Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields
Author | : Lisa Berger |
Publisher | : |
Total Pages | : |
Release | : 2020 |
ISBN-10 | : 1470462532 |
ISBN-13 | : 9781470462536 |
Rating | : 4/5 (536 Downloads) |
Download or read book Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields written by Lisa Berger and published by . This book was released on 2020 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: "We study the Jacobian J of the smooth projective curve C of genus r-1 with affine model yr = xr-1(x+ 1)(x + t) over the function field Fp(t), when p is prime and r [greater than or equal to] 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 Fq(t1/d) and show that the Birch and Swinnerton-Dyer conjecture holds for J over Fq(t1/d). When d is divisible by r and of the form p[nu] + 1, and Kd := Fp([mu]d, t1/d), we write down explicit points in J(Kd), show that they generate a subgroup V of rank (r-1)(d-2) whose index in J(Kd) is finite and a power of p, and show that the order of the Tate-Shafarevich group of J over Kd is [J(Kd) : V ]2. When r > 2, we prove that the "new" part of J is isogenous over Fp(t) to the square of a simple abelian variety of dimension [phi](r)/2 with endomorphism algebra Z[[mu]r]+. For a prime with pr, we prove that J[](L) = {0} for any abelian extension L of Fp(t)"--