# Tagged Questions

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### Non vanishing and cuspidality of the theta lift of trivial representation.

Hi! Let E/F be a quadratic number field extension. Then we make some hermitian and skew hermition vector spaces and define unitary group on it.(namely U(1) and U(3)) Then, I am wondering whether the ...
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### Is there a “right” proof of Riemann's Theta Relation?

Let $\theta$ denote the usual Jacobi Theta function (with auxiliary parameter $\tau = i$, for simplicity), i.e. $$\theta(z) = \sum_{n \in \mathbb{Z}} \exp(-\pi (a + n)^2 + 2 \pi i n z) \ .$$ I'm ...
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### Extension of the Jacobi triple product identity

The Jacobi triple product and the mathematical identity of it is: $$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2}$$ I would ...
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### Vector chromatic number and Lovasz theta

For $\alpha \ge 2$, an $\alpha$-vector coloring of a graph $X$ is an assignment of unit vectors to the vertices of $X$ such that vectors assigned to adjacent vertices have inner product less than or ...
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### What do theta functions have to do with quadratic reciprocity?

The theta function is the analytic function $\theta:U\to\mathbb{C}$ defined on the (open) right half-plane $U\subset\mathbb{C}$ by $\theta(\tau)=\sum_{n\in\mathbb{Z}}e^{-\pi n^2 \tau}$. It has the ...
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### Zero-free theta functions in the upper half plane

Problem $1$. Which full rank lattices $\Lambda \subset \mathbb R^d$ have their corresponding theta function $\theta_{\Lambda}(\tau):= \sum_{\bf n \in \Lambda } e^{\pi i \tau ||n||^2}$ zero-free in ...
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### Express Weierstrass' g_2 and g_3 in terms of theta-functions of the periods

If E is a complex elliptic curve defined as the quotient of C over a lattice generated by w_1 and w_2, then it can be also written in Weierstrass form y^2=4*x^3-g_2*x-g_3. The coefficients g_2 and g_3 ...
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### pairing theta functions for different complex structures

I apologize for my previous attempt to ask this, which was very badly written. Let us start with $\mathbb{C}\times\mathbb{C}$. To form an Hermitian line bundle over a complex torus with complex ...
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### Order of vanishing at the cusps for the modular theta function

I am trying to examine the behavior of the theta function $\theta(z)=\sum_{n\in\mathbb{Z}} e^{2\pi i n^2 z}$, which is modular for $\Gamma_0(4)$ of weight 1/2, at the cusps 0 and 1/2. My calculations ...
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### Convexity of Jacobi's theta function with zero argument

This question may be elementary, I have asked it on math.stackexchange.com but have not received any answer yet. Note that I am not an expert on theta/elliptic functions. Define Jacobi's theta ...
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### What is the modern understanding of the order of a mock theta function?

Ramanujan introduced mock theta functions and described them by an "order" which he did not define. As a result of the work of Zwegers and others we now have a better understanding of mock theta ...
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### minimum rank - lovasz function inequality

hey, does the following inequality holds for every graph? $d(G)\geq\theta(G)$ while $\theta$ is the lovasz theta function and $d(G)$ is the minimum rank over all the matrices that represent the ...
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### What's the difference between a Riemann theta and a Siegel theta function?

One of the things I'm working on has required me to look into the literature of multidimensional theta functions, and I've gotten a bit confused on a few naming details. A look at the DLMF says that "...
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### Can Fuchsian functions solve the general equation of degree n?

In the classic textbook Introduction to the Theory of Equations (Conkwright, 1941), on p. 85, the author writes that “the algebraic solution of the general equation of degree n is impossible if n is ...
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### Odd powers of the theta function as eigenforms

Is it "well-known" which odd powers of the theta function are eigenforms for the half-integral weight Hecke operators? If so, what is a good reference? Is there a slick algorithm for proving ...
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### Motivation of proof of Riemann-Roch for elliptic curve and generalizations

Given a lattice $L \subseteq \mathbb{C}$, Alain Robert defines a theta function as a meromorphic function such that $\theta(z+\omega)=a(\omega) e^{\pi h(\omega)(z+\frac{\omega}{2})} \theta(z)$ for all ...
The theta function of a lattice is defined to be $$\vartheta_\Lambda = \sum_{v\in\Lambda} q^{{\Vert v\Vert}^2}$$ which yields as a coefficient of qk the number of vectors of norm-squared k. On the ...
So I am (barely) familiar with the construction of the theta function of an integral lattice $L$. The theta function, as I understand it, is defined as the function which takes a variable $z$ and ...