# Questions tagged [theta-functions]

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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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### 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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### 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 ...
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: $$\Theta(\tau, ... 0 votes 0 answers 71 views ### Proving the Immersion part of an Embedding Trying to see the proof of embedding the Jacobian of a Compact Riemann Surface X using Theta functions. So, using the Theta divisor we have the corresponding line bundle say L, we want to prove ... 2 votes 1 answer 184 views ### Semidirect product of metaplectic group and Heisenberg group I know that Symplectic group has an action on Heisenberg group. I am wondering how to extend this to non-trivial two fold metaplectic covering? Thanks in advance! 1 vote 0 answers 85 views ### Construct a doubly periodic function on \mathbb{C} using Jacobi elliptic functions with anti-holomorphic involution I would like to find explicit examples of non-constant meromorphic doubly periodic f(z) on \mathbb{C}, i.e. a meromorphic function on \mathbb{C} / \Gamma for some lattice \Gamma, such that ... 3 votes 1 answer 116 views ### " Laurent expansion" of quasi-periodic complex complex function Suppose a complex function f(z) depends only on z, and satisfies the quasi-periodicity in both directions:$$f(z+ a_x)= e^{i \theta_{a_x}} f(z)f(z+ i a_y)= e^{i \theta_{a_y}} f(z)$$where \... 11 votes 1 answer 2k views ### Reference request: proof of Ramanujan's Cos/Cosh Identity The Ramanujan Cos/Cosh Identity, as stated here, is$$\left[1+2\sum_{n=1}^{\infty}\frac{\cos n\theta}{\cosh n\pi}\right]^{-2}+ \left[1+2\sum_{n=1}^{\infty}\frac{\cosh n\theta}{\cosh n\pi}\right]^{-2}=...
In this paper, Shanks uses the following formula: $$\sum_{s=0}^{n-1}q^{s(2n+1)} \times \left( \prod_{k=s+1}^{n} \dfrac{1-q^{2k}}{1-q^{2k-1}}\right) = \sum_{s=1}^{2n} q^{\frac{s(s-1)}{2}}$$ to get a ...