For questions related to modular forms

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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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1answer
32 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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0answers
12 views

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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24 views

Modular Equation and Modular Forms?

I'm reading Ramanujan's Notebook now and I see some kinds of "Modular Equation". At first I think Modular Equation is just a set of interesting fomulaes, but wikipedia says that Modular Equation is ...
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9 views

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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41 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 ...
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14 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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1answer
44 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 $\displaystyle\Lambda=\langle\omega_1,\omega_2\rangle=\omega_1\Bbb Z+\omega_2\Bbb Z\subset\Bbb C,$ ...
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46 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 ...
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55 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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29 views

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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35 views

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 ...
2
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1answer
30 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 $$ ...
2
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1answer
46 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 ...
2
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1answer
49 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 ...
3
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2answers
55 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 ...
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2answers
138 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
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0answers
36 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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0answers
31 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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50 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 ...
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0answers
14 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}} ...
3
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2answers
67 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 ...
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47 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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0answers
25 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
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1answer
78 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 ...
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0answers
46 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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0answers
83 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 ...
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58 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.
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0answers
9 views

Finding cusp classes of $\Gamma_{o}(p^2)$

$\Gamma_{o}(p^2)$ acts on $\mathbb{P}_1(\mathbb{Q})=\mathbb{Q} \cup \lbrace i \infty \rbrace$. I need to find the orbits of this action. Any hint would be helpful. My guess is that there are 3 ...
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0answers
16 views

Fundamental domain, local coordinates in $X_0(3)$

I am looking at the following problem (which has several subproblems! I have omitted some): Let $X_0(3) := \mathcal{H}^{\star}/\Gamma_0(3)$. Define explicit maps giving the local coordinate around ...
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1answer
32 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
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1answer
66 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 ...
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1answer
28 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
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1answer
31 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 ...
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0answers
74 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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0answers
40 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 ...
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1answer
55 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 $f\colon \mathbb{H} \rightarrow \mathbb{C}$ ...
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22 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 $\Gamma$. ...
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1answer
166 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.
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57 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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How to solve a pair of simultaneous linear congruences, using algebraic methods [closed]

What is the smallest whole number $x$ so that $x$ has remainder $14$ when divided by $400$, and $x$ has remainder $5$ when divided by $7572$? In other words: $$x \equiv 14 \pmod{400}$$ and $$ ...
3
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1answer
88 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 ...
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1answer
90 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 $$ ...
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0answers
41 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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0answers
65 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 ...
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39 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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1answer
56 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 ...
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19 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 $d(n)$ ...
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14 views

On discrete subgroups of modular group and quadratic forms

I'm trying to work my way through a couple of papers on product formulae associated with certain modular forms. In "Borcherds Products Associated with Certain Thompson Series" by Chang Heon Kim, the ...