For questions related to modular forms

learn more… | top users | synonyms

6
votes
1answer
246 views

Euler Product of Modular L-series

This question is probably really easy to someone that knows the theory of modular forms well, so I apologize if this is obvious. Suppose $E_1$ and $E_2$ are elliptic curves over $\mathbb{Q}$ and the ...
1
vote
1answer
63 views

Does the following define a point of the modular curve $X_1(n)$

Let $X$ be a compact connected Riemann surface of genus $1$ and suppose that there is a finite morphism $X\to \mathbf{P}^1$ of degree $n$ which ramifies totally over $0$ and $\infty$. Let $f^{-1}(0)$ ...
1
vote
1answer
189 views

Inferring Jacobi Triple product from $q$-binomial theorem?

Quite a while ago I asked a question about deducing the Jacobi triple product from the $q$-binomial theorem. In fact, from the $q$-binomial theorem $$ ...
1
vote
1answer
66 views

Reference for existence of quotients of modular Jacobians?

In the construction of Galois representations attached to modular forms of weight 2, the representation spaces can be found in Tate modules of certain quotients of Jacobians of modular curves by ...
2
votes
1answer
248 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 ...
0
votes
0answers
256 views

Modular Form Holomorphic at Cusps

Let $f$ be a weight $k$ modular form for $\Gamma$ and $f\vert_{\sigma}(z) = f(\sigma z)(cz + d)^{-k}$ where $\sigma = \begin{pmatrix} a & b\\c & d\end{pmatrix}$. Why is it true that $f$ is ...
6
votes
1answer
255 views

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 $$ ...
0
votes
1answer
230 views

solving this laplacian

given the laplacian $ -y^{2}( \partial _{x}^{2} + \partial _{y}^{2} )f(x,y)=Ef(x,y) $ can we find a solution in the form $ f(x,y)= \sum_{n,m}C_{n,m} \phi (x,E) g(y,E)$ if we impose the extra ...
3
votes
1answer
252 views

Preparation for a modular forms course

I'm trying to get a sense of what type of math to brush up on in order to take a course based on the book A First Course in Modular Forms by F. Diamond and J. Shurman. The text claims to be accessible ...
1
vote
1answer
237 views

Eisenstein series estimate from Stein-Shakarchi

I need help with this one from Stein & Shakarchi. Let $$E_{4}(\tau)=\sum_{(n,m)\neq (0,0)}{\frac{1}{(n+m\tau)^{4}}}$$ be the Eisenstein series of order $4$. Show that ...
2
votes
0answers
151 views

A Dedekind eta function sum of form $y_0^k+y_1^k+y_2^k+y_3^k+… = 0$

Given the Dedekind eta function $\eta(\tau)$. Define, $y_p = e^{\pi i p/6}\,\eta(\tfrac{\tau+2p}{5})$ Prove the multi-grade identity [1], $y_0^k + y_1^k + y_2^k + y_3^k + y_4^k + ...
8
votes
2answers
492 views

Ramanujan's Tau function, an arithmetic property

The problem: Let $\tau(n)$ denote the Ramanujan $\tau$-function and $\sigma(n)$ be the sum of the positive divisors of $n$. Show that $$ (1-n)\tau(n) = 24\sum_{j=1}^{n-1} \sigma(j)\tau(n-j).$$ ...
1
vote
1answer
404 views

On the Fourier Coefficients of a Modular Form

Let $f \in S_2(\Gamma(N))$ be a weight 2 cusp form of full level $N > 5$. It has a Fourier expansion: $$ f(z) = a_1 q^{1/N} + a_2q^{2/N} + \cdots. $$ Now let $S = \left(\begin{array}{cc} 0 & ...
2
votes
1answer
74 views

How could $Y(1)$ be the quotient of the upper half plane?

This is a silly question, but I can't resolve this: $Y(1)$ is defined to be $\mathbb{H}/PSL_2(\mathbb{Z})$. So it seems that its universal cover should be $\mathbb{H}$. On the other hand $Y(1)$ is ...
1
vote
1answer
77 views

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 ...
0
votes
1answer
249 views

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 ...
1
vote
0answers
93 views

The lambda invariant

Consider the strip $\{x+iy: -1\leq x < 1 , y>1/2\}$ in the complex upper half plane and let $\lambda$ be the usual $\Gamma(2)$-invariant modular function on the complex upper-half plane. ...
2
votes
1answer
116 views

Real elliptic curves in the fundamental domain of $\Gamma(2)$

An elliptic curve (over $\mathbf{C}$) is real if its j-invariant is real. The set of real elliptic curves in the standard fundamental domain of $\mathrm{SL}_2(\mathbf{Z})$ can be explicitly ...
1
vote
1answer
65 views

Is the image of this strip equal to the modular curve $Y(2)$

Let $B$ be the image of the strip $$\{x+iy : -1\leq x< 1, y>\frac{1}{2} \} \subset \mathbf{H}$$ in the modular curve $Y(2)$ under the quotient map $\mathbf{H} \to Y(2)$. Is $B=Y(2)$? Is ...
9
votes
4answers
383 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 + ...
9
votes
1answer
312 views

Relation between congruence subgroups. $\Gamma(M)\Gamma(N) = \Gamma(\gcd(M,N))$

I'm hoping there's a pleasant way to solve this one. Prove that $\Gamma(M)\Gamma(N) = \Gamma(\gcd(M,N))$. Showing that $\Gamma(M)\Gamma(N) \subset \Gamma(\gcd(M,N))$ is rather straight forward, ...
1
vote
1answer
220 views

Modular functions of weight zero

The following question was suggested by Sasha's answer to the following question : Is the derivative of a modular function a modular function . Question. What are the modular functions with respect ...
8
votes
1answer
192 views

Converting an infinite product to sum; Ramanujan $\tau$ function

I've gotten what seems most of the way, but I'm quite stuck at this point. Define $\tau(n)$ by \begin{align*} q\prod_{n=1}^\infty (1-q^n)^{24} = \sum_{n=1}^\infty\tau(n)q^n. \end{align*} ...
3
votes
1answer
175 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
votes
0answers
90 views

A specific point on the modular curve X(2)

Let $\lambda$ be the lambda modular function on the complex upper half plane. It induces an analytic isomorphism $\lambda:X(2) \longrightarrow \mathbf{P}^1$ and sends $X(2) - Y(2)$ to $\{0,1,\infty\}$ ...
4
votes
1answer
64 views

Why is this map a translation?

Let $\Gamma$ be a congruence subgroup, $s$ be a cusp of $\Gamma$, $\sigma \in SL_{2}(\mathbb{Z})$ be the map which sends $\infty$ to $s$, and $\Gamma_{s} = \{\gamma \in \Gamma \mid \gamma(s) = s\}$. ...
3
votes
2answers
378 views

Learning about partitions and modular forms

I'm interested in learning about partitions and modular forms. I already know algebra and analysis (complex and real). Can any one suggest me books or other materials from where I can learn these ...
3
votes
1answer
169 views

A few questions about $\mathbb{Q}$-models of modular curves (curves given by congruence groups)

I'm just now beginning to learn about descending the curves $X(N)$ to $\mathbb{Q}$, and I have a few questions: Does $X(N)$ have a $\mathbb{Q}$-point for every $N$? What is ...
2
votes
1answer
85 views

Complex line integral of modular function

Let $f$ be a modular function on the upper half-plane, $\rho = e^{2\pi i/3}$ and $v_\rho(f)$ the order of $f$ at $\rho$, i.e. the integer $n$ such that $f/(z-\rho)^n$ is holomorphic and non-zero at ...
2
votes
1answer
101 views

Are there any relations between the two statements about the poles and zeros of a modular form?

I learned two pieces of info on the relations of poles and zeros of a modular form $f$ as follows: $\mathbb{H}$ is the upper half plane, $G$ is the modular group $\text{SL}_2(\mathbb{Z})$, ...
33
votes
1answer
705 views

How does one see Hecke Operators as helping to generalize Quadratic Reciprocity?

My question is really about how to think of Hecke operators as helping to generalize quadratic reciprocity. Quadratic reciprocity can be stated like this: Let $\rho: Gal(\mathbb{Q})\rightarrow ...
4
votes
1answer
128 views

Are there any interpretations for the weights of modular forms?

To be specific, do the weights have some geometric meanings? A modular form $f$ satisfies $f(\frac{az+b}{cz+d})(cz+d)^{-2k}=f(z)$, where $z\in \mathbb{C}$. $k$ or $2k$ is called the weight of $f$.
25
votes
2answers
1k views

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 ...
5
votes
1answer
183 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
votes
1answer
171 views

Presentations of the ring of modular forms

I have seen two ways of describing the ring structure of modular (with respect to the full $SL (2, \mathbb Z)$) forms: $\mathbb C [E_4, E_6]$ and $\mathbb C [E_4,E_6, \Delta]/(E_4^3-E_6^2-1728\Delta ...
3
votes
2answers
145 views

Bounding the Number of Zeros of Modular Forms

Given a (meromorphic or holomorphic) modular form $f$ of weight $k$ on some genus zero congruence subgroup $\Gamma$, are there known bounds for the number of zeros and poles that $f$ has on the ...
2
votes
1answer
498 views

Eisenstein series associated with the cusps of a congruence subgroup

While reading a paper I came across the phrase "the Eisenstein series of weight 2 associated with the cusps of $\Gamma_{0}(6)$". Can anyone give me the definition of Eisenstein series of weight $k$ ...
0
votes
2answers
183 views

When are two modular functions equal?

Suppose $f, g \in M_{k}(\Gamma)$ for some $\Gamma$ a congruence subgroup of $PSL_{2}(\mathbb{Z})$. If $k \geq 1$ then it is a finite computation to determine whether or not $f = g$. What if $k = 0$, ...
3
votes
1answer
223 views

Finding a pencil of elliptic curves parametrized by a given modular surface

The following is an attempt to formulate a couple of questions which have been lurking in the back of my mind for a while. I'm sorry if this is long, or if my terminology is not correct, or if my ...
6
votes
1answer
263 views

Do general automorphic factors arise in some canonical way?

Disclaimer: none of what I'm about to say I understand particularly well; corrections and clarifications are not only welcome but will be accepted with great gratitude. The wikipedia article on ...
2
votes
1answer
308 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 ...
3
votes
0answers
181 views

Hauptmoduls for modular curves

If I have a modular curve, how does one in general find a Hauptmodul for this curve?
6
votes
1answer
219 views

Modular forms database

Suppose one was given a sequence and $a_{0}, a_{1}, a_{2}, \ldots$. Is there a searchable database somewhere to see if $a_{0} + a_{1}q + a_{2}q^{2} + \cdots$ is expressible as modular form (or some ...
3
votes
1answer
182 views

Theta series representation of lattices (Leech lattice, especially)

I'm reading now about theta series and its significance in researching self-dual, even lattices. Now, I found the wikipedia article about the Leech lattice, but I'm having trouble understanding where ...
4
votes
1answer
109 views

Reference request: theta functions for lattices which are not unimodular or even

In A Course in Arithmetic, Serre works out some of the theory of theta functions for even, self-dual lattices, e.g. such theta functions are modular forms of weight $n/2$, where $n$ is the dimension ...
2
votes
1answer
253 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. ...
3
votes
1answer
164 views

Bounds for Fourier coefficients of cusp forms

I've asked the background question here, which still left unanswered. Now I have a more precise question. In my homework I've been asked to prove that $$\left| \sum_{1\leq n \leq N} a_f (n)e^{2\pi i ...
5
votes
1answer
302 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
votes
1answer
225 views

j-invariant Fourier expansion

I'm reading about Fourier expansion of modular functions, but I have trouble understanding the following equation: Is it some inherent property of the denominator, as it is?
1
vote
1answer
203 views

Congruence subgroup of $SL_2(\mathbb{Z})$ definition

Congruence subgroups of $SL_2(\mathbb{Z})$ usually seem to be defined as a subgroup that contains $\Gamma(N) = \left\{\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \mod N \right\}$ for some ...