AI 中文总结
该论文研究d>4的单参数超椭圆族的约化性质，证明扎里斯基一般系数对应的超椭圆族在大素数处有一般平凡约化，且平凡素数的自然密度在基变换后为1。
AI 中文摘要
设d=2g+1>4，A^{d-1}是参数化超椭圆族C_a: y^2 = x^d + a_{d-1}x^{d-1} +... + a_1x+t在t直线上的系数空间。我们证明，对于\bar{Z}^{d-1}中扎里斯基一般的系数向量a=(a_1,...,a_{d-1})，每个足够大的素数p，C_a在p上方的每个素数处都有一般平凡约化。对于\bar{Z}^{d-1}中的任意a，Q(a)(t)上超椭圆曲线C: y^2=x^d+a_{d-1}x^{d-1}+...+a_1x+t的平凡素数集合的自然密度至少为1/[Q(a,ζ_d):Q(a)]；在基变换到Q(a,ζ_d)(t)后，其自然密度为1。
英文摘要
Let d=2g+1>4 and let A^{d-1} be the coefficient space parametrizing hyperelliptic families C_a: y^2 = x^d + a_{d-1}x^{d-1} + ... + a_1x+t over the t-line. We show that for a Zariski-generic coefficient vector a=(a_1,...,a_{d-1}) in \bar{Z}^{d-1}, for every prime p large enough, C_a has generically ordinary reduction at every prime above p. For arbitrary a in \bar{Z}^{d-1}, the set of ordinary primes of the hyperelliptic curve C: y^2=x^d+a_{d-1}x^{d-1}+...+a_1x+t over Q(a)(t) has natural density at least 1/[Q(a,ζ_d):Q(a)]; after base change to Q(a,ζ_d)(t), it has natural density 1.
Comments16 pages
