论文标题
在$ \ mathbb {f} _p $ - 希尔伯特类多项式模型的根
On $\mathbb{F}_p$-roots of the Hilbert class polynomial modulo $p$
论文作者
论文摘要
The Hilbert class polynomial $H_{\mathcal{O}}(x)\in \mathbb{Z}[x]$ attached to an order $\mathcal{O}$ in an imaginary quadratic field $K$ is the monic polynomial whose roots are precisely the distinct $j$-invariants of elliptic curves over $ \ mathbb {c} $带有$ \ mathcal {o} $的复杂乘法。令$ p $为$ k $中的Prime惰性，严格大于$ | \ operatatorName {disc}（\ Mathcal {o}）| $。我们表明$ \ mathbb {f} _p $ - $ h_ \ mathcal {o}（x）\！\！的数量\ pmod {p} $是零或$ | \ operatorName {pic}（\ Mathcal {o}）[2] | $，通过展示$ \ operatotorname {pic}（\ nathcal {o}）的免费和传递动作，对$ \ mathbbbb {f - _p a_p， $ H_ \ Mathcal {O}（x）\！\！ \ pmod p $无论何时是非空的。我们还为$ \ mathbb {f} _p $ -roots的非空置提供了具体标准。 Xiao等人首先获得了类似的结果。 J.数字理论，doi：10.1142/s1793042122500555]，并通过Li等人〜[Arxiv：2108.00168]（以不同的方法涵盖当前的结果）进行了更大的概括。
The Hilbert class polynomial $H_{\mathcal{O}}(x)\in \mathbb{Z}[x]$ attached to an order $\mathcal{O}$ in an imaginary quadratic field $K$ is the monic polynomial whose roots are precisely the distinct $j$-invariants of elliptic curves over $\mathbb{C}$ with complex multiplication by $\mathcal{O}$. Let $p$ be a prime inert in $K$ and strictly greater than $|\operatorname{disc}(\mathcal{O})|$. We show that the number of $\mathbb{F}_p$-roots of $H_\mathcal{O}(x)\!\! \pmod{p}$ is either zero or $|\operatorname{Pic}(\mathcal{O})[2]|$ by exhibiting a free and transitive action of $\operatorname{Pic}(\mathcal{O})[2]$ on the set of $\mathbb{F}_p$-roots of $H_\mathcal{O}(x)\!\! \pmod p$ whenever it is nonempty. We also provide a concrete criterion for the nonemptiness of the set of $\mathbb{F}_p$-roots. A similar result was first obtained by Xiao et al.~[Int. J. Number Theory, DOI: 10.1142/S1793042122500555] and generalized much further by Li et al.~[arXiv:2108.00168] (that covers the current result) with a different approach.
