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A Lower Bound for the Height of a Rational Function atS-unit Points

✍ Scribed by Pietro Corvaja; Umberto Zannier

Publisher
Springer Vienna
Year
2004
Tongue
English
Weight
220 KB
Volume
144
Category
Article
ISSN
0026-9255

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We produce an absolute lower bound for the height of the algebraic numbers (different from zero and from the roots of unity) lying in an abelian extension of the rationals. The proof rests on elementary congruences in cyclotomic fields and on Kronecker Weber theorem.

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Let E=K be an elliptic curve defined over a number field, let Δ₯ be the canonical height on E; and let K ab =K be the maximal abelian extension of K: Extending work of M. Baker (IMRN 29 (2003) 1571-1582), we prove that there is a constant CΓ°E=KÞ40 so that every nontorsion point PAEΓ°K ab Þ satisfies Δ₯