Answer by Alexey Ustinov for What do theta functions have to do with...
This is not an answer but a comment concerning the Landsberg-Schaar relation (LS). It admits not only analytic proof. The article A proof of the Landsberg-Schaar relation by finite methods by Ben Moore...
View ArticleAnswer by paul garrett for What do theta functions have to do with quadratic...
One way a person could stumble on quadratic reciprocity while looking at theta functions is by trying to prove that Weil's construction of an adelic "Segal-Shale-Weil/oscillator representation" really...
View ArticleAnswer by Stopple for What do theta functions have to do with quadratic...
Going in the direction of more generality:With $\theta(\tau)=\sum_n\exp(\pi i n^2 \tau)$, theta reciprocity describes how the function behaves under the linear fractional transformation...
View ArticleAnswer by Jonah Sinick for What do theta functions have to do with quadratic...
Hecke generalized the argument that you mention to prove quadratic reciprocity relative to any given number field $K$ (see, e.g. his Lectures on the Theory of Algebraic Numbers).In The Fourier-Analytic...
View ArticleWhat 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...
View ArticleAnswer by LeSserafimRuleTheKPOPSky for What do theta functions have to do...
I think you're looking for the work of Tomio Kubota.The square of the theta function is a modular form. For a while, and still today, the theta function itself is sometimes considered a modular form...
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