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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Zeroes of Jacobi Theta Functions

Based on wolfram alpha: $$\sum_{i=0}^{\infty}[x^{i^2}] = \frac{1}{2}(v_3(0,x) +1) $$ Whereas $v_3$ is the third Jacobi Theta function. See: http://bit.ly/1FIyUTq I am curious for what values in ...
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Modular parametrization of elliptic curve

Let $f$ be a cusp form of weight $2$ on $\Gamma_0(N)$ and assume that $f$ is a Hecke form and a newform. Then, we easily see that $$C(\gamma)=2i\pi \int_{\tau}^{\gamma \tau}{f(\tau')d\tau'} \quad ...
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Elementary proof of Ramanujan's “most beautiful identity”

Ramanujan presented many identities, Hardy chose one which for him represented the best of Ramanujan. There are many proofs for this identity. (for example, H. H. Chan’s proof, M. Hirschhorn's ...
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When is a modular curve defined over Q?

Let $X(N)$ and $Y(N)$ be respectively the compactified and uncompactified modular curves parametrising elliptic curves with full level $N$ structure. In other words, a point on $Y(N)$ is (essentially) ...
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Introduction to newforms and oldforms

Looking for an introductory text on newforms and oldforms. I have only found http://sage.math.washington.edu/edu/Fall2003/252/lectures/11-12-03/11-12-03.pdf which is not introductory in nature. ...
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checking if a Hecke eigenform is a smiultaneous Hecke eigenform

Let $f$ be a modular form of weight $k$ with respect to a congruence subgroup. Assuming that $f$ is a normalized Hecke eigenfunction for exactly one Hecke operator $T(l)$. How can one usually check, ...
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28 views

Proving that The set of limit points is not empty in an infinite group of linear fractional transformation.

suppose S is an infinite group of linear fractional transformation , show that the set of limit points of S is not empty . I'm studying a modular form course , and I got stuck in this question ,i ...
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dimension of space of modular functions using the Riemann-Roch theorem?

Let $H$ be the upper half-plane, and $M_k$ be the space of modular forms of weight $k$ on $H$ under the action of $SL(2,\mathbb{Z})$. I have read (Koblitz, Introduction to Elliptic Curves and Modular ...
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Does this function appear in theory of modular forms?

Does the following function $\phi(q,z)$ appear in theory of modular forms $$ \phi(q,z):=\prod_{n=-\infty}^{\infty}(1+q^nz) $$ for $(q,z)$ in some domain in $\mathbb{C}^2$? Any reference will be ...
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Relation between elliptic curves and Dirichlet L-series

I have read that an elliptic curve $E$ is modular if $a(n) = c(n)$ for all $n$, where $a(n)$ is the $n$-th coefficient in the Dirichlet series of $E$, $L(E,s)$, and $c(n)$ is the $n$-th coefficient in ...
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Infinite sum involving ascending powers

I was working on a problem involving finite differences and came across the following sum as a general solution to a problem I was working with $$ 1 + x + \frac{1}{1 + a}x^2 + \frac{1}{1 + a} ...
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coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$

I want to prove that $\forall n \in \mathbb{N}$ at least one of the Fourier coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$ is greater than 1( The ...
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What's the “exact depth” of a quasi modular form?

In https://archive.org/details/arxiv-math0509653 (p2, Remark 4) the Rankin Cohen bracket $[f,g]_n$ is defined as $\Phi_{n,k,s,l,t}(f,g)$ where $s$, $t$ are the exact depths of $f$, $g$. What does ...
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32 views

Congruence and first odd primes

I tried to find the solutions to this modular equation: $3^{(5+7+11+13+17+19+\dots +p(m-3)+p(m-2))} \equiv p(m-1) \bmod p(m) $ where $p(m)$ is the m-th odd prime number(note that it's three to the ...
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Why is $f(z)y^k$ bounded for $f$ a cusp form?

For $f$ is a cusp form of weight $2k, k>0$ ($f(z)=(cz+d)^{-2k}f(\frac{az+b}{cz+d}$)), then why is $f(z)y^k$ bounded? If expanded $f$ in $\sum a_nq^n$, it's domain is a open disc, hence I can't ...
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45 views

Action of automorphisms on Eisenstein series

Let $ f \in \mathcal{M}_{k}(\Gamma) $ and $ \sigma \in \textit{Aut}(\mathbb{C}) $. Suppose $$ f = \sum_{n=0}^{\infty}a_{n}q^{n} .$$ Then we define the action of $ \textit{Aut}(\mathbb{C}) $ on $ ...
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Some clarifications regarding Deligne's paper on $\ell$-adic representations arising from modular forms

In Deligne's article in Séminaire Bourbaki "Formes modulaires et représentation $\ell$-adiques" ...
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Residue theorem and Angle of modular function

Let $f$ be a meromorphic function on the region $Im(z)>0$, $v_p(f)$ be the order of $p$. (The number $n$ such that $\frac{f(z)}{(z-p)^n}$ is holomorphic and non-zero at $p$.) Moreover, assume $f$ ...
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$\left\langle\begin{pmatrix}0&-1\\1&0\end{pmatrix},\begin{pmatrix}1&1\\0&1\end{pmatrix}\right\rangle$ is the free product of two cyclic groups.

Let $G$ be a group generated by two matrices $ S=\left( \begin {array}{cc} 0&{-1}\\1&0\end{array}\right),T=\left( \begin{array}{cc}1&1\\0&1\end{array}\right) $ in $SL_2(\mathbb Z)/ \{ ...
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Dirichlet Series of weakly multiplicative characters

Let $$ \sum_{n=1}^{\infty} a(n)n^{-s} = \prod_{p}\left((1-\alpha_{p}p^{-s})(1-\alpha_{p}'p^{-s})\right)^{-1},$$ where $ a(n) $ is weakly multiplicative (i.e $ a(n)a(m) = a(n,m) $ if $ ...
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normalised eigenforms question

I'm looking at the questions on this sheet: https://www0.maths.ox.ac.uk/system/files/coursematerial/2014/3116/5/MF_2014_Sheet4.pdf and The Q is Q2 a). Now, assuming this is true, the rest of the ...
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Generators of $\Gamma_0(N)$

Let $\textbf{T}:=\bigl(\begin{smallmatrix} 1&1\\ 0&1 \end{smallmatrix} \bigr)$, $\textbf{S}:=\bigl(\begin{smallmatrix} 0&1/\sqrt{N}\\ -\sqrt{N}&0 \end{smallmatrix} \bigr)$ and $H$ the ...
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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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36 views

Boundedness of a modular form in $\mathbb{H}$

Let be $k>0$ and $f \in S_k(\Gamma)$. I want to show that the function $h(z)=Im(z)^{\frac{k}{2}}\cdot |f(z)|, \; z\in\mathbb{H}$ is bounded in $\mathbb{H}$. I have already shown, that $h$ is ...
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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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Why do modular curves parametrise elliptic curves?

Let $Y_1(N)=\Gamma_1(N)/H$, where $H$ is the upper half plane. In these lecture notes http://math.uga.edu/~pete/modularandshimura.pdf , the author makes the following statement: "$Y_1(N)$ ...
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Space of functions in the upper half-plane

Let $f(\tau)$ be a (say, holomorphic) function in the upper half-plane. Consider $$ A={\rm Span}_{\mathbb{C}} \{ f(\gamma \cdot \tau) : \gamma \in SL_2 (\mathbb{Z}) \}. $$ Is there a standard name ...
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Regarding the derivative of the $j$-invariant

Is anyone aware of a formula for the derivative of the $j$-invariant $j(\tau)$ with respect to $\tau$? Here, $\tau$ is in the upper half-plane. I would image there are probably quite a few formulae ...
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Whats the difference between modular forms of different levels?

We have a natural surjective group homomorphism: $\phi : SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}/(n\mathbb{Z}))$ from which, given any subgroup $H<SL_2(\mathbb{Z}/(n\mathbb{Z}))$, we may take the ...
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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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Simple question about lattices in $\mathbb{C}$

Milne's Modular Functions and Modular Forms states, at the bottom of page 10: We can normalize our lattices so they are of the form $$\Lambda(\tau) := \mathbb{Z} \cdot 1 + \mathbb{Z} \cdot ...
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How to construct this modular Galois representation?

I want to know how to construct this modular Galois representation: The $\bmod p$ representation is semi-simple. For any lattice $T$ , $\dfrac{T}{p^{3}T}$ does not have a $G$-stable cyclic subgroup ...
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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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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 ...
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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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47 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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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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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 = ...
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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 $ ...
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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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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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(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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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 ...
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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 ...
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57 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 ...
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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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Eichler-Shimura congruence

I'm trying to understand the Eichler-Shimura congruence which relates the Hecke operator $T_p$ to Frobenius at $p$ in characteristic $p$. Two possible ways to compute $T_p$ mod $p$ seem to be: A) ...
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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 ...
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21 views

Making sense of a summation simplification

(This is all related to the action of Hecke operators on modular functions) Given is the sum \begin{equation} n^{2k-1} \sum_{\substack{a \geq1,\\ ad=n,\\0 \leq b < d}} d^{-2k} \sum_{ m \in ...
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64 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 $\displaystyle\Lambda=\langle\omega_1,\omega_2\rangle=\omega_1\Bbb Z+\omega_2\Bbb Z\subset\Bbb C,$ ...