| Let C be the hyperelliptic curve of genus g given by the affine model y2 = x q + ¼ where q = 2g + 1 is prime, and let J be its Jacobian variety. Then J has complex multiplication by O , the ring of integers in the qth cyclotomic field. For alpha ∈ O , we let [alpha] denote the corresponding endomorphism of J. We define cyclotomic division polynomials psialpha( u) for alpha ∈ O in terms of theta functions attached to J that vanish at points Q ∈ C ⊆ J exactly when [alpha]*Q is on the theta divisor of J. We will use a recent generalized version of a formula of Frobenius and Stickelberger to derive special values of these polynomials, and use these to provide a product formula that generalizes a formula of Eisenstein for elliptic curves relating to elliptic units. We will then relate our formula to Jacobi sums. Additionally, the theory of formal groups will be used to show that the factors in the product produce a new class of S-units attached to the hyperelliptic curve. Finally, an algebraic definition of division polynomials psin( u) for n ∈ Z will be given and the lead and constant term of these will be computed algebraically in the hopes of extending this algebraic definition to cyclotomic division polynomials and providing an algebraic proof of the product formula. |
