## Algebra & Number Theory

### The distribution of $p$-torsion in degree $p$ cyclic fields

Jack Klys

#### Abstract

We compute all the moments of the $p$-torsion in the first step of a filtration of the class group defined by Gerth (1987) for cyclic fields of degree $p$, unconditionally for $p=3$ and under GRH in general. We show that it satisfies a distribution which Gerth conjectured as an extension of the Cohen–Lenstra–Martinet conjectures. In the $p=3$ case this gives the distribution of the $3$-torsion of the class group modulo the Galois invariant part. We follow the strategy used by Fouvry and Klüners (2007) in their proof of the distribution of the $4$-torsion in quadratic fields.

#### Article information

Source
Algebra Number Theory, Volume 14, Number 4 (2020), 815-854.

Dates
Revised: 30 October 2019
Accepted: 27 November 2019
First available in Project Euclid: 30 June 2020

https://projecteuclid.org/euclid.ant/1593482419

Digital Object Identifier
doi:10.2140/ant.2020.14.815

Mathematical Reviews number (MathSciNet)
MR4114057

Subjects
Primary: 11R29: Class numbers, class groups, discriminants
Secondary: 11R37: Class field theory 11R45: Density theorems

#### Citation

Klys, Jack. The distribution of $p$-torsion in degree $p$ cyclic fields. Algebra Number Theory 14 (2020), no. 4, 815--854. doi:10.2140/ant.2020.14.815. https://projecteuclid.org/euclid.ant/1593482419

#### References

• B. Alberts, “Cohen–Lenstra moments for some nonabelian groups”, preprint, 2016.
• S. Baier and M. P. Young, “Mean values with cubic characters”, J. Number Theory 130:4 (2010), 879–903.
• M. Bhargava, “The geometric sieve and the density of squarefree values of invariant polynomials”, preprint, 2014.
• N. Boston and M. M. Wood, “Non-abelian Cohen–Lenstra heuristics over function fields”, Compos. Math. 153:7 (2017), 1372–1390.
• H. Cohen and H. W. Lenstra, Jr., “Heuristics on class groups of number fields”, pp. 33–62 in Number theory (Noordwijkerhout, Netherlands, 1983), edited by H. Jager, Lecture Notes in Math. 1068, Springer, 1984.
• H. Cohen and J. Martinet, “Class groups of number fields: numerical heuristics”, Math. Comp. 48:177 (1987), 123–137.
• B. Datskovsky and D. J. Wright, “Density of discriminants of cubic extensions”, J. Reine Angew. Math. 386 (1988), 116–138.
• H. Davenport, Multiplicative number theory, 3rd ed., Graduate Texts in Math. 74, Springer, 2000.
• H. Davenport and H. Heilbronn, “On the density of discriminants of cubic fields, II”, Proc. Roy. Soc. Lond. Ser. A 322:1551 (1971), 405–420.
• J. S. Ellenberg, A. Venkatesh, and C. Westerland, “Homological stability for Hurwitz spaces and the Cohen–Lenstra conjecture over function fields”, Ann. of Math. $(2)$ 183:3 (2016), 729–786.
• É. Fouvry and J. Klüners, “Cohen–Lenstra heuristics of quadratic number fields”, pp. 40–55 in Algorithmic number theory, edited by F. Hess et al., Lecture Notes in Comput. Sci. 4076, Springer, 2006.
• É. Fouvry and J. Klüners, “On the 4-rank of class groups of quadratic number fields”, Invent. Math. 167:3 (2007), 455–513.
• F. Gerth, III, “The $4$-class ranks of quadratic fields”, Invent. Math. 77:3 (1984), 489–515.
• F. Gerth, III, “Densities for ranks of certain parts of $p$-class groups”, Proc. Amer. Math. Soc. 99:1 (1987), 1–8.
• L. J. Goldstein, “A generalization of the Siegel–Walfisz theorem”, Trans. Amer. Math. Soc. 149 (1970), 417–429.
• P. Hall, “A partition formula connected with Abelian groups”, Comment. Math. Helv. 11:1 (1938), 126–129.
• D. R. Heath-Brown, “Kummer's conjecture for cubic Gauss sums”, Israel J. Math. 120:part A (2000), 97–124.
• H. Iwaniec and E. Kowalski, Analytic number theory, Amer. Math. Soc. Colloq. Publ. 53, Amer. Math. Soc., Providence, RI, 2004.
• P. Koymans and C. Pagano, “On the distribution of ${\rm Cl}(K)[l^\infty]$ for degree $l$ cyclic fields”, preprint, 2018.
• F. Lemmermeyer, “The ambiguous class number formula revisited”, J. Ramanujan Math. Soc. 28:4 (2013), 415–421.
• D. C. Mayer, “Multiplicities of dihedral discriminants”, Math. Comp. 58:198 (1992), 831–847.
• D. Milovic, “On the $16$-rank of class groups of $\mathbb{Q}(\sqrt{-8p})$ for $p\equiv-1 \mod4$”, Geom. Funct. Anal. 27:4 (2017), 973–1016.
• D. Z. Milovic, “On the $8$-rank of narrow class groups of $\mathbb{Q}(\sqrt{-4pq})$, $\mathbb{Q}(\sqrt{-8pq})$, and $\mathbb{Q}(\sqrt{8pq})$”, Int. J. Number Theory 14:8 (2018), 2165–2193.
• J. Neukirch, Algebraic number theory, Grundlehren der Math. Wissenschaften 322, Springer, 1999.
• A. Smith, “$2^{\infty}$-Selmer groups, $2^{\infty}$-class groups, and Goldfeld's conjecture”, preprint, 2017.
• P. Stevenhagen, “Rédei-matrices and applications”, pp. 245–259 in Number theory (Paris, 1992-93), edited by S. David, Lond. Math. Soc. Lecture Note Ser. 215, Cambridge Univ. Press, 1995.
• M. M. Wood, “Nonabelian Cohen–Lenstra moments”, Duke Math. J. 168:3 (2019), 377–427.
• D. J. Wright, “Distribution of discriminants of abelian extensions”, Proc. Lond. Math. Soc. $(3)$ 58:1 (1989), 17–50.