## 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

Zbl 1030.11063
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### References:

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