An arithmetic analogue of Clifford's theorem

Richard P. Groenewegen

Journal de théorie des nombres de Bordeaux (2001)

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

Abstract

top
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 on metrized line bundles that resembles properties of the dimension of , where is a divisor on a curve . 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.

How to cite

top

Groenewegen, 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. [1] P. Francini, The function h° for quadratic number fields. These proceedings. Zbl1060.11076
  2. [2] W. Fulton, Algebraic Curves. Addison Wesley, 1989. Zbl0681.14011MR1042981
  3. [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. [4] R. Hartshorne, Algebraic Geometry. Springer-Verlag, 1977. Zbl0367.14001MR463157
  5. [5] J. Neukirch, Algebraische Zahlentheorie. Springer-Verlag, 1992. Zbl0747.11001

NotesEmbed ?

top

You must be logged in to post comments.