On the Malle conjecture and the Grunwald problem

We contribute to the Malle conjecture on the number N (K, G, y) of finite Galois extensions E of some number field K of finite group G and of discriminant of norm |N K/Q (d E)| $\le$ y. We prove the lower bound part of the conjecture for every group G and every number field K containing a certain number field K 0 depending on G : N (K, G, y) $\ge$ y $\alpha$(G) for y 1 and some specific exponent $\alpha$(G) depending on G. To achieve this goal, we start from a regular Galois extension F/K(T) that we specialize. We prove a strong version of the Hilbert Irreducibility Theorem which counts the number of specialized extensions F t0 /K and not only the specialization points t 0 , and which provides some control of |N K/Q (d Ft 0)|. We can also prescribe the local behaviour of the specialized extensions at some primes. Consequently, we deduce new results on the local-global Grunwald problem, in particular for some non-solvable groups G.