Functoriality and the Inverse Galois problem II: groups of type B n and G 2

Abstract

This paper contains an application of Langlands’ functoriality principle to the following classical problem: which finite groups, in particular which simple groups appear as Galois groups over ℚ ? Let ℓ be a prime and t a positive integer. We show that that the finite simple groups of Lie type B n ( ℓ k ) = 3 D S O 2 n + 1 ( 𝔽 ℓ k ) d e r if ℓ ≡ 3 , 5 ( mod 8 ) and G 2 ( ℓ k ) appear as Galois groups over ℚ , for some k divisible by t . In particular, for each of the two Lie types and fixed ℓ we construct infinitely many Galois groups but we do not have a precise control of k .