# Questions tagged [theta-functions]

The theta-functions tag has no usage guidance.

86
questions

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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, ...

**0**

votes

**0**answers

52 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

108 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

56 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

76 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 $\...

**8**

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}=...

**0**

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39 views

### Identity relation on Theta functions on 5 variables

I stumbled in an equation on the theta functions that is probably false itself, but promoted me a question. Let's define the notation
$$\theta_{p}\left(0\right)\theta_{q}\left(z_{1}\right)\theta_{r}\...

**15**

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436 views

### Finite version special case Jacobi triple product formula

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 ...

**2**

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**0**answers

126 views

### Rationality of Eisenstein series g2 and g3 for elliptic curves defined over numberfields

Let $K$ be a number field and let $E/K$ be an elliptic curve. (Fix an embedding of $K$ into the complex numbers $\mathbb{C}$). Let $\eta$ be the invariant differential of $E/K$. Let $\omega_1$ and $\...

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57 views

### Decomposing functions on the fundamental domain of the torus into cusp forms, eisenstein series

I am trying to do some elementary calculations to understand the properties of the following spectral resolution on $H/SL(2,\mathbf{Z})$. (Half plane mod modular group; fundamental domain of the torus)...

**3**

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68 views

### An elliptic function built from a log-theta-function integral?

I'm studying the apparent ellipticity of $$\Theta(z,a):=\frac{\exp(\tfrac2a\int_0^a\ln\vartheta(x+z)\,dx)}{\vartheta(z)\vartheta(z+a)},\tag1$$with $a$ is a free parameter and \begin{align}\vartheta(z)&...

**2**

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89 views

### theta function with a low bound in the sum

I encounter a sum similar to the Jacobi Theta function except there is a lower bound $-J$ with $J\geq 0$:
\begin{equation}
f(q,x)=\sum_{n=-J}^\infty q^{n^2}x^n.
\end{equation}
My question is whether ...

**5**

votes

**1**answer

195 views

### “One half of a theta-function” - is there something in the literature about it?

In an answer to my question Enumeration of lattice paths of a specific type (which was in turn caused by an older one, "Special" meanders) I encountered the series
$$
F(t,q):=\sum_{n=1}^\...

**6**

votes

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125 views

### Random walk on $\mathbf{Z}_d$ with Jacobi $\theta$ transition probabilities

In the context of a finite-dimensional quantum mechanical problem, I was led to study the random walk on $\mathbf{Z}_d$ (i.e the integers modulo $d$), $d$ odd with transition probabilities given by:
$...

**2**

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101 views

### Algebraic theta-functions of level $2$ on an elliptic curve

Consider an elliptic curve $E\!: y^2 = x(x-1)(x-\lambda)$, where $\lambda \in k \setminus \{0, 1\}$ for some field $k$ of $\mathrm{char}(k) \neq 2$. Also consider the ample divisor $D = 2P_0$, where $...

**2**

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95 views

### Understanding proof of q-theta function expression

In arXiv:math/0309252v4 at the bottom of page 11, the following result is proposed
$$ b_1 \theta(b_0 b_1^{\pm};p) \frac{z^{-1}\theta(z^2;p)}{\theta(b_0 z^{\pm};p)\theta(b_1 z^{\pm};p)} = (p;p)^{-2}\...

**3**

votes

**0**answers

90 views

### Evaluating a Fermi gas problem for a SO(2N+1) matrix integral

I have the following multiple integral derived from a random matrix calculation I wish to evaluate
$$\int_0^{\pi} dx_1 dx_2 \cdots dx_n \rho(x_1,x_2)\cdots \rho(x_n,x_1)$$
where the $\rho$ functions ...

**4**

votes

**1**answer

208 views

### Weight, Index, and Congruence Subgroup of Classical Jacobi Theta Functions

On the very first page in the Introduction of Eichler and Zagier's text on Jacobi forms, they mention that the theta function
$$\Theta_{x_{0}}(\tau, z) = \sum_{x \in \mathbb{Z}^{N}} q^{Q(x)} y^{B(x, ...

**0**

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71 views

### Deriving “quasi-theta” functions from theta functions' zeros

I've been trying to sift the zeros of \begin{align}\vartheta(z)&=e^{-\pi z^2}\prod_{k\ge1}(1+e^{-(2k-1)\pi+2\pi z})(1+e^{-(2k-1)\pi-2\pi z})\\
&=\prod_{k\ge1}(1+e^{-(2k-1)\pi+2i\pi z})(1+e^{-(...

**5**

votes

**1**answer

311 views

### $q$-analog of an integral from quantum field theory?

This question has been completely reformulated and a new property for the function $f_q$ has been added due to a series of helpful comments by fedja.
Consider the integral from quantum field theory ...

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vote

**1**answer

186 views

### How does $\prod_{\zeta}\varphi(q^{1/5}\zeta) = \varphi^6(q)/\varphi(q^5) $ hold?

While I check the proof for entry 10(vii) in Chapter 19 in Ramanujan's notebook, I couldn't understand one equality. It is
\begin{equation}
\prod_{\zeta}\varphi(q^{1/5}\zeta) = \varphi^6(q)/\varphi(q^...

**2**

votes

**1**answer

154 views

### “Sparse” Theta Series

The number of integer points with a given norm in the integer grid $\mathbb{Z} \times \mathbb{Z}$ may be calculated via the generating function
$$\theta_3(q)^2= \left(\sum_{n \in \mathbb{Z}} q^{n^2}\...

**4**

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164 views

### A statement in Siegel's paper on Riemann-Siegel Formula

Siegel left the following comment in the last paragraph of his famous "Über Riemanns Nachlaß zur analytischen Zahlentheorie", Section 1:
Das Integral $\Phi(u)$ ist ein Spezialfall des Integrales $$\...

**1**

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**0**answers

86 views

### Basic properties of prime forms

The prime form of a Riemann surface of genus $g$ can be defined in terms of multi-dimensional $\vartheta$-functions as follows:
$$E(z,w) = \frac{\vartheta\Big[\genfrac{}{}{0pt}{}{\vec{a}}{\vec{b}} \...

**4**

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232 views

### modularity of Theta functions attached to Hecke characters

Let $K/\mathbb{Q}$ be a quadratic imaginary field, and let $\chi$ be a Hecke character on $K$. Using Poisson summation, one can show that the theta function
$$
\theta(z):=\sum_{I\subseteq \mathcal{O}...

**6**

votes

**3**answers

526 views

### Alternating power series $\sum_{k\geq 0}(-1)^k z^{(2k+1)^2}$

Suppose that $f(x):=\sum_{k\geq 0}(-1)^ke^{-(2k+1)^2x}$ has a holomorphic continuation to a neighborhood of $0$, that is, $f(x)=\sum_{n\geq 0}a_n x^n$ for $x> 0$ small. I want to know the value of $...

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132 views

### Theta functions, a natural basis.

Let $L$ be the principal polarization of an abelian variety $A$. Some people say that the theta functions with characteristics form a natural basis of the global section $\Gamma(A,L^{\otimes n})$ for $...

**4**

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199 views

### Expansion of Jacobi theta function at $p$-torsion

I am aware of the formula $$\Theta(z,q)=z\exp\left( -2\sum_{k\geq 1} \frac{z^{2k}}{(2k)!}E_{2k}(q)\right)$$ for the Jacobi theta function at the origin $z=0$. The definition I am using for the theta ...

**15**

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**1**answer

923 views

### Theta functions, re-expressed

Recall the classical $\theta(q):=\prod_{k=1}^{\infty}(1-q^k)$ and
define the sequences $a_n$ and $b_n$ by
$$\frac{\theta^3(q)}{\theta(q^3)}=\sum_{n=0}^{\infty}a_nq^n \qquad \text{and} \qquad
F(q):=\...

**3**

votes

**1**answer

501 views

### On $x^k+y^k=1$ and the Dixonian elliptic functions

This solves this post and is also related to this MO post by involving $\tfrac12,\tfrac13,\tfrac14,\tfrac16$.
$p=2$
The singly-periodic trigonometric functions and the doubly-periodic Jacobi ...

**3**

votes

**2**answers

427 views

### Inverse Laplace transform of $sech(\sqrt{2\lambda})$ and Brownian motion occupation time

In dimension 3 we have that for $T=\int_{[0,\infty)}1_{B_{t}\in B(0,1)}dt$ has the Laplace transform
$$E[e^{-\lambda T}]=sech(\sqrt{2\lambda}).$$
And in dimension 1 we have the same for $\tau=\min\{t:...

**5**

votes

**1**answer

208 views

### Finding order of vanishing for Jacobi Theta function

From Rademacher's book (Topics in Analytic Number Theory) I'm using the functional equation of $\vartheta_2(0|\tau) = 2\sum_{m=0}^\infty q^{\left(m+\frac12\right)^2} = \vartheta_2(\tau)$ and the fact ...

**13**

votes

**2**answers

640 views

### A conjecture about algebraic values of $(-q;\,-q)_\infty/(q;\,q)_\infty$

Recall that $(a;\,q)_\infty$ is the $q$-Pochhammer symbol:
$$(a;\,q)_\infty=\prod_{n=0}^\infty(1-a \, q^n).\tag1$$
Its important special case $(q;\,q)_\infty=\prod_{n=1}^\infty(1-q^n)$ is sometimes ...

**4**

votes

**0**answers

90 views

### Triple Petersson Inner Products With Theta Functions

Our current work requires us to bound triple products of the form
$$\langle \Im(*)^{\frac{3}{2}}|\theta^3|^2,\mu_j \rangle,$$
where $\langle \cdot, \cdot \rangle$
is the Petersson inner product, ...

**7**

votes

**1**answer

290 views

### Modularity of certain theta series associated to hyperbolic lattice

Let $L$ be an even hyperbolic lattice, i.e. a free $\mathbb{Z}$-module with a non-degenerate inner product $\cdot$ valued in $\mathbb{Z}$ of signature $(1,n)$ such that the norm of every vector is ...

**4**

votes

**1**answer

120 views

### Modular S transformation on higher order $\vartheta$-functions using Poisson summation

I am trying to understand the action of a modular S transformation on a $\vartheta$-function. To do this for the problem I'm considering I first need to understand the following.
Given a $\vartheta$-...

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79 views

### Developing a functional equation for log-integral of theta function

I'm studying $u(z,q):=\exp(\left[\tfrac{\pi q}{12}-\ln2-\sum_{k\ge1}\ln(1-e^{-2k\pi q})\right]z+\int_1^z\ln\varphi(x,q)\,dx)$ , with $$\varphi(z,q)=2e^{-\pi qz^2-\pi q/4}\sinh \pi qz\prod_{k\ge1}(1-e^{...

**2**

votes

**2**answers

376 views

### Lower bound for the number of representations of integers as sum of squares

Let $k\geq 4$. As usual, let $r_k(n)$ denote the number of ways to represent $n$ as the sum of $k$ squares. Is this true that for every $\varepsilon>0$, one has $r_k(n) \gg n^{\frac{k}{2}-1-\...

**7**

votes

**2**answers

405 views

### How to estimate a specific infinite matrix sum

Let $M$ be an $n$ by $n$ matrix with each diagonal element equal to $k$ and each non-diagonal element equal to $k-1$ where $n$ and $k$ are positive integers. Let $k < n$ and we can assume both $k$ ...

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**0**answers

77 views

### Log-concavity of difference of theta functions

My knowledge on theta functions is limited, but I suspect that this is a quite challenging question. The 3rd Jacobian Theta function is given by
\begin{equation}
\theta_3(z,q)\,=\,\sum\limits_{n=-\...

**3**

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144 views

### Siegel's article “The volume of the fundamental domain for some infinite groups”: trouble with understanding computations

This is the article I mentioned. While the idea of what Siegel is doing in order to compute the volume of the fundamental domain described in the article (the very first one, for there are discussed ...

**5**

votes

**1**answer

405 views

### Summation of an infinite q-series

When calculating a Partition function, I encounter the following summation
$$\sum_{n=0}^{\infty} x^n q^{n^2}.$$
I know that the sum $\sum_{n=-\infty}^{\infty} x^n q^{n^2}$ is a Theta function, but I ...

**6**

votes

**1**answer

862 views

### Derivatives of theta functions at zero

Let $L$ be a line bundle over complex elliptic curve, $\deg L = k>0$. Theta functions
$$
\theta_s(z;\tau)_k=\sum_{r\in \mathbb{Z}} e^{\pi i [(\frac{s}{k} + r)^2 k \tau + 2kz(\frac{s}{k}+r)]}, \...

**8**

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551 views

### What is the $q$-analog of $\Gamma(z)\Gamma(1-z)=\frac\pi{\sin(\pi z)}$?

I would expect the $q$-Gamma function to have the property which would be the $q$-analog of the Euler reflection formula from my question title.
More concretely: $\Gamma(z)$ has simple poles at ...

**3**

votes

**1**answer

401 views

### Modular property of indefinite degenerate theta series

Is there anything known about the (mock)modular properties, if any, of the following theta series,
$\sum_{n\in {\mathbb Z}^r_+} e^{2\pi i \langle b, n\rangle} q^{\frac12 \langle n,n\rangle}$,
where $...

**6**

votes

**1**answer

317 views

### A Siegel modular form related to the product of two eta functions

I am looking for a Siegel modular form of genus $2$ (living on the Siegel modular 3-fold $A_2=\mathrm{Sp}(4,\mathbb{Z})\backslash \mathfrak H_2$) which becomes "roughly" the product of two eta ...

**7**

votes

**0**answers

100 views

### Equation of the curve corresponding to a polarization of an abelian surface

Let $\mathbb{C}^2/\Lambda$ be a polarized abelian surface. I think it is well-known how to write down the equation of the divisor corresponding to the polarization, in terms of theta functions etc. ...

**26**

votes

**1**answer

938 views

### Identities for power series like $\sum_n z^{n^3}$

Probably, one of the first power series that every mathematician encounter is the geometric series
$$\sum_{n=0}^\infty z^n = \frac1{1-z}, \quad z \in \mathbb{C},\; |z| < 1 .$$
Also, a particular ...

**3**

votes

**1**answer

292 views

### How can one parametrize a real elliptic normal curve such that four points are coplanar iff their parameters sum to zero?

Let $E \subset \mathbb{P}^3_{\mathbb{R}}$ be a real elliptic normal curve with two non-null-homotopic connected components. Is there a parametrization
$$ \chi: (\mathbb{R}/\mathbb{Z})\times (\mathbb{Z}...

**73**

votes

**1**answer

12k views

### What is an étale theta function?

Let me start out by urging you to take seriously that whatever I write about the papers surrounding IUTT really are questions. If you would like to use it as a guide to the mathematics in any way, ...