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.

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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. Here, $\sigma_k(n) =\sum_{d\mid n} d^k$, as usual. This ...
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207 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 ...
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40 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?
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51 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 ...
4
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118 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 ...
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98 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, ...
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64 views

Analogue of the Hecke operator for a general curve

Background: The eigenvalues of the Hecke opeator $T_p$ on the space of cusp forms $S_2(\Gamma_0(N))$ (of dimension $g=\text{genus}(X_0(N))$) are analysed using reduction mod $p$ and the use of the ...
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68 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 ...
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46 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 ...
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66 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
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51 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 ...
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88 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 ...
3
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60 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 - ...
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36 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 ...
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62 views

Modular forms on the theta group

The theta group $\Gamma$ is the subgroup of $SL(2;\mathbb{Z})$ generated by $T=\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ and $S=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.$ It ...
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54 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 $n$?
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48 views

Automorphisms of elliptic curves with a point of order N

Probably a stupid question, but I'm trying to figure out the following: Suppose we have an element of $X_1(N)$, i.e. an elliptic curve $E$ with a point $p$ of order $N$. If we have another such curve ...
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113 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 ...
3
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91 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 ...
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107 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 ...
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95 views

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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190 views

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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28 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 ...
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75 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): ...
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52 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() ...
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19 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, ...
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48 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\{ ...
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28 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 $\theta(\tau)=\sum_{n\in\Bbb Z}q^{n^2}$. ...
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80 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: ...
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47 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 ...
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39 views

Constructing modular forms for subgroups

If $\Gamma \le SL(2;\mathbb{Z})$ is a subgroup for which we know generators, is there a principle for constructing modular forms for $\Gamma$ out of those for $SL(2;\mathbb{Z})$? An example of what I ...
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44 views

Modular Forms Over Quadratic Number Fields

I'm trying to do a bit of reading and looking into mislay forms of weight one over quadratic number fields, but am finding it difficult to locate any books or papers. I've got a little material from ...
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48 views

eisenstein part of theta function

If $Q:\mathbb{Z}^{2k}\to \mathbb{Z}$ is any positive definite integer -valued quadratic form in $2k$ variables, then it is well known, that the $\textbf{theta series}$ ...
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20 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 + ...
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275 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 ...
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46 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? ...
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29 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 $\nu_\tau(f)$, ...
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61 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 ...
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120 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 ...
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57 views

What are the branch points of $X(n)\to X(1)$

Let $\Gamma \subset \mathrm{SL}_2(\mathbf{Z})$ be a finite index subgroup. Let $X_\Gamma \to X(1)$ be the corresponding morphism of compact connected Riemann surfaces (obtained by adding the cusps). ...
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158 views

A Dedekind eta function sum of form $y_0^k+y_1^k+y_2^k+y_3^k+… = 0$

Given the Dedekind eta function $\eta(\tau)$. Define, $y_p = e^{\pi i p/6}\,\eta(\tfrac{\tau+2p}{5})$ Prove the multi-grade identity [1], $y_0^k + y_1^k + y_2^k + y_3^k + y_4^k + ...
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$\exists C \in \mathbb{C}^{\times}$, $n\in\mathbb{N}$: $f = C\Delta^n$

Let $f \in M_k(\Gamma)$ not null in $\mathbb{H}$. I want to show that there exists a $C \in \mathbb{C}^{\times}$ and $n\in\mathbb{N}$ with $f = C\Delta^n$. I think one can show that for a $k>0$ ...
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23 views

Cuscs of a subgroup of $\Gamma$

I'm going to be completely honest about this: I need the solution of this to get permitted to the exam in complex analysis. The topic is not even relevant for the exam and I am absolutely not able to ...
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19 views

Why do modular curves parametrise elliptic curves?

Let $Y(n)=\Gamma(n)/H$, where $H$ is the upper half plane. In these lecture notes http://math.uga.edu/~pete/modularandshimura.pdf , the author makes the following statement: "$Y(n)$ parametrises ...
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32 views

Modular forms inducing functions on $SL_2(\mathbb{Q}_p)$

I am familiar with the classical thoery of modular forms. I've heard that a modular form on gives rise to a function on $SL_2(\mathbb{Q}_p)$ (or maybe $GL_2(\mathbb{Q}_p)$, or even $GL_2(\mathbb{A})$ ...
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38 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 ...
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28 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 ...
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33 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. ...
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41 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 $\eta(\tau)$?
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20 views

Weight half modular forms and theta functions

I had a question about weight 1/2 modular forms. In most places I've seen them defined, they are said to be functions which transform like the $\theta$ function $$\theta(\tau) = \sum_{n\in \mathbb{Z}} ...