Explicit arithmetic of Jacobians of generalized Legendre curves over global function fields

2015/04/30 by Berger, Lisa, Hall, Chris, Pannekoek, René +5
#11G10 #11G30 (primary) #11G40 #14G05 #14G25 #14K15 (secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1505.00021

Abstract

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 \mathbbF_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 \mathbbF_q(t^1/d) and show that the Birch and Swinnerton-Dyer conjecture holds for J over \mathbbF_q(t^1/d). When d is divisible by r and of the form p^ν+1, and K_d := \mathbbF_p(μ_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 \mathbbF_p(t) to the square of a simple abelian variety of dimension ϕ(r)/2 with endomorphism algebra ℤ[μ_r]⁺. For a prime ℓ with ℓ \nmid pr, we prove that J[ℓ](L)=\0\ for any abelian extension L of \mathbbF_p(t).

Related

• Explicit points on the Legendre curve
• On the arithmetic of a family of superelliptic curves
• Monodromy groups and exceptional Hodge classes, I: Fermat Jacobians
• Hyperelliptic jacobians without complex multiplication, doubly transitive permutation groups and projective representations
• Arithmetic of abelian varieties in Artin-Schreier extensions