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The size function \(h^0\) for quadratic number fields. (English) Zbl 1060.11076

Summary: We study the quadratic case of a conjecture made by G. van der Geer and R. Schoof [Sel. Math., New Ser. 6, No. 4, 377–398 (2000; Zbl 1030.11063)] 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.

MSC:

11R47 Other analytic theory
11R04 Algebraic numbers; rings of algebraic integers

Citations:

Zbl 1030.11063
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References:

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[3] Van Der Geer, G., Schoof, R., Effectivity of Arakelov divisors and the theta divisor of a number field. Math. AG/9802121 at http://xxx.lanl.gov, 1999. · Zbl 1030.11063
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