# On ℓ-torsion in class groups of number fields

@article{Ellenberg2016OnI, title={On ℓ-torsion in class groups of number fields}, author={Jordan S. Ellenberg and L. B. Pierce and Melanie Matchett Wood}, journal={Algebra \& Number Theory}, year={2016}, volume={11}, pages={1739-1778} }

For each integer $\ell \geq 1$, we prove an unconditional upper bound on the size of the $\ell$-torsion subgroup of the class group, which holds for all but a zero-density set of field extensions of $\mathbb{Q}$ of degree $d$, for any fixed $d \in \{2,3,4,5\}$ (with the additional restriction in the case $d=4$ that the field be non-$D_4$). For sufficiently large $\ell$ (specified explicitly), these results are as strong as a previously known bound that is conditional on GRH. As part of our…

## 34 Citations

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

- Mathematics
- 2019

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…

Average bounds for the $$\ell $$ℓ-torsion in class groups of cyclic extensions

- MathematicsResearch in Number Theory
- 2018

For all positive integers $$\ell $$ℓ, we prove non-trivial bounds for the $$\ell $$ℓ-torsion in the class group of K, which hold for almost all number fields K in certain families of cyclic…

The weak form of Malle’s conjecture and solvable groups

- MathematicsResearch in Number Theory
- 2020

For a fixed finite solvable group G and number field K , we prove an upper bound for the number of G -extensions L / K with restricted local behavior (at infinitely many places) and…

An effective Chebotarev density theorem for families of number fields, with an application to $$\ell $$-torsion in class groups

- MathematicsInventiones mathematicae
- 2019

We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of…

The p-rank $\epsilon$-conjecture on class groups is true for towers of p-extensions

- Mathematics
- 2020

Let p$\ge$2 be a given prime number. We prove, for any number field kappa and any integer e$\ge$1, the p-rank $\epsilon$-conjecture, on the p-class groups Cl\_F, for the family F\_kappa^p^e of towers…

Bounds for the ℓ ‐torsion in class groups

- Mathematics
- 2017

We prove for each integer ℓ⩾1 an unconditional upper bound for the size of the ℓ ‐torsion subgroup ClK[ℓ] of the class group of K , which holds for all but a zero density set of number fields K of…

Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions

- Mathematics
- 2020

Abstract Elementary abelian groups are finite groups in the form of A = ( ℤ / p ℤ ) r {A=(\mathbb{Z}/p\mathbb{Z})^{r}} for a prime number p. For every integer ℓ > 1 {\ell>1} and r > 1 {r>1} , we…

A sharp upper bound for the 2‐torsion of class groups of multiquadratic fields

- MathematicsMathematika
- 2022

Let K be a multiquadratic extension of Q$\mathbb {Q}$ and let Cl+(K)$\text{Cl}^{+}(K)$ be its narrow class group. Recently, the authors (Koymans and Pagano, Int. Math. Res. Not. 2022 (2022), no. 4,…

The average size of $3$-torsion in class groups of $2$-extensions

- Mathematics
- 2021

We determine the average size of the 3-torsion in class groups of G-extensions of a number field when G is any transitive 2-group containing a transposition, for example D4. It follows from the…

Pointwise Bound for $\ell$-torsion in Class Groups II: Nilpotent Extensions

- Mathematics
- 2020

For every finite $p$-group $G_p$ that is non-cyclic and non-quaternion and every positive integer $\ell\neq p$ that is greater than $2$, we prove the first non-trivial bound on $\ell$-torsion in…

## References

SHOWING 1-10 OF 51 REFERENCES

Averages and moments associated to class numbers of imaginary quadratic fields

- Mathematics, Computer ScienceCompositio Mathematica
- 2017

Nontrivial upper bounds for the average of the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$ for all primes is proved as well as nontrivialupper bounds for certain higher moments for allPrimes $\ell \geqslant 3$.

Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves

- Mathematics, Computer Science
- 2017

It is proved that the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class groups of cubic and higher degree number fields $K$ are proved.

On the 16-rank of class groups of $${\mathbb{Q}(\sqrt{-8p})}$$Q(-8p) for $${p \equiv -1 {\rm mod} 4}$$p≡-1mod4

- Mathematics
- 2017

We use a variant of Vinogradov’s method to show that the density of the set of prime numbers $${p \equiv -1 {\rm mod} 4}$$p≡-1mod4 for which the class group of the imaginary quadratic number field…

Governing fields and statistics for 4-Selmer groups and 8-class groups

- Mathematics
- 2016

Taking A to be an abelian variety with full 2-torsion over a number field k, we investigate how the 4-Selmer rank of the quadratic twist A^d changes with d. We show that this rank depends on the…

THE SECONDARY TERM IN THE COUNTING FUNCTION FOR CUBIC FIELDS

- Mathematics
- 2010

Work in progress, September 21, 2010. We prove asymptotic formulas for the number of cubic fields of positive or negative discriminant less than X. These formulas involve main terms of X and X…

ON THE 4-RANK OF CLASS GROUPS OF QUADRATIC NUMBER FIELDS

- Mathematics
- 2013

We give an overview of the work of É. Fouvry and J. Klüners on the 4-rank of quadratic number fields. We give a slight generalization of their work by proving that the Cohen-Lenstra conjectures for…

Central limit theorem for Artin $L$-functions

- Mathematics
- 2015

We show that the sum of the traces of Frobenius elements of Artin $L$-functions in a family of $G$-fields satisfies the Gaussian distribution under certain counting conjectures. We prove the counting…

The 3‐part of Class Numbers of Quadratic Fields

- Mathematics
- 2005

It is proved that the 3‐part of the class number of a quadratic field Q(√D) is O(|D|55/112 +ε) in general and O(|D|5/12+ε) if |D| has a divisor of size |D|5/6. These bounds follow as results of…

A SURVEY ON $K$-FREENESS

- Mathematics
- 2005

Abstract. We say that an integer n is k–free (k ≥ 2) if for every prime p the valuation vp(n) < k. If f : N → Z, we consider the enumerating function S f (x) defined as the number of positive…

Integral points on elliptic curves and 3-torsion in class groups

- Mathematics
- 2004

We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and methods based on…