A modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group.

learn more… | top users | synonyms

0
votes
2answers
83 views

Why is it so difficult to use geometric methods to construct weight one modular forms?

I am trying to understand why it is that geometric methods of constructing modular forms of weight one for $SL(2,\mathbb Z)$ fail. I have a relatively rudimentary understanding of it, but it would be ...
2
votes
0answers
34 views

Computing Kronecker's polynomial $\Phi_N(X,Y)$ modulo $2$ for $N=23,33$

Kronecker's polynomial $\Phi_N(X,Y)\in\mathbb{Z}[X,Y]$ is used to give a model for the modular curve $X_0(N)$ over $\mathbb{Q}$. I know that it is hard to compute this polynomial in general, but is it ...
1
vote
1answer
53 views

Complex Galois Representaions

I'm trying to understand 1 and 2 dimensional complex representations, induced representations and associated Artin L-functions for a project. I'm finding it hard to find appropriate material to help ...
1
vote
0answers
72 views

Eisenstein series of weight $2$ for in $M_2(\Gamma_0(N))$

Let $q$ be prime. By dimension formulas, $S_2(\Gamma_0(q))$ is of dimension $g$, while $M_2(\Gamma_0(q))$ is of dimension $g+1$. So the space of Eisenstein series is one dimensional. How to define ...
4
votes
1answer
134 views

Why any quasi modular form corresponds to some almost holomorphic form?

I am trying to understand this article. http://arxiv.org/pdf/math/0603268v1.pdf In particular Theorem 1. In particular it says that if $$ (c z + d)^{-k} f_0(\frac{a z + b} {c z + d}) = \sum_{j = ...
1
vote
1answer
32 views

The main involution on $ M_{2}(F) $ and it's extension to $ M_{2}(F_{\mathbb{A}}) $.

I'm presently reading through a paper of Shimura's; "Special Values of the Zeta Functions Associated with Hilbert Modular Forms". In the paper he defines $ \iota $ to be the main involution of $ ...
2
votes
0answers
86 views

Eigenforms for $\mathcal{S}_2(\Gamma_0(88))$

I'm having trouble with the following exercise (5.8.3) from Diamond and Shurman's text on modular forms (this isn't homework for class, I just wanted to work this out on my own): ...
2
votes
0answers
62 views

Finding newforms with Sage

I am new to Sage and modular forms. I have some conceptual questions. When I write sage: S = CuspForms(Gamma0(55),2,prec=14) sage: S.new_subspace().basis() ...
1
vote
0answers
110 views

(Eichler-Shimura Isomorphism) Proving c(f) is not a coboundary

I have seen a couple of questions related to the Eichler-Shimura Isomorphism, but almost all of them have to do with hodge theory (things I am unfamiliar with) and seem, to me, different/unrelated. ...
4
votes
1answer
84 views

Modular property of Weierstrass $\wp$ function

For $\tau\in \mathbb H=\{ x+iy\in \mathbb C \lvert x,y\in \mathbb R, \, y>0\}$ and $z\in \mathbb C$, let us define $$ \wp=\wp(\cdot,\tau): \mathbb C \rightarrow \mathbb P^1\, , \quad z\mapsto ...
5
votes
1answer
89 views

Bounds on eigenvalues of Hecke operator on the Jacobian

Let $p$ be a prime not dividing $N$. Consider the Hecke operator $T_p$ on the Jacobian $\text{Jac}(X_0(N))$. I'm thinking of $T_p$ as coming from a correspondence. If I understand correctly, the ...
1
vote
1answer
65 views

Why does the Hecke correspondence preserve principal divisors?

Let $p$ be prime not dividing $N$. Consider the Hecke correspondence $T_p$ inducing a set valued function on $X_0(N)$. I'd like to understand why it acts on $\text{Pic}^0$, and so I'd like to know why ...
2
votes
0answers
45 views

what is the significance of expressing Theta series as Eta-products?

I'm collection information for my upcoming project regarding eta-products and theta-series. I have the Günter Köhler's book, ...
12
votes
1answer
159 views

Is there a general way to prove series and products are modular?

The following$$\eta(q)=q^{1/24}(q)_\infty$$ $$E_{n}(z)=\sum_{z \in \Lambda\setminus \lbrace0\rbrace}z^{-n}$$ $$F(q)=q^{-1/60} \sum_{n \ge0} \frac{q^{n^2}}{(q;q)_\infty}$$ $$F(q)=q^{11/60} \sum_{n ...
5
votes
1answer
65 views

Recursive identity for elliptic lattice constants $\sum_{\lambda\in\Lambda\setminus0} \lambda^{-2k}$

I am stuck on Exercise 3 in these notes. To keep this question self-contained: we have $\displaystyle\Lambda=\langle\omega_1,\omega_2\rangle=\omega_1\Bbb Z+\omega_2\Bbb Z\subset\Bbb C,$ ...
4
votes
0answers
93 views

Hodge Bundles on Tropical Spaces

I am not sure that this question even makes sense, which I suppose is part of the questions itself. In any case, I attended a talk recently wherin there was some discussion about a "tropical ...
3
votes
0answers
107 views

Hasse's theorem on elliptic curves over finite fields

Suppose $\mathcal{E}$ is an elliptic curve defined over $\mathbb{Q}$, Then Hasse's theorem states that for any large characteristic $p$ there exists an algebraic number $\lambda_p$ of modulus ...
2
votes
1answer
50 views

Linear independence of Eisenstein Space Basis

I'm currently working throught Diamond and Shurman's book "A First Course on Modular Forms". On page 129 Theorem 4.5.2 states Let $ N $ be a positive integer and let $ k \geq 3 $. The set $$ ...
3
votes
1answer
85 views

The Fricke involution on arbitrary subgroups of $\operatorname{SL}_2(\textbf{Z})$

Let $\Gamma$ be a subgroup of $\operatorname{SL}_2(\textbf{Z})$. I want to understand how the Fricke involution $w_N : f \longrightarrow (\tau \longrightarrow N^{-k/2}\tau^{-k}f(-1/N\tau))$ acts on ...
4
votes
1answer
109 views

A sufficient condition for $\eta$-quotients to be modular forms

Let $f$ be the form : $$f(\tau)=\prod_{M\mid N}{\eta(M\tau)^{a_M}} \quad (\tau \in \mathcal{H})$$ Generally, we said that $f$ is an $\eta$-quotient when $(a_M)$ is a sequence of integers. One can ...
4
votes
2answers
175 views

Concrete and elementary applications of modular forms to elliptic curves

What are some useful facts/algorithms for elliptic curves that can be obtained (proved completely) using the theory of modular forms without heavy machinery? It's often been asked what elementary ...
8
votes
2answers
264 views

Concerning the classical normalized Eisenstein series

Earlier I asked this question. As of today, it has not been answered. Yet still, I have a follow-up question: In general, how does one express $E_4(\tau)$ and $E_6(\tau)$ in closed form for special ...
3
votes
0answers
81 views

Freeman Dyson's identity for the modular discriminant $\Delta$

In his 'Introduction to Modular Forms', Don Zagier states the Freeman Dyson's identity : $$\Delta(\tau)=\sum_{\substack{ (x_1,\ldots,x_5)\in \mathbb{Z}^5 \\ x_1+\cdots+x_5=0 \\ x_i \equiv i ...
1
vote
0answers
88 views

Special values of the Dedekind eta function

Does anyone know of an article or a book which contains a comprehensive list of some known explicit values of the Dedekind eta function $\eta(\tau)$?
6
votes
0answers
143 views

Special values of the classical normalized Eisenstein series

I am looking for a comprehensive list of some known special values of the classical normalized Eisenstein series $E_4(\tau)$ and $E_6(\tau)$. Does anyone know where I can find a table of some known ...
3
votes
2answers
116 views

Fundamental domains of infinite-index subgroups of $SL(2,\mathbb{Z})$

While discussing modular forms associated to different subgroups $\Gamma$ of $SL(2,\mathbb{Z})$, there appeared to be a heuristic relationship between the index $[SL(2,\mathbb{Z}) \colon \Gamma]$ and ...
2
votes
0answers
64 views

When are theta constants modular

I'm looking at $\theta$ constants with characteristic, defined by $$\theta\left(\begin{array}{c} \epsilon \\ \epsilon' \end{array}\right)(z,\tau) = \sum_{n\in \mathbb{Z}} \exp 2\pi i \left\{ ...
2
votes
0answers
58 views

Motivation behind theta function calculation in Diamond & Shurman

I am reading section 1.2 of A First Course in Modular Forms. Let $q=e^{2\pi i\tau}$, where $\tau\in\cal H$ is assumed to be in the upper half plane, and define $\theta(\tau)=\sum_{n\in\Bbb Z}q^{n^2}$. ...
3
votes
1answer
120 views

What is the residue of the reciprocal of Klein's $j$-invariant $1/j(\tau)$ at $e^{2\pi i /3}$

In Mathematica, Residue[1/KleinInvariantJ[t], {t,Exp[2*pi*I/3]}] results in a very ungainly expression involving complete elliptic integrals of the first and ...
3
votes
0answers
73 views

Klein's invariant and negative discriminant

Let $J = J(\tau)$ be Klein's invariant and let $0 < k < 1$ be the elliptic modulus. It is known that $$J = \frac{4}{27} \frac{(1 - \lambda + \lambda^2)^3}{\lambda^2 (1 - \lambda)^2},$$ where ...
3
votes
0answers
103 views

A question on the Prime number theorem

Let $N\geq1$. Could we infer $$\sum_{n\leq N}\mu(n)\ll N\exp(-c\sqrt{\log N})$$from $$\sum_{n\leq N}\Lambda(n)= N+O(N\exp( -c\sqrt{\log N})$$or $$\sum_{p \leq N}1=Li(x)$$ without resorting to the ...
1
vote
0answers
70 views

Klein's absolute invariant and eta-quotients

Could anyone please provide a reference for $(13)$, $(14)$, $(17)$, $(18)$, $(21)$, $(22)$, which are stated on MathWorld without proper citation? Thank you in advance.
0
votes
1answer
47 views

Reverse the twisting of modular form

It is known that the twisting of the Fourier expansion of a modular forms by a Dirichlet character produce a modular form. ...
3
votes
1answer
101 views

Hecke Operators

I'm currently working my way through the section on Hecke operators in Serre's book. In the proof that $ T(m)T(n) = T(mn) $ for all $ m,n \in \mathbb{Z}_{\geq 1} $ with $ \textit{gcd}(m,n) = 1 $. In ...
0
votes
1answer
48 views

A converse theorem for Hecke modular forms?

Let $f$ be a function which can be represented by a Dirichlet serie (for $\Re s$ big enough) $$f(s)=\sum_{n=1}^{\infty}{\frac{a(n)}{n^s}}$$ and which can be meromorphically extend to $\mathbb{C}$ by ...
2
votes
1answer
47 views

Period of a point $\tau \in \mathcal{H}$ (upper half plane) and elliptic points

This is probably a silly question but I'll ask it anyway. Here goes: Let $\Gamma$ be a congruence subgroup of SL$_2$($\mathbb{Z}$). To each point $\tau \in \mathcal{H}$ (where $\mathcal{H}$ is the ...
2
votes
0answers
109 views

Modular forms on $\Gamma_0(N)$ with character in Sage

I'm trying to work with modular forms on $\Gamma_0(N)$ with character in Sage. In particular, I've been using the following: ...
1
vote
0answers
53 views

Quick explanation of $\Gamma \tau$ notation?

I hope you can help me by quickly explaining the following notation: $\Gamma \tau$. This notation is encountered in A First Course in Modular Forms by Fred Diamond and Jerry Shurman (love the book by ...
2
votes
1answer
82 views

definition of half-integral weight modular forms

i start reading about modular forms of half-integral weight $k/2$ for $\Gamma' \subset \Gamma_0(4)$. As far as i understand these are holomorphic functions $f\colon \mathbb{H} \rightarrow \mathbb{C}$ ...
0
votes
0answers
47 views

Natural surjection from complex upper half plane into modular curve

I am considering the natural surjection $\pi : \mathcal{H} \to Y(\Gamma)$ where $\mathcal{H}$ is the complex upper half plane and $Y(\Gamma)$ the modular curve of the congruence subgroup $\Gamma$. ...
2
votes
1answer
178 views

Modular forms are arithmetic objects

What does arithmetic object exactly means? In an article, I found the following statement: modular forms are arithmetic objects. What this should means? Bests.
3
votes
0answers
75 views

Relations between the Eisenstein series and the hypergeometric series

It is known that $$E_4(\tau) = {}_{2}F_{1}\left(\frac{1}{12}, \frac{5}{12}; 1; \frac{1}{J(\tau)}\right)^4$$ and $$E_6(\tau) = {}_{2}F_{1}\left(\frac{1}{12}, \frac{7}{12}; 1; \frac{1}{1 - ...
3
votes
1answer
104 views

Eisenstein-type series

Is the series, $$1 - 24\sum_{n = 1}^\infty \frac{q^{2n}}{(1 - q^{2n})^2}, \quad q = e^{\pi i \tau}, \quad \textbf{I}[\tau] > 0,$$ somehow related to $$E_2(q) = 1 - 24\sum_{n = 1}^\infty ...
5
votes
1answer
117 views

On the square coeffecients of a modular form

Let $k\in \mathbb{N}$. Let $f\in M_k(\Gamma_0(N),\chi)$ be a modular form of weight $k$ on $\Gamma_0(N)$ with a Dirichlet character $\chi$. If $f$ has a Fourier expansion of the form $$ ...
2
votes
0answers
94 views

A problem in the Hecke's trick method

In his 'Introduction to modular forms', Don Zagier deals with the Hecke's trick which I don't really understand : Let $$G_2(\tau)=-\frac{1}{24}+\sum_{n=1}^{+\infty}{\sigma_1(n)q^n}$$ and ...
0
votes
0answers
258 views

Twisting modular forms by Dirichlet characters

Let $\chi,\chi_1$ be Dirichlet characters modulo $M$ and $N$. In Koblitz's book "Introduction to Elliptic Curves and Modular Forms", Proposition III.3.17, it is proved that if ...
5
votes
0answers
46 views

Do modular forms of higher weight occur in the proof of FLT?

Fermat's last theorem is a consequence of a statement about weight two modular forms. Going through the long proof, does one ever encounter modular forms of higher weight?
4
votes
1answer
103 views

Problem about Eisenstein series on $\Gamma_1(N)$

I'm learning about Eisenstein series on $\Gamma_1(N)$ and it seems to me that I have misunderstood something. I imagine the following situation : Let $\nu$ be a function on $(\mathbb{Z}/ N ...
1
vote
0answers
49 views

Is there an effective bound known for the coefficients of half integer weight cusp forms?

If $f(z)=\sum a_n q^n$ is a cusp form (of integer weight) normalized so that $a_1=1$, we have the inequality $$\vert a_n \vert \leq d(n) n^{(k-1)/2},$$ known as the Deligne bound (in which $d(n)$ ...
3
votes
0answers
42 views

Arithmetic properties of the partition function

Ramanujan mentioned in his paper in 1920 that "it appears that there are no equally simple properties for any moduli involving primes other than these three" $p(5n+4)\equiv0 \mod 5$ $p(7n+5)\equiv0 ...