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Zbl 1030.11063
van der Geer, Gerard; Schoof, René
Effectivity of Arakelov divisors and the theta divisor of a number field. (English)
[J] Sel. Math., New Ser. 6, No.4, 377-398 (2000). ISSN 1022-1824; ISSN 1420-9020/e

Let $F$ be a number field, with ring of integers $\Cal O_F$. Let $D$ be an Arakelov divisor on $\operatorname {Spec}\Cal O_F$; i.e., the formal sum of a divisor (in the usual sense) on the affine scheme $\operatorname {Spec}\Cal O_F$, and multiples $x_\sigma\cdot\sigma$, with $x_\sigma\in\Bbb R$, 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. \par 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 na\" ive definition $h^0(D)=\log\#\{f\in I: \text{$x_\sigma-\log\|f\|_\sigma\ge 0$ for all $\sigma\mid\infty$}\}$, this paper defines the effectivity of an Arakelov divisor $D$ to be a real number in the interval $[0,1)$ given by $e(D)=\exp(-\pi\sum_{\text{$\sigma$ real}}e^{-2x_\sigma} - 2\pi\sum_{\sigma \text{ complex}}e^{-x_\sigma})$. (Functions other than $\exp(-\pi e^{-x})$ may be used here; this choice was based on a letter of {\it 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 $H^0(D)=I$ and $h^0(D)=\log(\sum_{f\in I}e((f)+D))$ (where the summand is presumably $1$ when $f=0$). The latter is called the size of $H^0(D)$ and corresponds to the dimension of $H^0(D)$ in the case of a function field over a finite field. It depends only on the linear equivalence class of $D$. \par Also define a canonical divisor $\kappa$ on $\operatorname {Spec}\Cal O_F$ 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 $h^0(D)-h^0(\kappa-D)=\deg D-\frac 12\log|\Delta|$ 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 {\it J. Tate} [Thesis, printed in: J. W. S. Cassels (ed.) and A. Fröhlich (ed.), Algebraic Number Theory. Academic Press (1967; Zbl 0153.07403)]. \par 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 $h^0(D)\le\deg D+1$, and an analogue of the genus of $\operatorname {Spec}\Cal O_F$. \par The authors express a hope that this paper will stimulate others to continue investigating this definition of $h^0$.
[Paul Vojta (Berkeley)]

MSC 2000:: *11R58 Arithmetic theory of algebraic function fields
14G40 Arithmetic varieties and schemes
11R42 Zeta functions and L-functions of global number fields

Keywords: Arakelov divisor; effectivity; theta divisor; Riemann-Roch

Citations: Zbl 0835.11002; Zbl 0153.07403

Cited in: Zbl 1081.14034 Zbl 1059.11063 Zbl 1106.11036 Zbl 1043.11055 Zbl 1069.11044 Zbl 1060.11076 Zbl 1017.11047

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