无不可约小次数自同态的超奇异椭圆曲线
Supersingular Elliptic Curves Without Inseparable Small Degree Endomorphisms
- University of Waterloo（滑铁卢大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究将超奇异椭圆曲线的同源次数问题转化为二次型问题，利用Voronoi理论分析δ(p)的性质，得出大素数下δ(p)达上界的条件及几乎所有素数的δ(p)下界。
AI中文摘要：
对于固定特征p，令δ(p)表示保证任意超奇异曲线E到其弗罗贝尼乌斯共轭曲线E^(p)存在同源所需的最小次数。我们将关于δ(p)的存在性问题转化为正定整数三元二次型的问题，并利用三维完美型的Voronoi理论刻画δ(p)接近最大值的情形。对于足够大的素数，我们证明δ(p)达到其上界当且仅当p可由九个显式三次多项式之一表示，从而将这类素数的无穷性问题简化为这些三次多项式是否无穷多次取素数值。我们还证明，对于几乎所有素数，δ(p)至少比其上界低p^(1/6-o(1))。
英文摘要：
For a fixed characteristic $p$, let $δ(p)$ denote the smallest degree needed to guarantee the existence of an isogeny from any supersingular curve $E$ to its Frobenius conjugate $E^{(p)}$. We translate existence questions concerning $δ(p)$ into questions about positive definite integral ternary quadratic forms and use Voronoi's theory of perfect forms in dimension three to characterize when $δ(p)$ is nearly maximal. For sufficiently large primes, we show that $δ(p)$ attains its upper bound precisely when $p$ is represented by one of nine explicit cubic polynomials, reducing the infinitude of such primes to whether one of these cubics takes prime values infinitely often. We also prove that, for almost all primes, $δ(p)$ lies at least $p^{1/6-o(1)}$ below its upper bound.
