Some remarks on K-lattices and the Adelic Heisenberg Group for CM curves

TL;DR

This paper develops an adelic framework for CM elliptic curves, embedding them into the adelic Heisenberg group, and constructs theta functions that exhibit natural symmetries and actions, advancing the understanding of CM curves and their automorphisms.

Contribution

It introduces an adelic version of CM elliptic curves with actions of profinite endomorphism rings, embedding them into the adelic Heisenberg group and defining theta functions with symmetry properties.

Findings

Adelic theta functions are acted upon by the adelic Heisenberg group.

Theta functions behave well under complex automorphisms.

The embedding of CM elliptic curves into the adelic Heisenberg group is natural and canonical.

Abstract

We define an adelic version of a CM elliptic curve $E$ which is equipped with an action of the profinite completion of the endomorphism ring of $E$. The adelic elliptic curve so obtained is provided with a natural embedding into the adelic Heisenberg group. We embed into the adelic Heisenberg group the set of commensurability classes of arithmetic $1$-dimensional $\mathbb{K}$-lattices (here and subsequently, $\mathbb{K}$ denotes a quadratic imaginary number field) and define theta functions on it. We also embed the groupoid of commensurability modulo dilations into the union of adelic Heisenberg groups relative to a set of representatives of elliptic curves with $R$-multiplication ($R$ is the ring of algebraic integers of $\mathbb{K}$). We thus get adelic theta functions on the set of $1$-dimensional $\mathbb{K}$-lattices and on the groupoid of commensurability modulo dilations. Adelic theta functions turn out to be acted by the adelic Heisenberg group and behave nicely under complex automorphisms (Theorems 6.12 and 6.14).