The size function h 0 for quadratic number fields

Paolo Francini

Journal de théorie des nombres de Bordeaux (2001)

  • Volume: 13, Issue: 1, page 125-135
  • ISSN: 1246-7405

Abstract

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We study the quadratic case of a conjecture made by Van der Geer and Schoof about the behaviour of certain functions which are defined over the group of Arakelov divisors of a number field. These functions correspond to the standard function h 0 for divisors of algebraic curves and we prove that they reach their maximum value for principal Arakelov divisors and nowhere else. Moreover, we consider a function k 0 ˜ , which is an analogue of exp h 0 defined on the class group, and we show it also assumes its maximum at the trivial class.

How to cite

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Francini, Paolo. "The size function $h^0$ for quadratic number fields." Journal de théorie des nombres de Bordeaux 13.1 (2001): 125-135. <http://eudml.org/doc/248708>.

@article{Francini2001,
abstract = {We study the quadratic case of a conjecture made by Van der Geer and Schoof about the behaviour of certain functions which are defined over the group of Arakelov divisors of a number field. These functions correspond to the standard function $h^0$ for divisors of algebraic curves and we prove that they reach their maximum value for principal Arakelov divisors and nowhere else. Moreover, we consider a function $\widetilde\{k^0\}$, which is an analogue of exp $h^0$ defined on the class group, and we show it also assumes its maximum at the trivial class.},
author = {Francini, Paolo},
journal = {Journal de théorie des nombres de Bordeaux},
language = {eng},
number = {1},
pages = {125-135},
publisher = {Université Bordeaux I},
title = {The size function $h^0$ for quadratic number fields},
url = {http://eudml.org/doc/248708},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Francini, Paolo
TI - The size function $h^0$ for quadratic number fields
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 1
SP - 125
EP - 135
AB - We study the quadratic case of a conjecture made by Van der Geer and Schoof about the behaviour of certain functions which are defined over the group of Arakelov divisors of a number field. These functions correspond to the standard function $h^0$ for divisors of algebraic curves and we prove that they reach their maximum value for principal Arakelov divisors and nowhere else. Moreover, we consider a function $\widetilde{k^0}$, which is an analogue of exp $h^0$ defined on the class group, and we show it also assumes its maximum at the trivial class.
LA - eng
UR - http://eudml.org/doc/248708
ER -

References

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  1. [1] W. Banaszczyk, New bounds in some transference theorems in the geometry of numbers. Math. Ann.296 (1993), 625-635. Zbl0786.11035MR1233487
  2. [2] A. Borisov, Convolution structures and arithmetic cohomology. Math. AG/9807151 at http://xxx.lanl.gov, 1998. Zbl1158.11340
  3. [3] G. Van Der Geer, R. Schoof, Effectivity of Arakelov divisors and the theta divisor of a number field. Math. AG/9802121 at http://xxx.lanl.gov, 1999. Zbl1030.11063MR1847381
  4. [4] R.P. Groenewegen, The size function for number fields. Doctoraalscriptie, Universiteit van Amsterdam, 1999. 
  5. [5] H.E. Rose, A course in number theory. Oxford University Press, 1988. Zbl0637.10002MR952859

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