论文标题
单基因对$ 2 $ torsion的影响在奇数级数字的班级组中
Effect of monogenicity on $2$-torsion in the class group of number fields of odd degree
论文作者
论文摘要
我们在奇数学位的单基化领域的班级组中平均研究$ 2 $ torsion。 Bhargava-Hanke-Shankar最近表明，固定签名的单一立方场中的非平凡$ 2 $ torsion元素的平均数量是Cohen-Lenstra-Martinet-Malle-Malle Healistic for for Floce for for for for for for for for to for to for to for to for for to for to for for to for to for，固定签名的价值是两倍。修复任何奇数$ n $和签名$（r_ {1}，r_ {2}）$。在这项工作中，我们证明，$ n $ $ n $和签名$（r_ {1}，r_ {2}）$的单个单一化字段中的非平凡元素的平均数量是由Cohen-Lenstra-Martinet-Malle预测的价值两倍的界限。在$ n \ ge 5 $的尾部估计下有条件，这表明Bhargava-Hanke-Shankar在所有奇数和签名中都持续存在针对立方场发现的两倍现象。
We study average $2$-torsion in the class group of monogenised fields of odd degree. Bhargava-Hanke-Shankar have recently shown that the average number of non-trivial $2$-torsion elements in the class group of monogenised cubic fields of a fixed signature is twice the value predicted by the Cohen-Lenstra-Martinet-Malle heuristic for the full family. Fix any odd degree $n$ and signature $(r_{1},r_{2})$. In this work, we prove that the average number of non-trivial elements in the class group of monogenised fields of degree $n$ and signature $(r_{1},r_{2})$ is bounded by twice the value predicted by Cohen-Lenstra-Martinet-Malle. Conditional on a tail estimate for $n \ge 5$, this shows that the doubling phenomenon uncovered for cubic fields by Bhargava-Hanke-Shankar persists across all odd degrees and signatures.
