Mathematical Research Letters

Volume 28 (2021)

Number 2

On a conjecture for $\ell$-torsion in class groups of number fields: from the perspective of moments

Pages: 575 – 621



Lillian B. Pierce (Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.)

Caroline L. Turnage-Butterbaugh (Department of Mathematics and Statistics, Carleton College, Northfield, Minnesota, U.S.A.)

Melanie Matchett Wood (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)


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.”

The authors thank J. Ellenberg, J. Wang, B. Alberts, and the referee for insightful comments that improved and clarified the content and exposition. Pierce has been partially supported by NSF CAREER grant DMS-1652173, a Sloan Research Fellowship, a Birman Fellowship, and as a von Neumann Fellow at the Institute for Advanced Study, by the Charles Simonyi Endowment and NSF Grant No. 1128155. Pierce thanks the Hausdorff Center for Mathematics for hospitality during visits as a Bonn Research Fellow. Turnage-Butterbaugh is partially supported by NSF DMS-1902193 and NSF DMS-1854398 FRG, and thanks the Mathematical Sciences Research Institute (NSF DMS-1440140) and the Max Planck Institute for Mathematics for support and hospitality during portions of this work. Wood was partially supported by an American Institute of Mathematics Five-Year Fellowship, a Packard Fellowship for Science and Engineering, a Sloan Research Fellowship, National Science Foundation grant DMS-1301690 and CAREER grant DMS-1652116, and a Vilas Early Career Investigator Award. Wood thanks Princeton University for its hospitality during Fall 2018.

Received 5 February 2019

Accepted 24 December 2020

Published 13 May 2021