Questions tagged [theta-functions]
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On the Jacobi theta functions and the Borweins' cubic theta functions
The post has been divided into sections to show some patterns.
I. Parameter $s=\frac12$
Given the nome $q = e^{\pi i\tau}$ and the Jacobi theta functions $\vartheta_n(0,q)$. Define the modular lambda ...
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Why does this theta function value yield such a good Riemann sum approximation?
Let $T(q)$ denote the third Jacobi theta function at $z=0$; i.e.,
$$T(q) := \sum_{n\in\mathbb{Z}} q^{n^2}.$$
Then for any $x>0$, $T(\exp(-1/x)) \approx \sqrt{\pi x}$. For example, as the entry for ...
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Is the real part of the Eta function bounded by $2 \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}}{n^{\alpha}}} $ [closed]
Consider the series defined by
\begin{equation}
f(\alpha,\beta) := \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}}{n^{\alpha}}\cos(\beta\ln(n))}
\end{equation}
is it true that $$f(\alpha,\beta) \le 2\sum_{n=1}...
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Constant term arising in general exact expression for the canonical height of a Mordell-Weil generator point on an elliptic curve $E$
First I shall begin by laying out some notation (I shall be using the conventions that are used by both DLMF and Mathematica which occasionally differ from the standard literature):
Let $\Lambda:=\...
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Elliptic integral as quantity associated with Riemann surface?
There are many elliptic integrals, so to show my point let me
just pick one of them (complete elliptic integral of the first
kind [1]):
$$K(k) = \int_{0}^{1} \frac {dx} {\sqrt{(1-x^{2})(1-k^{2}x^{2})}}...
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How to write the division values of $\operatorname{sn}(u;k)$ as rational functions of theta functions with zero argument?
Define the "thetanulls" (theta functions (https://dlmf.nist.gov/20) with one argument equal to zero) as follows:
$$\vartheta_{00}(w) = \prod_{n = 1}^{\infty} (1-w^{2n})(1+w^{2n-1})^2,$$
$$\...
- 307
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Is the left derivative of this theta function zero at $1$?
Is it true that
$$f(x):=\sum_{j=1}^\infty(-1)^j j^2 x^{j^2}\to0$$
as $x\uparrow1$?
(One may note that $f(x)=xh'(x)$, where $h(x):=\vartheta _4(0,x)/2$ and $\vartheta _4$ is a theta function, so that $...
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Ratio of theta functions as roots of polynomials
I already asked the same question here, but received no answer. I did some little progress and so I'm asking again.
I was playing with the theta functions with argument $ z = 0 $
$ \vartheta_2(q) =\...
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Ratio of theta function derivatives with theta function
I have the following ratios I want to compute.
$$ \frac{ \left( \frac{\partial \vartheta_3(v, q)}{\partial v} \right)^2 }{C + \left(\vartheta_3(v, q)\right)^2 }, $$
where $C$ is a constant.
$$ \frac{ \...
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Reference for modularity of the Andrews–Gordon–Rogers–Ramanujan identities?
The right-hand side of the identity https://mathworld.wolfram.com/Andrews-GordonIdentity.html is a $q$-series $\frac{(q^i,q^{2k+1-i},q^{2k+1};q^{2k+1})_\infty}{(q;q)_\infty}$; is there a reference of ...
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Reference for the asymptotic mixing time of the random walk on the cycle
In Diaconis's book Group Representations in Probability and Statistics, Chapter 3C, there are explicit computations for the mixing time of the random walk on the cycle graph $\mathbb{Z}_{p}$, with $p$ ...
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Additional symmetries in Theta-like function
cross-posted from https://math.stackexchange.com/questions/4708694/curious-symmetry-in-a-theta-like-function
Let $\Theta : \mathfrak{h}\times \mathfrak{h} \to \mathbb{R}$ be defined as follows
$$ \...
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The theta function of an odd Dirichlet character
The theta function $\theta_\chi(t)$ of a Dirichlet character $\chi$ is defined to be $\theta_\chi(t) = \frac{1}{2} \sum\limits_{n=-\infty}^\infty \chi(n) e^{2\pi i n^2 t}$ if $\chi(-1) = 1$ (i.e., $\...
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2D lattice sum with numerator
I've been struggling a bit with a double sum that arose as the trace of an operator:
$$\sum_{(j,k)\in Z^2 \setminus (0,0)} \frac{(j+k)^n}{(j^2+k^2)^n},$$
where $n$ is an even natural number. Is there ...
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Theta characteristics of surfaces and theta functions
A theta-characteristic of a Riemann surface is commonly defined as a spinor bundle, i.e., a holomorphic line bundle whose square is the canonical line bundle. These are in a natural one-to-one ...
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Generalizing Klein's order 7 formula $y\left(y^2+7\Big(\tfrac{1-\sqrt{-7}}{2}\Big)y+7\Big(\tfrac{1+\sqrt{-7}}{2}\Big)^3\right)^3 = j$ to order 13?
I. Level 7
In Klein's "On the Order-Seven Transformations of Elliptic Functions", he gave two elegant resolvents of degrees 8 and 7 in pages 306 and 313. Translated to more understandable ...
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Theta lifting over function fields
Let $F$ be a number field and $\mathbb{A}$ its adele ring.
For a dual reductive pair $G$ and $H$, let $\pi$ be a cuspidal irreducible representation of $G(\mathbb{A})$. Let $\Theta(\pi)$ be the global ...
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Eta product of squared tau function
The Ramanujan tau function is the coefficient of the 24th power of the Dedekind eta function.
$$ \eta(x)^{24}= x\prod_{m=1}^\infty (1 - x^m)^{24} =\sum_{n=1}^\infty \tau(n)\,x^n , $$
I want to know ...
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Evaluation of mock modular forms at elliptic points
The holomorphic function
$$F(\tau)=-\frac{1}{\vartheta_4(\tau)}\sum_{n\in\mathbb Z}\frac{(-1)^nq^{\frac{n^2}{2}-\frac 18}}{1-q^{n-\frac12}}=2q^{\frac38}(1+3q^{\frac12}+7q+14q^{\frac32}+\dots),$$
is a ...
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294
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Siegel modular forms in Mathematica
Is there a convenient way to work with Siegel modular forms in Mathematica? I am interested in doing analytic computations using the $\chi_{10}(\Omega)$ Siegel modular form, where $\Omega$ is the $2\...
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3
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On the Klein quartic and the similar $a^2b+b^2c+c^2a$?
Given the Ramanujan theta function,
$$f(a,b) = \sum_{n=-\infty}^\infty a^{n(n+1)/2} \; b^{n(n-1)/2}$$
Let $q = e^{2\pi i \tau}$ and assume $\tau = \sqrt{-d}$.
I. Degree 5
\begin{align}
a &= q^{11/...
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What pre knowledge does Mumford's Tata collections on theta need?
I am a sophomore and have the foundation of complex analysis and abstract algebra. I have learned Legendre elliptic integral and Jacobian elliptic function, and it is through this that I know theta ...
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What is the relationship between the Leech lattice and Dedekind eta function?
Like this old question, A conceptual proof of Jacobi's product formula for $\Delta$ ?, I am asking again for a conceptual proof of Jacobi's miraculous product formula for $\Delta$ (the unique ...
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Entire function with almost periodic boundary condition?
Let $v_1 =\lambda_1 \zeta_1$ and $v_2 = \lambda_2 \zeta_2$ with $\zeta_1 = \frac{4\pi i\omega}{3}$ and $\zeta_2 = \frac{4\pi i\omega^2}{3}$ where $\omega = e^{2\pi i/3}$ is the third root of unity and ...
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Higher Cardano formulae in terms of $\Theta$
Consider a polynomial in one variable with complex coefficient
$$f(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$
we are interested in its roots. Babylonian solved for $n = 2$, and Cardano did it ...
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Equation about Jacobi Theta Functions
Reading some Conformal Field Theory, I came across the following equation
about the Jacobi Theta functions without any justification:
Let $$\theta_{2}(q)=\sum_{n \in \mathbb{Z}}q^{(n+\frac{1}{2})^{2}}$...
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How to prove Gauss's identities on the action of the theta operator $f' = x\frac{df}{dx}$ on Jacobi theta functions?
(I have previously posted this question on mathstackexchange, but after getting no response there I decided to ask it again here. Here is a link to the same question there, so if this question is not ...
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Geometric interpretation of Theta functions and the Jacobi inversion problem
A great part of complex geometry and, algebraic geometry has been developed to address the theory of abelian integrals/functions. A very special problem that kept many great mathematicians busy was ...
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How to obtain the harmonic theta series via the global theta correspondence explicitly?
I am interested in the following kind of modular forms: let $(L,q)$ be an even unimodular lattice inside $V:=L\otimes \mathbb{Q}$, and $P$ is a harmonic homogeneous polynomial of degree $d$ on $\...
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How to work out this elliptic function?
Let $f(x) = \sum\limits_{(n,m)\in\mathbb{Z}^2} \frac{1}{(x+ n + i m )^2}$
If feel it should be $1/E(x)$ where $E$ is some elliptic function, like $sn^2$. But Wolfram Alpha is giving me some strange ...
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Two pairings on the group $K(L)$ associated with a non-degenerate line bundle $L$ on an abelian variety
Let $A$ be an abelian variety over a field and let $L$ be a non-degenerate line bundle on $A$.
Then $L$ gives rise to a morphism $\lambda:A\to A^*$ from $A$ to its dual.
As usual, let $K(L):=\ker(\...
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Geometric explanation for coincidence in lengths of 16-dimensional even unimodular lattices?
Question
Up to equivalence, there are two positive-definite even unimodular lattices in $16$ dimensions: $D_{8}^{+}\oplus D_{8}^{+}$ and $D_{16}^{+}$. As observed by Witt in 1941, the theory of ...
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Estimating two dimensional theta function
My feeling is that this should be written somewhere but I don't know what to search for.
Let $Q(x,y)$ be a binary quadratic form over $\mathbb{C}$, with $\operatorname{Re}(Q)$ positive definite. Then ...
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Reference on the Chern-Simons theory and WZW models for mathematicians
I would like to ask if there are any beginner friendly references for learning CS theory and WZW models. It seems that most mathematical texts on the subjects begin with convenient definitions that ...
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Infinite product of $1-q^{n^2}$
Is there anything known about the following product? Is it a known function or related to a known function?
$$\prod_{n\geqslant1}(1-q^{n^2})$$
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Approximation for a series involving the derivative of a Jacobi theta function
I’ve considered the diffusion equation $$\frac{\partial f(x,t)}{\partial t}=\frac12 \frac{\partial^2 f(x,t)}{\partial x^2}$$ with the conditions $f(x,0)=\delta(x)$ and $f(-1,t)=f(1,t)=0\ \forall t>...
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operations on matrices preserving the property of being the Riemann matrix of a surface
I have heard about the Schottky problem and the related Novikov's conjecture about the characterization of matrices in the Siegel upper half-space which are indeed the Riemann matrix of a compact ...
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Positive integer solutions of $ab+ac+ad+bc+bd+cd=n$
Consider a quadratic form $Q(a,b,c,d)=ab+ac+ad+bc+bd+cd$ on $\mathbb Z^4$.
For some reason I am interested in the number of solutions $(a,b,c,d)\in\mathbb Z_{> 0}^4$ of $Q(a,b,c,d)=n$ as a function ...
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From a factor of automorphy on an abelian variety to a divisor
Given a complex abelian variety $A = V/\Gamma$ (for $\Gamma$ being a lattice in the complex vector space $V$), one knows how to describe a holomorphic line bundle in terms of factors of automorphy: By ...
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Computing theta functions of lattices in practice
I am motivated by a problem in 2d CFT to compute "generalized theta functions," expressions of the form
\begin{equation}
\vartheta_{L,u}(\tau) := \sum_{\alpha \in L} u(\alpha) q^{{\langle\...
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The loss of double periodicity (ellipticity)
Consider a meromorphic function $f(\mathfrak{a}_1, \mathfrak{a}_2)$, such that
$$
\begin{align}
f(\mathfrak{a}_1, \mathfrak{a}_2) = f (\mathfrak{a}_1 + 1, \mathfrak{a}_2) = f(\mathfrak{a}_1 + \tau, \...
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Obtain a series expansion of $a^2(q)a^2(q^4)$
Let $a(q)$ denote the Borwein function $$a(q)=\sum_{m,n=-\infty}^\infty q^{n^2+nm+m^2}.$$
In this research paper the author has obtained the series expansion for $a(q)a(q^4)$. I want the series ...
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Validity of $\ln z=\frac{\pi}{\operatorname{AGP}(\theta_2^2(1/z),\theta_3^2(1/z))}$
Definitions
For the definitions of $\operatorname{AGP}$ and $\operatorname{AGO}$, see here or here. $\theta_2(z)$ and $\theta_3(z)$ are defined as follows:
$$\theta_2(z)=\sum_{n=-\infty}^\infty z^{(n+...
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When exactly is the principal AGM equal to the optimal AGM?
Definitions
Following the terminology of SageMath, let the principal arithmetic-geometric mean, $\operatorname{AGP}$, of $(a,b)\in\mathbb{C}^2$ for $a\ne 0$, $b\ne 0$, $a\ne\pm b$ be defined as ...
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Work of Atkin on the 26th power of eta
The 26th power of the Dedekind $\eta$ function has been mentioned several times here on MO:
A 14th and 26th-power Dedekind eta function identity?
What's the status of the following relationship ...
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Can the following sum be counted or expressed in terms of special functions?
Let us define this sum as a function of $z \in \mathbb{C}$ with some positive parameter $a$
$$
f(z; a) = \sum\limits_{n = 0}^{\infty}\frac{|z|^{2n}}{n!}e^{-ian^2}.
$$
Probably, it can be expressed (or ...
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Jacobi forms and Kato's modular units
this is pretty much just a silly literature question; apologies in advance. Kato uses the following theta function (or slight variants thereof) in his construction of his Euler system:
$$\Theta(\tau, ...
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Proving the Immersion part of an Embedding
Trying to see the proof of embedding the Jacobian of a Compact Riemann Surface $X$ using Theta functions. So, using the Theta divisor we have the corresponding line bundle say $L$, we want to prove ...
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Semidirect product of metaplectic group and Heisenberg group
I know that Symplectic group has an action on Heisenberg group.
I am wondering how to extend this to non-trivial two fold metaplectic covering?
Thanks in advance!
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Construct a doubly periodic function on $\mathbb{C}$ using Jacobi elliptic functions with anti-holomorphic involution
I would like to find explicit examples of non-constant meromorphic doubly periodic $f(z)$ on $\mathbb{C}$, i.e. a meromorphic function on $\mathbb{C} / \Gamma$ for some lattice $\Gamma$, such that ...
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