Effectivity of Arakelov divisors and the theta divisor of a number field. (English) Zbl 1030.11063

Let \(F\) be a number field, with ring of integers \(\mathcal O_F\). Let \(D\) be an Arakelov divisor on \(\text{Spec}\,\mathcal O_F\); i.e., the formal sum of a divisor (in the usual sense) on the affine scheme \(\text{Spec}\,\mathcal O_F\), and multiples \(x_\sigma\cdot\sigma\), with \(x_\sigma\in\mathbb R\), for all infinite places \(\sigma\) of \(F\). This paper discusses a new definition of \(h^0(D)\) in the context of Arakelov theory, with the goal of proving results analogous to those that are true in the function field case.
Let \(I\) be the fractional ideal of \(F\) corresponding to \(D\) (in the sense that an element \(f\in F^{*}\) lies in \(I\) if and only if the divisor \((f)+D\) is effective, ignoring the infinite places). Instead of the naïve definition \(h^0(D)=\log\#\{f\in I: \text{}x_\sigma-\log\|f\|_\sigma\geq 0\) for all \(\sigma\mid\infty\)


11R58 Arithmetic theory of algebraic function fields
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
11R42 Zeta functions and \(L\)-functions of number fields
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