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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 ${𝒪}_{F}$. Let $D$ be an Arakelov divisor on $Spec{𝒪}_{F}$; i.e., the formal sum of a divisor (in the usual sense) on the affine scheme $Spec{𝒪}_{F}$, and multiples ${x}_{\sigma }·\sigma$, with ${x}_{\sigma }\in ℝ$, for all infinite places $\sigma$ of $F$. This paper discusses a new definition of ${h}^{0}\left(D\right)$ 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 $\left(f\right)+D$ is effective, ignoring the infinite places). Instead of the naïve definition ${h}^{0}\left(D\right)=log#\left\{f\in I:{x}_{\sigma }-log{\parallel f\parallel }_{\sigma }\ge 0\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}\sigma \mid \infty \right\}$, this paper defines the effectivity of an Arakelov divisor $D$ to be a real number in the interval $\left[0,1\right)$ given by $e\left(D\right)=exp\left(-\pi {\sum }_{\sigma \phantom{\rule{4.pt}{0ex}}\text{real}}{e}^{-2{x}_{\sigma }}-2\pi {\sum }_{\sigma \phantom{\rule{4.pt}{0ex}}\text{complex}}{e}^{-{x}_{\sigma }}\right)$. (Functions other than $exp\left(-\pi {e}^{-x}\right)$ may be used here; this choice was based on a letter of K. Iwasawa [in: N. Kurokawa et al. (ed.), Zeta functions in geometry. Tokyo, Kinokuniya, Adv. Stud. Pure Math. 21, 445-450 (1992; Zbl 0835.11002)].) The authors then define ${H}^{0}\left(D\right)=I$ and ${h}^{0}\left(D\right)=log\left({\sum }_{f\in I}e\left(\left(f\right)+D\right)\right)$ (where the summand is presumably 1 when $f=0$). The latter is called the size of ${H}^{0}\left(D\right)$ and corresponds to the dimension of ${H}^{0}\left(D\right)$ in the case of a function field over a finite field. It depends only on the linear equivalence class of $D$.

Also define a canonical divisor $\kappa$ on $Spec{𝒪}_{F}$ to be the Arakelov divisor whose finite part is the different of $F$ and whose infinite components are all zero. Then a Riemann-Roch theorem ${h}^{0}\left(D\right)-{h}^{0}\left(\kappa -D\right)=degD-\frac{1}{2}log|{\Delta }|$ is proved, where ${\Delta }$ is the discriminant of $F$. It is noted that this is a special case of a Riemann-Roch theorem due to J. Tate [Thesis, printed in: J. W. S. Cassels (ed.) and A. Fröhlich (ed.), Algebraic Number Theory. Academic Press (1967; Zbl 0153.07403)].

Additional results are given, again in the spirit of furthering the analogy with the function field case. These results include expressing the Riemann zeta function as an integral of the effectivity function, an analogue of the inequality ${h}^{0}\left(D\right)\le degD+1$, and an analogue of the genus of $Spec{𝒪}_{F}$.

The authors express a hope that this paper will stimulate others to continue investigating this definition of ${h}^{0}$.

##### MSC:
 11R58 Arithmetic theory of algebraic function fields 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 11R42 Zeta functions and $L$-functions of global number fields
##### Keywords:
Arakelov divisor; effectivity; theta divisor; Riemann-Roch