Duke Mathematical Journal

Hilbert irreducibility above algebraic groups

Umberto Zannier

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This article concerns Hilbert irreducibility for covers of algebraic groups, with results which appear to be difficult to treat by existing techniques. The present method works by first studying irreducibility above “torsion” specializations (e.g., over cyclotomic extensions) and then descending the field (by Chebotarev theorem). Among the results, we offer an irreducibility theorem for the fibers, above a cyclic dense subgroup, of a cover of Gmn (Theorem 1) and of a power En of an elliptic curve without CM (Theorem 2); this had not been treated before for n>1. As a further application, in the function field context, we obtain a kind of Bertini's theorem for algebraic subgroups of Gmn in place of linear spaces (Theorem 3). Along the way we shall prove other results, as a general lifting theorem above tori (Theorem 3.1)

Article information

Duke Math. J., Volume 153, Number 2 (2010), 397-425.

First available in Project Euclid: 26 May 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11D99: None of the above, but in this section
Secondary: 11G35: Varieties over global fields [See also 14G25]


Zannier, Umberto. Hilbert irreducibility above algebraic groups. Duke Math. J. 153 (2010), no. 2, 397--425. doi:10.1215/00127094-2010-027. https://projecteuclid.org/euclid.dmj/1274902084

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