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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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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30 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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37 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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40 views

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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32 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 $Y(n)=\Gamma(n)/H$, 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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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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45 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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63 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 ...
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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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48 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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Order finding (modular) - Kitaev's Factoring Algorithm

Show Ma is Reversible and Unitary. This is the solution I have found. I understand the proof for the most part, however I don't think it is right. If it isn't right to prove Ma is Reversible by ...
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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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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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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 $\displaystyle\Lambda=\langle\omega_1,\omega_2\rangle=\omega_1\Bbb Z+\omega_2\Bbb Z\subset\Bbb C,$ ...
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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 ...
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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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Definition of Hecke Operator on modular form of half-integral weight

We define, for $f\in M_{k/2}(\tilde\Gamma_1(N))$, $T_{p^2}f :=p^{k/2-2} f|[\tilde\Gamma_1(N)\zeta_{p^2}\tilde\Gamma_1(N)]_{k/2}$, where $\zeta_{p^2}$ is the lift of $\alpha=\begin{pmatrix} 1 & ...
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Counting cosets of matrices of determinant > 1 under the action of a congruence subgroup

More specifically, let $M_2(m,\mathbb{Z})$ be the $2\times 2$ matrices with integer entries of determinant $m$ and $\Gamma^0(N)=\left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}:ad-bc=1,\ b\equiv 0 ...
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37 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 $$ ...
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60 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 ...
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77 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 ...