Arithmetic invariants of Euclidean lattice

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 ${\rm Spec}(\mathbb{Z})$.