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Primitive divisors of elliptic divisibility sequences over function fields with constant $$j$$-invariant. (English) Zbl 07202823
Summary: We prove an optimal Zsigmondy bound for elliptic divisibility sequences over function fields in case the $$j$$-invariant of the elliptic curve is constant.
In more detail, given an elliptic curve $$E$$ with a point $$P$$ of infinite order over a global field, the sequence $$D_1, D_2, \ldots$$ of denominators of multiples $$P, 2P, \ldots$$ of $$P$$ is a strong divisibility sequence in the sense that $$\gcd( D_m, D_n) = D_{\gcd (m, n)}$$. This is the genus-one analogue of the genus-zero Fibonacci, Lucas and Lehmer sequences.
A number $$N$$ is called a Zsigmondy bound of the sequence if each term $$D_n$$ with $$n > N$$ presents a new prime factor. The optimal uniform Zsigmondy bound for the genus-zero sequences over Q is 30 by Bilu-Hanrot-Voutier [2], but finding such a bound remains an open problem in genus one, both over Q and over function fields.
We prove that the optimal Zsigmondy bound for ordinary elliptic divisibility sequences over function fields is 2 if the $$j$$-invariant is constant. In the supersingular case, we give a complete classification of which terms can and cannot have a new prime factor.
##### MSC:
 11G05 Elliptic curves over global fields 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 14H52 Elliptic curves 11G07 Elliptic curves over local fields 11B83 Special sequences and polynomials
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