Abelian constraints in inverse Galois theory |
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Authors: | Anna Cadoret Pierre Dèbes |
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Institution: | (1) I.M.B., Université Bordeaux 1, 351 Cours de la Libération, 33405 Talence Cedex, France;(2) Laboratoire Paul Painlevé, Mathématiques, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France |
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Abstract: | We show that if a finite group G is the Galois group of a Galois cover of over , then the orders p
n
of the abelianization of its p-Sylow subgroups are bounded in terms of their index m, of the branch point number r and the smallest prime of good reduction of the branch divisor. This is a new constraint for the regular inverse Galois problem: if p
n
is suitably large compared to r and m, the branch points must coalesce modulo small primes. We further conjecture that p
n
should be bounded only in terms of r and m. We use a connection with some rationality question on the torsion of abelian varieties. For example, our conjecture follows
from the so-called torsion conjectures. Our approach also provides a new viewpoint on Fried’s Modular Tower program and a
weak form of its main conjecture. |
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Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) Primary 12F12 14H30 11Gxx Secondary 14G32 14Kxx 14H10 |
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