Effectivity of Arakelov Divisors and the Theta Divisor of a Number Field

In the well known analogy between the theory of function fields of curves over finite fields and the arithmetic of algebraic number fields, the number theoretical analogue of a divisor on a curve is an Arakelov divisor. In this paper we introduce the notion of an effective Arakelov divisor; more precisely, we attach to every Arakelov divisor D its effectivity, a real number between 0 and 1. This notion naturally leads to another quantity associated to D. This is a positive real number h(D) which is the arithmetical analogue of the the dimension of the vector space H(D) of sections of the line bundle associated to a divisor D. It can be interpreted as the value of a theta function. The notions of effectivity and h can be extended to higher rank Arakelov bundles. The Poisson summation formula implies a Riemann-Roch Theorem involving the numbers h(D) and h(κ −D) with κ the canonical class; it can be seen as a special case of Tate’s Riemann-Roch formula. The notion of effectivity naturally leads to a definition of the zeta function of a number field which is closely analoguous to the zeta function of a curve over a finite field. In this way we recover the Dedekind zeta function as an integral over the Arakelov class group. We derive the finiteness of the class group and the unit theorem of Dirichlet from the functional equation of the theta function. Unfortunately, we do not have a definition of h(D) := h(κ −D) for an Arakelov divisor without recourse to duality. The quantity h(D) defines a real analytic function on the Arakelov divisor class group. Its restriction to the group of Arakelov divisors of degree 1 2 deg(κ) can be viewed as an analogue of the theta divisor of an algebraic curve over a finite field. Its value on the trivial bundle is an invariant of the number field. It is natural to try to obtain arithmetic analogues of various basic geometric facts like Clifford’s Theorem. This comes down to studying the properties of the function h(D), like its (local) maxima and minima.