论文标题
来自Cartier-Manin矩阵的PICARD曲线的L-PolyNomials
Computing L-Polynomials of Picard curves from Cartier-Manin matrices
论文作者
论文摘要
我们研究通用Picard曲线的Zeta函数$ z（C_P，T）$ c：y^3 = f（x）$在$ \ mathbb {q} $上定义的Primes $ p $ for $ c $的primes $ p $。我们定义一个度9多项式$ψ_f\ in \ mathbb {q} [x] $，使得$ψ_f（x^3/2）$的分裂字段是$ c $的jacobian的$ 2 $ torsion字段。我们证明，除了零密度零子集外，Zeta函数$ z（C_P，T）$由Cartier-Manin Matrix $ a_p $ a_p $ a_p $ c $ c $ modulo $ p $和拆卸行为modulo modulo $ p $ p $ f $ f $ f $ and $ c $;我们还表明，对于Primes $ \ equiv 1 \ pmod {3} $，矩阵$ a_p $足够，而对于Primes $ \ equiv 2 \ pmod {3} $，$ C $上的通用假设是不必要的。可能具有独立关注的证据元素是确定普通PICARD曲线普通降低素数的密度。通过将其与Sutherland的最新工作相结合，我们获得了一种实用的确定性算法，该算法使用$ n \ log（n）^{3+o（1）} $ bit操作计算几乎所有Primes $ p \ le n $计算$ z（c_p，t）$。这是大于2的属曲线的第一个实际结果。
We study the sequence of zeta functions $Z(C_p,T)$ of a generic Picard curve $C:y^3=f(x)$ defined over $\mathbb{Q}$ at primes $p$ of good reduction for $C$. We define a degree 9 polynomial $ψ_f\in \mathbb{Q}[x]$ such that the splitting field of $ψ_f(x^3/2)$ is the $2$-torsion field of the Jacobian of $C$. We prove that, for all but a density zero subset of primes, the zeta function $Z(C_p,T)$ is uniquely determined by the Cartier-Manin matrix $A_p$ of $C$ modulo $p$ and the splitting behavior modulo $p$ of $f$ and $ψ_f$; we also show that for primes $\equiv 1 \pmod{3}$ the matrix $A_p$ suffices and that for primes $\equiv 2 \pmod{3}$ the genericity assumption on $C$ is unnecessary. An element of the proof, which may be of independent interest, is the determination of the density of the set of primes of ordinary reduction for a generic Picard curve. By combining this with recent work of Sutherland, we obtain a practical deterministic algorithm that computes $Z(C_p,T)$ for almost all primes $p \le N$ using $N\log(N)^{3+o(1)}$ bit operations. This is the first practical result of this type for curves of genus greater than 2.
