## 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 $$\sigma\mid\infty$$

### MSC:

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

### Keywords:

Arakelov divisor; effectivity; theta divisor; Riemann-Roch

### Citations:

Zbl 0835.11002; Zbl 0153.07403
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