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A new pointwise bound for 3-torsion of class groups

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arxiv 2505.00611 v1 pith:Z57MFWR3 submitted 2025-05-01 math.NT

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

classification math.NT
keywords epsilontorsionboundclasskappaapproxaveragebounds
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abstract

Ellenberg--Venkatesh proved in 2007 that $h_3(d) \ll_\epsilon |d|^{1/3 + \epsilon}$, 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 $h_3(d) \ll_\epsilon |d|^{\kappa + \epsilon}$ with $\kappa \approx 0.3193 \cdots$. We also combine our methods with work of Heath-Brown--Pierce to give new bounds for average $\ell$-torsion of real quadratic fields.

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