Arithmetic invariants of Euclidean lattice

TL;DR

This paper explores arithmetic invariants of Euclidean lattices within Arakelov geometry, establishing connections between Riemann-Roch analogues, quadratic form finiteness, and Mellin transforms over divisor class groups.

Contribution

It introduces new perspectives on arithmetic invariants, linking them to fundamental principles like the Heisenberg uncertainty and finiteness theorems in Arakelov theory.

Findings

Heisenberg uncertainty principle limits Riemann-Roch for $h^0_{Ar}$

Finiteness of quadratic form classes relates to hermitian vector bundle classes

Mellin transform expressed as an integral over Arakelov divisor class group

Abstract

In this paper we study the arithmetic invariants of Euclidean lattice in the context of Arakelov geometry. We regard a Euclidean lattice as a hermitian vector bundle $\bar E$ on ${\rm Spec}(\mathbb{Z})$ and consider two typical arithmetic analogues of the dimension of the space of global sections of a vector bundle on an algebraic curve. One is $$h^0_{\rm Ar}(\bar E):=\log \vert E\cap B_1 \vert$$ where $B_1$ is the unit ball, and the other is $$h^0_{\theta}(\bar{E}):=\log\sum_{v\in E}e^{-\pi\Vert v\Vert^2}$$ where $\sum_{v\in E}e^{-\pi\Vert v\Vert^2}$ is the theta function of $\bar E$. In this paper, we shall prove the following three statements: (i) the fact that one can not reach an absolute Riemann-Roch theorem for $h^0_{\rm Ar}(\bar E)$ is an instance of the Heissenberg uncertainty principle; (ii) the finiteness of equivalence classes in the genus of a positive quadratic form defined over $\mathbb{Z}$ is equivalent to the finiteness of certain isometry classes of hermitian vector bundles on ${\rm Spec}(\mathbb{Z})$, and it can be deduced from a finiteness theorem in Arakelov theory of ${\rm Spec}(\mathbb{Z})$; (iii) for any smooth function $f$ on $\mathbb{R}_{+}$ such that $f>0$ and that $f\circ {\rm exp}$ is a Schwartz function on $\mathbb{R}$, the Mellin transform of $f$ can be written as an integral over the Arakelov divisor class group of ${\rm Spec}(\mathbb{Z})$.