Arithmetic invariants of Euclidean lattice
Shun Tang
TL;DR
The paper develops an Arakelov-geometric framework for Euclidean lattices by viewing them as Hermitian vector bundles over Spec(Z) and by studying two arithmetic dimensions, $h^0_{Ar}$ and $h^0_{\theta}$. It shows that an absolute Riemann–Roch theorem cannot hold for $h^0_{Ar}$ due to a Heisenberg-type uncertainty principle, while $h^0_{\theta}$ satisfies an absolute RR through duality and Poisson summation. It connects the finiteness of genus-class numbers for positive quadratic forms to finiteness statements about isometry classes of Hermitian bundles via adelic moduli, and establishes a finiteness theorem in Arakelov theory to underpin this correspondence. The work further proves that the Mellin transform of suitable positive functions can be expressed as integrals over the Arakelov divisor class group, providing a general analytic bridge between Mellin transforms, zeta functions, and automorphic data. Together, these results illuminate arithmetic invariants of Euclidean lattices, finiteness phenomena, and analytic representations in the Arakelov setting of Spec(Z).
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 $$h^0_θ(\bar{E}):=\log\sum_{v\in E}e^{-π\Vert v\Vert^2}$$ where $\sum_{v\in E}e^{-π\Vert v\Vert^2}$ 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 ${\rm Spec}(\mathbb{Z})$.