All Questions
Tagged with analytic-number-theory modular-forms
140 questions
38
votes
4
answers
6k
views
Modular forms and the Riemann Hypothesis
Is there any statement directly about modular forms that is equivalent to the Riemann Hypothesis for L-functions?
What I'm thinking of is this: under the Mellin transform, the Riemann zeta function $...
- 889
30
votes
2
answers
9k
views
Why Is $e^{\pi\sqrt{232}}$ an Almost Integer?
We have already discussed why $e^{(\pi\sqrt{163})}$ is an almost integer.
Why are powers of $\exp(\pi\sqrt{163})$ almost integers?
Basically $j(\frac{1+\sqrt{-163}}{2} ) \simeq 744 - e^{\pi\sqrt{163}...
- 1,089
28
votes
1
answer
2k
views
Intuitive reason why the $j$-invariant is a cube?
Let $\tau$ be a CM point of discriminant $D$. Assume that $D$ is not divisible by $3$. Then $j(\tau)$ is an algebraic integer of degree equal to the class number $h(D)$. Let $ \gamma_2(\tau)=j(\tau)^{...
- 2,375
27
votes
4
answers
2k
views
Which quaternary quadratic form represents $n$ the greatest number of times?
Let $Q$ be a four-variable positive-definite quadratic form with integer coefficients and let $r_{Q}(n)$ be the number of representations of $n$ by $Q$. The theory of modular forms explains how $r_{Q}(...
- 20.4k
23
votes
3
answers
2k
views
Why are values of Eisenstein $E_2^*$ algebraic integers?
I'm looking for a proof that the following term is an algebraic integer whenever $\tau_N=\frac{N+\sqrt{-N}}{2}$ is a quadratic irrationality with class number $1$:
$$A_N:=\sqrt{-N}\cdot\frac{E_2(\...
- 598
21
votes
1
answer
1k
views
Why does this quasi-modular function have integral values?
It is a well-known result that the modular function $1728J(\tau) := \frac{1728E_4(\tau)^3}{E_4(\tau)^3-E_6(\tau)^2}$ has integral values if $\tau$ has class number 1 - for example at $\tau_{163}:=\...
- 598
17
votes
1
answer
987
views
Theta functions, re-expressed
Recall the classical $\theta(q):=\prod_{k=1}^{\infty}(1-q^k)$ and
define the sequences $a_n$ and $b_n$ by
$$\frac{\theta^3(q)}{\theta(q^3)}=\sum_{n=0}^{\infty}a_nq^n \qquad \text{and} \qquad
F(q):=\...
- 43.2k
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
15
votes
5
answers
2k
views
Very strong multiplicity one for Hecke eigenforms
In Invent. math. 116, 645-649 (1994) Dinakar Ramakrishnan proves a theorem which I understand to imply that the following statement (in light of the fact that elliptic curves over $\mathbb{Q}$ are ...
- 7,062
15
votes
1
answer
680
views
Is $\eta(\tau)^2$ a modular form of weight 1 on $\Gamma(12)$?
As we know, the Dedekind eta function $\eta(\tau)$ acquires a phase $\exp(2\pi i/24)$ under the modular transformation: $\tau \rightarrow \tau+1$. Therefore $\eta(\tau)^2$ is invariant under $\tau \...
- 153
14
votes
4
answers
3k
views
Jacobi's theorem on sums of two squares (reference request)
One of Jacobi's theorems states that the number of representations of a positive integer $n$ as a sum of two squares of integers equals
$$4(d_1(n)-d_3(n)),$$
where the function $d_i$ counts the number ...
- 3,062
13
votes
3
answers
2k
views
Density of a set of integers
EDIT: this question of mine has received little attention, perhaps in part because it was stated in a too general and complicated way. So let me give it a second chance:
Fix an integer $r \geq 0$. ...
- 26k
12
votes
3
answers
2k
views
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 ...
- 1,905
12
votes
1
answer
997
views
How much can an Eisenstein series be truncated?
For ease of exposition, I will stick to the simplest case: consider the Eisenstein series for $SL_2(\bf R)$
$$E(z,s)=\sum_{\gamma\in P_{\bf Z}\backslash SL_2(\bf Z)}\text{Im}(\gamma z)^s=\sum_{(c,d)\...
- 3,799
12
votes
0
answers
508
views
Weyl law for Maass forms with nontrivial character
The classical Weyl law for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$ counts the number of Maass cusp forms on $\Gamma \backslash \mathbb{H}$ with Laplace eigenvalue less than $T$. This is originally due to ...
- 8,374
11
votes
2
answers
551
views
Effective bound on the expansion of the $j$-invariant
The $j$-ivariant has the following Fourier expansion
$$j(\tau)=\frac 1q +\sum_{n=0}^{\infty}a_nq^n=\frac{1}{q}+744+196884q+21493760q^2+\cdots.$$
Here is $q=e^{2\pi i \tau}$.
Is there some simple ...
- 2,375
11
votes
1
answer
699
views
"strange" diophantine and parity of the partition function
Let $\{x_i\}:=\{x_1=5, x_2=13, x_3=29, x_4=37, x_5=45, \dots \}$
be the sequence of those positive integers of the form
$$
p^{4\alpha+1}n^2$$
in increasing order where $p\equiv 5\pmod 8$ is prime ...
- 43.2k
11
votes
1
answer
563
views
Is $j(\tau)^{1/3}$ the hauptmodul for the congruence subgroup generated by $\tau\rightarrow\tau+3, \tau\rightarrow-1/\tau$?
The 3rd root of the modular invariant $j$ is
$$ j(\tau)^{1/3}=q^{-1/3}(1+ 248q+ 4124q^2+ 34752q^3+\cdots),$$
where $q=e^{2\pi i \tau}$.
I was wondering if $j(\tau)^{1/3}$ the hauptmodul for the ...
- 143
11
votes
1
answer
262
views
The best bound on the growth of $\sum_{n\le X}a(n)$
Let $f(z) = \sum_{n=1}^{\infty} a(n)n^{(k-1)/2} q^n$ be a cusp form, I am interested to know what is currently the best bound on the growth of $\sum_{n\le X}a(n)$ in the following two case :
When $f$ ...
- 155
11
votes
1
answer
660
views
What is the motivation behind Ramanujan's conjecture?
One motivation I have seen given for Ramanujan's conjecture for the estimate
$$ |a_p|< C p^{k - \frac{1}{2}} $$
for the Fourier coefficients of a cusp form of weight $2k$ is that it allows one to ...
- 6,224
10
votes
2
answers
1k
views
Modular forms with finitely many or very few non-zero Fourier coefficients
I have an elementary question on modular forms, but which I don't know how to solve.
a) Is there a congruence subgroup $\Gamma \leq \mathrm{SL}_2(\Bbb Z)$, an integer $k \in \Bbb Z$ and a non-...
- 255
10
votes
1
answer
716
views
Why are the coefficients of the modular equation so large?
The modular equation $\Phi_n(X,Y)$ is a polynomial in $\mathbf Z[X,Y]$ relating the modular invariant $j$ and the functions
$j\left(\frac{a\tau+b}{c\tau+d}\right)$, where $ad-bc=n$.
For example, we ...
- 2,375
10
votes
2
answers
705
views
Averages over integer points of the sphere
A paper of William Duke proves that integer points on the sphere are equidistributed:
$$ V_n = \{ (x,y,z) \in \mathbb{Z}^2 : x^2 + y^2 + z^2 = n \}. $$
Up to reflections across the $x$, $y$ and $z$ ...
- 22.8k
10
votes
1
answer
314
views
Coefficient bounds on cusp forms, half-integer weight
Let $f(\tau) = \sum_{n=1}^{\infty} a(n) q^n$ be a cusp form on $\Gamma_0(4N)$ of half-integer weight $k \ge 5/2.$ The Ramanujan-Petersson conjecture in this case is that $$a(n) \ll n^{(k-1)/2 + \...
- 101
10
votes
1
answer
1k
views
Reference for the odd dihedral case of Artin's conjecture
The example that Matt Emerton cited here prompted me to become interested in how one proves that odd two dimensional dihedral Galois representation are modular. This is the first case of the strong ...
- 7,062
9
votes
2
answers
556
views
Holomorphic Hoffstein-Lockhart
In the article Hoffstein, Jeffrey; Lockhart, Paul "Coefficients of Maass forms and the Siegel zero." Ann. of Math. (2) 140 (1994), no. 1, 161–181, it is stablished a good bound for the Petersson norm ...
- 243
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
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
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
3
answers
1k
views
Effective detection of CM modular forms
Say $f$ is a newform of weight $k$ and level $\Gamma_1(N)$. $f$ is called CM if, for example, there is an imaginary quadratic field $K$ such that for all $p\nmid N$ which are inert in $K$, the $p$th ...
- 4,807
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
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
$$L(s,f\times\bar f)=\sum \frac{a(n)\overline{a(n)}}...
- 585
8
votes
1
answer
739
views
Average of Fourier coefficients of a cusp form of half integral weight
Suppose $f$ is a cusp form of half integral weight $k$ w.r.t. the group $\Gamma_0(4)$ ($k$ is not very low, can assume $k \ge 11/2$), and $a_n$ is its Fourier coefficient. The Linnik bound says that ...
- 5,169
8
votes
1
answer
356
views
Average bounds on Rankin-Selberg coefficients for modular forms
Let $f$ be a cuspidal Hecke newform of weight $k$ and level $N$, and denote by $a_f(n)$ its $n$-th Fourier coefficient. The newform $f$ is normalized so that $a_f(1) = 1$. As a consequence of Rankin-...
- 5,424
7
votes
1
answer
501
views
Smallest Mazur's good prime
Let $p$ and $\ell$ be primes $\geq 5$ such that $\ell$ divides $p-1$. Following Mazur, we say that a prime $q$ is a $\textit{good prime}$ if $\ell$ does not divide $q-1$ and $q$ is not a $\ell$th ...
- 1,109
7
votes
1
answer
435
views
Fricke involution and Atkin operator
Let $f\in S_k(\Gamma_0(N))$ be a cusp form for $N>1$. Consider the following operators acting on $f$ via the natural action of $GL_2^{+}(\mathbb{R})$ :
$$ W_N=\begin{pmatrix}
0 & -1\\
N & 0
...
- 145
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
$$\mathcal L(s,F\times u)=\int_{{SL}(2,\mathbb{Z})\...
- 3,804
7
votes
1
answer
175
views
Is every eigenvector sequence for the Hecke operators a eigenform?
Let $f(n) = a_n$ be a sequence taking values in $\mathbb C$ for $n=1,2,...$. Let $T_m$ be the Hecke operators (of a fixed weight $k$) defined as usual in terms of the $a_n$. That is:
$$T_m(f)(n) = \...
- 7,746
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
7
votes
1
answer
811
views
Alternative way to prove the functional equation for Eisenstein series?
Let $E(z,s):=\pi^{-s}\Gamma (s) \sum_{(m,n)=1}\frac{y^s}{|mz+n|^{2s}}$ be the real-analytic Eisenstein series.
It satisfies the functional equation $E(z,s)=E(z,1-s)$ with two poles at $s=0,1$.
The ...
- 3,804
7
votes
1
answer
353
views
Numerically double-checking formula with L-values
I'm working with a special case of Ichino's triple product formula, which for classical holomorphic newforms $f$, $g$ ,$h$ of weights $k$, $m-k$, $m$ (and central characters satisfying $\chi_f \chi_g =...
- 131
7
votes
1
answer
264
views
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 \...
- 1,364
7
votes
0
answers
673
views
Mock modular forms and (indefinite) quadratic forms
Define the function
$$f(q,z,y) = \sum_{n \ge 0,m,l} c(n,m,l) q^n z^m y^l$$
where $c(n,m,l)$ is defined by
$$ c(n,m,l) =
\begin{cases}
(-1)^{s+l} & \text{if } 4n - m^2 + l^2 = 2s(s+1)\\
0 & \...
- 1,298
6
votes
1
answer
305
views
Are the Fourier coefficients of $\eta(q^m)^m / \eta(q)$ non-negative?
In this paper, the following result is proved.
For any prime $p$, all the Fourier coefficients of
$$\eta(q^p)^p / \eta(q) = q^{\frac{p^2-1}{12}} \prod_{n=1}^\infty (1 - q^{pn})^p (1 - q^{n})^{-1}$$
...
- 427
6
votes
1
answer
943
views
Voronoi formula for the symmetric $L$-function with level $N $
Sorry to disturb. Does any experts here know something upon the Voronoi type for the symmetric $L$-functions$$\sum_{n\le X} A_F(1,n)e\left ( \frac{an}{c}\right)=?$$
Here $F$ is a symmetric-lift of a $...
- 353
6
votes
1
answer
2k
views
The connection between the Weil conjectures and Ramanujan's conjecture
I'm writing an essay about Ramanujan's conjecture and have some questions:
1 How is Ramanujan's conjecture connected with the Weil conjectures?
2 How could Ramanujan's conjecture be assumed true or ...
user108974
6
votes
1
answer
600
views
How to compute Coefficients in Chudnovsky's Formula?
My aim is to understand all three coefficients arising in the Chudnovsky-Formula (see also Question 300385). Two of them are easily computed, but I failed with the third:
It is known that for all $\...
- 598
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
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 $\...
- 63