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: ...
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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
166 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.
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218 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 $$ ...
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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, ...
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102 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} ...
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5answers
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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 ...
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329 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 $$|\operatorname {SL}_2(\mathbb Z/p^e\mathbb Z)|=p^{3e}(1-{1 /p^2})$$
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444 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 of the square of ramanujan's octic continued fraction which I discovered using certain three term ...
4
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1answer
170 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 ...
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311 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.
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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
196 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 ...
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580 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
113 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 ...
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1answer
294 views

Modular functions and elliptic functions

Does anybody know of an equation formally equating modular functions and elliptic functions similar to Euler's equation for exponential and trigonometric functions? Any advice much appreciated. ...
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250 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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213 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 ...
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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), ...
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In what sense is Taniyama-Shimura the $n=2$ case of Langlands?

As I understand it (If I'm imprecise, as I will likely be, please correct me), Langlands says roughly as follows: For every representation $Gal(\mathbb{Q}) \rightarrow GL_n(\mathbb{C})$ we can form a ...
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1answer
613 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 ...
14
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1answer
252 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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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 ...
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Proving that $\sigma_7(n) = \sigma_3(n) + 120 \sum_{m=1}^{n-1} \sigma_3(m)\sigma_3(n-m)$ without using modular forms?

This problem appears as a (starred!) exercise in D. Zagier's notes on modular forms. I have to admit that I have no idea how to do it. Here, $\sigma_k(n) =\sum_{d\mid n} d^k$, as usual. This ...
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370 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} ...
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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 ...
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1answer
261 views

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 ...
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201 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 $$J(\tau)=\frac{4}{27} \frac{(1-\lambda+\lambda^2)^3}{\lambda^2(1-\lambda)^2} $$ ...
5
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346 views

Klein's j-invariant and Ford circles

Klein's j-invariant has structure which seems to resemble Ford circles: The latter show up all over number theory (continued fractions, Rademacher's expansion for p(n), etc.) Can someone explain ...
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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 $\eta(\tau)$?
7
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1answer
391 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 ...
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 ...
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1answer
210 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 ...
3
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1answer
255 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 ...
6
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1answer
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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 $$ ...
6
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1answer
323 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 ...
4
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1answer
358 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 ...
2
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1answer
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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 ...
2
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1answer
84 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 $a_1,a_2,a_3,a_4$ ...
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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 ...
4
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1answer
145 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 ...
4
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1answer
101 views

Expressing Eisenstein series E_k in terms of E_4 and E_6

Given an Eisenstein series $E_k$ (of level 1), it is a polynomial $P_k(E_4,E_6)$ in $E_4$ and $E_6$, and http://en.wikipedia.org/wiki/Eisenstein_series#Recurrence_relation should give a finite ...
3
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1answer
312 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 ...
3
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285 views

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
65 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 ...
2
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1answer
98 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
69 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 ...