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Abstract

We prove many cases of the Inverse Galois Problem for those simple groups arising from orthogonal groups over finite fields. For example, we show that the finite simple groups $\Omega _{2n+1}(p)$ and $\operatorname {P}\!\Omega _{4n}^+(p)$ both occur as the Galois group of a Galois extension of the rationals for all integers $n\geq 2$ and all primes $p\geq 5$. We obtain our representations by studying families of twists of elliptic curves and using some known cases of the Birch and Swinnerton-Dyer conjecture along with a big monodromy result of Hall.

Locations

• Transactions of the American Mathematical Society Series B - View - PDF