Progress towards counting $D_5$ quintic fields

Abstract

Let $N\left(5,D_5,X\right)$ 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 $N\left(5,D_5,X\right)\sim C\cdot X^{\frac{1}{2}}$ for some constant $C$. The best upper bound currently known is $N\left(5,D_5,X\right)\ll X^{\frac{3}{4}+\epsilon }$, 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 $\ll X^{\frac{2}{3}}$. 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.