ma+n \overline{a}}{c}\right),$$where $\overline{a}$
denotes the multiplicative inverse of $a\bmod c$.
Does ...
- 43.7k