Questions tagged [modular-function]

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29 views

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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1answer
59 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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1answer
43 views

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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4answers
180 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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1answer
20 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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30 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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23 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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21 views

How is the bijection obtained?

I am reading in Serge Lang's book " Introduction to Modular Forms " . On page 8 there is written that there is a bijection between functions of lattices , homogenous of degree -k and functions g on H (...
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1answer
57 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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49 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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1answer
83 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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2answers
39 views

Weight zero modular functions and the J-invarient

I know that every weight zero modular function can be written as a rational polynomial in the J-invariant, but I'm not sure how to explicitly calculate the rational polynomial for a given weight zero ...
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1answer
39 views

modular group, prime ideals

I'm probably in over my head, but I came across the following sentence in a thesis by Evan Oliver entitled "Congruence Subarrangements of the Schmidt Arrangement": "The modular group is the set of ...
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2answers
91 views

unexplained modulo, rs ≡ 1 (mod m) where r = 3, s= 59 and m = 176

I'm trying to work through the math behind public key encryption, I'm a computer programmer, but not a mathematician. I came across this wonderful example, but I'm confused about the use of (mod x) my ...
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23 views

Inverse of modular exponential

We have two numbers $a$ and $b$ and three very large prime numbers $p_1, p_2$ and $n$. We compute $r_1=$ $a^{p_1}$ $mod$ $n$ and $r_2=$ $b^{p_2}$ $mod$ $n$. Now we forget about $a$ and $b$. Is there ...
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30 views

Solutions for a modular expression.

If I have a modular expression of type: \begin{equation}\nonumber w(c)=-\frac{\pi}{2}\frac{\left|-1+c\right|(1+c)+\left|1+c\right|(-1+c)}{(1-c^2)} \end{equation} How can I express the solutions to $w$ ...
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73 views

Finding equivalent submodular function

Let $V$ be a finite set of points in $\mathbb{R}^n$ with $d(x,y)$ denoting the usual Euclidean distance between two points $x$ and $y$. I am trying to order points from the set $V$ iteratively, at ...
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1answer
90 views

Algebraic singular moduli

I am trying to understand a point made by Zagier in the paper Elliptic Modular Forms and Their Applications. Proposition 22 is the statement that if $\tau$ in the upper half plane $\mathcal{H}$ is a ...
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1answer
22 views

Function that describes certain modulo function values

Im searching for a function $f(x),\ x\in\Bbb {N}$ which displays every value where: $h(x) =\big(\frac{3 \cdot x}{32}+33825\big)\mod 802 = 23$ i.e: $h(7296) = 23$ and $f(1) = 7296$ How can I get to ...
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65 views

calculator with modular function

I am looking to buy a calculator with Modular exponentiation functions and vendors do not mention this function in their websites I would really appreciate it if anyone could help me find a good ...
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71 views

Showing $E_4(z)$ and $E_6(z)$ are algebraically independent over $\mathbb{C}$

Let $E_4(z)= - \frac{B_4}{8}+ \sum_{n=1}^\infty \sigma_3(n) q^n$ and $E_6(z)= - \frac{B_6}{12}+ \sum_{n=1}^\infty \sigma_5(n) q^n$ How does one show they are algebraically independant over $\...
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1answer
125 views

Modular function for $SL_2(\mathbb{Z})$ is a field

How do I prove 1) Modular functions for $SL_2(\mathbb{Z})$ is a field K with addition and multiplication defined pointwise. 2) $K= \mathbb{C}(j)$, where $j(z)= \frac{(240 E_4)^3}{\bigtriangleup}$?
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2answers
132 views

Relationship between the Picard-Fuchs differential equation and the hypergeometric differential equation

Consider the Picard-Fuchs differential equation: $$\frac{d^2 \Omega}{dJ^2} + \frac{1}{J} \frac{d\Omega}{dJ} + \frac{31J - 4}{144J^2 (1 - J)^2} \Omega = 0.$$ The author of this article claims that the ...
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4answers
117 views

Solving the functional equation $\tau \left(\frac{-1}{z}\right) = - \tau(z)$

Let $\mathbb{H} \subset \mathbb{C}$ be the upper half plane. Find $\tau: \mathbb{H} \to \mathbb{C} $, holomorphic and non-constant, satisfying $\tau \left( \frac{-1}{z} \right) = - \tau(z)$. There ...
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1answer
342 views

Finding solutions to polynomial modular equations like $(x+1)^6 - x^6 \equiv 0\ (\textrm{mod}\ 19)$

I'm more or less okay with linear modulo equations, and was wondering how to solve polynomial modulo equations like $(x+1)^6 - x^6 \equiv 0\ (\textrm{mod}\ 19)$. The possible solutions are 2, 7, 9, ...
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195 views

On a proof of the Big Picard Theorem using covering spaces

I'm studying the paper The Big Picard Theorem and Other Results on Riemann Surfaces by P. Arés-Gastesi and T. Venkataramana, which presents a new proof of the Big Picard Theorem using covering spaces ...
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1answer
40 views

Let $R = \Bbb Z/6\Bbb Z = \{\bar 0,…\bar5\}$. In the polynomial ring $R[x]$ compute the product of the polynomials

Let $R = \Bbb Z/6\Bbb Z = \{\bar 0,...\bar5\}$. In the polynomial ring $R[x]$ compute the product of the polynomials $f = \bar2x^2+\bar3x +\bar1\,,\, g = \bar3x^3 +\bar4x^2 +\bar2$ My attempt: $$\...
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1answer
102 views

Generators of the Modular group (Motivation)

I would like to know the motivation of the geometric argument applied on a 'fundamental domain' to get the generators of $SL_2(\mathbb{Z})$. In particular, are there easier examples where this sort of ...
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71 views

Trigonometric system of 3 modular equations

Question: If $$\big(\cos x \cos y\big)^{2/3}+\big(\sin x \sin y \big)^{2/3}=1\tag1$$ $$\big(\cos y \cos z\big)^{2/3}+\big(\sin y \sin z \big)^{2/3}=1\tag2$$ then $$\sqrt{3}\big(\sin y \cos y \big)^{1/...
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73 views

Algebras of modular forms

It is well known that the algebra of modular forms of integral weight, trivial multiplier and $\mathrm{SL}(2,\mathbb{Z})$ is generated by $E_4$ and $E_6,$ and is isomorphic to the following weighted ...
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161 views

What is the derivative of Rogers-Ramanujan Continued Fraction?

We have many useful formulae about the derivatives of modular functions. For example, \begin{eqnarray} &&j'(\tau)=-\frac{E_6}{E_4} j(\tau), \\ &&\eta'(\tau)=\frac{1}{24}E_2 \eta(\tau), ...
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1answer
110 views

How to calculate singular moduli $\alpha_{3,n}$ of Ramanujan' s “$q_{3}$” theory?

Ramanujan's theory "$q_{3}$" How to calculate singular moduli $t=\alpha_{3,n}$ explicitly of this? $$\frac{\,_2F_1\big(\tfrac13,\tfrac23,1,1-t\big)}{\,_2F_1\big(\tfrac13,\tfrac23,1,t\big)} =\sqrt{n}...
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2answers
344 views

Ramanujan's Class Invariant $G_{625}$

How to calculate the Ramanujan Class Invariant $G_{625}$? Equation is: $x^6-2\varphi^5x^5+2\varphi x+\varphi^6=0$. $\varphi$ is the golden ratio.
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81 views

Examples of modular forms of irrational or complex weights

I think that it is possible to define modular forms of complex weight. But I have never seen modular forms of irrational weight, much less of complex (unreal) weight. If you know any examples of ...
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1answer
67 views

Hecke Operator being a modular function

I have a question on http://www.personal.psu.edu/rcv4/567c10.pdf I do not understand the proof of Theorem 10.6. I get that from Theorem 10.5, we get that Tn(f) satisfies the weakly modular equation....
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1answer
203 views

Why does genus 0 imply that the function field has one generator?

I have read that given a genus 0 modular curve $X$ defined over a base field $k$. The corresponding function field k(X) can be generated by one modular function. I have also read that converse, that ...
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1answer
93 views

How are Dedikind$\eta$ quotients connected to modular curves?

Im trying to understand the connection between the Dedekind $\eta$ function and its products/quotients and certain modular curves. Why is that the modular function $j_2 = \left( \frac{\eta(\tau)}{\...
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1answer
200 views

Encrypt a short message

I'm supposed to encrypt a short message, OAHU, using f(p) = (3p + 7) mod 26. I've tried to understand how to use a modulo, but frankly I just can't seem to wrap my head around it. If someone could ...
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0answers
69 views

Applications of inverse of Klein's $j$-invariant

The Klein $j$-invariant $$j(\tau) = q^{-1} + 744 + 196884q + \cdots$$ is a weight $0$ modular function holomorphic for $\tau$ in the upper-half plane $\mathbb{H}$. I understand that $j$ is important, ...
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54 views

The roots of $J' = \frac{dJ}{d\tau}$

I was wondering if anything is known about the roots of $J' = \frac{dJ}{d\tau}$. Here, $J(\tau) = j(\tau)/12^3$ is Klein's absolute invariant. Some roots can be calculated (at least I know how to), ...
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1answer
111 views

References on Hauptmoduln

Here it is said that a Hauptmodul (a generator of a modular function field) is unique up to a Möbius transformation. My impression is that it is really hard to find references on Hauptmoduln and ...
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1answer
128 views

Question on a proof of $\zeta(3)\notin\mathbb{Q}$

I have a question on this article proving $\zeta(3)\notin\mathbb{Q}.$ by using modular forms. This is theorem 1 at page 275 (page 5 in the pdf). Most things in the proof are clear but I don't get the ...
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1answer
73 views

Haar measure, can image of modular function be any subgroup of $(0,\infty)$?

It is easy to find examples of locally compact second countable Hausdorff topological groups $G$ whose modular function $\Delta$ has image $\{1\}$ or $(0,\infty)$. Are there groups $G$ of this kind ...