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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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 ...
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When is a modular curve defined over Q?

Let $X(N)$ and $Y(N)$ be respectively the compactified and uncompactified modular curves parametrising elliptic curves with full level $N$ structure. In other words, a point on $Y(N)$ is (essentially) ...
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field of rational functions of a curve

Let $C$ be the algebraic curve defined by the modular polynomial $\phi_N$ of order $N>1$ over the rational numbers, i.e. \begin{equation}C:=\text{specm}(\mathbb{Q}[X,Y]/\phi_N(X,Y)). \end{equation} ...
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Regarding the connected component of $|1/J| < 1$ containing $\infty$

How does one explicitly describe the connected component of $|1/J| < 1$ containing $\infty$? Here, $J = J(\tau) = j(\tau)/12^3$ is the normalized $j$-invariant so that $J(i) = 1$, and $\tau$ is in ...
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Integral relations in Fricke and Klein

Can someone please explain how Fricke and Klein obtain the integral relationa stated at the top of p. 34 in this book? The entire book can be previewed on Google Books. It is an old book and I do not ...
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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 ...
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Weierstrass product expression for Klein's j-invariant

The first sentence of @ccorn's answer to a previous question of mine was: “Because of the modular symmetries of $j(\tau)$, the zeros of $j(\tau)$ are precisely the ...
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Fourier Series of Eisenstein series [closed]

$$G_{2k}(\tau)= 2\zeta(2k)+2\frac{(2\pi i)^{2k}}{(2k-1)!}\sum_{n\geq 1}\frac{n^{2k-1}q^n}{1-q^n}$$ where $q =e^{2\pi i \tau}$ and $G_k=\sum_{\omega \in L , \omega \neq 0}\frac{1}{\omega^k} $ $L(\tau ...
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Arithmatic on modular curves

I had tried to read the first few pages of Glenn Stevens's$\,$ Arithmetic on Modular Curves, but it is somehow extremely unreadable to me, the text format is odd and stating too much facts without ...
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24 views

Conjecture about modular functions

I read this conjecture: ( If I'm right it is still unproved) Conjecture: Let $f(\tau)~=~ q^{c}\sum\limits_{n=0}^{\infty} a_{n}q^{\frac{n}{b}}\not\equiv 0$ be a modular function for some subgroup G of ...
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Modular forms: What is $\mathbb{H} / SL_2(\mathbb{Z})$?

I am beginning to understand the very basics of modular forms, in that I understand the concept of a weakly modular function, I have seen the examples of $G_k(z)$ and $E_k(z)$ as weakly modular ...
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37 views

Tau Function expressed as a Cauchy Product

I'm reading about the Ramanujan tau function, defined in my lecture notes by $q \prod_{n=1}^{\infty} (1-q^n)^{24} = \sum_{n=1}^{\infty} \tau(n) q^n$. Wolfram provides the identity $\tau(n) = ...
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What are the zeros of the j-function?

Recall that, for a complex number $\tau$ with positive imaginary part, the $j$-invariant is given by $j(\tau)=1728 \frac{g_2(\tau)^3}{g_2(\tau)^3-27g_3(\tau)^2}$ where $g_2(\tau)=60 ...
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139 views

Why do modular curves parametrise elliptic curves?

Let $Y_1(N)=\Gamma_1(N)/H$, where $H$ is the upper half plane. In these lecture notes http://math.uga.edu/~pete/modularandshimura.pdf , the author makes the following statement: "$Y_1(N)$ ...
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79 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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Modular parametrization of elliptic curve

Let $f$ be a cusp form of weight $2$ on $\Gamma_0(N)$ and assume that $f$ is a Hecke form and a newform. Then, we easily see that $$C(\gamma)=2i\pi \int_{\tau}^{\gamma \tau}{f(\tau')d\tau'} \quad ...
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25 views

Zeroes of Jacobi Theta Functions

Based on wolfram alpha: $$\sum_{i=0}^{\infty}[x^{i^2}] = \frac{1}{2}(v_3(0,x) +1) $$ Whereas $v_3$ is the third Jacobi Theta function. See: http://bit.ly/1FIyUTq I am curious for what values in ...
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1answer
17 views

Introduction to newforms and oldforms

Looking for an introductory text on newforms and oldforms. I have only found http://sage.math.washington.edu/edu/Fall2003/252/lectures/11-12-03/11-12-03.pdf which is not introductory in nature. ...
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Hecke operators on modular forms

Would you please explain the importance of Hecke operators on modular forms? I am studying modular forms mostly on my own and I have a pretty good understanding up to Hecke operators. So, I just ...
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410 views

Proving addition and multiplication

(1)Show that addition and multiplication mod n are associative operations. (2)Show that there are both an additive and a multiplicative identity. (3)Show that multiplication distributes over ...
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Elementary proof of Ramanujan's “most beautiful identity”

Ramanujan presented many identities, Hardy chose one which for him represented the best of Ramanujan. There are many proofs for this identity. (for example, H. H. Chan’s proof, M. Hirschhorn's ...
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checking if a Hecke eigenform is a smiultaneous Hecke eigenform

Let $f$ be a modular form of weight $k$ with respect to a congruence subgroup. Assuming that $f$ is a normalized Hecke eigenfunction for exactly one Hecke operator $T(l)$. How can one usually check, ...
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27 views

Proving that The set of limit points is not empty in an infinite group of linear fractional transformation.

suppose S is an infinite group of linear fractional transformation , show that the set of limit points of S is not empty . I'm studying a modular form course , and I got stuck in this question ,i ...
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dimension of space of modular functions using the Riemann-Roch theorem?

Let $H$ be the upper half-plane, and $M_k$ be the space of modular forms of weight $k$ on $H$ under the action of $SL(2,\mathbb{Z})$. I have read (Koblitz, Introduction to Elliptic Curves and Modular ...
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Does this function appear in theory of modular forms?

Does the following function $\phi(q,z)$ appear in theory of modular forms $$ \phi(q,z):=\prod_{n=-\infty}^{\infty}(1+q^nz) $$ for $(q,z)$ in some domain in $\mathbb{C}^2$? Any reference will be ...
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Relation between elliptic curves and Dirichlet L-series

I have read that an elliptic curve $E$ is modular if $a(n) = c(n)$ for all $n$, where $a(n)$ is the $n$-th coefficient in the Dirichlet series of $E$, $L(E,s)$, and $c(n)$ is the $n$-th coefficient in ...
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28 views

coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$

I want to prove that $\forall n \in \mathbb{N}$ at least one of the Fourier coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$ is greater than 1( The ...
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What's the “exact depth” of a quasi modular form?

In https://archive.org/details/arxiv-math0509653 (p2, Remark 4) the Rankin Cohen bracket $[f,g]_n$ is defined as $\Phi_{n,k,s,l,t}(f,g)$ where $s$, $t$ are the exact depths of $f$, $g$. What does ...
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32 views

Congruence and first odd primes

I tried to find the solutions to this modular equation: $3^{(5+7+11+13+17+19+\dots +p(m-3)+p(m-2))} \equiv p(m-1) \bmod p(m) $ where $p(m)$ is the m-th odd prime number(note that it's three to the ...
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29 views

Why is $f(z)y^k$ bounded for $f$ a cusp form?

For $f$ is a cusp form of weight $2k, k>0$ ($f(z)=(cz+d)^{-2k}f(\frac{az+b}{cz+d}$)), then why is $f(z)y^k$ bounded? If expanded $f$ in $\sum a_nq^n$, it's domain is a open disc, hence I can't ...
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Action of automorphisms on Eisenstein series

Let $ f \in \mathcal{M}_{k}(\Gamma) $ and $ \sigma \in \textit{Aut}(\mathbb{C}) $. Suppose $$ f = \sum_{n=0}^{\infty}a_{n}q^{n} .$$ Then we define the action of $ \textit{Aut}(\mathbb{C}) $ on $ ...
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30 views

Residue theorem and Angle of modular function

Let $f$ be a meromorphic function on the region $Im(z)>0$, $v_p(f)$ be the order of $p$. (The number $n$ such that $\frac{f(z)}{(z-p)^n}$ is holomorphic and non-zero at $p$.) Moreover, assume $f$ ...
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$\left\langle\begin{pmatrix}0&-1\\1&0\end{pmatrix},\begin{pmatrix}1&1\\0&1\end{pmatrix}\right\rangle$ is the free product of two cyclic groups.

Let $G$ be a group generated by two matrices $ S=\left( \begin {array}{cc} 0&{-1}\\1&0\end{array}\right),T=\left( \begin{array}{cc}1&1\\0&1\end{array}\right) $ in $SL_2(\mathbb Z)/ \{ ...
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1answer
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Complex Galois Representaions

I'm trying to understand 1 and 2 dimensional complex representations, induced representations and associated Artin L-functions for a project. I'm finding it hard to find appropriate material to help ...
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Dirichlet Series of weakly multiplicative characters

Let $$ \sum_{n=1}^{\infty} a(n)n^{-s} = \prod_{p}\left((1-\alpha_{p}p^{-s})(1-\alpha_{p}'p^{-s})\right)^{-1},$$ where $ a(n) $ is weakly multiplicative (i.e $ a(n)a(m) = a(n,m) $ if $ ...
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51 views

Generators of $\Gamma_0(N)$

Let $\textbf{T}:=\bigl(\begin{smallmatrix} 1&1\\ 0&1 \end{smallmatrix} \bigr)$, $\textbf{S}:=\bigl(\begin{smallmatrix} 0&1/\sqrt{N}\\ -\sqrt{N}&0 \end{smallmatrix} \bigr)$ and $H$ the ...
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normalised eigenforms question

I'm looking at the questions on this sheet: https://www0.maths.ox.ac.uk/system/files/coursematerial/2014/3116/5/MF_2014_Sheet4.pdf and The Q is Q2 a). Now, assuming this is true, the rest of the ...
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$\exists C \in \mathbb{C}^{\times}$, $n\in\mathbb{N}$: $f = C\Delta^n$

Let $f \in M_k(\Gamma)$ not null in $\mathbb{H}$. I want to show that there exists a $C \in \mathbb{C}^{\times}$ and $n\in\mathbb{N}$ with $f = C\Delta^n$. I think one can show that for a $k>0$ ...
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Boundedness of a modular form in $\mathbb{H}$

Let be $k>0$ and $f \in S_k(\Gamma)$. I want to show that the function $h(z)=Im(z)^{\frac{k}{2}}\cdot |f(z)|, \; z\in\mathbb{H}$ is bounded in $\mathbb{H}$. I have already shown, that $h$ is ...
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Cuscs of a subgroup of $\Gamma$

I'm going to be completely honest about this: I need the solution of this to get permitted to the exam in complex analysis. The topic is not even relevant for the exam and I am absolutely not able to ...
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Whats the difference between modular forms of different levels?

We have a natural surjective group homomorphism: $\phi : SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}/(n\mathbb{Z}))$ from which, given any subgroup $H<SL_2(\mathbb{Z}/(n\mathbb{Z}))$, we may take the ...
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Residue of Rankin-Selberg L-function for non-trivial nebentypus

Let $f\in S_k(\Gamma_0(N),\chi)$ be a normalized holomorphic newform (i.e. weight $k$, level $N$, nebentypus $\chi$) and write its Fourier expansion as $$ f(z)=\sum_{n\ge 1} ...
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Modular property of Weierstrass $\wp$ function

For $\tau\in \mathbb H=\{ x+iy\in \mathbb C \lvert x,y\in \mathbb R, \, y>0\}$ and $z\in \mathbb C$, let us define $$ \wp=\wp(\cdot,\tau): \mathbb C \rightarrow \mathbb P^1\, , \quad z\mapsto ...
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Why any quasi modular form corresponds to some almost holomorphic form?

I am trying to understand this article. http://arxiv.org/pdf/math/0603268v1.pdf In particular Theorem 1. In particular it says that if $$ (c z + d)^{-k} f_0(\frac{a z + b} {c z + d}) = \sum_{j = ...
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56 views

Regarding the derivative of the $j$-invariant

Is anyone aware of a formula for the derivative of the $j$-invariant $j(\tau)$ with respect to $\tau$? Here, $\tau$ is in the upper half-plane. I would image there are probably quite a few formulae ...
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Space of functions in the upper half-plane

Let $f(\tau)$ be a (say, holomorphic) function in the upper half-plane. Consider $$ A={\rm Span}_{\mathbb{C}} \{ f(\gamma \cdot \tau) : \gamma \in SL_2 (\mathbb{Z}) \}. $$ Is there a standard name ...
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40 views

Modular forms inducing functions on $SL_2(\mathbb{Q}_p)$

I am familiar with the classical thoery of modular forms. I've heard that a modular form on gives rise to a function on $SL_2(\mathbb{Q}_p)$ (or maybe $GL_2(\mathbb{Q}_p)$, or even $GL_2(\mathbb{A})$ ...
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50 views

Simple question about lattices in $\mathbb{C}$

Milne's Modular Functions and Modular Forms states, at the bottom of page 10: We can normalize our lattices so they are of the form $$\Lambda(\tau) := \mathbb{Z} \cdot 1 + \mathbb{Z} \cdot ...
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Is there a general way to prove series and products are modular?

The following$$\eta(q)=q^{1/24}(q)_\infty$$ $$E_{n}(z)=\sum_{z \in \Lambda\setminus \lbrace0\rbrace}z^{-n}$$ $$F(q)=q^{-1/60} \sum_{n \ge0} \frac{q^{n^2}}{(q;q)_\infty}$$ $$F(q)=q^{11/60} \sum_{n ...