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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Cusps in the Compactification of Modular Curves $Y(\Gamma)$

I'm currently reading some of the geometric theory behind the theory of modular forms, Diamond and Shurman's book is my main reference. If $\Gamma$ is a congruence subgroup of $SL_2(\mathbb{Z})$, the ...
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$f\in M_k (\Gamma_1(N)$ and $g(z)=f(dz)$, then $g\in M_k(\Gamma_1 (dN)$

Let $N,d\in\mathbb{N}$ and $\Gamma_1(N)=\{\begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL_2(\mathbb{Z}):a\equiv d\equiv 1\mod N,~c\equiv 0\mod N\}$. I need a proof (references or your ideas) of: ...
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Right cosets $\Gamma_1(N) \cap\alpha^{-1} \Gamma_1(N)\alpha$ in $\Gamma_1$

In the book Introduction to Elliptic Curves and Modular Forms from Neal Koblitz there is a certain step in the solution of an exercise that is left to the reader (IV.3, Problem 5 and solution). Since ...
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Automorphic forms and the Rankin Selberg method

I was just solving an Exercise, where we looked at an analytic function $\phi : \mathbf{H} \to \mathbf{C}$, which is automorphic and $\phi (z) = \mathcal{O}(y^{-C})$ for all $C > 0$ as $z \to i \...
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Modular curve, $G$-Galois branched cover

I was reading this answer by Pete L. Clark: "For "most" finite simple groups $G$ it is indeed the case that $G=⟨x,y⟩$ where $x$ has order $2$ and $y$ has order $3$. Equivalently, $G$ is a ...
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Why are $L$-functions a big deal?

I've been studying modular forms this semester and we did a lot of calculations of $L$-functions, e.g. $L$-functions of Dirichlet-characters and $L$-functions of cusp-forms. But I somehow don't see, ...
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References on Hauptmoduln

Here it is said that a Hauptmodul (a generator of a modular function field) is unique up to a Möbius transformation. My impression is that it is really hard to find references on Hauptmoduln and ...
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What do mathematicians mean when they say “form”?

As in differential form, modular form, quadratic form? I'm sorry if this is a really silly question.
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Modular forms under isogenies

Let $f$ be a modular form of level 1, say an Eisenstein series to simplify. If $n$ is a natural number, what can we say about $f(n\tau)$ in terms of $f(\tau)$?
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Calculation of a limit including the Dedekind eta function

I do not understand this attached equation, where $\eta$ means the Dedekind eta function and $E_2$ the Eisenstein series of weight $2$. Has someone an idea what happens here?
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How to do this integral? $\int_{-i\infty}^{i\infty}F_{\psi}(z) dz$?

Let $\psi$ be a character with conductor $f_\psi$. Define $$F_\psi(z)=\begin{cases}\sum_{n=1}^{\infty}\psi(n)e^{2\pi i nz}&\text{ if }& \text{Im}(z)>0\\ -\sum_{n=1}^{\infty}\psi(-n)e^{-2 \...
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Question on a proof of $\zeta(3)\notin\mathbb{Q}$

I have a question on this article proving $\zeta(3)\notin\mathbb{Q}.$ by using modular forms. This is theorem 1 at page 275 (page 5 in the pdf). Most things in the proof are clear but I don't get the ...
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$\left(\frac{\Delta(6z)\Delta(z)}{\Delta(3z)\Delta(2z)}\right)^{1/2}$ generates the field of modular functions on $\Gamma_1(6)$

I have already shown that the function, defined on the upper half plane $\{z\in\mathbb{C}:\Im(z)>0\}$, $$t(z)=\left(\frac{\Delta(6z)\Delta(z)}{\Delta(3z)\Delta(2z)}\right)^{1/2}$$ is a modular ...
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How did Hecke come up with Hecke-operators?

I'm currently studying Hecke-operators and I'm curious how Hecke came up with them. The original definition he gave in his paper is $$\left( f \mid T_n\right) (z) = n^{k - 1} \sum_{ad = n, \, b \mod d,...
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$A(z)$ is a automorphic form $\Rightarrow$ $A(-1/z)z^{-k}$ is a form

I've found this: Let A be an automorphic form (holomorphic) of degree $-k$, where $k\ge 2$. Call $a_0$ the constant term in its expansion at $i\infty$, $q$ the period of the form $A(-1/z)z^{-k}$ ...
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Reference/Literature on Eichler integrals

Has anyone reference or literature advices on Eichler integrals? I want to know what the Eichler integral associated to a modular form is. It should help to understand this, page 65, 4th sentence in ...
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Why is 12 the smallest weight for which a cusp forms exists

On wikipedia (here) I have read the following: Twelve is the smallest weight for which a cusp form exists. [...] This fact is related to a constellation of interesting appearances of the number ...
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Is $\eta^{24}(\tau)\,j(\tau) = {E_4}^3(q)$?

Given the j-function $j(\tau)$, $j(\tau) = 1728J(\tau)$, where $J(\tau)$ is Klein’s absolute invariant, the Dedekind eta function $\eta(\tau)$, and the following Eisenstein series, $\begin{align} ...
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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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Modular transformations of $\eta(\tau)$

Under a modular transformation the Dedekind $\eta$ function transforms as $$\eta(-1/\tau) = \sqrt{-i}\eta(\tau).\tag*{$(*)$}$$Siegel gives a proof in this paper here that uses complex analytic ...
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Values of modular functions on the cusps

I have the meromorphic modular function (weight $0$) on $\Gamma_1(6)$ $$y(z)=\frac{\eta(6z)^8\eta(z)^4}{\eta(2z)^8\eta(3z)^4},$$whereby $\eta$ is the Dedekind eta function. The set of cusps of $\...
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Does viewing an Eisenstein series as a sum over cusps explain the antagonism between Eisenstein serieses and cusp forms?

I'm trying to understand the relationship between various aspects of the concept of "Eisenstein series" (as discussed for example in Diamond & Shurman's "A First Course in Modular Forms"), in ...
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calculation with modular forms

Let $\Delta(z)=(2\pi)^{12} q\prod_{n=1}^{\infty}(1-q^n)^{24}$ with $q=e^{2\pi i z}$ be the modular discriminant. I found $$\left(\dfrac{\Delta(6z)\Delta(z)}{\Delta(3z)\Delta(2z)}\right)^{1/2}=q\prod_{...
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roots of modular forms in the complex field

For $\begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL_2(\mathbb{Z})$ the modular discriminant $$\Delta(z)=(2\pi)^{12}\eta(z)^{24}\qquad(1)$$ holds $$\Delta\left(\dfrac{az+b}{cz+d}\right)=(cz+d)^{12}\...
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What are these theta functions appearing in Sloane's database

Looking at Sloane's database, I found a neat formula for the lambda-invariant. Let $q:\tau \mapsto \exp(\pi i \tau)$ on the complex upper-half plane. Then $$\lambda(q) = 16q\;\prod_{k>0} \frac{(1 +...
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What is an L-Function and how can it be used for categorization?

On hackernews, I came across this article, describing the LMFDB, the database of L-functions, modular forms, and related objects which aims at the categorization of mathematical objects where ...
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Demystifying modular forms

I am really struggling to understand what modular forms are and how I should think of them. Unfortunately I often see others being in the same shoes as me when it comes to modular forms, I imagine ...
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Nice formulas for the lambda invariant of an elliptic curve

Where can I find some nice formulas for the lambda invariant of an elliptic curve? I vaguely recall there's a nice product formula in terms of $q$, but a Google search didn't give me much. Also, are ...
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contradiction to $M_k(\Gamma_1(6))=\mathbb{C}$

I just read the fact $M_0(\Gamma)=\mathbb{C}$ (constant functions) where $M_0(\Gamma)$ is the space of modular forms of weight $0$ on the congruence subgroup $\Gamma$. But in this article at page ...
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Is a modular form on $\text{SL}_2(\mathbb{Z})$ also a modular form on congruence subgroups?

Is a modular form $f$ of weight $k$ with respect to $\text{SL}_2(\mathbb{Z})$ always a modular form to a congruence subgroup $\Gamma$ (for example $\Gamma_1(N)$)? If the transformation law $f|_k\...
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Proving $\zeta(3)\in\mathbb{R}\setminus\mathbb{Q}$ using modular forms

I am looking for references proving $\zeta(3)\in\mathbb{R}\setminus\mathbb{Q}$ using modular forms, like this paper written by F. Beukers. Does anybody know some different papers or books? Thanks.
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Geometric interpretation of the width (of cusps)

Given a congruence subgroup $\Gamma$ and a cusp $\alpha$. Choose $\gamma\in SL_2(\mathbb{Z})$ such that $\gamma\infty=\alpha$. The minimal $h$ such that $\begin{pmatrix}1&h\\0&1\end{pmatrix}\...
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two congruence subgroups with the same set of cusps

Are there two congruence subgroups $\Gamma, \Gamma'$ of $SL_2(\mathbb{Z})$ with the same set of cusps? Thanks in advance.
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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 ...
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Need of cusps (with respect to Modular Forms)

In every introduction about Modular Forms (on $SL_2(\mathbb{Z})$ and congruence subgroups) one reads the term 'cusps'. A Modular Form should be holomorphic in the cusps. Can anybody explain to me, ...
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Hecke $L$-function of cusp form is entire

Let $f=\sum a(n)q^n\in S_k(N,\chi)$ be a cusp form of integral weight. Can someone give me the proof of the fact that : the Hecke $L$-function $L(f,s)=\sum\frac{a(n)}{n^s}$ is entire. I searched in ...
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Derivative of the Klein j-invariant

To prove that the field of meromorphic functions on $X(1)$ is generated by the Klein j-invariant, we need to show that the derivative of $j(\tau)$ does not vanish. (It only has simple zeroes). But I ...
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Regarding the weight of the given modular form

I am following this course. A question regarding the modular form of weight two that is constructed in lecture 33 emerged. Let me briefly tell you what's going on there. Let $\mathbb{H}$ denote the ...
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Convergence behaviour of Eichler integral

Consiger $g : \mathbb H \to \mathbb C$ a modular form of weight $2-k, k \in \frac{1}{2}\mathbb Z$. Let $z \in \mathbb H$ and consider the following integral: $$\int_{-\overline{z}}^{i\infty}(\omega+z)...
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Generators of the congruence subgroup $\Gamma (5) \subset SL(2,\mathbb Z)$

Recall that $SL(2,\mathbb Z)=\left\{A=\begin{pmatrix} a&b \\ c&d \end{pmatrix}: \det(A)=1; a,b,c,d \in \mathbb Z \right\} $ and $\Gamma(5)=\left\{A=\begin{pmatrix} a&b \\ c&d \end{...
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Modular properties of weight $\frac{3}{2}$ theta functions

Define the theta-functions $g(z) = - \sum_{n \in \mathbf{Z}} \left(n + \frac{1}{6} \right) e^{3 \pi i \left( n + \frac{1}{6} \right)^2 z}$. How can I show that $$g(z + 1) = e^{\frac{1}{24}2 \pi i} g(z)...
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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 ...
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What is an exponential singularity?

In Ken Ono's Lecture notes on Harmonic Maass forms and Mock Modular Functions (here) the author uses the term "exponential singularity". What does this mean? Thanks a lot!
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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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Discriminant cusp form and generator of an ideal in the Hilbert class field (proof of prop. 14.1 Heegner points on $X_0(N)$, Gross)

Let $K$ be a quadratic imaginary field where the rational prime $N$ splits: $\mathfrak{n}\cdot\bar{\mathfrak{n}}=(N)$ and denote with $H_K$ the Hilbert class field of $K$. At the beginnig of the ...
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$p$ adic modular forms and wide open neighbourhood (e.g. Coleman primitive): is it possible to obtain a holomorphic function?

It is well known that a modular form (of weight $k$ and level $N$) is in particular also a classical modular form; this can be seen both using Serre's definition with $q$-expansion and the Katz's one, ...
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Hauptmodul for $\Gamma_1(3)$

I'm looking for a nice formula for a Hauptmodul for $\Gamma_1(3)$. This is certainly very classical.A reference would be great.
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How to compute the Hecke operator on an Eisenstein series?

In my current course on Modular Forms we are now discussing Hecke operators and we are asked the following: Prove that for any even integer $k \geq 4$ and prime $p$ we have $T_pG_k = \sigma_{k−1}(p)...
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show that $\sum P(x,y) e^{x^2 + y^2}$ is a modular form over $\Gamma_0(4)$ where $P(x,y) = x^4 - 6 x^2 y^2 + y^4$

In these lecture notes of Zagier, I read that generalized theta functions are still modular forms. Let $q = e^{2\pi i z}$ $$\theta(z) = \sum_{(x,y) \in \mathbb{Z}} \Big[ x^4 - 6 x^2 y^2 + y^4 \Big] \...
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How can we show the equality $[SL_2(\mathbb Z): \Gamma_0(N)]=N\prod_{p\mid N}\left(1+\frac1p\right)$?

If you want to read question directly, please go to down. Firstly, let's give the definitions of $\Gamma(N)$, $\Gamma_1(N)$ ve $\Gamma_0(N)$: $$\Gamma(N) : = \left \{\left[ \begin{array}{cc}a & b ...