2025/12/03 by Tang, Shun
#11E12 #14C40 #14G40 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2512.03488
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 E on \rm Spec(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 h0\rm Ar( E):=log \vert E∩ B1 \vert where B1 is the unit ball, and the other is h0θ(E):=log∑v∈ Ee-π\Vert v\Vert2 where ∑v∈ Ee-π\Vert v\Vert2 is the theta function of 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 h0\rm Ar( 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 ℤ is equivalent to the finiteness of certain isometry classes of hermitian vector bundles on \rm Spec(ℤ), and it can be deduced from a finiteness theorem in Arakelov theory of \rm Spec(ℤ); (iii) for any smooth function f on ℝ+ such that f>0 and that f∘ \rm exp is a Schwartz function on ℝ, the Mellin transform of f can be written as an integral over the Arakelov divisor class group of \rm Spec(ℤ).