turkmath.org

The Inverse Galois problem asks whether every finite group G is the Galois group of a Galois extension of the field of rational numbers Q. The Malle conjecture is a quantitative version of it: it predicts the number of Galois extensions of Q (or any other number field) with the Galois group G of bounded "size" (such as the discriminant).  In this talk, we will present a generalization of the conjecture to finite étale group schemes. We show how the generalization helps explain inconsistencies of the Malle conjecture found by Klüners. We also discuss the case of the conjecture when G is a commutative finite étale group scheme, which generalizes the classical work of Wright on the number of abelian extensions of bounded discriminant. The talk is based on a joint work with Takehiko Yasuda.