ON THE MALLE CONJECTURE AND THE GRUNWALD PROBLEM

Abstract : 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)| ≤ 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) ≥ y α(G) for y 1 and some specific exponent α(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.
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https://hal.archives-ouvertes.fr/hal-01965547
Contributor : François Motte <>
Submitted on : Monday, February 25, 2019 - 12:05:04 PM
Last modification on : Saturday, March 2, 2019 - 1:22:32 AM

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  • HAL Id : hal-01965547, version 5
  • ARXIV : 1812.11376

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François Motte. ON THE MALLE CONJECTURE AND THE GRUNWALD PROBLEM. 2019. ⟨hal-01965547v5⟩

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