短算术项表示有限域上魏尔斯特拉斯标准形式椭圆曲线的基数
Short arithmetic terms for the cardinality of elliptic curves over rings of remainder classes and over finite fields
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中文总结 AI 辅助
该研究改进了椭圆曲线计数的算术项构造方法，降低了中间整数位数，对素模p≥17时大幅减少了运算次数，提升了有限域上魏尔斯特拉斯标准形式椭圆曲线基数的计算效率。
中文摘要 AI 辅助
算术项是加法、乘法、减法、带余除法和整数幂运算的固定有限复合。通过改进第二作者（arXiv:2608.22049）针对魏尔斯特拉斯标准形式椭圆曲线的通用方法，得到自然数A、B、n上的算术项，可对每个模n≥1计数(ℤ/nℤ)²中的解，其中间整数的二进制位数约为2n⁵，而非约2n¹¹。对于素模p≥17，基于哈塞不变量和弗罗贝尼乌斯迹的第二种构造，给出了约30次运算的项，替代了通用方法中广义几何级数的51次乘积。
英文摘要
Arithmetic terms are fixed finite compositions of addition, truncated subtraction, multiplication, integer division and exponentiation on natural numbers. We construct such terms for the number of affine solutions of $ y^2 = x^3 + Ax + B $. For arbitrary moduli $ n \geq 1 $, a specialization of Prunescu's general construction reduces the number of monomial contributions from fourty-nine to fifteen and the size of the packed integer from approximately $ 2n^{11} $ to approximately $ 2n^5 $ binary digits. For prime moduli $ p \geq 17 $, the Hasse invariant and the trace of Frobenius give a term of about thirty operations. For curves defined over $ \mathbb F_p $, a further term counts the points over $ \mathbb F_{p^k} $ with $ k $ as a variable. For primes $ p \geq 5 $, we also obtain arithmetic terms for the counts over $ \mathbb Z / p^k \mathbb Z $ with $ k $ variable, covering good reduction, nodes and cusps, including non-minimal equations and zero discriminant.
发表机构
- University of Bucharest（布加勒斯特大学）
- Simion Stoilow Institute of Mathematics of the Romanian Academy（罗马尼亚科学院西蒙·斯托伊洛数学研究所）
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