A new pointwise bound for $3$-torsion of class groups

TL;DR

The paper proves a new pointwise bound for 3-torsion in class groups of real quadratic fields, establishing $h_3(d) \\ll_\varepsilon |d|^{\kappa+\varepsilon}$ with $\kappa \approx 0.3193$, by synthesizing a flexible Ellenberg–Venkatesh framework with refined small-split-prime analysis and elliptic-curve height methods, aided by Scholz reflection. It also extends the analysis to average bounds for $\ell$-torsion with $\ell\ge5$ in real quadratic fields, using large-sieve techniques to handle most fields and the pointwise machinery for the remaining ones. The approach combines ideas from KT, HV, HBP, FW, and rank-height controls on elliptic curves to control torsion via the distribution of small primes and relations in the class group. The results improve on the trivial bound and contribute to arithmetic statistics by providing new pointwise and averaged bounds for torsion in real quadratic fields, with potential implications for related invariants and heuristics. All bounds are stated with explicit dependence on $\varepsilon$ and use $|d|$ in the natural logarithmic/size scales common in this area.

Abstract

Ellenberg--Venkatesh proved in 2007 that $h_3(d) \ll_ε|d|^{1/3 + ε}$, where $h_3(d)$ denotes the size of the $3$-torsion of the class group of $\mathbb{Q}(\sqrt{d})$. We improve this bound to $h_3(d) \ll_ε|d|^{κ+ ε}$ with $κ\approx 0.3193 \cdots$. We also combine our methods with work of Heath-Brown--Pierce to give new bounds for average $\ell$-torsion of real quadratic fields.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• proof
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Proposition 3.4: KL