Progress towards counting D5 quintic fields

Larson, Eric; Rolen, Larry

doi:10.2140/involve.2012.5.91

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Progress towards counting $D_5$ quintic fields

Eric Larson and Larry Rolen

Vol. 5 (2012), No. 1, 91–97

DOI: 10.2140/involve.2012.5.91

Abstract

Let $N (5, D_{5}, X)$ be the number of quintic number fields whose Galois closure has Galois group $D_{5}$ and whose discriminant is bounded by $X$ . By a conjecture of Malle, we expect that $N (5, D_{5}, X) \sim C \cdot X^{\frac{1}{2}}$ for some constant $C$ . The best upper bound currently known is $N (5, D_{5}, X) ≪ X^{\frac{3}{4} + ε}$ , and we show this could be improved by counting points on a certain variety defined by a norm equation; computer calculations give strong evidence that this number is $≪ X^{\frac{2}{3}}$ . Finally, we show how such norm equations can be helpful by reinterpreting an earlier proof of Wong on upper bounds for $A_{4}$ quartic fields in terms of a similar norm equation.

Keywords

quintic dihedral number fields, Cohen–Lenstra heuristics for $p = 5$

Mathematical Subject Classification 2010

Primary: 11R45

Secondary: 11R29, 14G05

Milestones

Received: 20 July 2011

Accepted: 4 August 2011

Published: 28 April 2012

Communicated by Ken Ono

Authors

Eric Larson
Department of Mathematics Harvard Cambridge, MA 02138 United States

Larry Rolen
Department of Mathematics and Computer Science Emory University Atlanta, GA 30322 United States

Vol. 5, No. 1, 2012