For questions related to modular forms
2
votes
0answers
32 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 ...
41
votes
3answers
554 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
34 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 ...
0
votes
0answers
14 views
Is this function a modular function?
Is the following function a modular function?
$$\xi(x) _q=\sum_{k=1}^q \frac{1}{2q}( exp(\frac{-i2\pi (k-1)x}{q})+exp(\frac{+i2\pi (k-1)x}{q}) )$$
with $x\in\Bbb{R}$ and $q\in\Bbb{N}$ and $q\neq0$.
...
4
votes
1answer
72 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
63 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
52 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
36 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
53 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
72 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 $q=e^{2i\pi\tau}$)
$$ j(\tau) = ...
4
votes
2answers
119 views
How to calculate $|\operatorname {SL}_2(\mathbb Z/N\mathbb Z)|$?
the answer should be
$$|\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})$$
4
votes
0answers
75 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 ...
2
votes
1answer
41 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 ...
0
votes
0answers
27 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
39 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 ...
21
votes
3answers
359 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
30 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.
3
votes
1answer
71 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
77 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
41 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
104 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
39 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 ...
0
votes
0answers
47 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
0answers
25 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
81 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
31 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
71 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 ...
2
votes
1answer
78 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
59 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
25 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
103 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
32 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
55 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
50 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
88 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
63 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 ...
4
votes
2answers
114 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 ...
1
vote
1answer
43 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 ...
0
votes
0answers
89 views
finding torsion points of elliptic curves on MAGMA
I'm trying to learn how to use MAGMA for computing torsion points (i.e. n-torsion subgroups) of elliptic curves in various field. So far, I've looked in the documentation and other resources and I'm ...
0
votes
0answers
27 views
Transcendence of map induced from modular parametrization
Let $E$ be an elliptic curve defined over $\mathbb{Q}$ having a modular parametrization $\Phi_N:X_0(N)\to E(\mathbb{C})$.
In reading through an article by Henri Darmon, I came across a statement ...
1
vote
1answer
47 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
39 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
133 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
74 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
80 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:
...
8
votes
1answer
116 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 ...
14
votes
1answer
238 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
68 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
59 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)}$$ where $q = e^{2\pi i\tau}$ and $\left(\frac{n}{5}\right)$ is ...
8
votes
2answers
224 views
Eisenstein Series and cusps of $\Gamma_{1}(N)$
Let $q = e^{2\pi i\tau}$, $\operatorname{Im}\tau > 0$ and let $$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 ...
