Malle's conjecture for Galois octic fields over ℚ

Abstract We compute the asymptotic number of octic number fields whose Galois groups over ℚ are isomorphic to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>D</m:mi> <m:mn>4</m:mn> </m:msub> </m:math> D_{4} , the symmetries of a square, when ordering such fields by their absolute discriminants. In particular, we verify the strong form of Malle’s conjecture for such octic <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>D</m:mi> <m:mn>4</m:mn> </m:msub> </m:math> D_{4} -fields and obtain the constant of proportionality. We further demonstrate that the constant of proportionality satisfies the Malle–Bhargava principle of being a product of local masses, despite the fact that this principle does not hold for discriminants of quartic <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>D</m:mi> <m:mn>4</m:mn> </m:msub> </m:math> D_{4} -fields. This is the first instance of asymptotics being recovered for a non-concentrated family (in the sense of Alberts–Lemke Oliver–Wang–Wood) of number fields of Galois group neither abelian nor symmetric. Previously, this was only known for abelian fields, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>S</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> S_{n} -fields with degree 𝑛 for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>3</m:mn> <m:mo>,</m:mo> <m:mn>4</m:mn> <m:mo>,</m:mo> <m:mn>5</m:mn> </m:mrow> </m:mrow> </m:math> n=3,4,5 , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>S</m:mi> <m:mn>3</m:mn> </m:msub> </m:math> S_{3} -fields with degree 6.