Title: Counting $D_4$ extensions by multiple invariants
Abstract: Malle’s conjecture is a central question in arithmetic statistics, predicting an asymptotic formula for the number of number fields with a prescribed Galois group and bounded discriminant. Recently, such counting problems have received considerable attention. In 2022 Gundlach introduced the notion of ordering number fields by multiple height functions instead of by discriminant. He gives a conjectural asymptotic for this counting problem and proves it for all abelian groups.
We establish a variant of Gundlach’s conjecture for the dihedral group of order 8, D4. We do so, by giving a parametrization of D4 octics, interpreting the multi-heights in this parametrization and using tools from analytic number theory to deal with the resulting counting problem. Finally, we discuss the role of the leading constant in regards to recent work of Loughran and Santens and the connection to Manin's conjecture. This is joint work with Anna Zanoli.