Questions tagged [modular-forms]
Questions about modular forms and related areas
1,416 questions
4 votes
0 answers
65 views
A question on the spectral large sieve
Let $q\in \mathbb{N}$. Denote by ${B}(q,\chi)$ an orthogonal basis of $GL(2)$-Maass cusp forms of level $q$ with nebentypus $\chi\bmod q$. Does there exist a corresponding version for the following ...
- 101
4 votes
0 answers
80 views
Sturm bound for Katz modular forms
Let $\mathfrak{P}$ be a place of $\bar{\mathbb{Q}}$ above a prime number $p$, $k \geq 1$, $N \geq 1$ prime to $p$ and $\varepsilon$ a Dirichlet character mod $N$. It is well known that a modular form ...
9 votes
0 answers
300 views
Is there an operator-theoretic foundation for Quantum Modular Forms?
As far as I know it seems that quantum modular forms lack a clean operator-theoretic foundation parallel to that of classical and Maass forms. The spectral theory of automorphic Laplacians and the ...
- 91
1 vote
0 answers
90 views
Question about formula for Siegel-Eisenstein series with character
What's the definition for a Siegel-Eisenstein series with 2 characters? I know in the $\mathrm{SL}_2$ case it's something like this: let $\psi, \chi$ be two characters with modulus $u,v$ respectively, ...
- 115
2 votes
0 answers
109 views
Linear relations between critical $L$-values of cusp forms
Let $k\geq 3$ be an integer, fix a level $N$, consider the critical $L$-values of all cusp forms in this level and weight with algebraic Fourier coefficients: $$\mathbb{L}_{N,k}:= \{(2\pi)^{k-1-s} L(f,...
- 768
0 votes
0 answers
105 views
Determining if a set intersects the orbit of another in Siegel upper half space
Consider Siegel upper half space, consisting of symmetric matrices $X+iY$ such that $Y$ is positive definite. This has an action of $\operatorname{Sp}_{2n}(\mathbb{Z})$ on it by generalized Möbius ...
- 29
5 votes
1 answer
667 views
The Bring quintic and the Baby Monster?
I. Quintic The general quintic was reduced to the Bring form $x^5+ax+b=0$ in the 1790s, while the Baby Monster was found in the 1970s. We combine the two together using the McKay-Thompson series (...
- 13k
2 votes
1 answer
294 views
Show $\alpha(n)=0$ when $n \equiv 3\mod4$
Consider the modular forms $$\theta(q):=\sum_{n\in\mathbb Z}q^{n^2},\qquad E_4(q):=1+240\sum_{n\ge1}\sigma_3(n)q^n,$$ $$\sum_{n\ge1}\alpha(n)q^n:=\frac1{16}(\theta^{10}(q)-\theta^2(q)E_4(q^2)).\tag{$\...
- 341
1 vote
0 answers
185 views
Regarding Artin images
My first goal is trying to understand the congruence between Fourier coefficients of $\Delta(z)$ and weight 1 modular form $\eta(z) \eta(23z) \pmod{23}$ given by the following: $a(p) \equiv$ \begin{...
- 95
4 votes
1 answer
502 views
Given the Ramanujan $G_n$ function, why is the quintic $x^5+5x^4+40x^3 = 4^3\left(\frac{4}{G_n^{16}}-G_n^{8}\right)^3$ solvable in radicals?
I. Definitions. Given the nome $q = e^{\pi i\tau}\,$ and $\tau=\sqrt{-n}\,$ for positive integer $n$, then the Ramanujan G and g functions are, $$\begin{align}2^{1/4}G_n &= q^{-\frac{1}{24}}\prod_{...
- 13k
3 votes
1 answer
230 views
Why does the general quintic factor over the Rogers-Ramanujan continued fraction $R(q)$?
I. Let $q = e^{2\pi i\tau}$ and $r=R(q)$ be the Rogers-Ramanujan continued fraction. Then the j-function $j(\tau)$ has the formula using polynomial invariants of the icosahedron, $$j(\tau) = -\frac{(r^...
- 13k
-2 votes
1 answer
284 views
Why is every modular form for $SL_2(\mathbb{Z})$ of weight $0$ constant? [closed]
There is a powerful theorem in modular form, which says Valence theorem Let $f \neq 0$ be a modular form of weight $k \geqslant 0$. We have $$ m_f(\infty) + \frac{1}{2}m_f(i) + \frac{1}{3}m_f(\rho) + \...
- 97
-2 votes
1 answer
175 views
Dimension of spaces of automorphic forms
find $\rm{dim} (\Theta_k)$ : $\mathbb H=\{x+iy|y>0;x,y\in \mathbb R\}$is the upper half plane. $z=x+iy$ and $k$ can be any positive real number If $f:\mathbb H\to\mathbb C$ ,which satisfies: 1.$...
- 341
9 votes
1 answer
342 views
Soft arguments for modularity of eta quotients
I've been trying to understand these 1959 results of Newman giving necessary and sufficient conditions for an eta quotient $f(z) = \prod_{0 < d \mid N} \eta(dz)^{r_d} $ of integer weight $k = \...
- 2,853
1 vote
0 answers
138 views
Level of twists of holomorphic cusp forms
Let $f(z)= \sum_{n \geq 1} \lambda_f(n)n^{(k-1)/2}e^{2\pi i nz}$ be a holomorphic cusp newform of level one, weight $k$ and trivial nebentypus. Let $p$ be an odd prime and $\chi$ be the principal ...
- 459
2 votes
0 answers
130 views
On the old forms of the vector space of holomorphic cusp forms in ${S}_k(N,\chi)$
I have a puzzle which perhaps looks naive for many experts here. Let $k \ge 2$ be an even integer and $N > 0$ be an integer. Let $\chi$ be a primitive character to modulus $q$ such that $N|q$, ...
- 415
2 votes
0 answers
174 views
Non vanishing of modular L functions on real line
In M. Ram Murty's paper "Oscillations of Fourier coefficients of modular forms", Math. Ann. 262, 431-446 (1983), MR696516, Zbl 0489.10020 (an offprint can be found here), I see a conjecture (...
- 121
2 votes
0 answers
192 views
Zero periods of old modular forms
Let K be an imaginary quadratic field $Q(\sqrt{-3})$. Let $p$ be a prime which split in $K$ and Let $\varpi$ be one of the factors of $p$ in $K$. Let $E$ be an elliptic curve $y^2=x^3+\varpi^2/4$ with ...
- 478
5 votes
0 answers
157 views
Regarding exceptional primes
I am reading Swinnerton Dyer's paper on "On $\ell$-adic representations and congruences for coefficients of modular forms". It defines a prime $\ell$ to be exceptional for an eigenform $f \...
- 95
5 votes
0 answers
457 views
A curious irrational series for $\pi$
In 1914 Ramanujan [Quart. J. Math. (Oxford) 45 (1914)] discovered the following irrational series for $1/\pi$: $$\sum_{k=0}^\infty\left(k+\frac{31}{270+48\sqrt5}\right)\binom{2k}k^3\left(\frac{(\sqrt5-...
- 17.5k
6 votes
1 answer
585 views
A Ramanujan style identity involving Bessel sum?
I was reading a paper on Bessel functions appearing in number theory including modular forms, and I found an identity reminiscent of Guinand's formula: $$ \frac{1}{s}+4\sum_{n=1}^\infty \frac{n}{\...
- 621
3 votes
1 answer
258 views
Does $\Phi$ satisfy this modular-type functional equation?
Does $\Phi(s) := 4 \sum_{t=1}^\infty \frac{t}{\sqrt{s}} K_1(2t\sqrt{s})$ satisfy $\Phi(1/s)=s^4\Phi(s)?$ Here $K_1$ is the modified Bessel function. I'm interested in this because I want to further ...
- 621
2 votes
0 answers
76 views
S-ramified extensions and Hilbert modular forms
Let $F$ be a totally real field of even degree and let $f$ be a Hilbert cusp form of level $\mathfrak{n}$ that is an eigenform for Hecke operators. Let $p$ be an odd prime number and $\mathfrak{p}$ a ...
- 385
1 vote
0 answers
120 views
Vertical sign distributions among the Fourier coefficients of modular forms
I am curious about a certain sign distribution question regarding the Fourier coefficients of modular forms. I will describe a particular concrete case of my curiosity, from where variants and ...
- 731
4 votes
0 answers
126 views
counting integral points on grassmannians by modular forms
An integral point $P$ of a Grassmannian $Gr(k,n)$ is a $k$-dimensional subspace such that $P \cap \mathbb{Z}^n$ is a rank $k$ sublattice of $\mathbb{Z}^n$. Its height $H$ is given by the determinant ...
- 439
9 votes
1 answer
804 views
Short proof that SL(2, Z)' is a congruence subgroup of level 12
As part of my investigations into modular forms, I want to prove that the commutator subgroup $\mathrm{SL}_2(\mathbb Z)'$ is a congruence subgroup of level $12$. That is: $\Gamma(12)\subseteq\mathrm{...
- 451
1 vote
0 answers
177 views
Etale fundamental group of modular curve
What is the étale-$\pi_1$ of the modular curve $Y$ with level structure $\Gamma \subset \operatorname{PSL}(2,\mathbb{Z})$? How is that related to $\Gamma$? I have found some related discussions about ...
4 votes
0 answers
112 views
Number of branches in a Hida family
Let $\bar{\rho}:\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}_2(\bar{\mathbb{F}}_p)$ be an irreducible modular Galois representation and consider the set $S(\bar{\rho})$ ...
- 765
4 votes
0 answers
173 views
References on Eichler--Shimura theory, congruences, and the use of schemes over $\mathbb{Z}[1/N]$
I'm trying to understand the theorem predicting the existence of a normalized newform $f \in S_2(\Gamma_0(N))$ associated to an elliptic curve $E/\mathbb{Q}$, such that there is a modular ...
- 359
3 votes
1 answer
408 views
Ramanujan-Petersson conjecture over SL(3, Z)
Recently, I am reading the paper *"Twisted moments of L-functions and spectral reciprocity" by Blomer, Valentin and Khan, Rizwanur in 2019 (arXiv, DOI). I wonder what is the Ramanujan-...
- 97
1 vote
1 answer
305 views
Modular forms of weight 24 and level one
Consider the $\mathbb{Z}$-lattice consisting of modular forms $f=\sum_{n\geq0}a_n q^n$ of weight $24$ and level $1$ such that $a_n$ is an integer for $n\geq1$. This is a full rank lattice of the space ...
- 385
6 votes
0 answers
187 views
Shared large partial quotient and modular-like relations in continued fractions of ζ(3)/(m² log π)
Update: Two additional unusual continued fraction behaviors have been observed for expressions involving zeta(3) and log(pi) and are documented in the second script below. I've recently encountered ...
- 61
2 votes
0 answers
121 views
Spectral theory for 1/2-integral weight forms
Has anyone devloped the spectral theory of $L^{2}$-integrable $1/2$-integral weight automorphic forms? I know that "The subconvexity problem for Artin L–functions" discusses the spectral ...
- 183
3 votes
0 answers
150 views
The cubic root of modular functions
Assume $p\neq 3$ is a prime. Let $f$ be a modular function on $X_0(3p)$ whose divisor is support on the upper half plane (i.e. $f$ has no zeros and poles at the cusps). If the divisor $D(f)=3D$ where $...
- 478
4 votes
2 answers
267 views
Transformation law for Dedekind $\eta$ from that of Jacobi's $\vartheta$
The Dedekind $\eta$ function has the transformation law $$\eta(-1/\tau) = \sqrt{-i\tau}\,\eta(\tau)\ .\tag{1}$$ The Jacobi $\vartheta$ functions obey very similar laws, e.g. $$\vartheta_{01}(z; -1/\...
- 1,293
1 vote
0 answers
152 views
Regarding operators in modular forms
If $f \in M_k(N, \chi)$ is a normalized integer weight Hecke eigenform for all $T_p$, is there a way to obtain $g \in M_k \left(4N, \chi \left( \frac{-4}{\cdot} \right) \right)$ via the use of ...
- 95
1 vote
0 answers
133 views
Functional equation of L-function with associated squared representation function by quadratic form
Let $r_Q(n)=\#\{x\in \mathbb{Z}^m|Q(x)=n\}$ be the number of representations of an integer $n$ by a definite quadratic form Q, and define the L-function $$\zeta_Q(s)=\sum_{n=1}^\infty r_Q(n)n^{-s}$$ ...
- 365
1 vote
0 answers
133 views
Effective equidistribution of roots of quadratic polynomials modulo a prime
Let $f(X) = a X^2 + b X + c \in \mathbb{Z}[X]$ be an irreducible quadratic polynomial with integer coefficients. For a large prime $p$, the polynomial $f$ has either two or zero roots in $\mathbb{F}_p$...
- 1,944
2 votes
0 answers
129 views
Period lattice of CM modular forms on $\Gamma_0(N)$ and $\Gamma_1(N)$
Let K be an imaginary quadratic field $Q(\sqrt{-3})$. Let $p$ be a prime which split in $K$ and Let $\varpi$ be one of the factors of $p$ in $K$. Let $E$ be an elliptic curve $y^2=x^3+\varpi^2/4$ with ...
- 478
12 votes
1 answer
500 views
The Chudnovskys' original proof of their $1/\pi$ formula
I am trying to understand the famous paper by the Chudnovsky brothers, "Approximations and complex multiplication according to Ramanujan" (reprinted in Pi: A Source Book), which (among other ...
- 87.7k
0 votes
0 answers
108 views
Regarding Hecke eigenforms
Let $f = \displaystyle{\sum_{n}} a(n) q^{n} \in M_{k}(N, \chi)$ be a normalized Hecke eigenform. We know that the coefficients a(n) follow a multiplicative relation, $a(n) a(m) = \displaystyle{\sum_{d ...
- 95
6 votes
1 answer
296 views
Existence of holomorphic automorphic functions non-vanishing at a given point
I am interested in modular forms on classical groups e.g. Hilbert, Siegel and Hermitian modular forms. The general question is the following: Let $G$ be a reductive group over $\mathbb Q$, and $g_\...
- 7,442
2 votes
0 answers
139 views
Regarding Rankin-Cohen bracket of two modular forms
How does U-operator act on Rankin-Cohen brackets of two modular forms $[f_{1},f_{2}]$ where $f_{1}, f_{2} \in M_{k_{i}}(N, \chi)$ for $i \in \{1,2\}$? Here, $U_{m}(f) = \frac{1}{m} \displaystyle{\...
- 95
0 votes
2 answers
359 views
Perfect cube coefficients of 8-th power of Euler pentagonal function
Thanks for your reading. For Euler's pentagonal numbers, we know they can be attached to modular form $\begin{align} \prod_{n=1}^{\infty}\left(1-x^n\right)&=\sum_{k=-\infty}^{\infty}(-1)^k x^{k(3 ...
- 473
3 votes
1 answer
244 views
Regarding normalized Hecke eigenforms
Consider the space of modular forms $M_{k}(\Gamma_{0}(tN), \chi_{t} \chi_{N})$ where $t, N > 1, \: (t, N) = 1$ and both are relatively prime to 6. Does the space always has a basis of normalized ...
- 95
1 vote
0 answers
134 views
Order 5 cyclotomic unit and the Rogers-Ramanujan identity
I asked this on math.SE two weeks ago, with no feedback except 43 views and an upvote. I would understand if this will be closed here as too vague and/or shallow. Still, here it is. The question is ...
- 19.4k
19 votes
2 answers
835 views
CM zeros of unary theta series
Show that $$\sum_{n\ge1}\left(\dfrac{n}{5}\right)\exp(2i\pi n^2(18+i)/50)=0$$ where $(n/5)=1,-1,-1,1,0,...$ is the Legendre symbol. The series converges extremely rapidly, so this can be checked ...
- 13.9k
1 vote
0 answers
200 views
A question on the normalisation of the Fourier coefficients of the Voronoi formula in Corbett's paper
every expert on MO. Really sorry to disturb. Recently, I was reading the paper of Corbett--Voronoi summation for $\rm{GL}_n$: collusion between level and modulus ( https://arxiv.org/pdf/1807.00716). I ...
- 101
4 votes
0 answers
107 views
Coefficients of noncongruence modular cuspforms
I'm interested in the best-known bounds on coefficients of a weight $k$ holomorphic cusp form on a noncongruence subgroup. Write $$ f(z) = \sum_{n \geq 1} a(n) \exp((n - \alpha)z/q).$$ The best result ...
- 1,642
1 vote
0 answers
55 views
On the p-valuation of singular moduli: the case for $\chi(l)=-1$
I was reading the Zagier's letter to Gross, and in this letter Zagier wanted to calculate a sum $$ \nu_l = \sum_{\substack{n \geq 1\\ n \text{ odd}}} \sum_{d\geq 1}\chi(d) \cdot \#\{k \in \mathbb{Z} \...
- 11