PreprintCC BY 4.0arXiv
Counting Nilpotent Galois extensions with prescribed genus number
Abstract
Let G be a nontrivial finite nilpotent group. In this paper, we study the distribution of genus numbers of Galois extensions of Q with Galois group isomorphic to G, ordered by absolute discriminant. We show that the proportion of fields with any prescribed genus number tends to zero, and obtain a quantitative bound for this proportion. Moreover, if every element of order ℓ in G is central, where ℓ is the smallest prime divisor of |G|, we determine the orders of magnitude of the average genus number and its higher moments.
Related
- Maximal Unramified p-Extensions with Prescribed Galois Groups: A Quantitative Refinement of Ozaki's Theorem
- Counting number fields with symplectic Galois group
- Galois groups of polynomials over a fixed number field: determinant methods in general boxes, resolvents, and quantitative counting
- Higher Cotorsion Pairs on Image-Nilpotent η-Extensions
- Hochschild cohomology and AS-Gorenstein property of weak Hopf Galois extensions
BibTeX
@misc{dixit2026counting,
title = {{Counting Nilpotent Galois extensions with prescribed genus number}},
author = {Anup B. Dixit and Anwesh Ray},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01556},
eprint = {2611.01556},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01556},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Counting Nilpotent Galois extensions with prescribed genus number
AU - Anup B. Dixit
AU - Anwesh Ray
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01556
UR - https://arxiv.org/abs/2611.01556
AB - Let G be a nontrivial finite nilpotent group. In this paper, we study the distribution of genus numbers of Galois extensions of Q with Galois group isomorphic to G, ordered by absolute discriminant. We show that the proportion of fields with any prescribed genus number tends to zero, and obtain a quantitative bound for this proportion. Moreover, if every element of order ℓ in G is central, where ℓ is the smallest prime divisor of |G|, we determine the orders of magnitude of the average genus number and its higher moments.
ER -