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For every finite field F and every positive integer r, there exists a finiteextension F' of F such that either SO(2r+1,F') or its simple derived group canbe realized as a Galois group over Q. If the characteristic of F is 3 or 5 (mod8), then we can guarantee that the derived group of SO(2r+1,F') can berealized. Likewise, for every finite field F, there exists a finite extensionF' of F such that the finite simple group G_2(F') can be realized a Galoisgroup over Q.  The proof uses automorphic forms to construct Galois representations whichcut out Galois extensions of the desired type.

Functoriality and the Inverse Galois problem II: groups of type B_n and G_2