A refined Malle conjecture for Heisenberg groups

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 $2$.

Disclosure

“ially supported by NSF DMS-2140043, the James Mills Peirce Fellowship at Harvard University, and a Churchill Scholarship. The second author was supported by the Herschel Smith Fund. The authors acknowledge the use of AI tools, including GPT-5.5 and Codex by OpenAI, for autonomously writing initial Magma code that was used to search for small permutation groups admitting a Brauer–Manin obstruction, and suggesting the statements of Proposition 6.11 and Corollary 5.6. The authors ar”

PDF page 5

Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-24T00:34:12.480Z

Structural counts