Arithmetical Properties of Polynomials Associated with the Lemniscate Elliptic Functions

The arithmetical properties of the polynomials Pµ closely parallel the properties of Lucas' Un. The main new feature of interest (not occurring in the real multiplication case) is a genuine double numerical periodicity when the free variables z and w are given fixed values in G, and the residues of the resulting sequence in G are considered for moduli in G. Indeed Lucas claimed in his fundamental paper and elsewhere to have discovered doubly periodic numerical functions connected with the elliptic functions, but he apparently published nothing on this subject. [4]