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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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: ...
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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 ...
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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} ...
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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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248 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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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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31 views

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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167 views

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 = ...
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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 ...
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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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Entring in the world of Modular Forms

Following wikipedia, a modular form of weight $k$ for the modular group $\mathrm{SL}_2(\mathbb{Z})$ is a complex-valued function  $f$  on the upper half-plane $H = \{z\in\mathbb{C}\colon Im(z)>0\}$ ...
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334 views

Finiteness of Dimension of $M_{k}(\Gamma)$

Let $M_{k}(\Gamma)$ denote the space of weight $k$ modular forms for the congruence subgroup $\Gamma$. Are there any proofs of the finiteness of the dimension of $M_{k}(\Gamma)$ that don't rely on ...
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About regular/irregular cusps and dimension formula for modular forms and cusp forms

I'm reading Diamond and Shurman's book about modular forms, in particular I'm reading the proof of the dimension formula of the vector spaces $\mathcal{M}_{k}(\Gamma)$ and $\mathcal{S}_{k}(\Gamma)$ ...
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23 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 ...
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How to visualize the region $\mathbb{H}/\Gamma_0(4)$ and its cusps?

In number theory we learn that $\theta(z) = \sum q^{n^2}$ is a modular form with respect to $\Gamma = \Gamma_0(4)$. This boils down to two properties: $\theta(z)= \theta(z+1)$ this shift symmetry ...
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Zeros of the second derivative of the modular $j$-function

I am interested in the zeros of $j''(z)$, where $j:\mathbb{H}\rightarrow\mathbb{C}$ is the classic modular function. Specifically I am interested in knowing if the zeros of $j''$ are algebraic over ...
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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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47 views

$p$-depletion of a modular form

Let $p$ a prime and $N$ an integer such that $p\not\mid N$. I will denote with $X_0(m)$ the modular curve with respect to the congruence subgroup $\Gamma_0(m)$. Let $f$ be a modular form with ...
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38 views

About period of elliptic points

I'm reading about modular forms and I've seen some results which only make use of elliptic points of period 2 and 3 (and cusps). I would like to know if it's possible to find elliptic points of period ...
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Counting points on the Klein quartic

In Moreno's book "Algebraic Curves over Finite Fields", he mentions the following in passing with no further comments ($K$ denotes the Klein quartic defined by $X^3 Y + Y^3 Z + Z^3 X = 0$): The ...
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Modular Discriminant and Pentagonal Numbers

I am asked to show $$(2\pi)^{-12}\Delta(\tau) = q \cdot \Big (\sum_{n\in \mathbb{Z}} (-1)^n \cdot q^{(3n^2+n)/2} \Big)^{24}$$ where $\Delta:\mathbb{H} \to \mathbb{C}$ is the modular discriminant, ...
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Every point in the upper half plane is congruent to a point in the fundamental region of $\lambda$

In Ahlfors complex analysis book, 2nd edition Pg. 273-274 Chapter 7 Section 3.5 we have the theorem $\textbf{Theorem 8.}$ Every point $\tau$ in the upper half plane is equivalent under the ...
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The j-function of a point on the boundary of the fundamental domain is real valued.

I am trying to prove the following statement: Let $z \in \mathbb{D} $ (the standard fundamental domain for $SL_{2}(\mathbb{Z})$). Prove that if $z$ lies on the boundary of $\mathbb{D}$, or if ...
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35 views

Weight 2 Newforms of large level computations.

I am stuck with some weight $2$ newform computations of large level. For example I want to compute newforms of level $11520$. Can anyone suggest me a way to do it? I need it to solve some diophantine ...
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Residue of Rankin-Selberg Dirichlet series

Let $f\in S_k(N,\chi)$ be a cusp forms, and let $R_f(s)=\sum_{n=1}^{\infty}\frac{a(n)^2}{n^s}$ the Rankin-Selberg Dirichlet series then $R_f(s)$ hase a pole at $s=k.$ Can someone suggest to me a ...
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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 ...
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Eisenstein series of congruence subgroup

Let $P$ be a prime number, and let $k >2$ be a positive even integer. Now I want to compute the Fourier expansion of the Eisenstein series of $\Gamma_0(p)$ at the cusp $0$, i.e., $$ E_{k, 0}(z) = ...
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Elliptic points of modular group

For $SL_2(Z)$ we know the elliptic point is either equivalent to $i$ or $1/2+\sqrt{3}/2i$. Then for finite index subgroup of $SL_2(Z)$, $\Gamma$, is this still true? There will be fewer points fixed ...
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Abstract regular curve over non-algebraically closed field

In Hartshorne chapter I.6 is discussed the construction of the abstract nonsingular curve as part of the proof for the well known correspondence between complete regular irreducible algrebaic curves ...
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eigenbasis of old forms

Let $S_k(N)$ be the space of weight $k$ cusp forms with respect to $\Gamma_0(N)$ for $1\leq N\leq 100$. We have a decomposition: $$S_k(N)= S_k^{\text{new}}\oplus S_k^{\text{old}}$$ Suppose that we ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 \\ ...
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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 $N$. ...
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How to show $\exists C \in \mathbb{C}^{\times}$, $n\in\mathbb{N}$: $f = C\Delta^n$?

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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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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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 = ...
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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 ...
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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 ...
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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 ...