Geometric Aspects of Lucas Sequences, I

Abstract

We present a way of viewing Lucas sequences in the framework of group scheme theory. This enables us to treat the Lucas sequences from a geometric and functorial viewpoint, which was suggested by Laxton ⟨On groups of linear recurrences, I⟩ and by Aoki-Sakai ⟨Mod p equivalence classes of linear recurrence sequences of degree two⟩. Introduction The Lucas sequences, including the Fibonacci sequence, have been studied widely for a long time, and there is left an enormous accumulation of research. Particularly the divisibility problem is a main subject in the study on Lucas sequences. More explicitly, let P and Q be non-zero integers, and let (wk)k≥0 be the sequence defined by the linear recurrence relation wk+2 = Pwk+1 −Qwk with the intial terms w0, w1 ∈ Z. If w0 = 0 and w1 = 1, then (wk)k≥0 is nothing but the Lucas sequnces (Lk)k≥0 associated to (P,Q). The divisibility problem asks to describe the set {k ∈ N ; wk ≡ 0 mod m} for a positive integer m. The first step was certainly taken forward by Edouard Lucas [6] as the laws of apparition and repetition in the case where m is a prime number and (wk)k≥0 is the Lucas sequence, and there have been piled up various kinds of results after then. In this article we study the divisibility problem for Lucas sequences from a geometirc viewpoint, translating several descriptions on Lucas sequences into the language of affine group schemes. For example, the laws of apparition and repetition is formulated in our context as follows: Theorem(=Proposition 3.23+Theorem 3.25) Let P and Q be non-zero integers with (P,Q) = 1, and let w0, w1 ∈ Z with (w0, w1) = 1. Define the sequence (wk)k≥0 by the recurrence relation wk+2 = Pwk+1−Qwk with initial terms w0 and w1, and put μ = ordp(w 1−Pw0w1+Qw 0). Let p be an odd prime with (p,Q) = 1 and n a positive integer. Then we have the length of the orbit (w0 : w1)Θ in P(Z/pZ) = 1 (n ≤ μ) r(pn−μ) (n > μ) . Furthermore, there exists k ≥ 0 such that wk ≡ 0 mod pn if and only if (w0 : w1) ∈ (0 : 1).Θ in P1(Z/pnZ). Here Θ denotes the subgroup of G(D)(Z(p)) generated by β(θ) = (P/4Q, 1/4Q), and r(pν) denotes the rank mod pν of the Lucas sequence associated to (P,Q). ∗) Partially supported by Grant-in-Aid for Scientific Research No.26400024 2010 Mathematics Subject Classification Primary 13B05; Secondary 14L15, 12G05. 1