Hyperelliptic curves

Consider two families of hyperelliptic curves (over Q) C^n,a : y^2 = x^n + a and C_n,a : y^2 = x(x^n + a), and their Jacobians J^n,a, J_n,a respectively. We give the partial characterization of the torsion part of J^n,a (Q) and J_n,a (Q). More precisely, we show that the only prime factors of the orders of such groups are 2 and prime divisors of n (we also give the upper bounds for the exponents). Moreover, we give the complete description of the torsion part of J_8,a (Q). Namely, we show that J_8,a (Q)tors = J_8,a (Q) [2]. In addition, we characterize the torsion parts of J_p,a (Q), where p is an odd prime, and of J^n,a (Q), where n = 4, 6, 8. The main ingredients in the proofs are explicit computations of zeta functions of the title curves in some cases, which are of independent interest, and applications of the Chebotarev Density Theorem.

Abstract: In this paper we show how polynomial mappings of degree $\ mathfrak {K} $ from a union of disjoint intervals onto $[-1, 1] $ generate a countable number of special cases of generalizations of Chebyshev polynomials. We also derive a new expression for these generalized Chebyshev polynomials for any genus $ g $, from which the coefficients of $ x^ n $ can be found explicitly in terms of the branch points and the recurrence coefficients. We find that this representation is useful for specializing to polynomial mapping cases for ...