Quadratic fields with a class group of large 3-rank
Quadratic fields with a class group of large 3-rank
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DOI：
10.4064/aa191027-22-6
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发表时间：
2019-10
期刊：
影响因子：
0.7
通讯作者：
A. Levin;Shengkuan Yan;Luke Wiljanen
中科院分区：
文献类型：
--
作者：
A. Levin;Shengkuan Yan;Luke Wiljanen
证明了存在>>X^{1/30}/（log X）的虚二次数域，其理想类群的秩至少为5，且判别式绝对值有界于X.这改善了早期的结果克雷格，谁证明了无限的虚二次领域的理想类组的3-秩至少为4。的证明依赖于建设的梅斯特为j-不变0椭圆曲线的大Mordell-Weil秩，和方法的第一作者和Gillibert的建设扭转理想类群的数域从有理挠的Jacobian曲线。我们也考虑类似的问题，合理的3-扭转超椭圆雅可比。
We prove that there are >>X^{1/30}/(log X) imaginary quadratic number fields with an ideal class group of 3-rank at least 5 and discriminant bounded in absolute value by X. This improves on an earlier result of Craig, who proved the infinitude of imaginary quadratic fields with an ideal class group of 3-rank at least 4. The proofs rely on constructions of Mestre for j-invariant 0 elliptic curves of large Mordell-Weil rank, and a method of the first author and Gillibert for constructing torsion in ideal class groups of number fields from rational torsion in Jacobians of curves. We also consider analogous questions concerning rational 3-torsion in hyperelliptic Jacobians.