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The inverse Galois problem for orthogonal groups

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arxiv 1409.1151 v1 pith:S3ASAQCG submitted 2014-09-03 math.NT

classification math.NT
keywords galoisgroupscasesfiniteinverseorthogonalproblemsimple
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We prove many new 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 POmega_{4n}^+(p) both occur as the Galois group of a Galois extension of the rationals for all integers n>1 and all primes p>3. 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.

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  1. Lifting $G$-irreducible but $\mathrm{GL}_n$-reducible Galois representations

    math.NT 2019-08 accept novelty 6.0 of 10

    For every even N at least 6, the paper produces infinitely many SO_{N+1}-valued residual Galois representations that are SO_{N+1}-irreducible but GL_{N+1}-reducible, each with a geometric lift of Zariski-dense image.

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