A modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group.

learn more… | top users | synonyms

1
vote
0answers
15 views

Stabilizer of a point in R$\cup \infty$ in a discrete subgroup $\Gamma \subset SL_2(R)$

Let $\Gamma $ be a discrete subgroup of $SL_2(R)$. Let $\Gamma_z$={$\gamma \in \Gamma : \gamma(z)=z$}. Let $Z_{\Gamma}$={$\pm I\cap \Gamma$}. Then how to prove for z $\in R\cup \infty$, $\Gamma_z /...
4
votes
2answers
80 views

Is $\phi :SL_2(Z) \to SL_2(Z/NZ)$ still surjective if we replace Z with some ring of integers?

It is well known that the natural map $\phi :SL_2(Z) \to SL_2(Z/NZ)$ is surjective. So that the kernel, i.e. the principal congruence subgroup is of finite index. But what if we replace Z with some ...
1
vote
1answer
54 views

finding generators for spaces of half-integral weight modulars of level 8

I'm trying to realize the spaces of half-integral weight modular forms for $\Gamma_{0}(8)$ as the spaces of polynomials in some modular forms of level 8. For every integer $k$, it is known that every ...
0
votes
1answer
51 views

Exercise 3 on page 5 and exercise 7 on page 6 in Koblitz's Introduction to modular forms.

I want to prove $1$ cannot be a congruent number, by using the fact that if it were congruent then the equation $x^4-y^4=u^2$ would have a solution in integers with $u$ being odd. I proved this last ...
1
vote
1answer
34 views

Fundamental domain in Poinaré disc and cusps.

We usually choose the fundamental domain of full modular group $\Gamma$ to be $\mathcal{F} = \{z\in \mathbb{H} \, | \, |z|>1, \, |\text{Re}(z)|<\frac{1}{2} \}$. Using the change of variables $z ...
2
votes
1answer
52 views

On ramification index of a congruence subgroups at cusps

I am trying to solve this problem from Koblitz "Introduction to Elliptic Curves and modular forms" page 144, where $\Gamma=\mathrm{SL}_2(\mathbb{Z}),$ and $T=\left(% \begin{array}{cc} 1 & 1 \\ ...
0
votes
0answers
30 views

Products of fourier coefficients of distinct eigenforms.

Let $f(z)=\sum\limits_{n=1}^{\infty}a_f(n)q^n$ and $g(z)=\sum\limits_{m=1}^{\infty}b_g(m)q^m$ be two distinct eigenforms in $S(k,N)$, the space of holomorphic cusp forms of weight $k$ and level $N$. ...
1
vote
2answers
51 views

How to prove that $\frac{1}{n}L/L\simeq (\mathbb{Z}/n\mathbb{Z})^2$?

Let $L$ be any lattice in $\mathbb{C},$ and $L'$ a lattice containing $L$ with index $n$ (i.e $n=\sharp L'/L$) I found this statement "The lattice $L'$ must be contained in $\frac{1}{n}L = \{\...
1
vote
0answers
38 views

Generalized elliptic curves over cusps and orbits of $\mathbb{Q}\cup\infty$

In the post http://mathoverflow.net/questions/51147/what-objects-do-the-cusps-of-modular-curve-classify, it says that the fibers over the cusps of a modular curve are n-gons. Wikipedia (https://en.m....
4
votes
0answers
70 views

Tate curve and cusps

I know this is a naive question, but what is the relation between the Tate curve and cusps on a modular curve? Naive googling seems to suggest that level structures on the Tate curve (up to ...
1
vote
1answer
67 views

The lattice of $y^2=x(x-1)(x-λ)$

We know that every elliptic curve is associated with a lattice. So is the lattice of $y^2=x(x-1)(x-λ)$ just the lattice spanned by $\{0,1,λ\}$? If yes, is there some direct explanation? (Do not ...
3
votes
1answer
45 views

Elliptic curves over $\mathbb C$ have same endomorphism ring but not isomorphic

I am finding elliptic curves over $\mathbb C$ have same endomorphism ring but not isomorphic. Elliptic curves over $\mathbb C$ can be identified with $\mathbb C/\land$ for some lattice $\land$. And ...
1
vote
1answer
28 views

Laurent expansion at infinity for a weakly modular function with respect to a congruence subgroups

Let $\Gamma\subset \mathrm{SL}_2(\mathbb Z)$ be a congruence subgroup and $h$ the fan width of $\Gamma$ (i.e; the minimum $h>0$ such that $\left(% \begin{array}{cc} 1 & h \\ 0 & 1 \\ ...
2
votes
0answers
32 views

Archimedean Hecke Algebra for number fields

Suppose we have a $GL_1$ over a number field $F$. I am interested in a description for the archimedean Hecke algebra (always taking the maximal compact subgroup). We know will be the tensor product ...
7
votes
1answer
152 views

The elliptic integral $\frac{K'}{K}=\sqrt{2}-1$ is known in closed form?

Has anybody computed in closed form the elliptic integral of the first kind $K(k)$ when $\frac{K'}{K}=\sqrt{2}-1$? I tried to search the literature, but nothing has turned up. This page http://...
2
votes
0answers
29 views

Maass forms as sections of line bundles?

One way of obtained modular forms are as global sections of line bundle on the modular surface. On the other hand, I have not seen Maass forms constructed this way. (Though there is some work does ...
2
votes
1answer
53 views

How many coefficients do you need to determine a modular form

Here is the question i'm particularly interested in : Let $f$ be a modular form, suppose we know the $a_p(f)$ for all but finitely many prime $p$. Is this enough to know the modular forms i.e. to ...
2
votes
0answers
29 views

Differentiation formula in Miyake's “Modular Forms”

Miyake proves this lemma (and subsequently uses it to show the area of a hyperbolic triangle is the angle deficit): $$ (y^{-1}dz)\circ \alpha - y^{-1}dz = -2i d[log(j(\alpha,z)],$$ where $\alpha = \...
1
vote
1answer
65 views

a conjecture of two equivalent q-continued fractions related to the reciprocal of the Göllnitz-Gordon continued fraction A111374-OEIS

Given the square of the nome $q=e^{2i\pi\tau}$ and ramanujan theta function $f(a,b)=\sum_{k=-\infty}^{\infty}a^{k(k+1)/2}b^{k(k-1)/2}$ with $|q|\lt1$, define, $$\begin{aligned}M(q)=\cfrac{1-q^3}{1-q^...
0
votes
0answers
29 views

Genus of a modular curve mod p

I am reading a paper that states the following without reference: The modular curve $X_0(\mathcal{l})/p$, $l \neq p$ has genus $[\mathcal{l}/12]$. Does anyone know where this comes from? I am ...
4
votes
0answers
48 views

Motivation for the definition of modular forms

I've just started to study modular forms and I was wondering about how one would motivate the definition. I agree that $f\left( \frac{az + b}{c z + d} \right) = (cz + d)^k f(z)$ is an interesting ...
1
vote
1answer
70 views

proof of a lemma in Serre's book “A Course in Arithmetic” p 82

Im trying to understuand the proof of this Lemma Let $\Omega$ be a lattice in $\Bbb C.$ The series :$$L:=\underset{0\neq\rho\in\Omega}{\sum}\frac{1}{|\rho|^t}$$ is absolutely convergent for $t&...
2
votes
0answers
43 views

How are holomorphic and real-analytic Eisenstein series related?

The holomorphic Eisenstein series can be given as $$G_{2k}(z)=\sum_{(c,d)\in{\bf Z}^2\backslash(0,0)}\frac{1}{(cz+d)^{2k}}$$ while the real-analytic Eisenstein series is $$E(z,s)=\frac{1}{2}\sum_{(c,d)...
4
votes
1answer
53 views

Interpretation of enhanced elliptic curves

In "A first course in modular forms" (Diamond-Shurman) the author defines something called an 'enhanced elliptic curve' for the congruence subgroups $\Gamma_0(N), \Gamma_1(N)$ and $\Gamma(N)$. For ...
0
votes
1answer
36 views

$f$ weakly modular. How to prove that the order of vanishing $v_p(f)$ is well defined for $p\in SL_2(\mathbb{Z}) \setminus H$

I'm studying the proof of Lemma 1.6.6 in this Lecture notes page 16 Lemma If $f$ is weakly modular of weight $k,$ the order of vanishing $ v_p(f)$ is well defined for $p\in SL_2(\mathbb{Z}) ...
3
votes
2answers
54 views

Good description of orbits of upper half plane under $SL_2 (Z)$

It's known that $SL_2(Z)$ acts on $H=\{z\, |\, Im(z)>0\}$, is there a good description of orbits of $i$ and $w$, other than directly write down $=\{ \frac{ac|z|^2+bc\bar z+adz+bd}{c^2|z|^2+dc\bar z ...
2
votes
1answer
100 views

Using Modularity Theorem and Ribet's Theorem to disprove existence of rational solutions

This is likely overly optimistic, but can one take the results from the Modularity theorem and Ribet's theorem, and distill these down to an undergrad math level of a way to check if certain rational ...
1
vote
1answer
60 views

Dimension of the space of cusp forms

I guess this is rather basic, and as much a reality check as a question. We have the classical equations for the dimension of the space of modular forms of weight $k$ for $\Gamma(1)$: $$\dim M_{12j+...
4
votes
1answer
218 views

How to solve the Brioschi quintic in terms of elliptic functions?

Given the Brioschi quintic $$w^{5}-10cw^{3}+45c^{2}w-c^2=0$$ I'm interested in seeing different ways of solving it in terms of elliptic functions or theta functions.
5
votes
1answer
131 views

Rogers-Ramanujan continued fraction in terms of theta functions?

The Rogers-Ramanujan cfrac is, $$r = r(\tau)= \cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\ddots}}}$$ If $q = \exp(2\pi i \tau)$, then it is known that, $$\frac{1}{r}-r =\frac{\eta(\tau/5)}{\eta(5\...
5
votes
2answers
57 views

On the j-function $j\big(\frac{1+\sqrt{-31}}{2}\big)$ and special cubics

Given the j-function, $$j(\tau) = {1 \over q} + 744 + 196884 q + 21493760 q^2 + \dots$$ It seems that for $\tau = \frac{1+\sqrt{-d}}{2}$ and $d=12n+7$, then it has the form, $$j(\tau) = 12^3(1-x^2)^...
1
vote
0answers
37 views

Some special Hecke groups

Any reference for the Hecke group generated by $z \mapsto -\frac 1 z, \quad z \mapsto z + \frac 2 p$ for $p$ an odd prime? I am conscious that these groups are not discrete. I am interested by the ...
1
vote
1answer
60 views

How to prove that the Peterson inner product of $E_k$ with a cusp eigenform is $0$?

How to prove that the Peterson inner product of $E_k$ and $f$ is $0$ where $f \in S_k$ is an eigenform for all of the level $1$ Hecke operators?
2
votes
1answer
38 views

Why does the valence formula imply $M_k=0$ for $k <0$?

I'm studying Modular Forms and in the notes I'm reading the author states the following result, known as the valence formula: "Let $f$ be a non-zero weakly modular meromorphic form of weight $k$ ...
2
votes
0answers
33 views

Finding the value of a sum of two Hecke eigenvalues

I did some computations but I am stuck in finding the exression of the sum $$\lambda_f(n^2)+\lambda_f(n)^2 $$ in terms of $\lambda_f(n),$ where $f$ is a modular form for the full modular group. Any ...
2
votes
2answers
42 views

Transitive action of $SL_2(\mathbb{R})$ on $\mathbb{H}$

I'm studying Modular Forms and I'm not understanding why the action of $SL_2(\mathbb{R})$ on $\mathbb{H}$ defined by $\begin{pmatrix} a & b \\ c & d \end{pmatrix}z=\frac{az+b}{cd+d}$ is ...
2
votes
1answer
72 views

connection between odd primes and a certain q-series

I posed a conjecture about odd primes and a certain q-serieshere.I thought it would be more appropriate ,if I could ask the converse of the aforementioned problem . Is $p$ an odd prime iff $$\prod_{n=...
1
vote
0answers
20 views

Question about the sign of a certain sum

Given a modular form $f$ of an even weight $k$ for the full modular group. Let $\lambda_f(n)$ the $n$-th normalized Fourier coefficient of $f.$ For a fixed positive integers $a$ and $b,$ I want to ...
4
votes
1answer
152 views

conjecture about primes and a certain q-series.

Using wolfram Mathematica ,I observed an interesting and surprising property concerning prime numbers and q-series which I could not prove.Yet there is strong evidence supporting it. I would be happy ...
2
votes
2answers
263 views

A theta function around its natural boundary

Let $q = e^{2\pi i\tau}$, if $$\psi(q^2)=\sum_{n=0}^{\infty} q^{n(n+1)}$$ is one of ramanujan theta functions,is it possible to evaluate the limit $$\lim_{q\rightarrow 1} (1-q){\psi^2(q^2)}$$ In fact ...
2
votes
1answer
105 views

What is the limit $\lim_{q\to 1} \vartheta_{2}^{2}(q)(1-q)$

If $\vartheta_{2}(q)$ is jacobi's theta function, what is the limit $$\lim_{q\to 1} \vartheta_{2}^{2}(q)(1-q)$$ for the nome $q$. I would like to know whether the limit exists or not. If it does, ...
7
votes
2answers
342 views

Ramanujan theta function and its continued fraction

I believe Ramanujan would have loved this kind of identity. After deriving the identity, I wanted to share it with the mathematical community. If it's well known, please inform me and give me some ...
18
votes
1answer
505 views

a new continued fraction for $\sqrt{2}$

In a q-continued fraction related to the octahedral group I defined a new q-continued fraction for the square of ramanujan's octic continued fraction which I discovered using certain three term ...
0
votes
1answer
92 views

the zeros of theta function?

Recall that the theta function with character $(a,b)\in \mathbb{R}^2$ is defined by $$ \vartheta_{a,b}(z, \tau) :=\sum^\infty_{n=-\infty} e^{\pi i (n + a)^{2} \tau + 2 \pi i(n + a)(z + b)},\quad a,b \...
2
votes
1answer
140 views

Can Eisenstein Series output complex numbers?

The 2nd Eisenstein Series, defined by, $$E_{2}(\tau)=1-24\sum_{n=0}^{\infty} \frac{nq^{2n}}{1-q^{2n}},$$ where $q=e^{i\pi \tau}$ is the nome acts on upper-half plane. Must it always output real ...
0
votes
1answer
40 views

Explicit descriptions of the modular curves $Y(5)$ and $Y_1(5)$

The modular curves $Y(5)$ and $Y_1(5)$ associated to the congruence subgroups $\Gamma(5)$ and $\Gamma_1(5)$ are of genus 0. As such, they are nothing but the Riemann sphere $\mathbb P^1$ minus a ...
2
votes
0answers
40 views

Why are rational numbers required in cusps of congruence subgroups?

While we consider the action of congruence subgroups on $\mathbb{H}$ (the upper half plane), we compactify using an additional point at infinity, that is fine. But why do we add even all rational ...
1
vote
1answer
52 views

Why only congruence subgroups for modular forms?

When we define modular forms, why do we restrict ourselves to congrumence subgroups? Why not any subgroup of finite index? Or, even more generally why not any subgroup? Is it just a matter of ...
3
votes
3answers
311 views

Ramanujan's tau function identity

While studying Ramanujan's tau function, I observed that the function satisfies a beautiful identity that I had not seen previously in the literature. Let $\tau(n)$ be Ramanujan's tau function, such ...
5
votes
2answers
247 views

The ratio of jacobi theta functions

Let $q=e^{2\pi i\tau}$. If $\theta_2$ and $\theta_3$ are jacobi theta functions , is it true that the ratio of the two functions can be expressed as a continued fraction of the form $$ \frac{\theta_2(...