Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields

We study the Jacobian J of the smooth projective curve C of genus r -1 with affine model y(r) = x(r -1) (x + 1) (x + t) over the function field F-p(t), when p is prime and r >= 2 is an integer prime to p. When q is a power of p and d is a positive integer, we compute the L-function of J over F-q(t(1/d)) and show that the Birch and Swinnerton-Dyer conjecture holds for J over F-q(t(1/d)). When d is divisible by r and of the form p(nu) + 1, and K-d := F-p(mu(d), t(1/d)), we write down explicit points in J(K-d), show that they generate a subgroup V of rank (r - 1)(d - 2) whose index in J(K-d) is finite and a power of p, and show that the order of the Tate-Shafarevich group of J over K-d is [J(K-d) : V](2). When r > 2, we prove that the "new" part of J is isogenous over <(F-p(t))over bar> to the square of a simple abelian variety of dimension phi(r)/2 with endomorphism algebra Z[mu(r)](+). For a prime with l (sic) pr, we prove that J[l](L) = {0} for any abelian extension L of (F) over bar (p)(t).