It is conjectured that within the class group of any number field, for every integer $\ell \geq 1$, the $\ell$-torsion subgroup is very small (in an appropriate sense, relative to the discriminant of the field). In nearly all settings, the full strength of this conjecture remains open, and even partial progress is limited. Significant recent progress toward average versions of the $\ell$-torsion conjecture has relied crucially on counts for number fields, raising interest in how these two types of question relate. In this paper we make explicit the quantitative relationships between the $\ell$-torsion conjecture and other well-known conjectures: the Cohen–Lenstra heuristics, counts for number fields of fixed discriminant, counts for number fields of bounded discriminant (or related invariants), counts for elliptic curves with fixed conductor. All of these considerations reinforce that we expect the $\ell$-torsion conjecture to be true, despite limited progress toward it. Our perspective focuses on the relation between pointwise bounds, averages, and higher moments, and demonstrates the broad utility of the “method of moments.”

中文翻译：

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关于数字字段类组中的$ \ ell $ -torsion的猜想：从时刻的角度

可以推测，在任何数字字段的类组中，对于每个整数$ \ ell \ geq 1 $，$ \ ell $ -torsion子组都非常小（在适当的意义上，相对于该字段的判别而言）。在几乎所有情况下，这种猜想的全部力量仍然是敞开的，甚至部分进展也受到限制。最近在朝向平均水平的“扭曲”猜想方面取得的重大进展主要依赖于数字字段的计数，这引起了人们对这两种类型问题的联系的兴趣。在本文中，我们明确说明了$ \ ell $ -torsion猜想与其他著名猜想之间的定量关系：Cohen-Lenstra启发式方法，固定判别式的数量字段计数，有界判别式（或相关不变量）的数量字段计数），计算固定导体的椭圆曲线。所有这些考虑因素都强化了我们的预期，即扭转扭曲猜想是正确的，尽管取得的进展有限。我们的观点集中于点限，平均数和较高矩之间的关系，并论证了“矩方法”的广泛用途。