论文标题
跨越有限磁场上普通椭圆曲线的功率的等值类别。申请属属的曲线的理性点$ \ leq 4 $
Spanning the isogeny class of a power of an ordinary elliptic curve over a finite field. Application to the number of rational points of curves of genus $\leq 4$
论文作者
论文摘要
让$ e $成为有限场上的普通椭圆曲线，而$ g $为正整数。在某些技术假设下，我们给出了一种算法，以跨越$ e^g $的等级类别中主要两极化的阿贝尔品种的同构类别。这些品种首先被描述为（不一定是最大）二次订单，然后从几何形式上说，就其代数theta null点而言。我们还展示了如何通过仔细选择theta null点的仿射升力来计算在theta常数中以多项式赋予的偶数重量的Siegel模块化形式。然后，我们使用这些结果对SERRE的障碍物进行代数计算，用于对主要极化的Abelian三倍为$ e^3 $以及Igusa模块化形式（尺寸为$ 4 $）。我们用有限领域的许多合理点的曲线示例来说明我们的算法。
Let $E$ be an ordinary elliptic curve over a finite field and $g$ be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of $E^g$. The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null point. We then use these results to give an algebraic computation of Serre's obstruction for principally polarized abelian threefolds isogenous to $E^3$ and of the Igusa modular form in dimension $4$. We illustrate our algorithms with examples of curves with many rational points over finite fields.
