Elliptic curves and related sequences

A Somos 4 sequence is a sequence (hn) of rational numbers defined by the quadratic recursion hm+2 hm−2 = λ1 hm+1 hm−1 + λ2 h 2 m for all m ∈ Z for some rational constants λ1, λ2. Elliptic divisibility sequences or EDSs are an important special case where λ1 = h 2 2, λ2 = −h1 h3, the hn are integers and hn divides hm whenever n divides m. Somos (4) is the particular Somos 4 sequence whose coefficients λi and initial values are all 1. In this thesis we study the properties of EDSs and Somos 4 sequences reduced modulo a prime power p. In chapter 2 we collect some results from number theory, and in chapter 3 we give a brief introduction to elliptic curves. In chapter 4 we introduce elliptic divisibility sequences, describe their relationship with elliptic curves, and outline what is known about the properties of an EDS modulo a prime power p (work by Morgan Ward and Rachel Shipsey). In chapter 5 we extend the EDS “symmetry formulae” of Ward and Shipsey to higher powers of p. We use this to find the period of (hn mod p ) in terms of the period of (hn mod p), confirming a conjecture by Shipsey for the r = 2 case. In chapter 6 we give an introduction to Somos 4 sequences and list several conjectures by Raphael Robinson on the modulo p periodicity properties of Somos(4). In chapter 7 we describe a recent result by Nelson Stephens relating a Somos 4 sequence (hn) to the sequence of points Q+ [n]P on an elliptic curve. We use this to prove that most of Robinson’s conjectures on the pattern of zeroes hold in the sequence (hn mod p ). In chapter 8 we consider prime powers p dividing some term of a given Somos 4 sequence (hn), and we find conditions under which Robinson’s periodicity conjectures hold for (hn mod p ) (although we do not always know if Somos(4) satisfies them). We do this by defining an equivalent sequence (`n), finding an EDS congruent to (`n) modulo p r and using our EDS results from chapter 4. Finally, in chapter 9 we use elliptic curves to prove that a weakened version of Robinson’s periodicity conjecture holds for prime powers which are coprime to λ1 and to every term of a given Somos 4 sequence.