## Duke Mathematical Journal

### Hilbert irreducibility above algebraic groups

Umberto Zannier

#### Abstract

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 ${\mathbb G}_{\rm m}^n$ (Theorem 1) and of a power $E^n$ 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 ${\mathbb G}_{\rm m}^n$ 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

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

Dates
First available in Project Euclid: 26 May 2010

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1274902084

Digital Object Identifier
doi:10.1215/00127094-2010-027

Mathematical Reviews number (MathSciNet)
MR2667137

Zentralblatt MATH identifier
1208.11080

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

#### Citation

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

#### References

• D. Bertrand, Kummer theory on the product of an elliptic curve by the multiplicative group, Glasgow Math. J. 22 (1981), 83--88.
• E. Bombieri and W. Gubler, Heights in Diophantine Geometry, Cambridge Univ. Press, Cambridge, 2006.
• J.-L. Colliot-ThéLèNe and J.-J. Sansuc, Principal homogeneous spaces under Flasque tori: applications, J. Algebra 106 (1987), 148--205.
• P. Corvaja, Rational fixed points for linear group actions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), 561--597.
• P. Corvaja and U. Zannier, Some new applications of the subspace theorem, Compositio Math. 131 (2002), 319--340.
• —, On the integral points on certain surfaces, Int. Math. Res. Not. 2006, 98623.
• —, Some cases of Vojta's conjecture on integral points over function fields, J. Algebraic Geom. 17 (2008), 295--333.
• P. DèBes, On the irreducibility of the polynomials $P(t\sp m,Y)$, J. Number Theory 42 (1992), 141--157.
• R. Dvornicich and U. Zannier, Cyclotomic diophantine problems (Hilbert irreducibility and invariant sets for polynomial maps), Duke Math. J. 139 (2007), 527--554.
• A. Ferretti and U. Zannier, Equations in the Hadamard ring of rational functions, Ann. Scu. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), 457--475.
• M. Fried, On Hilbert's irreducibility theorem, J. Number Theory 6 (1974), 211--231.
• M. Fried and M. Jarden, Field arithmetic, Springer, Berlin, 2008.
• A. E. Ingham, The Distribution of Prime Numbers, Cambridge Univ. Press, Cambridge, 1932, reprinted 1992.
• S. L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287--297.
• S. Lang, Elliptic curves, diophantine analysis, Springer, Berlin, 1978.
• —, Number Theory, III: Diophantine Geometry, Enc. of Math. Sci. 60, Springer, Berlin, 1991.
• M. Raynaud, Courbes sur une variété abélienne et points de torsion, Invent. Math. 71 (1983), 207--233.
• A. Schinzel, On Hilbert's irreducibility theorem, Ann. Polon. Math. 16 (1965), 333--340.
• —, An analogue of Hilbert's irreducibility theorem, Number Theory (Banff, AB, 1988), 509--514., de Gruyter, Berlin, 1990.
• —, Polynomials with special regard to reducibility, Cambridge Univ. Press, Cambridge, 2000.
• J.-P. Serre, Corps locaux, Hermann, Paris, 1968.
• —, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259--331.
• —, Lectures on the Mordell-Weil Theorem, $2$nd ed., Vieweg, Braunschweig, 1989.
• —, Topics in Galois Theory, Jones and Bartlett, Boston, 1992.
• —, On a theorem of Jordan, Bull. Amer. Math. Sec. (N.S.) 40 (2003), 429--440.
• J. Silverman, The Arithmetic of Elliptic Curves, Grad. Texts Math. 106, Springer, New York, 1986.
• U. Zannier, A proof of Pisot $d$th root conjecture, Ann. of Math. 2 (2000), 375--383.