On reduced Arakelov divisors of real quadratic fields

Ha Thanh Nguyen Tran¹

¹Department of Mathematics and Systems Analysis Aalto University School of Science Otakaari 1, 02150 Espoo, Finland

Acta Arithmetica, Tome 173 (2016) no. 4, pp. 297-315

Résumé

We generalize the concept of reduced Arakelov divisors and define $C$-reduced divisors for a given number $C \geq 1$. These $C$-reduced divisors have remarkable properties, similar to the properties of reduced ones. We describe an algorithm to test whether an Arakelov divisor of a real quadratic field $F$ is $C$-reduced in time polynomial in $\log|\varDelta_F|$ with $\varDelta_F$ the discriminant of $F$. Moreover, we give an example of a cubic field for which our algorithm does not work.

Détail

DOI : 10.4064/aa8007-2-2016

Keywords: generalize concept reduced arakelov divisors define c reduced divisors given number geq these c reduced divisors have remarkable properties similar properties reduced describe algorithm test whether arakelov divisor real quadratic field c reduced time polynomial log vardelta vardelta discriminant moreover example cubic field which algorithm does work