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.

learn more… | top users | synonyms

3
votes
1answer
46 views

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 ...
3
votes
0answers
48 views

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 ...
0
votes
1answer
42 views

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} ...
1
vote
2answers
66 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 ...
-1
votes
1answer
32 views

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 ...
0
votes
0answers
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 ...
1
vote
2answers
52 views

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 ...
2
votes
1answer
43 views

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 ...
0
votes
0answers
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) = ...
2
votes
1answer
47 views

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 ...
5
votes
1answer
100 views

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 ...
0
votes
0answers
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 ...
9
votes
1answer
128 views

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 ...
3
votes
0answers
75 views

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 ...
3
votes
1answer
58 views

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) ...
1
vote
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. ...
1
vote
0answers
31 views

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, ...
0
votes
1answer
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 ...
0
votes
1answer
41 views

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 ...
2
votes
0answers
19 views

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 ...
3
votes
1answer
43 views

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 ...
2
votes
1answer
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} ...
0
votes
0answers
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 ...
0
votes
0answers
16 views

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 ...
0
votes
1answer
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 ...
2
votes
1answer
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 ...
0
votes
1answer
44 views

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 $ ...
3
votes
0answers
34 views

Some clarifications regarding Deligne's paper on $\ell$-adic representations arising from modular forms

In Deligne's article in Séminaire Bourbaki "Formes modulaires et représentation $\ell$-adiques" ...
0
votes
1answer
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 $f$ ...
3
votes
1answer
66 views

$\left\langle\begin{pmatrix}0&-1\\1&0\end{pmatrix},\begin{pmatrix}1&1\\0&1\end{pmatrix}\right\rangle$ 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)/ \{ ...
0
votes
0answers
17 views

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 $ ...
2
votes
0answers
19 views

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 ...
2
votes
1answer
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 ...
1
vote
0answers
13 views

$\exists C \in \mathbb{C}^{\times}$, $n\in\mathbb{N}$: $f = C\Delta^n$

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 $k>0$ ...
1
vote
1answer
35 views

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 ...
1
vote
0answers
29 views

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 ...
6
votes
1answer
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: "$Y_1(N)$ ...
0
votes
0answers
28 views

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 ...
2
votes
1answer
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 ...
9
votes
1answer
85 views

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 ...
1
vote
0answers
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 $GL_2(\mathbb{A})$ ...
2
votes
1answer
49 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 ...
0
votes
0answers
56 views

How to construct this modular Galois representation?

I want to know how to construct this modular Galois representation: The $\bmod p$ representation is semi-simple. For any lattice $T$ , $\dfrac{T}{p^{3}T}$ does not have a $G$-stable cyclic subgroup ...
3
votes
0answers
66 views

Analogue of the Hecke operator for a general curve

Background: The eigenvalues of the Hecke opeator $T_p$ on the space of cusp forms $S_2(\Gamma_0(N))$ (of dimension $g=\text{genus}(X_0(N))$) are analysed using reduction mod $p$ and the use of the ...
0
votes
2answers
70 views

Why is it so difficult to use geometric methods to construct weight one modular forms?

I am trying to understand why it is that geometric methods of constructing modular forms of weight one for $SL(2,\mathbb Z)$ fail. I have a relatively rudimentary understanding of it, but it would be ...
2
votes
0answers
29 views

Computing Kronecker's polynomial $\Phi_N(X,Y)$ modulo $2$ for $N=23,33$

Kronecker's polynomial $\Phi_N(X,Y)\in\mathbb{Z}[X,Y]$ is used to give a model for the modular curve $X_0(N)$ over $\mathbb{Q}$. I know that it is hard to compute this polynomial in general, but is it ...
1
vote
1answer
47 views

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 ...
1
vote
0answers
39 views

Eisenstein series of weight $2$ for in $M_2(\Gamma_0(N))$

Let $q$ be prime. By dimension formulas, $S_2(\Gamma_0(q))$ is of dimension $g$, while $M_2(\Gamma_0(q))$ is of dimension $g+1$. So the space of Eisenstein series is one dimensional. How to define ...
4
votes
1answer
58 views

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 = ...
1
vote
1answer
28 views

The main involution on $ M_{2}(F) $ and it's extension to $ M_{2}(F_{\mathbb{A}}) $.

I'm presently reading through a paper of Shimura's; "Special Values of the Zeta Functions Associated with Hilbert Modular Forms". In the paper he defines $ \iota $ to be the main involution of $ ...