论文标题
椭圆曲线和深神经网络等级
Ranks of elliptic curves and deep neural networks
论文作者
论文摘要
确定椭圆曲线E/Q的等级是一个困难的问题，在某些应用中（例如，在搜索高级曲线时）必须依靠旨在估计分析等级的启发式方法（这等于Birch和Swinnerton-Dyer猜测下的等级）。 在本文中，我们开发了由深卷积神经网络（CNN）建模的等级分类启发式方法。与广泛使用的Mestre-Nagao总和类似，它作为输入E和一系列标准化的A_P-S（其中A_P = P+1-＃E（F_P），如果P是某个范围内的良好量）（p <10^k = = 3,4,5），并试图预测等级的排名（或检测级别curve curve curve of Curve of Curve）。该模型已在两个数据集上进行了训练和测试：LMFDB和一个自定义数据集由椭圆曲线组成，具有微不足道的扭转，导体最高为10^30，最高为10。为了进行比较，还开发了八个简单的默斯特雷 - nagao总和的简单神经网络模型。 实验表明，CNN的性能要比LMFDB数据集上的Mestre-nagao总和（有趣的是，有趣的神经网络都作为输入，所有Mestre-Nagao总和的性能都比每个单独的总和要好得多），而它们在自定义制造的数据集上大致相等。
Determining the rank of an elliptic curve E/Q is a hard problem, and in some applications (e.g. when searching for curves of high rank) one has to rely on heuristics aimed at estimating the analytic rank (which is equal to the rank under the Birch and Swinnerton-Dyer conjecture). In this paper, we develop rank classification heuristics modeled by deep convolutional neural networks (CNN). Similarly to widely used Mestre-Nagao sums, it takes as an input the conductor of E and a sequence of normalized a_p-s (where a_p=p+1-#E(F_p) if p is a prime of good reduction) in some range (p<10^k for k=3,4,5), and tries to predict rank (or detect curves of ``high'' rank). The model has been trained and tested on two datasets: the LMFDB and a custom dataset consisting of elliptic curves with trivial torsion, conductor up to 10^30, and rank up to 10. For comparison, eight simple neural network models of Mestre-Nagao sums have also been developed. Experiments showed that CNN performed better than Mestre-Nagao sums on the LMFDB dataset (interestingly neural network that took as an input all Mestre-Nagao sums performed much better than each sum individually), while they were roughly equal on custom made dataset.
