# The size function ${h}^{0}$ for quadratic number fields

Journal de théorie des nombres de Bordeaux (2001)

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

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topFrancini, 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

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

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