3
votes
0answers
33 views

Proving the Uniformization Theorem for Elliptic Curves (An Exercise from Silverman's Advanced Topics on Elliptic Curves )

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves there are two demonstrations of the Uniformization Theorem for the Elliptic Curves (It says that, given an Elliptic Curve $E$, ...
2
votes
1answer
47 views

Rewriting $\tau(p)\Delta(\tau)$ when $p$ is prime

$p$ is a prime, and $\tau$ is Ramanujan's tau function: $$p^{11}\Delta(p\tau)+\frac{1}{p}\sum_{k=0}^{p-1}\Delta\bigg(\frac{\tau + ...
2
votes
1answer
51 views

Modular Form, but Cusp Form in Disguise

Suppose I have a holomorphic modular form $f \in M_k(\Gamma_0(N), \chi)$ with $k \in \mathbb Z^+$ and $f = \sum_{n=1}^\infty a(n)q^n$. Considering the $q$-expansion, one may suspicious that $f$ is ...
2
votes
1answer
40 views

Why a modular form is a highest weight vector of a discrete series summand of $L_2(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}))$?

It is said that a modular form is a highest weight vector of a discrete series summand of $L_2(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}))$. Why a modular form is a highest weight vector of a ...
0
votes
1answer
18 views

Diamond operator on a meromorphic function

Is there a standard way in which the diamond operator for modular forms is defined for an arbitrary meromorphic function?. I know the definition for a weight k modular form, but since I usually wont ...
2
votes
1answer
58 views

Equidistribution of lattice points on spheres in dimensions $d \ge 4$ (Pommerenke's theorem)

I am looking for either an English translation of: Pommerenke, C.: Über die Gleichverteilung von Gitterpunkten auf m-dimensionalen Ellipsoiden. Acta Arith. 5, 227-257 (1959) Or a reference in ...
1
vote
0answers
9 views

Is it true that $\theta_{1,1}^{4N} \in J_{2N,2N}(2N)$?

I need examples of Jacobi forms for full congruence subgroups $\Gamma(N) $ of $SL(2,Z)$. As a particular case, take the theta function $\theta_{1,1}(t,z) := \sum_{n\in\mathbb{Z}} exp(\pi it(n + ...
1
vote
0answers
118 views

The conjecture $A159634(n) = \psi(2n)/3$ where $\psi$ is the Dedekind psi function.

Steven Finch is the author of the OEIS-sequence A159634 which is defined as "Coefficient for dimensions of spaces of modular & cusp forms of weight $k/2$, level $4n$ >and trivial character, ...
3
votes
0answers
67 views

Proving that $\sigma_7(n) = \sigma_3(n) + 120 \sum_{m=1}^{n-1} \sigma_3(m)\sigma_3(n-m)$ without using modular forms

This problem appears as a (starred!) exercise in D. Zagier's notes on modular forms. I have to admit that I have no idea how to do it. As usual, $\sigma_k(n) =\sum_{d\mid n} d^k$. This identity is ...
2
votes
0answers
32 views

Learning roadmap for p-adic modular forms and eigenvarieties.

What are some good sources for breaking into the field of p-adic modular forms? It was suggested to me to read Katz' paper "P-adic Properties of Modular Schemes and Modular Forms". I have found this ...
1
vote
1answer
57 views

Definition of Automorphic form

I am looking at how one maps the space of cusp forms $S_k(N,\psi)$ into the space of automorphic forms on $GL_2$ over the Adeles. And I have a question about the definition of $K$-finiteness. I am ...
3
votes
0answers
39 views

Are the Fourier coefficients of a new form real?

Let $f\in S^\text{new}_k(Γ_0(N))$ be a $\text{newform}$ . Are all its Fourier coefficients real? Of course the Hecke operators $T_n$ are selfadjoint for $(n,N)=1$, but is it also true for all $n$?
24
votes
5answers
402 views

Other interesting consequences of $d=163$?

Question: Any other interesting consequences of $d=163$ having class number $h(-d)=1$ aside from the list below? Let $\tau = \tfrac{1+\sqrt{-163}}{2}$. We have (see notes at end of list), ...
8
votes
3answers
90 views

How to show that $\Delta\left(\frac{az+b}{cz+d}\right)=(cz+d)^{12}\Delta(z)$?

Let $(a, b; c, d) \in SL_2(\mathbb{Z})$ and $\Delta(z)=q\prod_{n=1}^{\infty} (1-q^n)^{24}$, $q=e^{2\pi i z}$. How to show that $\Delta(\frac{az+b}{cz+d})=(cz+d)^{12}\Delta(z)$? Thank you very much. I ...
0
votes
0answers
19 views

The filtration of mod l modular form

I'm reading a paper of Swinnerton-Dyer, "On $l$-adic representations for coefficients of modular forms." He defines the notion of "filtration" for mod $l$ modular forms: If $\tilde{f} \in \tilde{M}$ ...
3
votes
1answer
92 views

Real life applications of Maass wave forms

Explaining my work on Maass wave forms to friends and family (all non-mathematician) typically earns me blank faces. So I wonder whether there is some good example to explain their meaning to laymen. ...
8
votes
4answers
319 views

What are applications of number theory in physics?

I was reading Goro Shimura's The Map of My Life. He wrote the following quote in the book. It made me come up with the title question. In particular, is there any application of the theory of modular ...
4
votes
0answers
93 views

Modular Functions with Rational Fourier Expansions

I have been reading the paper of Cox, McKay and Stevenhagen "Principal Moduli and Class Fields", http://arxiv.org/pdf/math/0311202v1.pdf, and I have a question regarding the nature of the function ...
0
votes
1answer
34 views

A quick fact check about filtrations of modular forms and $E_{p+1}$

Suppose $f$ is a mod $p$ modular form of level $N$ with $p>3$ and $p$ not dividing $N$. Is it true that $w(f E_{p+1})=w(f)+p+1$, where $w(f)$ is the filtration of $f$? (The filtration of $f$ is the ...
3
votes
1answer
101 views

modular form -Petersson inner product

my question is about Petersson inner product. i need to prove that $(E_k,f) =0 $ $\forall f \in S_k(SL_2(\mathbb{Z}))$ the only thing that i think that should help me is that the space of cusp form ...
4
votes
2answers
65 views

Other names for $E_{p+1}$ $\pmod{p}$?

If I want to know properties of $E_{p+1}$ modulo $p$, do you know a name for this modular form, so that it is easier to search via the internet? So far, what I know is that $E_{p-1}$ is the Hasse ...
5
votes
3answers
188 views

Inverting the modular $J$ function

Klein's modular function $J(z)$ is defined and studied in e.g. Apostol's book Modular functions and Dirichlet series in number theory. Certain specific evaluations are available, for example, ...
6
votes
1answer
107 views

Recognizing if a power series is a $q$-expansion of a modular form

Given a power series in $q$, is it possible to tell if it is the $q$-expansion of a modular form (of level $N$ say)? I don't need to show results of this sort, but it has come up enough that I'm ...
4
votes
1answer
63 views

Expressing Eisenstein series E_k in terms of E_4 and E_6

Given an Eisenstein series $E_k$ (of level 1), it is a polynomial $P_k(E_4,E_6)$ in $E_4$ and $E_6$, and http://en.wikipedia.org/wiki/Eisenstein_series#Recurrence_relation should give a finite ...
6
votes
0answers
254 views

Can $ 262537412640768743.99999999999925 $ be beaten with simple expressions? [closed]

We know: $$\begin{align}e^{\pi \sqrt{163}} &= 262537412640768743.9999999999992500726\dots\\ x^{24} - 24&=262537412640768743.9999999999992511239\dots\end{align}$$ where $x$ is the real solution ...
4
votes
1answer
146 views

Elementary tools for proving congruences of modular forms

My impression is that the specialists in the field use geometric modular forms when proving congruences of modular forms. While this is probably the right way, I don't think I will be able to get a ...
5
votes
2answers
181 views

Definition of Modular Forms over finite Fields

I'm still severely lacking in background at the moment, but I'm interested in doing something with congruence properties of modular forms (relations between coefficients of the q-expansions that hold ...
1
vote
2answers
73 views

Routine question about derivatives of automorphic forms being L^2

I consider Automorphic forms on $G = SL_2 (\mathbb{R})$, which are $\Gamma$-invariant, $K$-finite, $Z(g)$ finite, and of moderate growth. If I have such an automorphic form, which happens to be in ...
6
votes
1answer
120 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
1answer
82 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
1answer
107 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
201 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 $$|\operatorname {SL}_2(\mathbb Z/p^e\mathbb Z)|=p^{3e}(1-{1 /p^2})$$
5
votes
0answers
142 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 ...
33
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 ...
1
vote
1answer
255 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
81 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
1answer
137 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
53 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
66 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
69 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
209 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
80 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 ...
1
vote
1answer
133 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
2answers
342 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 ...
2
votes
1answer
133 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. ...
9
votes
1answer
598 views

How does one graduate from Hecke Operators to Hecke Correspondences?

I've read (skimmed heartily) basic books on the topic of modular forms. (The last being Silverman's Advanced Topics in the Arithmetic of Elliptic Curves.) I strive for an understanding which is as ...
3
votes
1answer
129 views

The filtration of a mod l modular form

I'm reading a paper of Swinnerton-Dyer, "On l-adic representations for coefficients of modular forms." He defines the notion of "filtration" for mod $\ell$ modular forms: If $\tilde{f} \in ...
1
vote
1answer
69 views

Algebraic modular form values depending only on coset reps

We have a reductive group $G/\mathbb{Q}$ and a representation space $V$ of this group. Let $K$ be an open subgroup of $G(\mathbb{A}_{\mathbb{Q}})$ (where $\mathbb{A}_{\mathbb{Q}}$ are adeles of ...
3
votes
3answers
433 views

Ramanujan 691 congruence

I know how to prove this congruence in two ways (one using basics of modular forms and the other using Hecke operators) and I will be working on proving other such congruences soon. The congruence ...
3
votes
0answers
104 views

What is a cusp parameter?

I was reading this paper, and on the first page they define a cusp form as $$ f(z) = \sum_{n > -\alpha} a(n) e^{2\pi i (n + \alpha)z}. $$ Is this equivalent to the usual definition of a cusp form ...