Groenewegen, Richard P. An arithmetic analogue of Clifford’s theorem. (English) Zbl 1069.11044 J. Théor. Nombres Bordx. 13, No. 1, 143-154 (2001). 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. Cited in 6 Documents MSC: 11R04 Algebraic numbers; rings of algebraic integers 14G40 Arithmetic varieties and schemes; Arakelov theory; heights Citations:Zbl 1030.11063 PDF BibTeX XML Cite \textit{R. P. Groenewegen}, J. Théor. Nombres Bordx. 13, No. 1, 143--154 (2001; Zbl 1069.11044) Full Text: DOI Numdam EuDML EMIS OpenURL References: [1] Francini, P., The function h° for quadratic number fields. These proceedings. · Zbl 1060.11076 [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 [4] Hartshorne, R., Algebraic Geometry. Springer-Verlag, 1977. · Zbl 0367.14001 [5] Neukirch, J., Algebraische Zahlentheorie. Springer-Verlag, 1992. · Zbl 0747.11001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.