0
votes
0answers
21 views

On the local L-parameter of U(3) obtained by the theta lift from U(1)

Let $E/F$ be a quadratic extension of number fields and $v$ a place of $F$. Let $\sigma$ be a automorphic character of $U(1)(\mathbb{A}_F)$ and $\Theta(\sigma)$ the theta lift of $\sigma$ to ...
3
votes
3answers
571 views

Mock Theta Functions

I am studying about Mock modular forms and Mock theta functions. I wonder how Zwegers connected mock theta functions with Harmonic Maass Forms? I mean, what was the philosophy/idea of Mock Theta ...
28
votes
3answers
1k views

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 ...
6
votes
4answers
830 views

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 ...
1
vote
1answer
206 views

Are there three variable generalizations of Ramanujans theta function?

The Ramanujan theta function $$f(a,b) = \sum_{n \in \mathbb Z} a^{n(n+1)/2} b^{n(n-1)/2}$$ satisfies the following Jacobi triple product identity $f(a,b) = (-a;ab)\_\infty (-b;ab)\_\infty ...
9
votes
1answer
464 views

Niemeier lattices and theta functions

I have an extremely elementary question. Let's say someone randomly hands you a theta function associated to a Niemeier lattice (unimodular even, n=24). What can you say about which Niemeier lattice ...
35
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 ...
3
votes
1answer
461 views

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 ...