Counting $D_4$-field extensions by multi-invariants

par Anna Zanoli (IST Austria)

lundi 8 déc. 2025, 11:00 → 12:00 Europe/Paris

Salle Yvette Cauchois (IHP - Bâtiment Perrin)

In joint work with W. Hansen, we count the number of Galois extensions $M/\mathbb{Q}$ with fixed Galois group $\text{Gal}(M/\mathbb{Q})=D_4$ ordered by multi-invariants introduced by Gundlach. Our results verify the asymptotic behavior predicted by Gundlach’s refinement of Malle’s conjecture, and we further compare the leading constant to recent predictions of Loughran and Santens.
In our work, we combine a parametrization of $D_4$-extensions as quadratic extensions of biquadratic fields with analytic techniques for handling character sums. This approach allows us to adapt the methods of Friedlander–Iwaniec and of Rome to the setting of $D_4$-octic fields, ultimately leading to a precise asymptotic and an explicit description of the leading constant.