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.

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Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$

I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$. I found a solution by myself 10 hours after I posted it, here it is: $$f(x)=\...
7
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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 ...
4
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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
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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(...
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Other interesting consequences of $d=163$?

Question: Any other interesting consequences of $d=163$ having class number $h(-d)=1$ aside from the list below? Let $\tau = \tfrac{1+\sqrt{-163}}{2}$. We have (see notes at end of list), $$e^{\pi\...
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Intuition for the Importance of Modular Forms

I am learning about modular forms for the first time this term and am just starting to wrap my head around what might be the big picture of things. I was wondering if the following interpretation of ...
5
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3answers
547 views

Inverting the modular $J$ function

Klein's modular function $J(z)$ is defined and studied in e.g. Apostol's book Modular functions and Dirichlet series in number theory. Certain specific evaluations are available, for example, $$J(i)...
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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 ...
18
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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 ...
4
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1answer
175 views

visualizing functions invariant (or almost) under modular transformation

In the spirit of Möbius Transformations Revealed, I'd like to make a pair of movies depicting how Klein's absolute invariant $j(\tau)$ and the Dedekind eta function $\eta(\tau)$ transform when $\tau\...
4
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2answers
341 views

Is there a precise mathematical connection between hypergeometric functions and modular forms

I've been playing around with Gauss' hypergeometric series $F(a,b,c,z)$ these days and I was wondering if there is some relation with the theory of modular forms.
3
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a q-continued fraction related to the octahedral group

Let $q=e^{2\pi i\tau}$. If $u(\tau)$ is Ramanujan's octic continued fraction, $$u(\tau)=\cfrac{\sqrt{2}\,q^{1/8}}{1+\cfrac{q}{1+q+\cfrac{q^2}{1+q^2+\cfrac{q^3}{1+q^3+\ddots}}}}$$ is it true that ...
7
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1answer
216 views

Ideal of cusp forms for $\Gamma_0(4)$ is principal

Let $\Gamma_0(4)$ be a congruence subgroup of $SL(2,\mathbb{Z})$ defined as $$\Gamma_0(4)=\{M=\begin{pmatrix} a &b\\ c& d \end{pmatrix}\in SL(2,\mathbb{Z})\mid c=0\bmod 4\}.$$ Dedekind eta-...
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619 views

Representing a number as a sum of at most $k$ squares

Fix an integer $k >0 $ and would like to know the maximum number of different ways that a number $n$ can be expressed as a sum of $k$ squares, i.e. the number of integer solutions to $$ n = x_1^2 + ...
5
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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\...
8
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1answer
258 views

Hypergeometric formulas for the Rogers-Ramanujan identities?

Let $q = e^{2\pi i \tau}$. Given the j-function, $$j = j(q) = 1/q + 744 + 196884q + 21493760q^2 + \dots$$ and define, $$k = j-1728$$ Let $\tau =\sqrt{-N}$, where $N > 1$. Anybody knows how ...
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Modular Forms: Find a set of representatives for the cusps of $\Gamma_0(4)$

We write $SL_2(\mathbb{Z})$ as $\coprod\alpha_i\Gamma_0(4)$, where $\coprod$ means the disjoint union, and the $\alpha_i$ are the coset representatives of $SL_2(\mathbb{Z})\diagup\Gamma_0(4)$. We ...
2
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1answer
111 views

Infinite sum involving ascending powers

I was working on a problem involving finite differences and came across the following sum as a general solution to a problem I was working with $$ 1 + x + \frac{1}{1 + a}x^2 + \frac{1}{1 + a} \frac{...
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1answer
664 views

The importance of modular forms

I'm studying modular forms and my professor started the course talking about elliptic functions. These particular functions form a field (once that the lattice $\Lambda$ is fixed) called $E(\Lambda)=\...
5
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343 views

How to calculate $|\operatorname {SL}_2(\mathbb Z/N\mathbb Z)|$?

the answer should be $$|\operatorname {SL}_2(\mathbb Z/N\mathbb Z)|=N^{3}\prod_{p|N}(1-{1 /p^2})$$ But first how to prove $$|\operatorname {SL}_2(\mathbb Z/p^e\mathbb Z)|=p^{3e}(1-{1 /p^2})$$
14
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1answer
268 views

Modular interpretation of modular curves

Let $\Gamma$ be a congruence subgroup of level $N$. What is the modular interpretation of $\Gamma\backslash \mathcal{H}^*$, by that I mean what are the elliptic curves + additional structure it ...
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What is the exact value of $\eta(6i)$?

Let $\eta(\tau)$ be the Dedekind eta function. In his Lost Notebook, Ramanujan played around with a related function and came up with some of the nice evaluations, $$\begin{aligned} \eta(i) &= \...
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How to prove that $\frac{\eta^{14}(q^4)}{\eta^{4}(q^8)}=4\eta^4(q^2)\eta^2(q^4)\eta^4(q^8)+\eta^4(q)\eta^2(q^2)\eta^4(q^4)$?

How can we prove that $$\frac{\eta^{14}(q^4)}{\eta^{4}(q^8)}=4\eta^4(q^2)\eta^2(q^4)\eta^4(q^8)+\eta^4(q)\eta^2(q^2)\eta^4(q^4) \ ?$$ Here, $\eta(q)$ is the Dedekind Eta Function, which is defined by ...
7
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1answer
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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://...
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428 views

An infinite product for $\left(\frac{\eta(13\tau)}{\eta(\tau)}\right)^2$?

Given the Dedekind eta function, $$\eta(\tau) = q^{1/24} \prod_{n=1}^\infty (1-q^n)$$ where $q = \exp(2\pi i\tau)$. Consider the following "family", $\begin{align} \left(\frac{\eta(2\tau)}{\eta(\...
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Special values of the classical normalized Eisenstein series

I am looking for a comprehensive list of some known special values of the classical normalized Eisenstein series $E_4(\tau)$ and $E_6(\tau)$. Does anyone know where I can find a table of some known ...
5
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1answer
144 views

Approximation of $e^{\pi\sqrt{n}}$ using Ramanujan's Class Invariants

From Wikipedia article we get $\displaystyle \begin{aligned}e^{\pi\sqrt{43}} &\approx 884736743.999777466 \\ e^{\pi\sqrt{67}} &\approx 147197952743.999998662454\\ e^{\pi\sqrt{163}}&\...
4
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1answer
224 views

Where does Klein's j-invariant take the values 0 and 1, and with what multiplicities?

I tried to solve the following problem from Ahlfors' text, please verify my solution: Where does the function $$J(\tau)=\frac{4}{27} \frac{(1-\lambda+\lambda^2)^3}{\lambda^2(1-\lambda)^2} $$ ...
7
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1answer
404 views

why does a certain formula in Lang's book on modular forms hold?

Background: Let $k$ be an even integer. The Eisenstein series are defined by $$E_{k} = 1 - \frac{2k}{B_{k}}\sum_{n=1}^{\infty} \sigma_{k-1}(n)q^{n}$$ where $$\sigma_{k-1}(n)= \sum\limits_{d \mid n,\;...
7
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A question on FLT and Taniyama Shimura

Sometime back i watched the documentary of Andrew Wiles proving the Fermat's Last theorem. A truly inspiring video and i still watch it whenever i am in a depressed mood. There are certain things(...
5
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1answer
218 views

Silly question about weakly modular functions

This is so far the most naive of my questions. Weakly modular functions of weight $2k$ correspond to $k$-forms on $X(1)$, right? But $X(1)$ is a curve. So shouldn't there not be any $k$-forms for $k\...
3
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1answer
318 views

modular form -Petersson inner product

my question is about Petersson inner product. i need to prove that $(E_k,f) =0 $ $\forall f \in S_k(SL_2(\mathbb{Z}))$ the only thing that i think that should help me is that the space of cusp form ...
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Special values of the Dedekind eta function

Does anyone know of an article or a book which contains a comprehensive list of some known explicit values of the Dedekind eta function $\eta(\tau)$?
6
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326 views

Cusp forms' Fourier coefficients sign changes

I need some clarification on the following, if possible: I have seen in that for every $ f \in S_k$ which Fourier transform is $\sum_{n=1}^\infty a(n)q^n$ there is an upper bound $\sum_{n=1}^N \|a(n)\...
6
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1answer
276 views

Intermediate step in deducing Jacobi's triple product identity.

An intermediate step deduces Jacobi's triple product identity by taking the $q$-binomial theorem $$ \prod_{i=1}^{m-1}(1+xq^i)=\sum_{j=0}^m\binom{m}{j}_q q^{\binom{j}{2}}x^j $$ and deducing $$ \...
4
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1answer
370 views

Agreement of $q$-expansion of modular forms

If I have modular functions $f$ and $g$ with $f = a_{1} + a_{2}q + \cdots$ and $g = b_{1} + b_{2}a + \cdots$ both $q$-expansions, why does/how does it follow $f = g$ after checking only finitely many ...
3
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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 ...
3
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1answer
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Formula for index of $\Gamma_1(n)$ in SL$_2(\mathbf Z)$

Is there a precise formula for the index of the congruence group $\Gamma_1(n)$ in SL$_2(\mathbf Z)$? I couldn't find it in Diamond and Shurman, and neither could I find an explicit formula with a ...
2
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1answer
41 views

Modular equation for a quotient of eta functions

In this article at page $4$ F. Beukers introduces a function $$y(\tau) = \frac{\eta^{8}(6\tau)\eta^{4}(\tau)}{\eta^{8}(2\tau)\eta^{4}(3\tau)}\tag{1}$$ where $\tau$ is a complex number with positive ...
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100 views

A special modular function: $ j $-invariant.

It is known that j invariant $$j(\tau)= 1728 \frac{g_2^3(\tau)}{\Delta(\tau)} $$ $\tau \in \mathbb{H}$ attains every complex value , Can someone guide me its proof.?? where $L(\tau ) = \{\tau m +...
2
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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 ...
4
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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 ...
3
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1answer
375 views

Is the derivative of a modular function a modular function

Fix a positive integer $n$. Let $f:\mathbf{H}\longrightarrow \mathbf{C}$ be a modular function with respect to the group $\Gamma(n)$. Is the derivative $$\frac{df}{d\tau}:\mathbf{H}\longrightarrow \...
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Ramanujan congruences and étale cohomology

What is a good reference for the story of congruences such as $$\displaystyle \tau(n) \equiv \sigma(11)(n) \mod\ 691$$ with a conceptual explanation with connections to étale cohomology, etc?
2
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1answer
86 views

Coefficient in the Fourier expansion of the cusp form

Ideal of cusp for $\Gamma_{0}(4)$ is principal and generated by $f(z)=η(2z)^{12}=q+\sum a(n)q^n $, this is discussed here. How one can compute the coefficient $a(n)$ when $n$ is rather large ? for ...
2
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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, ...
2
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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=...
2
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1answer
88 views

History of the Coefficients of Elliptic Curves — Why $a_6$? [duplicate]

I would like to know what is the motivation behind the naming convention of the Weierstrass form of elliptic curves given as $$E:y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ I can see that $a_1,a_2,a_3,a_4$ ...
2
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0answers
106 views

Show the Dedekind $\eta(\tau)$ function is a modular form of weight $\frac{1}{2}$ for $\Gamma_0(6)$

Given that $\Gamma_0(6)$ is generated by matrices in the set $$G = \left\{\left(\begin{array}{cc} 1 & 1\\ 0 & 1 \end{array}\right), \left(\begin{array}{cc} 7 & -3\\ 12 & -5 \end{array}...
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1answer
24 views

Are all Dirichlet coefficients of any element of the Selberg class necessarily algebraic?

The title says it all: do we know at least one element of the Selberg class having at least one transcendental coefficient in its development in a Dirichlet series for $\Re(s)>1$? Or are such ...