Questions tagged [modular-function]
This tag is for questions relating to Modular Function or, Elliptic Modular Function.
118 questions
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Modular integrals and Poisson resummation
Consider the integral
\begin{equation}
\int_0^\infty \dfrac{dT}{T} \sum_{(m,n)\in\mathbb{Z}^2}e^{-\alpha Q(m,n) T}
\end{equation}
This integral is divergent both at $T\to0$ logarithmically and $T\to\...
- 43
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Fourier coefficients of rational powers of the Klein-j invariant
Consider $j(q) = E_4^3(q)/\Delta(q)$ to be the Klein $j$-function with $E_4$ being the weight $4$ normalised Eisenstein series and $\Delta(q)$ being the modular discriminant. It is known that $j^{1/3}(...
- 33
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Closed form of the integral $\int_{0}^{i\infty} \frac{E_4(z)}{\sqrt{j(z)}}dz$
When playing around with modular form integrations, one may accidentally come across the following mysterious integral:
\begin{align*}
I=\int_0^{i\infty}\frac{E_4(z)}{\sqrt{j(z)}}dz\approx 0....
- 1,064
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The issue of defining the action of the double coset operator on the divisor group of a modular curve
This problem comes from the end of Section 5.1 in GTM 228 A First Course in Modular Forms. Let $\Gamma_1$ and $\Gamma_2$ be congruence congruence subgroups of SL$_2(\mathbb{Z})$ and $\alpha\in$ GL$_2^+...
- 562
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Eisenstein's series
I started reading Silverman advanced topics on elliptic curves, I have some questions.
Why are cuspidal form in a certain way special?
By dimension arguments, one may argue that there are relations ...
- 19
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41
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The disk is a covering of the plane minus 2 points
I am studying the little Picard theorem from Rudin. In theroem 16.20 he constructs a function $\lambda$ wich satisfies the following:
-$\lambda$ is holomorphic in the upper plane $\Pi^+$
-$\lambda$ is ...
- 79
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$q$-expansion of the Weber functions
Let $\mathfrak{f}_1(\tau) = \frac{\eta(\tau/2)}{\eta(\tau)}$ where
$$ \eta(\tau) = q^{1/24}\prod_{n=1}^\infty (1-q^n) $$
and $q = e^{2\pi i \tau}$
I have trouble deriving the following $q$-expansion ...
- 435
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On the elliptic modular equation of weight 5
Schläfli's form of elliptic modular equation of weight $5$ is
$$\frac{u^3}{v^3}+\frac{v^3}{u^3}=2\left(u^2v^2-\frac{1}{u^2v^2}\right)$$
Where $$\begin{cases}u=2^{-1/4}q^{-1/24}(1+q)(1+q^3)(1+q^5)\...
user1018345
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Sum over ideals in $2$-part of class group for imaginary quadratic $K$, $j$-invariant
Let $K$ be an imaginary quadratic number field and suppose the $2$-part of $\mathcal{Cl}(K)$ is $G$ of order $2^k$.
Is it possible to compute the following sum efficiently:
$$\alpha = \sum_{\mathfrak{...
- 335
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Proving that an entire function is exponential. [duplicate]
I was doing exercises of this document of ENS Paris (in french) and I am stuck at this one (the last one in the document):
Let $\Lambda = \mathbb{Z} \omega_1 + \mathbb{Z} \omega_2 \subset \mathbb{C}$ ...
- 155
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Laurent expansion of modular functions
I am studying "A Course in Arithmetic", by Serre, and he says that, since modular functions are periodic, we can express them as functions of $q=e^{2\pi iz}$, which are denoted now by $\...
- 301
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Fundamental domain of $PSL_2(\mathbb{Z})$
Serre, in the book A Course in Arithmetic, shows that $D=\{ z\in H;\ |z|\geq 1\ \wedge \ |Re(z)|\leq 1/2 \}$ is a fundamental domain for the action of $PSL_2(\mathbb{Z})$ in the complex upper half ...
- 301
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The golden ratio $\phi$ for $_2F_1\big(\frac16,\frac16,\frac23,-2^7\phi^9\big)$ and $_2F_1\big(\frac16,\frac56,1,\frac{\phi^{-5}}{5\sqrt5}\big)$?
I. Context
Given the golden ratio $\phi$, then we have the nice closed-forms,
\begin{align}
_2F_1\left(\frac16,\frac16,\frac23,-2^7\phi^9\right) &= \frac{3}{5^{5/6}}\phi^{-1}\\[6pt]
_2F_1\left(\...
- 61.1k
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If $f$ is a modular function of weight $2k$, then $v_p(f)=v_{g(p)}(f).$
I'm studying the Serre's book A Course in Arithmetic and at the beginning of the page 85 he says: "when $f$ is a modular function of weight $2k$, the identity $f(z)=(cz+d)^{-2k} f(g(z))$ shows ...
- 301
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Reference request: When is the modular lambda function algebraic?
I read somewhere that said that the modular lambda function, defined as the ratio of two theta functions is algebraic whenever the parameter $\tau$ is defined over a number field, but I cannot find a ...
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