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arXiv · 2607.06476

A refined Malle conjecture for Heisenberg groups

Jack B. Miller·Tim Santens
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Abstract

Based on a conjecture of Loughran and the second author, we give an explicit prediction for the leading constant in Malle's conjecture for Galois $\mathcal{H}$-extensions of $\mathbb{Q}$ ordered by discriminant, where $\mathcal{H}$ is the $3\times 3$ Heisenberg group over $\mathbb{F}_4$. The predicted leading constant is not a single Euler product, but rather a sum of two distinct Euler products. Our methods also give an efficient algorithm for computing the conjectural Loughran-Santens leading constant for many $2$-groups of nilpotency class $2$.

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Jack B. Miller, Tim Santens. 2026-07-07. A refined Malle conjecture for Heisenberg groups. https://arxiv.org/abs/2607.06476

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