Matrix divisibility sequences

Abstract

The plan for the paper is as follows: we shall first introduce the general notion of a matrix divisibility sequence indexed by a semigroup. Then we will see how a faithful representation of the semigroup by endomorphisms of an affine space gives rise to a matrix divisibility sequence, by considering the Jacobian matrices of the endomorphisms. We will show how most of the commonly known divisibility sequences (mentioned brie y above) arise as determinants of matrix divisibility sequences through interesting semigroups of endomorphisms of affine spaces, often associated to a representation of addition in an algebraic group. For example, Lucas sequences are associated to the 2 x 2 Borel group. We also construct the elliptic matrix divisibility sequence that underlies the usual elliptic divisibility sequences, and prove that it has primitive right matrix divisor classes.