# An arithmetic analogue of Clifford's theorem

Journal de théorie des nombres de Bordeaux (2001)

- Volume: 13, Issue: 1, page 143-156
- ISSN: 1246-7405

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topGroenewegen, Richard P.. "An arithmetic analogue of Clifford's theorem." Journal de théorie des nombres de Bordeaux 13.1 (2001): 143-156. <http://eudml.org/doc/93793>.

@article{Groenewegen2001,

abstract = {Number fields can be viewed as analogues of curves over fields. Here we use metrized line bundles as analogues of divisors on curves. Van der Geer and Schoof 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 Rieman-Roch theorem. For three theorems of curves, notably Clifford’s theorem, we will propose arithmetic analogues.},

author = {Groenewegen, Richard P.},

journal = {Journal de théorie des nombres de Bordeaux},

language = {eng},

number = {1},

pages = {143-156},

publisher = {Université Bordeaux I},

title = {An arithmetic analogue of Clifford's theorem},

url = {http://eudml.org/doc/93793},

volume = {13},

year = {2001},

}

TY - JOUR

AU - Groenewegen, Richard P.

TI - An arithmetic analogue of Clifford's theorem

JO - Journal de théorie des nombres de Bordeaux

PY - 2001

PB - Université Bordeaux I

VL - 13

IS - 1

SP - 143

EP - 156

AB - Number fields can be viewed as analogues of curves over fields. Here we use metrized line bundles as analogues of divisors on curves. Van der Geer and Schoof 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 Rieman-Roch theorem. For three theorems of curves, notably Clifford’s theorem, we will propose arithmetic analogues.

LA - eng

UR - http://eudml.org/doc/93793

ER -

## References

top- [1] P. Francini, The function h° for quadratic number fields. These proceedings. Zbl1060.11076
- [2] W. Fulton, Algebraic Curves. Addison Wesley, 1989. Zbl0681.14011MR1042981
- [3] G. Van Der Geer, R. Schoof, Effectivity of Arakelov Divisors and the Theta Divisor of a Number Field. Preprint 1999, version 3. URL: "http://xxx.lanl.gov/abs/math/9802121" . Zbl1030.11063MR1847381
- [4] R. Hartshorne, Algebraic Geometry. Springer-Verlag, 1977. Zbl0367.14001MR463157
- [5] J. Neukirch, Algebraische Zahlentheorie. Springer-Verlag, 1992. Zbl0747.11001

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