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An arithmetic analogue of Clifford’s theorem. (English) Zbl 1069.11044

Summary: Number fields can be viewed as analogues of curves over fields. Here we use metrized line bundles as analogues of divisors on curves. G. Van der Geer and R. Schoof [Sel. Math., New Ser. 6, No. 4, 377–398 (2000; Zbl 1030.11063)] gave a definition of a function \(h^0\) on metrized line bundles that resembles properties of the dimension \(l(D)\) of \(H^0(X,{\mathcal L}(D))\), where \(D\) is a divisor on a curve \(X\). In particular, they get a direct analogue of the Riemann-Roch theorem. For three theorems of curves notably Clifford’s theorem we will propose arithmetic analogues.

MSC:

11R04 Algebraic numbers; rings of algebraic integers
14G40 Arithmetic varieties and schemes; Arakelov theory; heights

Citations:

Zbl 1030.11063
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References:

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[2] Fulton, W., Algebraic Curves. Addison Wesley, 1989. · Zbl 0681.14011
[3] Van Der Geer, G., Schoof, R., Effectivity of Arakelov Divisors and the Theta Divisor of a Number Field. Preprint 1999, version 3. URL: “http://xxx.lanl.gov/abs/math/9802121” . · Zbl 1030.11063
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