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Erschienen in:
Raynaud’s group-scheme and reduction of coverings Kapitel lesen Erstes Kapitel lesen

2012 | OriginalPaper | Buchkapitel

Uniform estimates for primitive divisors in elliptic divisibility sequences

verfasst von : Patrick Ingram, Joseph H. Silverman

Erschienen in: Number Theory, Analysis and Geometry

Verlag: Springer US

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Abstract

Let P be a nontorsion rational point on an elliptic curve E, given by a minimal Weierstrass equation, and write the first coordinate of nP as A n ∕ D n 2, a fraction in lowest terms. The sequence of values D n is the elliptic divisibility sequence (EDS) associated to P. A prime p is a primitive divisor of D n if p divides D n , and p does not divide any earlier term in the sequence. The Zsigmondy set for P is the set of n such that D n has no primitive divisors. It is known that Z is finite. In the first part of the paper we prove various uniform bounds for the size of the Zsigmondy set, including (1) if the j-invariant of E is integral, then the size of the Zsigmondy set is bounded independently of E and P, and (2) if the abc Conjecture is true, then the size of the Zsigmondy set is bounded independently of E and P for all curves and points. In the second part of the paper, we derive upper bounds for the maximum element in the Zsigmondy set for points on twists of a fixed elliptic curve.

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Vorheriges Kapitel On the local divisibility of Heegner points
Nächstes Kapitel The heat kernel, theta inversion and zetas on Г∖G∕K
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Metadaten
Titel
Uniform estimates for primitive divisors in elliptic divisibility sequences
verfasst von
Patrick Ingram
Joseph H. Silverman
Copyright-Jahr
2012
Verlag
Springer US
Buch
Number Theory, Analysis and Geometry

Print ISBN: 978-1-4614-1259-5

Electronic ISBN: 978-1-4614-1260-1

Copyright-Jahr: 2012

DOI
https://doi.org/10.1007/978-1-4614-1260-1_12