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34
votes
0answers
1k views

What does the theta divisor of a number field know about its arithmetic?

This question is about a remark made by van der Geer and Schoof in their beautiful article "Effectivity of Arakelov divisors and the theta divisor of a number field" (from '98) (link). Let me first ...
9
votes
0answers
576 views

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 ...
8
votes
0answers
386 views

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 ...
5
votes
0answers
223 views

Why does the Rogers-Ramanujan continued fraction $R(q)$ appear in Emma Lehmer's quintic?

Define the Ramanujan theta function $f(a,b)$ as, $$f(a,b) = \sum_{n=-\infty}^{\infty} a^{n(n+1)/2}\,b^{n(n-1)/2}$$ and the Dedekind eta function, $$\eta(\tau) = q^{1/24}\prod_{n=1}^{\infty} ...
1
vote
0answers
45 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. ...
1
vote
0answers
161 views

Can the series $\sum\limits_{n=0}^\infty q^{F_n}$ be expressed in terms of theta functions?

Let $F_0=0,F_1=1,...$ be the Fibonacci numbers. Is there a known closed form for the sum $\sum\limits_{n=0}^\infty q^{F_n}$? By closed form, I mean in terms of well-known functions, the first ones to ...
1
vote
0answers
74 views

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 ...
1
vote
0answers
102 views

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 ...
0
votes
0answers
90 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 ...
0
votes
0answers
177 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: ...
0
votes
0answers
424 views

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 ...