Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields
Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields
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全局函数域上广义勒让德曲线雅可比行列式的显式算术
DOI：
10.1090/memo/1295
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Ulmer, Douglas
中科院分区：
文献类型：
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作者：
Berger, Lisa;Hall, Chris;Pannekoek, René;Park, Jennifer;Pries, Rachel;Sharif, Shahed;Silverberg, Alice;Ulmer, Douglas
我们研究光滑投影曲线C的雅可比矩阵J属与仿射模型r−1年= xr−1 (x + 1) (x + t)在功能领域Fp (t),当p是质数和r≥2是一个整数' p。当q是一个p和d是一个正整数,计算J的l函数Fq (t1 / d)和显示,桦树和Swinnerton-Dyer猜想适用于J / Fq (t1 / d)。当d能被r整除，且形式为pν+ 1，且Kd:= Fp （μd, t1/d）时，我们写出了J （Kd）上的显式点，证明了它们生成了秩为（r−1）（d−2）的子群V，其在J （Kd）上的索引是有限的p的幂，并证明了J / Kd上的Tate-Shafarevich群的阶为[J (Kd): V] 2。当r> 2时，我们证明了J的“新”部分在Fp (t)上是具有自同态代数Z [μr]+的φ (r)/2维的简单阿贝尔变的平方的同构。对于具有l pr的素数l，证明了J [l](l)对Fp (t)的任意阿贝尔扩展l ={0}。
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≥ 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 pν+ 1, and Kd:= Fp (μd, 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 φ (r)/2 with endomorphism algebra Z [μr]+. For a prime l with l pr, we prove that J [l](L)={0} for any abelian extension L of Fp (t).