## 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

Zbl 1030.11063
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### References:

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