Questions tagged [theta-series]
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30
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Question on the relation of global theta lifting and local theta lift
Let $F$ be a number field and $G$ (resp. $H$) an odd orthogonal (resp. metaplectic group) over $F$.
Let $v$ be a finite place of $F$ and $\sigma_v$ a supercuspidal representation of $G_v(F_v)$. Let $\...
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1
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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 ...
- 87
3
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1
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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 $$\theta_{2}(q)=\sum_{n \in \mathbb{Z}}q^{(n+\frac{1}{2})^{2}}$...
- 31
5
votes
1
answer
250
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Approximation for a series involving the derivative of a Jacobi theta function
I’ve considered the diffusion equation $$\frac{\partial f(x,t)}{\partial t}=\frac12 \frac{\partial^2 f(x,t)}{\partial x^2}$$ with the conditions $f(x,0)=\delta(x)$ and $f(-1,t)=f(1,t)=0\ \forall t>...
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Positive integer solutions of $ab+ac+ad+bc+bd+cd=n$
Consider a quadratic form $Q(a,b,c,d)=ab+ac+ad+bc+bd+cd$ on $\mathbb Z^4$.
For some reason I am interested in the number of solutions $(a,b,c,d)\in\mathbb Z_{> 0}^4$ of $Q(a,b,c,d)=n$ as a function ...
- 4,761
4
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1
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270
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Closed formula for reversion of Jacobi theta series
Considering the Jacobi theta: $\theta_3(z) = \sum_{n\in\mathbb{Z}} q^{n^2}$,
we can invert $\theta_3-1$ in a small enough neighbourhood of 0.
Routine computation with Lagrange-Burmann inversion gives ...
- 313
4
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1
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What are the known number-theoretic functions, that are related to "the number of ideals of norm $n$, that belong to the class $[c]$"?
Let $L$ be a number field, $\mathcal{O}_L$ its ring of integers, and $\mathcal{Cl(O}_L)$ its ideal class group. Let's fix an arbitrary class $[c] \in \mathcal{Cl(O}_L)$. By $r(n)=r([c], n)$, I mean ...
4
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0
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$3$-variable Jacobi style identity linked to generalised Frobenius partitions
I was fiddling around with a family of probabilistic models and came across two "identities", which appear to be linked to generalized Frobenius partitions (more on this below). I would be ...
- 562
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Jacobi forms and Kato's modular units
this is pretty much just a silly literature question; apologies in advance. Kato uses the following theta function (or slight variants thereof) in his construction of his Euler system:
$$\Theta(\tau, ...
- 1,964
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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 ...
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0
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"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 ...
- 3,680
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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:
$$(i)\ 1 + 80q^3 ...
- 3,177
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Understanding proof of q-theta function expression
In arXiv:math/0309252v4 at the bottom of page 11, the following result is proposed
$$ 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}\...
- 181
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(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 ...
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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 ...
- 313
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votes
1
answer
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"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
$$\theta_3(q)^2= \left(\sum_{n \in \mathbb{Z}} q^{n^2}\...
- 800
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Expansion of Jacobi theta function at $p$-torsion
I am aware of the formula $$\Theta(z,q)=z\exp\left( -2\sum_{k\geq 1} \frac{z^{2k}}{(2k)!}E_{2k}(q)\right)$$ for the Jacobi theta function at the origin $z=0$. The definition I am using for the theta ...
- 113
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Modularity of certain theta series associated to hyperbolic lattice
Let $L$ be an even hyperbolic lattice, i.e. a free $\mathbb{Z}$-module with a non-degenerate inner product $\cdot$ valued in $\mathbb{Z}$ of signature $(1,n)$ such that the norm of every vector is ...
- 1,493
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Are all weight 5/2 modular forms theta series?
Given a quadratic form in $5$ variables, the theta series is a weight $5/2$ modular form. For a weight $3/2$ modular form we know that there always is a linear combination of theta series that work ...
- 2,419
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Numerical coincidence? Why is $\sum(x^{k^2}) = \sum(x^{(k+1/2)^2})$ for $x = 0.8$?
A well-known formula for the logarithm is given by
$$\log x = -\frac{\pi}{\operatorname{AGM}(a^2,b^2)}, \qquad x < 1$$
where AGM is the arithmetic-geometric mean, and $a$ and $b$ are given by
$$a ...
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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 $...
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1
answer
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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:
$$
r_k(P)=\textrm{cardinality of }\{v\in \...
- 2,324
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votes
0
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87
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Log-concavity of difference of theta functions
My knowledge on theta functions is limited, but I suspect that this is a quite challenging question. The 3rd Jacobian Theta function is given by
\begin{equation}
\theta_3(z,q)\,=\,\sum\limits_{n=-\...
- 31
3
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0
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Siegel's article “The volume of the fundamental domain for some infinite groups”: trouble with understanding computations
This is the article I mentioned. While the idea of what Siegel is doing in order to compute the volume of the fundamental domain described in the article (the very first one, for there are discussed ...
- 31
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Integral quaternary forms and theta functions
The following question arises when I attempt to understand the modular parameterization of the elliptic curve $$E:y^2-y=x^3-x$$
In Mazur-Swinnerton-Dyer and Zagier's construction, a theta function ...
- 3,317
9
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2
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Determining the Lambert series for $xq+x^2q^4+x^3q^9+...+x^nq^{n^2}+...$
I am trying to determine the polynomials $P_n(x)$ from
$$
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^...
- 17.5k
3
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1
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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 $...
- 190
3
votes
0
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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 ...
- 2,419
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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 ...
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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 $k$...
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