Questions tagged [modular-function]

This tag is for questions relating to Modular Function or, Elliptic Modular Function.

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$F=\{z\in\Bbb{H}:\ |z|>1,\ 2|\Re(z)|<\lambda\}$ is fundamental domain for $G_\lambda$(the subgroup of $SL(2,\Bbb{R})$ generated by $S$ and $T_\lambda$

Let $0<\lambda<2$ be a real number and $G_\lambda$ be the subgroup of $SL(2,\Bbb{R})$ generated by $S=\begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}$ and $T_\lambda=\begin{pmatrix}1 & \...
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1 vote
1 answer
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Modular function for certain upper triangular matrices (Anton Deitmar Automorphic Forms Exercise 3.4)

I am doing the following exercise from Anton Deitmar's Automorphic Forms Let $B$ be the group of all real matrices of the form $\begin{bmatrix}1 & x\\ 0 & y\end{bmatrix}$ with $y\neq 0$. Show ...
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5 votes
2 answers
259 views

Plotting graphs of Modular Forms

After watching all the 8 parts of "“Introduction to Modular Forms,” by Keith Conrad" on YouTube, I got "extremely intrigued" by plotting graphs of Modular Forms ( on SL(2,Z) ). So ...
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1 vote
2 answers
140 views

Prove that $(3/2)^n\bmod 1\equiv (2^n-1)/2^n $ has infinitely many solutions as $n \to\infty$ [closed]

The first part of the question is false and was proved by the answer vefore. The second part of the question asks to prove that $(3/2)^n \bmod 1\equiv (2^n-1)/2^n$ also has infinite solutions as $n \...
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3 votes
0 answers
175 views

Determine whether $\frac{K\left(\sqrt{1-x^2}\right)^2}{K(x)^2}$ is positive rational (given $x$)

Denote the complete elliptic integral of the first kind by $$K(x)=\int_0^{\pi /2}\frac{d\varphi}{\sqrt{1-x^2\sin^2\varphi}}$$ and $$f(x)=\frac{K\left(\sqrt{1-x^2}\right)^2}{K(x)^2}$$ Question: Given a ...
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2 votes
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$\lambda (\tau)$ as a rational function of $j(\tau)$

It is known that $$j(\tau)=\frac{256(1-x)^3}{x^2}$$ where $x=\lambda (\tau)(1-\lambda (\tau))$ and $\lambda (\tau)$ is the modular lambda function. But I came across the following statement (https://...
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  • 275
-1 votes
1 answer
63 views

Approximations of real numbers on $(0,1)$ with powers of the form $(3/2)^n \pmod 1$

Let $0<r<1$ a real number which is not a fraction of the form $p/2^n$ for any integers $p,n$. Now, for every integer $n\ge 1$ we can find the closest fraction of the form $p/2^n$ to $r$, which ...
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2 votes
0 answers
39 views

What are examples of modular form of level $1$ (i.e. modular form on $SL_2(\mathbb{Z})$) with poles?

I am thinking of Eisenstein series, $$G_k(\tau) = \sum_{(c,d)\in {\mathbb{Z}}^2-\{(0,0)\}}\frac{1}{(c\tau+d)^k}, \tau \in \mathbb{H} $$ because we don't sum over $(0,0)$, so I'd like to call $(0,0)$ a ...
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0 votes
1 answer
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Point Pair invariants

I have been reading Iwaniec's spectral theory of Automorphic forms, and in one of its definition, its defined that a function $k: \mathbb{H}× \mathbb{H} \rightarrow C$ is point pair invariant if $k(gz,...
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1 vote
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Integration on the boundary of fundamental domain of Full modular group

Suppose $f(x+iy)$ is a function satisfying exponential decay for every fixed x. Then can I conclude that integration on the boundary of fundamental domain of Full modular group (i.e. $SL_2(\mathbb{Z})$...
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  • 317
3 votes
2 answers
117 views

Derive singular value $\lambda(\sqrt{2}i)=(\sqrt{2}-1)^2$

Does anyone know how to prove that the following special value of the Modular Lambda Function is correct? $$\lambda(\sqrt{2}i)=(\sqrt{2}-1)^2$$ I have a somewhat promising observation that might help ...
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6 votes
1 answer
136 views

Is $\prod_{n=0}^\infty \left(1-\frac{1}{\cosh ^2((n+1/2)\pi)}\right)=\frac{1}{\sqrt[4]{2}}$ true?

The infinite product $$\prod_{n=0}^\infty \left(1-\frac{1}{\cosh  ^2((n+1/2)\pi)}\right)$$ agrees with $\frac{1}{\sqrt[4]{2}}$ to at least 100 decimal places. The "identity" is reminiscent ...
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  • 327
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0 answers
130 views

Square of Jacobi theta function is sum of hyperbolic secant?

I'm presently reading Henri Cohen's Introduction to Modular Forms (https://arxiv.org/pdf/1809.10907.pdf) and I'm trying to do exercise 1.5, which partially entails showing that: $T_2(a)\equiv\sum_{n=-\...
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1 vote
2 answers
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Pole order of $\frac{4}{27}\frac{\left(\lambda^2-\lambda+1\right)^3}{\lambda^2\left(1-\lambda\right)^2}\left(=j(\tau)\right)$ [closed]

Concerning the relation $$j=\frac{4}{27}\frac{\left(\lambda^2-\lambda+1\right)^3}{\lambda^2\left(1-\lambda\right)^2},$$ I understand, that the RHS is an element of $\mathbb{C}(j)$, and thus the LHS ...
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  • 23
-1 votes
1 answer
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A question based on proving a result Related to Klein J function

I was unable to solve some questions asked in my mid term of number theory exam and so I am asking it here. If $\tau \in H$ and $x= e^{2\pi i \tau}$, prove that $[{504 \sum_{n=0}^{\infty}\sigma_{5}(n)...
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  • 1,426
3 votes
0 answers
108 views

Non-modular coverings of modular curve

Let $Y$ be a Riemann surface, and $$T=\sum_{k=0}^n a_k y^k \in K(Y)[y]$$ be a polynomial in $y$ of degree $n$ over the field $K(Y)$ of meromorphic functions on $Y$. If we denote by $O$ the discrete ...
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  • 414
28 votes
2 answers
849 views

Closed form of $\frac{e^{-\frac{\pi}{5}}}{1+\frac{e^{-\pi}}{1+\frac{e^{-2\pi}}{1+\frac{e^{-3\pi}}{1+\ddots}}}}$

It is well known that $$\operatorname{R}(-e^{-\pi})=-\cfrac{e^{-\frac{\pi}{5}}}{1-\cfrac{e^{-\pi}}{1+\cfrac{e^{-2\pi}}{1-\cfrac{e^{-3\pi}}{1+\ddots}}}}=\frac{\sqrt{5}-1}{2}-\sqrt{\frac{5-\sqrt{5}}{2}}$...
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  • 946
3 votes
1 answer
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$q$-expansion of Klein's absolute invariant using infinite products

Given that $$j=\frac{1}{13824q^2}\left(2^8q^2\prod_{k\gt 0}(1+q^{2k})^{16}+\prod_{k\gt 0}(1+q^{2k-1})^{16}+\prod_{k\gt 0}(1-q^{2k-1})^{16}\right)^3,$$ how can I show that $$j=\frac{1}{1728q^2}(1+c_1 q^...
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  • 209
4 votes
1 answer
101 views

Explicit equations for $Y(N)$ for small $N$

Consider the congruence subgroup $$\Gamma(N) = \left\{\left(\begin{array}{cc} a & b \\ c & d\end{array} \right) \in SL_2(\mathbb{Z})\ ;\...
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5 votes
1 answer
178 views

Ramanujan Identity related to JacobiFunction [duplicate]

The following identity is allegedly due to Ramanujan $$\int_0^\infty \frac{{\rm d}x}{(1+x^2)(1+r^2x^2)(1+r^4x^2)\cdots} = \frac{\pi/2}{\sum_{n=0}^\infty r^{\frac{n(n+1)}{2}}} \, $$but how do you prove ...
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1 vote
1 answer
111 views

Square root of Eisenstein series

I am interested in square roots of Eisenstein series, such as $\sqrt{E_4}$ (see also this question). If we define the square root through its Taylor series, $$\sqrt{E_4}(\tau)=1 + 120 q - 6120 q^2 + ...
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  • 414
1 vote
0 answers
66 views

Radius of convergence of Eisenstein series

The Eisenstein series $E_k$ on the upper half plane $\mathbb H$ are holomorphic, $1$-periodic and therefore allow a Fourier series $$E_k(\tau)=1-\frac{2k}{B_k}\sum_{n=1}^{\infty}\sigma_{k-1}(n)q^n, \...
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  • 414
3 votes
0 answers
68 views

Modular Forms and Hypergeometric Functions

In "The 1-2-3 of Modular Forms" Zagier gives, as an example of proposition 21 (pg 61), the identity $$\vartheta_3(z)^2 = \sum_{n=0}^{\infty}\begin{pmatrix}2n\\n\end{pmatrix}\left(\frac{\...
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2 votes
0 answers
164 views

Fundamental domain for nonmodular functions

A fundamental domain for the modular group can be viewed a subset of the upper half plane $\mathbb H$ that contains exactly one point from every orbit of the $SL(2,\mathbb Z)$ action on $\mathbb H$. ...
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  • 414
0 votes
0 answers
60 views

What is the meaning of points $\tau$ in the upper half-plane which are not CM-points but with $j(\tau)$ an algebraic integer?

Due to the relation between imaginary quadratic fields and complex multiplication ($CM$) one knows that for all $CM$-points in the upper half-plane $$\mathbb{H}:=\{z \in \mathbb{C}:\ Im\ z>0\}$$ ...
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1 vote
0 answers
51 views

Understanding a proof about mapping properties of Klein's modular function $J$ (Apostol's "Modular Functions and Dirichlet Series in Number Theory")

I'm studying Tom Apostol's "Modular Functions and Dirichlet Series in Number Theory", and there's one statement in Chapter 2 that I do not understand. Let $R_\Gamma$ be the usual fundamental ...
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2 votes
0 answers
58 views

Hauptmoduln with non-negative Fourier coefficients

It is well-known that the $j$-invariant is a Hauptmodul for $SL(2,\mathbb Z)$ and has nonnegative integer Fourier coefficients. A reason for this is that the coefficients can be written as non-...
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0 votes
0 answers
28 views

Where (or How) can I find Fourier expansion of antiholomorphic (antianalytic) version of modular J-invariant, i.e. Fourier[ J( ̅t ) ].

It is known that Fourier expansion of the modular J-invariant is J(q)= 1/q + 196884q + ... It would be greatly appreciated if someone advise me where (or how) can I find the Fourier expansion of ...
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3 votes
1 answer
166 views

$N$-th root of modular forms

In my research, I have encountered many $q$-series of functions that turn out to be the Fourier expansions of roots of modular forms. Examples are $n$-th roots of Eisenstein series and the $j$-...
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  • 414
2 votes
1 answer
236 views

how to prove these formulas about infinite product?

Recently , I read one paper titled 'Modular equations and approximations to π' by Ramanujan, in which there are some formulas for $q=\pi i \tau$( where $\tau=x+yi, y>0$, hence $|q|<1)$ : $$\...
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1 vote
0 answers
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How to prove that $E_6(e^{\frac{2\pi i}{3}})\ne 0$ where $E_6$ is the Eisenstein series of weight 6?

The fact that $E_6(e^{\frac{2\pi i}{3}})\ne 0$ is considered to be obvious in some proofs of the facts about the Eisenstein series that I read, but I can't see it myself. Please, help me to make it ...
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  • 329
0 votes
1 answer
101 views

How the concept of space is defined for Modular Form?

The set of all functions of modular forms of weight $k$ is denoted by $M_k$. It is said in a document that $M_k$ is "clearly a vector space over $C$". My question is if $M_k$ is a set of functions, ...
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-1 votes
2 answers
97 views

Doubt in zeroes of Klein J Function

While studying Analytic number theory from Tom M Apostol Modular functions and Dirichlet series in number theory I am unable to think about a conclusion of theorem 2.7 . It's image - I have doubt in ...
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1 vote
0 answers
50 views

Equation for $X_0(3)$

I'm a number theorist, not an algebraic geometer, but I was trying to figure out an equation for $X_0(3)$ from scratch using my low-level techniques. I first wrote out the $q$-expansions for $j(\tau)$...
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  • 2,185
0 votes
2 answers
88 views

Doubt in deducing property of modular function

While studying number theory from Apostol's Modular functions and Dirichlet Series in Number Theory, I am having problem in deducing 1 statement. Statement is - An entire modular form of weight 0 ...
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0 votes
2 answers
98 views

Regarding expressing $j_p $ as a polynomial in $\Phi $

I am self studying apostol Modular functions and Dirichlet series in Number Theory and I have a doubt in an argument of theorem 4.11 . I have posted two images and I hope both of them are clear. I ...
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  • 1,426
0 votes
1 answer
219 views

Doubts in proof of a theorem related to modular functions from Tom Apostol 's Modular functions and Dirichlet series in Number Theory

I am self studying Apostol book and could not think about doubts in this theorem . Images of proof are at bottom of this page. EDIT 1 -> in later part of theorem I have some more doubts which I am ...
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0 votes
0 answers
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Regarding a property of Klein modular function J

I am self studying analytic number theory and I could not deduce this statement given in Tom M Apostol. Statement is - The function J takes every value exactly once in closure of fundamental region. ...
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  • 1,426
1 vote
2 answers
109 views

Regarding doubt in proof that every modular function can be represented as rational function of J.

I am self studying analytic number theory from Tom M. Apostol Modular Functions and Dirichlet Series in Number Theory and I am stuck on this theorem on page 40. Theorem 2.8. Every rational ...
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  • 1,426
3 votes
0 answers
176 views

Questions related to the Dedekind psi function $\psi(n)$

The Wikipedia article Dedekind psi function indicates the Dedekind psi function defined in formula (1) below was introduced by Richard Dedekind in connection with modular functions. (1) $\quad\psi(n)=...
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  • 4,668
0 votes
0 answers
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Help me understand the translation between homogeneous elliptic functions and modular functions

I am trying to follow Iwaniec's book on automorphic forms and am getting hung up on the basic definitions. He seems to construct modular forms from elliptic functions and I can't really follow the ...
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  • 578
2 votes
0 answers
170 views

Regarding definitions of modular functions and modular forms

I am self studying Analytic number theory from Apostol Dirichlet series and modular functions in number theory and I have a doubt regarding defintions of modular forms and modular functions. My ...
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  • 1,426
0 votes
1 answer
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Simplifying an expression with modulus

I'm looking for a method to simplify this expression with modulus: $$A\: mod\: 10\: +\:\left \lfloor\frac{A}{10}\right \rfloor\: mod\: 10\: +\: \left \lfloor\frac{A}{100}\right \rfloor\: mod\: 10\: +...
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  • 5,950
1 vote
4 answers
1k views

Sum of number's digits

I'm looking for a math function that taking an integer returns the sum of it's digits. I tried with this method: -I've got a digit sequence $d_{k+1},d_{k},...,d_{1}$ with $d_{k+1}\ne 0$ and $d_i \in \...
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  • 5,950
2 votes
1 answer
58 views

To prove that the Elliptic modular function is invariant under the modular transformation

I am not being able to understand that the Elliptic modular function $J(\tau)=\frac{g_2(w_1,w_2)^3}{g_2(w_1,w_2)^3-27g_3(w_1,w_2)^2}$ is invariant under the modular transformation $\tau\mapsto \frac{a\...
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  • 1,217
2 votes
0 answers
45 views

When the value of modular $j$ function is cubic number

It seems to be well known that when the class number of $K=\mathbb{Q}(\sqrt{-d})$ is one, the value of modular $j$ function at $\mathbb{Z}$ basis of integer ring of $K$ is cubic number. But I couldn'...
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0 votes
0 answers
68 views

Modular function and left Haar measure

In what follows, $G$ is a locally compact group with left Haar measure $\lambda$ and modular function $\Delta$. I came across this statement : It follows from a careful application of Hölder's ...
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  • 3,524
2 votes
1 answer
260 views

When $J(\tau)\in\mathbb{R}$? ($J$ is Klein's $j$-invariant.)

I would like to know the set $X=J^{-1}(\mathbb{R})$, where $J$ is Klein's $J$-invariant. Since $J$ is a modular function, it suffices to know the intersection of $X$ and the fundamental domain $\{\tau\...
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0 votes
1 answer
185 views

Constant term of weakly modular form of weight 2 vanishes

I stumbled upon this fascinating statement while browsing through old exercise sheets and don't find a fruitful approach to tackle the problem. Statement The constant term of the Fourier expansion ...
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4 votes
1 answer
120 views

What are the simplest known classes of bijections in $\mathbb{Z}/n\mathbb{Z}$, where n is a power of 2?

What are some of the simplest known bijections in $\mathbb{Z}/n\mathbb{Z}$? Offhand, the following classes of primitive bijections come to mind: Addition/subtraction (+/–) of any constant ...
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