Unconditional pointwise bounds for $\ell$-torsion in class groups of number fields toward the $\ell$-torsion ($\varepsilon$-)conjecture: the regimes left open by the Ellenberg–Venkatesh reflection method

The ideal class group $\mathrm{Cl}_K$ of a number field $K$ is finite, and its $\ell$-torsion subgroup $\mathrm{Cl}_K[\ell]$ is conjectured, for fixed degree $n=[K:\mathbb{Q}]$, fixed prime $\ell$, and absolute discriminant $D_K$, to be negligible compared with the class group itself: $|\mathrm{Cl}_K[\ell]|\ll_{n,\ell,\varepsilon}D_K^{\varepsilon}$ for every $\varepsilon>0$ (the $\ell$-torsion or $\varepsilon$-conjecture, suggested by Duke, Zhang, and Brumer–Silverman). The trivial bound is $D_K^{1/2+\varepsilon}$, and outside genus theory no case of the conjecture is known for all fields of a fixed degree. Ellenberg and Venkatesh (2007) introduced a reflection-principle method — a production of many small split primes, control of multiplicative relations in the Arakelov class group, and Scholz-type reflection — that produced the first unconditional pointwise power-saving bounds for $3$-torsion in cubic and quartic fields, such as $|\mathrm{Cl}_K[3]|\ll D_K^{1/2-\delta}$ for every field of degree at most $4$, and bounds for arbitrary $\ell$ in certain higher-degree families; but it applies only under an intermediate-subfield hypothesis and fails entirely for fields containing a primitive $\ell$-th root of unity $\zeta_\ell$. The open problem is to extend unconditional pointwise power-saving bounds beyond these restrictions: to the pairs $(n,\ell)$ with $n\in\{2,3,4\}$ and prime $\ell\ge 5$, and with $n\ge 5$ and odd $\ell$, where no power saving at all is known for all degree-$n$ fields; to fields containing $\zeta_\ell$; and to improve the known exponents toward $D_K^{\varepsilon}$ — for quadratic $3$-torsion the current record is $|D_K|^{0.3194}$ (Chan–Koymans, 2025), against the conjectural exponent $0$.

Background

A number field $K$ is a finite-degree extension of $\mathbb{Q}$; its ring of integers $\mathcal{O}_K$ consists of the elements of $K$ satisfying a monic polynomial equation with integer coefficients. In $\mathcal{O}_K$ every nonzero ideal factors uniquely into prime ideals, but unique factorization of elements can fail, and the ideal class group $\mathrm{Cl}_K$ — the quotient of the group of nonzero fractional ideals by the subgroup of principal ideals — is a finite abelian group measuring this failure; it is trivial exactly when $\mathcal{O}_K$ has unique factorization of elements. The absolute discriminant $D_K$ is the positive integer invariant measuring the arithmetic complexity of $K$ (its prime divisors are exactly the ramified primes). By the Brauer–Siegel theorem (the asymptotic theory of the class number together with the regulator, i.e. the covolume of the logarithmic embedding of the unit group), fields of fixed degree $n$ satisfy the trivial bound $|\mathrm{Cl}_K|\ll_{n,\varepsilon}D_K^{1/2+\varepsilon}$, and the $\ell$-torsion subgroup $\mathrm{Cl}_K[\ell]=\{c\in\mathrm{Cl}_K:\ell c=0\}$ (elements whose order divides $\ell$) can only be smaller.

For fixed degree $n$ and prime $\ell$, the $\ell$-torsion conjecture — also called the $\varepsilon$-conjecture — predicts that torsion in a single class group is negligible:

$$|\mathrm{Cl}_K[\ell]|\ll_{n,\ell,\varepsilon}D_K^{\varepsilon}\quad\text{for every }\varepsilon>0$$

and every degree-$n$ field $K$. The conjecture was suggested by Duke, by Zhang (who formulated it as the "$\varepsilon$-conjecture" for CM fields, i.e. totally imaginary quadratic extensions of totally real fields), and in a stronger uniform form by Brumer and Silverman, whose "Question CL($\ell,d$)" was motivated by the arithmetic of elliptic curves: as Ellenberg–Venkatesh note, the conjecture would imply strong bounds for Mordell–Weil ranks (the ranks of the finitely generated groups of rational points of elliptic curves). The only case known for all fields of a fixed degree is genus-theoretic: $n=2$, $\ell=2$, where Gauss's genus theory (the classical description of $2$-torsion via quadratic characters modulo $D_K$) gives $|\mathrm{Cl}_K[2]|\ll_\varepsilon D_K^{\varepsilon}$.

Ellenberg and Venkatesh (2007) introduced the first general method for unconditional pointwise bounds — valid for each field individually, not merely on average over a family. Their Key Lemma (Lemma 2.3 in the published version) shows that $\ell$-torsion is small whenever $K$ has available many small prime ideals that are non-inert (i.e. not remaining prime) in $K$: if there are $M$ unramified prime ideals of $K$ of degree $1$ (norm a rational prime) and norm at most $D_K^{\varpi}$, with $\varpi<1/(2\ell(n-1))$, then $|\mathrm{Cl}_K[\ell]|\ll_{n,\ell,\varepsilon,\varpi}D_K^{1/2+\varepsilon}/M$, so any $M$ growing like a positive power of $D_K$ yields a power saving $D_K^{1/2-\delta}$ (the general form of the lemma allows prime ideals not extended from any proper subfield). Under the Generalized Riemann Hypothesis (GRH) for the Dedekind zeta function $\zeta_K(s)=\sum_{\mathfrak{a}}N(\mathfrak{a})^{-s}$, whose analytic properties encode the distribution of prime ideals, such primes exist for every field and one obtains $|\mathrm{Cl}_K[\ell]|\ll_{n,\ell,\varepsilon}D_K^{1/2-1/(2\ell(n-1))+\varepsilon}$. The unconditional part of the paper escapes GRH in two steps. First, multiplicative relations among the small elements produced are controlled through the Arakelov class group, an enlargement of the class group that also records the archimedean (real and complex embedding) sizes of elements and thereby neutralizes obstructions coming from the units, the invertible elements of $\mathcal{O}_K$ forming an infinite group of rank up to $n-1$. Second, a Scholz-type reflection principle — generalizing the classical theorem of Scholz that the $3$-ranks of the class groups of $\mathbb{Q}(\sqrt{d})$ and $\mathbb{Q}(\sqrt{-3d})$ differ by at most one — relates $\ell$-torsion of $K$ to that of the reflect extension $K(\zeta_\ell)$, where $\zeta_\ell$ is a primitive $\ell$-th root of unity, through the eigenspace decomposition of $\mathrm{Cl}_{K(\zeta_\ell)}[\ell]$ under $\mathrm{Gal}(K(\zeta_\ell)/K)$.

Unconditionally this yields: $|\mathrm{Cl}_K[3]|\ll_{\varepsilon}|D_K|^{1/3+\varepsilon}$ for quadratic $K$ (improving bounds of Pierce and of Helfgott–Venkatesh, and implying, for example, that there are at most $|D|^{1/3+\varepsilon}$ cubic extensions of $\mathbb{Q}$ with discriminant $D$); the first power-saving pointwise bounds for $3$-torsion for all cubic and all quartic fields (published exponent $1/2-1/168$ in the $A_4$/$S_4$-quartic case); and, for arbitrary $\ell$, bounds such as $|\mathrm{Cl}_K[\ell]|\ll(\mathrm{disc}\,K)^{1/2-1/(2\ell d(2d-1))+\varepsilon}$ for fields $K$ of even degree $d>2$ over $K_0=\mathbb{Q}(\zeta_\ell+\zeta_\ell^{-1})$ with $\zeta_\ell\notin K$ such that $K(\zeta_\ell)/K_0$ has no proper subextensions apart from $K$ and $\mathbb{Q}(\zeta_\ell)$ (Proposition 3.6) — the paper's introduction notes that this entails, e.g., a nontrivial bound for the $5$-torsion of any quadratic extension of $\mathbb{Q}(\sqrt{5})$, and such quadratic extensions are treated directly by the preceding proposition, with the better exponent $1/2-1/(2\ell)+\varepsilon$. The paper is explicit about the method's limits: the intermediate-subfield hypothesis "might be significantly weakened", but "the method of this paper fails entirely if $\zeta_\ell\in K$" — when $K$ already contains $\zeta_\ell$ there is no reflect extension and the reflection principle degenerates. The quartic case with $\zeta_3\in K$ is instead handled through the integral-point method of Helfgott–Venkatesh, whose extension from $\mathbb{Q}$ to a general number field the authors state is not written. The exponents obtained are also far from the conjectural $D_K^{\varepsilon}$.

The source-faithful open question is therefore: how far can unconditional pointwise bounds for class-group $\ell$-torsion be pushed beyond these restrictions — to the pairs $(n,\ell)$ for which no power saving is known at all, to fields containing $\zeta_\ell$, and in exponent toward the $\varepsilon$-conjecture?

Problem Statement

Let $n\ge 2$, let $\ell$ be a prime, and for a number field $K$ of degree $n$ with absolute discriminant $D_K$ let $\mathrm{Cl}_K[\ell]$ denote the $\ell$-torsion subgroup of its ideal class group. Say that property $\mathbf{C}_{n,\ell}(\Delta)$ holds if $|\mathrm{Cl}_K[\ell]|\ll_{n,\ell,\Delta,\varepsilon}D_K^{\Delta+\varepsilon}$ for every degree-$n$ field $K$ and every $\varepsilon>0$ (Pierce's notation). The trivial bound gives $\mathbf{C}_{n,\ell}(1/2)$ for all $n,\ell$; the $\ell$-torsion ($\varepsilon$-)conjecture predicts $\mathbf{C}_{n,\ell}(0)$, i.e. $|\mathrm{Cl}_K[\ell]|\ll_{n,\ell,\varepsilon}D_K^{\varepsilon}$ for every $\varepsilon>0$.

The problem is to establish unconditional pointwise power-saving bounds for class-group $\ell$-torsion — that is, $\mathbf{C}_{n,\ell}(\Delta)$ for some $\Delta<1/2$, valid for every field in the stated class, with no unproved hypothesis (no GRH, no subconvexity assumption, no averaging over a family) — in the regimes left open by the Ellenberg–Venkatesh reflection method. The common objective uniting the following deliverables is a fixed power saving below the trivial exponent $1/2$, holding field-by-field; the ultimate target is the $\varepsilon$-conjecture itself.

1. Uncovered pairs $(n,\ell)$: prove that $\mathbf{C}_{n,\ell}(\Delta)$ holds for some $\Delta<1/2$, for the first time, either (a) for some $n\in\{2,3,4\}$ and a prime $\ell\ge 5$, or (b) for some $n\ge 5$ and an odd prime $\ell$ — in both cases for all degree-$n$ fields, with no restriction on the Galois group of the normal closure, on subfield structure, or on the prime factorization of the discriminant.

2. Removing the reflection-method restrictions: in the setting of Ellenberg–Venkatesh's Proposition 3.6 ($K_0=\mathbb{Q}(\zeta_\ell+\zeta_\ell^{-1})$, $K/K_0$ of even degree $d>2$, $\zeta_\ell\notin K$), weaken or remove the hypothesis that $K(\zeta_\ell)/K_0$ has no proper subextensions apart from $K$ and $\mathbb{Q}(\zeta_\ell)$ (a weakening must be stated as an explicit, strictly weaker hypothesis that is satisfied by infinitely many such fields $K$, with the power-saving bound proved for those fields); and prove, for every odd prime $\ell$ and every degree $n\ge\ell$ with $\ell-1\mid n$, power-saving bounds for all degree-$n$ fields $K$ containing $\zeta_\ell$, for which the reflection method fails entirely.

3. Exponents toward the conjecture: improve the known pointwise exponents toward $0$; in the first instance for quadratic $3$-torsion, where the current record is $|\mathrm{Cl}_K[3]|\le C|D_K|^{0.3194}$ for every quadratic field $K$ (Chan–Koymans, who write $h_3(d)\le C|d|^{0.3194}$ for squarefree $d$, where $h_3(d)=|\mathrm{Cl}_{\mathbb{Q}(\sqrt{d})}[3]|$), conditional results give $|D_K|^{1/4+\varepsilon}$, and the conjecture predicts $|D_K|^{\varepsilon}$.

Each bound sought is an explicit power saving with $\delta=1/2-\Delta>0$, for the class of fields and prime $\ell$ of the regime in question, and holds unconditionally and pointwise; the effectivity of the bound and of its implied constant is part of the result. The three regimes are independent open questions.

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