The inverse Galois problem for orthogonal groups

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 ratio...

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Published in:Transactions of the American Mathematical Society. Series B 2023-08, Vol.10 (33), p.1173-1211
Main Author: Zywina, David
Format: Article
Language:English
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Research article
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Summary: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.
ISSN:2330-0000
2330-0000
DOI:10.1090/btran/124