For questions related to modular forms

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5
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1answer
100 views

Proving convergence of a Hilbert modular theta function $\vartheta(z):= \sum\limits_{x \in \mathcal{O}_F} e^{\pi i \operatorname{Tr}(x^2 z)}$

I'm trying to understand a somewhat sketchy proof that I found online of the convergence of the analog of Jacobi's theta function $\displaystyle{\theta(\tau) := \sum_{n = -\infty}^{\infty} e^{2 \pi i ...
3
votes
1answer
104 views

What are modular forms used for?

I have seen the definition of a modular form, but it seems obscure to me. I get the impression that if I were to read a lot about them, eventually I would see how they can be used. I am curious about ...
71
votes
4answers
2k views

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: ...
1
vote
1answer
93 views

Modularity theorem and some results

Let $C$ be an elliptic curve over rationals. Then we can attach to $C$ an L-series $L(C,s)$. I read about the Modularity theorem http://en.wikipedia.org/wiki/Modularity_theorem In the section ...
6
votes
1answer
140 views

p-adic modular form example

In Serre's paper on $p$-adic modular forms, he gives the example (in the case of $p = 2,3,5$) of $\frac{1}{Q}$ and $\frac{1}{j}$ as $p$-adic modular forms, where $Q = E_4 = 1 + 540\sum ...
1
vote
2answers
72 views

High-order elements of $SL_2(\mathbb{Z})$ have no real eigenvalues

Let $\gamma=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbb{Z})$, $k$ the order of $\gamma$, i.e. $\gamma^k=1$ and $k=\min\{ l : \gamma^l = 1 \}$. I have to show that $\gamma$ ...
1
vote
1answer
89 views

Coefficients of powers of the theta function

Let $q=\exp(2 \pi i z)$ and $$\theta(z)=\sum_{n=-\infty}^\infty q^{n^2}.$$ Now, I shall show that the powers of $\theta$ are given by $$\theta(z)^r = \sum_{n=0}^\infty S_r(n) q^n$$ where $S_r(n)$ ...
3
votes
0answers
85 views

Good source of problems for Knapp's Elliptic Curves?

I'm studying elliptic curves (and eventually modular forms) out of Knapp's book because of the softer algebraic geometry prereqs. It's incredibly accessible but the problem is that I don't know I can ...
1
vote
0answers
60 views

Show that $\sum_{d\in\mathbb{Z}}\sum_{c\ne0}\frac{1}{(c\tau+d)(c\tau+d+1)}=-{2\pi i\over \tau}$?

It is an exercise about Eisenstein Series $G_2(\tau)$, to prove that $$(G_2[\gamma]_2)(\tau)=G_2(\tau)-{2\pi i\over\tau}$$ where $\gamma=\begin{pmatrix} 0&1\\ -1 &0 \end{pmatrix}$ I just ...
3
votes
1answer
119 views

Stark's formula for the j-invariant

In his paper On the "gap" in Heegner's proof (which you can find : here) Stark gives the following formula for the $j$-invariant (for some $\tau \in \mathcal{H}$ and $q=e^{2i\pi\tau}$) $$ j(\tau) = ...
5
votes
2answers
224 views

How to calculate $|\operatorname {SL}_2(\mathbb Z/N\mathbb Z)|$?

the answer should be $$|\operatorname {SL}_2(\mathbb Z/N\mathbb Z)|=N^{3}\prod_{p|N}(1-{1 /p^2})$$ But first how to prove $$|\operatorname {SL}_2(\mathbb Z/p^e\mathbb Z)|=p^{3e}(1-{1 /p^2})$$
6
votes
0answers
159 views

When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ for $\Gamma$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least ...
3
votes
1answer
125 views

visualizing functions invariant (or almost) under modular transformation

In the spirit of Möbius Transformations Revealed, I'd like to make a pair of movies depicting how Klein's absolute invariant $j(\tau)$ and the Dedekind eta function $\eta(\tau)$ transform when ...
2
votes
0answers
42 views

$f,f',…,f^{(j)}$ is $\mathbb C$-linearly independent if $f$ is a modular form

Conjecture: Let be $f$ a modular form of weight $k$ and $j$ a strictly positive integer, then the set $f,f',...,f^{(j)}$ is $\mathbb C$-linearly independent in $A$. Is that conjecture true or false? ...
3
votes
1answer
80 views

Formula for index of $\Gamma_1(n)$ in SL$_2(\mathbf Z)$

Is there a precise formula for the index of the congruence group $\Gamma_1(n)$ in SL$_2(\mathbf Z)$? I couldn't find it in Diamond and Shurman, and neither could I find an explicit formula with a ...
36
votes
5answers
1k views

Intuition for the Importance of Modular Forms

I am learning about modular forms for the first time this term and am just starting to wrap my head around what might be the big picture of things. I was wondering if the following interpretation of ...
0
votes
0answers
41 views

References for the conformal equivalence of the space of complex 1-tori and C?

What are some good references with proofs of the conformal equivalence of the space of complex tori and $\mathbb{C}$? So far I only have the book by Jones and Singerman.
4
votes
1answer
270 views

General questions about Eisenstein series and modular forms

I am in a situation where I am trying to get a feel for modular forms type stuff, but don't have anyone to talk to about it (I'm not in academia at the moment). I would like to test my understanding ...
5
votes
1answer
121 views

Direct proof of the non-zeroness of an Eisenstein series

Question: Can you show directly from its formula that $G_4(i)\neq0$? Recall that the holomorphic Eisenstein series of weight $2k$ is defined by: $$G_{2k}(\tau)= \sum_{(m,n)\in\mathbb{Z}^2\setminus ...
4
votes
1answer
93 views

Condition on infinite component for a cuspidal automorphic representation to be attached to a Hilbert newform

Let $F$ be a totally real field with real primes $\tau_1,\ldots,\tau_n$ and $\pi$ a cuspidal automorphic representation of $\mathrm{GL}_2(\mathbf{A}_F)$. My rough understanding is that in order for ...
1
vote
1answer
318 views

how to calculate $q$-expansion coefficients of a modular form?

Can you please explain for me how we can calculate the coefficients of $q$-expansion series of a modular form function $f$? I am really confused.
0
votes
1answer
115 views

Are the coefficients of this eigenform totally real?

Let $f \in S_2(\Gamma_0(N))$ (cusp forms) be a normalized Hecke eigenform, and let $K_f$ be the number field obtained by adjoining all its Fourier coefficients to $\mathbb{Q}$. Then is $K_f$ totally ...
1
vote
1answer
99 views

Why is the Petersson inner product positive definite?

The Petersson inner product is defined on the space $\mathcal{S}_k(\Gamma)$ of weight $k$ cusp forms of level $\Gamma$, and takes values in $\mathbb{C}$. First of all, I wonder: what does it mean for ...
1
vote
1answer
48 views

A certain combination of fourier coefficients of modular forms

Let $f$ be a modular form of weight 2 for the group $\Gamma_0(N)$, with Fourier expansion $$ \sum_{n\geq 0} a(n)\ q^n. $$ Let $d$ be an integer dividing $N$, and consider the Fourier series $\sum ...
3
votes
0answers
87 views

Number of $SL_2(\mathbb{Z})$-translates lying within a fixed ray in $\mathfrak{h}$

This is probably an easy question, but I can't seem get the right idea so here goes: Let $\tau \in \mathfrak{h}$ (where $\mathfrak{h}$ denotes the upper halfplane) be given. For any two positive real ...
0
votes
1answer
55 views

Problem about decomposition of modular space into eigenspace

It is said we can use the operator $$ \pi_\chi=\frac{1}{\phi(N)}\sum_{d\in\mathbb{Z}_N^*}\chi(d)^{-1}\langle d\rangle $$ to project function in $\mathcal{M}_k(\Gamma_1(N))$ into the $\chi-$eigenspace ...
0
votes
1answer
86 views

What is the difference between the dimension of $\mathcal{M}_1(\Gamma)$ and $\mathcal{S}_1(\Gamma)$?

There is a formula given in the book saying that $$ \dim{\mathcal{M}_1(\Gamma)}=\dim\mathcal{S}_1(\Gamma)+\frac{\varepsilon^{\text{reg}}_\infty}{2} $$ Where $\varepsilon^{\text{reg}}_\infty$ means ...
3
votes
2answers
158 views

Lambert series help

How can one go about proving lambert series identities like, $$\left(1+240\sum_{n=1}^\infty \frac{n^3q^n}{1-q^n} \right)^2=1+480\sum_{n=1}^\infty \frac{n^7q^n}{1-q^n}$$ All the papers I have looked ...
1
vote
1answer
68 views

Why $\Delta(N\tau)\in\mathcal{S}_{12}(\Gamma_0(N))$

There is an argument saying that because $\Delta(\tau)\in\mathcal{S}_{12}(\text{SL}_2(\mathbb{Z}))$, then we have $\Delta(N\tau)\in\mathcal{S}_{12}(\Gamma_0(N))$. I don't understand the logic here. ...
2
votes
0answers
28 views

Problem about the order of a automorphic function on an elliptic curve.

Let $f:X(\Gamma)\rightarrow\mathbb{C}$ be an automorphic function of weight 0, saying that $f$ is $\Gamma$-invariant. Now at each noncusp point $\tau\in\mathbb{C}$, $f$ has an order $\nu_\tau(f)$, ...
3
votes
1answer
142 views

In one version of the Modularity Theorem, what does “arise from modular forms” mean?

One version of Modularity Theorem says that The elliptic curves with rational $j$-values arise from modular forms. Where $$j(\tau)=1728\frac{g_2^3(\tau)}{\Delta(\tau)}$$ I know every ...
1
vote
1answer
56 views

Question about a property of elliptic function

Let $E=\mathbb{C}/\Lambda$ be a complex elliptic curve where $\Lambda=\omega_1\mathbb{Z}\oplus\omega_2\mathbb{Z}$ Let $f$ be a nonconstant elliptic function with respect to $\Lambda$. Let ...
5
votes
1answer
68 views

A question about complex tori.

Let $\Lambda$,$\Lambda'$ be two lattice in $\mathbb{C}$ and $m\neq 0\in\mathbb{C}$ satisfying $$ m\Lambda\subset\Lambda' $$ The, the book I'm reading says that by the theory of finite Abelian groups ...
1
vote
1answer
87 views

Why the kernel of isogeny is finite?

It is said that the kernel of a isogeny is finite because it is discrete and complex tori are compact. I have some questions about this. 1. Following is my reason for the kernel is discrete. ...
2
votes
1answer
241 views

Question about weight 2 Eisenstein series

I'm new to modular form, reading the book A First Course in Modular Forms We have the weight 2 Eisenstein series $$ G_2(\tau)=\sum_{c\in\mathbb{Z}}\sum_{d\in\mathbb{Z}_c'}\frac{1}{(c\tau+d)^2} $$ ...
2
votes
1answer
83 views

Why does a certain equality in Iwaniec's Topics in classical automorphic forms hold?

I have been reading the book Topics in classical automorphic forms for a while, where I encountered a formula in the middle of Sec 8.6 (p143) $$F(z;\beta,\gamma)=\frac ...
5
votes
2answers
238 views

Klein's j-invariant and Ford circles

Klein's j-invariant has structure which seems to resemble Ford circles: The latter show up all over number theory (continued fractions, Rademacher's expansion for p(n), etc.) Can someone explain ...
2
votes
1answer
70 views

Divisor of a modular form

I am trying to compute the divisor of $\Delta(z)/\Delta(pz)$ on the modular curve $X_0(p)$ where $p$ is a prime. I know that as a function on the full modular group, the $\Delta$ function has only a ...
1
vote
1answer
99 views

Cusps for higher dimensional hyperbolic spaces

Take your favourite model of the hyperbolic plane $\mathfrak h$. It is well known how to define cusps, and we know those are $\mathbb Q \cup \{ \infty \}$. Now pick a higher dimensional hyperbolic ...
2
votes
0answers
52 views

How to prove that $G_3>0$ in this case?

Let $\Lambda=\{a+be^{2\pi i/3}|a,b\in Z\}$, then $G_{3}(\Lambda)=\sum_{\omega\in\Lambda-\{0\}}\frac{1}{\omega^{6}}$ should be real and nonzero, but how can one prove that it's positive? Moreover, in ...
3
votes
1answer
178 views

Elliptic Points of Modular Group in Upper Half Plane

This is a very small question. Let $\mathbb{\Gamma} = \mathrm{SL_2}(\mathbb{Z})$ be the modular group, $\mathcal{F} = \{z \in \mathbb{C} ;\; \lvert z \rvert \geq 1,\; \lvert \Re (z) \rvert \leq ...
4
votes
0answers
91 views

Connection between modular forms and line bundles

I need a good reference about the connection between modular forms and line bundles. I found only Milne's note that treats briefly this argument. I've already checked, but without finding anything, ...
1
vote
1answer
157 views

Dimension of modular forms for a congruence subgroup using contour integration.

There is a well known proof for finding the dimension of modular forms for the full modular group using the residue formula. It is on page 71 of these online notes: ...
9
votes
1answer
180 views

Connection between the $L$-function of a modular form and the $L$-function of its associated $\ell$-adic representation as defined by Bloch-Kato

Let $f\in\mathcal{S}_k(\Gamma_1(N),\epsilon)$ is a normalized eigenform for all the Hecke operators, with character $\epsilon$, $k\geq 2$, and assume the $q$-expansion of $f$ has rational ...
18
votes
1answer
391 views

The importance of modular forms

I'm studying modular forms and my professor started the course talking about elliptic functions. These particular functions form a field (once that the lattice $\Lambda$ is fixed) called ...
4
votes
1answer
83 views

Local invariants of the discrete Galois module associated to a $p$-ordinary newform

Let $f=\sum_{n=1}^\infty a_nq^n$ be a $p$-ordinary newform of weight $k\geq 2$, level $N$, and character $\chi$, and let $\rho_f:G_\mathbf{Q}\rightarrow\mathrm{GL}_2(K_f)$ be the associated $p$-adic ...
2
votes
0answers
107 views

Evaluating the Hauptmodul of $X_{1}(5)$

The function field of $X_{1}(5)$ is generated by the function $$t(\tau) = q\prod_{n = 1}^{\infty}(1 - q^{n})^{5\left(\frac{n}{5}\right)}$$ where $q = e^{2\pi i\tau}$ and $\left(\frac{n}{5}\right)$ is ...
8
votes
2answers
399 views

Eisenstein Series and cusps of $\Gamma_{1}(N)$

Let $q = e^{2\pi i\tau}$, $\operatorname{Im}\tau > 0$ and let $$G(\tau) = 1 - 24\sum_{n = 1}^{\infty}\frac{nq^{n}}{1 - q^{n}}$$ be the weight 2 Eisenstein series for $\Gamma(1)$. Consider the ...
1
vote
1answer
117 views

Eisenstein series solution

Denote the function $$ \Psi (x,y)= y^{1/2+ik}+ \sum_{g\in SL(2,\Bbb Z)} \frac{y^{1/2+ik}} {|c_{g}z+d_{g}|^{1/2+ik}}\tag{1}$$ My question is if I can write the wave function in terms of the Eisenstein ...
2
votes
1answer
149 views

Classical Hecke eigenforms for $\Gamma_0(p)$.

I am in need of classical eigenforms for $\Gamma_0(p)$. In particular I need these to be newforms for each of the even weights 8 to 26 and would settle just for cases $p=2,3$ at this moment in time. ...