Questions tagged [theta-functions]

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10
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0answers
83 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 ...
1
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0answers
51 views

How to generalise the moderate growth condition in theta series?

I am considering the definition of moderate growth in the non-reductive group. (For example, theta series in such automorphic form defined in $N \rtimes G$, where $N$ is an unipotent group and $G$ is ...
2
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0answers
111 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 $\...
1
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0answers
33 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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0answers
59 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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0answers
78 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
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1answer
186 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
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0answers
124 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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0answers
92 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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0answers
92 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}\...
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0answers
88 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
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1answer
173 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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0answers
70 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^{-(...
4
votes
1answer
279 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 ...
1
vote
1answer
180 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
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1answer
145 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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0answers
152 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 $$\...
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73 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}} \...
3
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0answers
176 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
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3answers
492 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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0answers
124 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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0answers
181 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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1answer
907 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
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1answer
462 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
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2answers
338 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
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1answer
178 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
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2answers
625 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
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0answers
81 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
1answer
274 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
1answer
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 $\vartheta$-...
1
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0answers
76 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
2answers
284 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
2answers
372 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 $k$ ...
3
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0answers
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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0answers
138 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 ...
4
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1answer
365 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
1answer
703 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)]}, \...
7
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0answers
492 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
1answer
395 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
1answer
311 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
0answers
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. ...
24
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0answers
741 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
1answer
282 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}...
71
votes
1answer
11k 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, ...
0
votes
1answer
219 views

holomorphic continuation

consider the function given by $f(t):=\sum\limits_{n=0}^{\infty}e^{-\left(n+\frac{1}{2}\right)^2t}$ for $t\in (0,\infty)$. This function can be continued holomorphically for all complex numbers with ...
0
votes
0answers
222 views

Is the Jacobi theta function invertible?

Let $\theta$ denote the Jacobi theta function: $$\theta=\sum_{k=0}^{\infty}{(-1)^kq^{k(k+1)}sin((2k+1)\frac{2\pi}{\omega_1}Re(z))},$$ and we have a complex number $t$. Suppose that we know there ...
1
vote
0answers
126 views

symmetric theta structures and arithmetic subgroups

A symmetric theta structure is a theta structure that commutes with (a lift of) the natural involution $\imath: A \to A$ an an abelian variety. For simplicity I will assume that $A$ is a surface. Now,...
5
votes
1answer
321 views

Index of congruence modular subgroup of level (1,d)

Let $D = \text{diag}(1,d)\in M_{2}(\mathbb{Z})$ be a $2\times 2$ matrix, where $d$ is an odd integer. We define the subgroup $\Gamma_D\subset M_{4}(\mathbb{Z})$ as: $$\Gamma_D := \left\lbrace R\in M_{...
0
votes
0answers
173 views

Bound on integral of elliptic theta function

I need to prove that the following bound is true. I thought this might follow from the inversion property of the theta function, as the infinite sum in the integrand is precisely $\theta_3(0,\mathrm{e}...
5
votes
0answers
413 views

Understanding Umemura's Theorem for roots of algebraic equations

I am trying to understand Umemura's Theorem for expressing the roots of any algebraic equation by higher genus theta functions. The original paper can be found here: Umemura, H.: Resolution of ...