Sato-Tate groups of y^2=x^8+c and y^2=x^7-cx
Sato-Tate groups of y^2=x^8+c and y^2=x^7-cx
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y^2=x^8 c 和 y^2=x^7-cx 的 Sato-Tate 群
DOI：
10.1090/conm/663/13351
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Andrew V. Sutherland
中科院分区：
文献类型：
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作者：
Francesc Fit'e;Andrew V. Sutherland
我们考虑了Q上两族具有较大自同构群的亏格为3的超椭圆曲线的归一化Frobenius迹的分布：y^2=x^8+c和y^2=x^7-cx，其中c在Q*中.我们给出了在良好约化素数下计算这些族中曲线的Frobenius迹的有效算法。利用由这些算法产生的数据，我们得到了出现的Sato-Tate群的启发式描述，无论是一般的还是对于c的特定值，然后我们通过Sato-Tate群和Galois自同态类型之间的对应来显式地计算Sato-Tate群来证明这些启发式描述是正确的。
We consider the distribution of normalized Frobenius traces for two families of genus 3 hyperelliptic curves over Q that have large automorphism groups: y^2=x^8+c and y^2=x^7-cx with c in Q*. We give efficient algorithms to compute the trace of Frobenius for curves in these families at primes of good reduction. Using data generated by these algorithms, we obtain a heuristic description of the Sato-Tate groups that arise, both generically and for particular values of c. We then prove that these heuristic descriptions are correct by explicitly computing the Sato-Tate groups via the correspondence between Sato-Tate groups and Galois endomorphism types.