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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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