For questions related to modular forms

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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 & ...
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1answer
73 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 ...
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1answer
76 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 ...
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239 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 ...
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91 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. ...
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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 ...
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1answer
64 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 ...
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369 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 + ...
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310 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, ...
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213 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 ...
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187 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*} ...
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1answer
170 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 ...
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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 $\{0,1,\infty\}$ ...
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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 $\Gamma_{s} = \{\gamma \in \Gamma \mid \gamma(s) = s\}$. ...
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2answers
371 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 ...
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168 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 ...
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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 ...
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1answer
99 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 $\text{SL}_2(\mathbb{Z})$, ...
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1answer
695 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 ...
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1answer
124 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 $f$.
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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 ...
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1answer
181 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 ...
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163 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 ...
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2answers
144 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 ...
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1answer
479 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 $k$ ...
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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 $k = 0$, ...
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1answer
214 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 ...
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1answer
255 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 ...
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1answer
303 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 ...
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Hauptmoduls for modular curves

If I have a modular curve, how does one in general find a Hauptmodul for this curve?
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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 ...
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1answer
180 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 ...
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1answer
107 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 ...
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1answer
249 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. ...
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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 ...
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1answer
301 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 ...
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1answer
222 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?
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196 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 ...
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360 views

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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1answer
133 views

Bounded denominators for modular forms

I recently saw a conjecture that a modular form is a congruence modular form if and only if it has bounded denominators. I wonder if one direction or the other is already known to be true? EDIT: For ...
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2answers
530 views

Why is $\tau(n) \equiv \sigma_{11}(n) \pmod{691}$?

If $n$ is a natural number, let $\displaystyle \sigma_{11}(n) = \sum_{d \mid n} d^{11}$. The modular form $\Delta$ is defined by $\displaystyle \Delta(q) = q \prod_{n=1}^{\infty}(1 - q^n)^{24}$. ...
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282 views

Proving finite dimensionality of modular forms using representation theory?

It is well known how to use algebraic geometry (differentials, divisors, and Riemann-Roch) in order to prove the finite dimensionality of the vector space of modular forms of some fixed weight and ...
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276 views

Connection between Hecke operators and Hecke algebras

Hecke operators are things that act on modular forms and give rise to a lot of interesting arithmetical results: http://en.wikipedia.org/wiki/Hecke_operator On the other hand on the wikpedia page ...
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Ramanujan congruences and étale cohomology

What is a good reference for the story of congruences such as $$\displaystyle \tau(n) \equiv \sigma(11)(n) \mod\ 691$$ with a conceptual explanation with connections to étale cohomology, etc?
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171 views

Reference for Atkin-Lehner theorem

For my understanding of modular forms, I have "properly" read the last chapter of Serre's book on arithmetic. I have not "properly" read the setup of Hecke operators for congruence subgroups. I was ...
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344 views

why does a certain formula in Lang's book on modular forms hold?

Background: Let $k$ be an even integer. The Eisenstein series are defined by $$E_{k} = 1 - \frac{2k}{B_{k}}\sum_{n=1}^{\infty} \sigma_{k-1}(n)q^{n}$$ where $$\sigma_{k-1}(n)= \sum\limits_{d \mid ...
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2answers
868 views

Has Deligne-Rapoport been translated?

The classic text of Deligne and Rapoport giving a model for the mod p reduction of the modular curve is found in a "Lecture Notes in Mathematics" volume. Has it been translated from the original ...
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1answer
652 views

How are modular forms fundamental operations?

The German mathematician Martin Eichler once stated that there were five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms. This was also ...
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529 views

A question on FLT and Taniyama Shimura

Sometime back i watched the documentary of Andrew Wiles proving the Fermat's Last theorem. A truly inspiring video and i still watch it whenever i am in a depressed mood. There are certain ...
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1answer
400 views

Definition of cusp of a congruence group

I am reading p.22 of Dan Bump's Automorphihic forms and representations. A cusp of the congruence group acting on the upper half plane is defined to be an orbit of the action of the congruence ...