Communications in Number Theory and Physics

Volume 9 (2015)

Number 4

Some remarks on $\mathbb{K}$-lattices and the Adelic Heisenberg group for CM curves

Pages: 763 – 797

DOI: http://dx.doi.org/10.4310/CNTP.2015.v9.n4.a5

Authors

Francesco D’Andrea (Dipartimento di Matematica e Applicazioni, Università di Napoli, Italy; and I.N.F.N. Sezione di Napoli, Italy)

Davide Franco (Dipartimento di Matematica e Applicazioni, Università di Napoli, Italy)

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).

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Published 13 May 2016