Questions tagged [theta-series]

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21
votes
2answers
650 views

Which even lattices have a theta series with this property?

This is a slight generalization of a question I made in Math StackExchange, which is still unanswered after a month, so I decided to post it here. I am sorry in advance if it is inappropriate for this ...
1
vote
0answers
129 views

“Modularity” of generalized theta series

The most basic theta series is defined by $\theta(z)=\sum_{n=-\infty}^{\infty}q^{n^2}$, where $q=e^{2\pi iz}$. This is connected to the question of how many representatations does an integer $n$ have ...
1
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0answers
84 views

Name for a pair of lattices one of which having theta series with coefficients a subsequence of another lattice's theta series coefficients

Is there a name for a pair of lattices which have the property given in the title (up to a change of variable)? The following example of a pair captures the property mentioned above: $$(i)\ 1 + 80q^3 ...
2
votes
0answers
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}\...
7
votes
1answer
141 views

(Binary) Theta functions and Atkin's U-operator

Let $D<0$ be a fundamental discriminant and consider the theta series $$\vartheta_Q(\tau)=\sum_{v\in\mathbb{Z}^2} q^{Q(v)}$$ associated to a quadratic form $Q$ of discriminant $D$. It appears to be ...
2
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0answers
71 views

Generalised theta series for fixed-rank sublattices

The theta series for a lattice $\Lambda$ is defined by $$\displaystyle \Theta_\Lambda(q) = \sum_{x \in \Lambda} q^{x \cdot x}.$$ Setting $q=e^{-\pi\tau}$ yields the (maybe more usual) related theta ...
2
votes
1answer
153 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
195 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 ...
7
votes
1answer
285 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
0answers
280 views

Are all weight 5/2 modular forms theta series?

Given a quadratic form in $5$ variables, the theta series is a weight $5/2$ modular form. For a weight $3/2$ modular form we know that there always is a linear combination of theta series that work ...
8
votes
2answers
976 views

Numerical coincidence? Why is sum(x^(k^2)) = sum(x^((k+1/2)^2)) for x = 0.8?

A well-known formula for the logarithm is given by $$\log x = -\frac{\pi}{AGM(a^2,b^2)}, \qquad x < 1$$ where AGM is the arithmetic-geometric mean, and $a$ and $b$ are given by $$a = \sum_{k\...
5
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0answers
90 views

On a particular case of Dirichlet series [closed]

I've this series: $$ \sum_{\ell = 1}^{+ \infty} e^{-t \ \ell^2} \sin{(k\ell)} = f(k, t) $$ where $ t \in [0,\infty]$ , $ k \in [0,2\pi] $. I need the limit of series like an analytic function of $...
3
votes
1answer
108 views

Number of vectors of fixed norm

Let $P$ and $Q$ be two even, unimodular, positive definite quadratic forms of rank $n$. Let $r_{k}(P)$ be the number of vectors of norm $k$, in symbols: $$ r_k(P)=\textrm{cardinality of }\{v\in \...
3
votes
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
votes
0answers
143 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
3answers
397 views

Integral quaternary forms and theta functions

The following question arises when I attempt to understand the modular parameterization of the elliptic curve $$E:y^2-y=x^3-x$$ In Mazur-Swinnerton-Dyer and Zagier's construction, a theta function ...
8
votes
2answers
468 views

Determining the Lambert series for $xq+x^2q^4+x^3q^9+…+x^nq^{n^2}+…$

I am trying to determine the polynomials $P_n(x)$ from $$ xq+x^2q^4+x^3q^9+...+\ x^nq^{n^2}+...=\sum_{n\geqslant1}\frac{P_n(x)q^n}{1-xq^n}; $$ that is, $$ \sum_{d|n}x^{\frac nd-1}P_d(x)=\begin{cases}x^...
3
votes
1answer
400 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 $...
3
votes
0answers
75 views

Noncommutivity of various lifts

Let $D$ be a definite quaternion algebra over $\mathbb{Q}$ ramified at $p$ and $\infty$ for simplicity. Then an automorphic form on $D^{x}/F^{x}$ has several different interpretations. First, through ...
4
votes
0answers
85 views

Global theta series on GL

Let $E/F$ be a quadratic extension of number field and let $V$ be a Hermitian space over $E$. Then we have Weil representations for the dual pair $U(n,n)\times U(V)$, and we can consider the theta ...
6
votes
1answer
194 views

Recursions for some binary theta series in characteristic 3

Define $A(0), A(1), A(2) \dots$ in ${\bf Z}/3[[x]]$ as follows. For $n$ in $\bf N$ let $s=3^{2n+1}$. Then $A(n) = \sum a_kx^k$ where $a_k$ is the mod 3 reduction of the number of representations of $k$...