Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2501.18574

Inductive methods for counting number fields

Brandon Alberts·Robert J. Lemke Oliver·Jiuya Wang·Melanie Matchett Wood
View original source ↗Read source document ↗Export metadata ↓

Abstract

We give a new method for counting extensions of a number field asymptotically by discriminant, which we employ to prove many new cases of Malle's Conjecture and counterexamples to Malle's Conjecture. We consider families of extensions whose Galois closure is a fixed permutation group $G$. Our method relies on having asymptotic counts for $T$-extensions for some normal subgroup $T$ of $G$, uniform bounds for the number of such $T$-extensions, and possibly weak bounds on the asymptotic number of $G/T$-extensions. However, we do not require that most $T$-extensions of a $G/T$-extension are $G$-extensions. Our new results use $T$ either abelian or $S_3^m$, though our framework is general.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Brandon Alberts, Robert J. Lemke Oliver, Jiuya Wang, Melanie Matchett Wood. 2025-01-30. Inductive methods for counting number fields. https://arxiv.org/abs/2501.18574

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the factorisation of the $p$-adic Rankin-Selberg $L$-function in the supersingular case

Given a cusp form $f$ which is supersingular at a fixed prime $p$ away from the level, and a Coleman family $F$ through one of its $p$-stabilisations, we construct a $2$-variable meromorphic $p$-adic $L$-function for the symmetric square of $F$. We prove that this new $p$-adic $L$-function interpolates values of complex imprimitive symmetric square $L$-functions, for the various specialisations of the family $F$. We use this $p$-adic $L$-function to prove a $p$-adic factorisation formula, expressing the geometric $p$-adic $L$-function attached to the Rankin--Selberg convolution of $f$ with itself as a the product of the $p$-adic symmetric square $L$-function of $f$ and a Kubota-Leopoldt $L$-function. This extends a result of Dasgupta in the ordinary case.

math.NT↗
arXiv2026

Exceptional poles of archimedean Rankin-Selberg L-functions for irreducible generic representations of GL(n,R)

For irreducible generic representations $π_1$ and $π_2$ of $\operatorname{GL}_n(\mathbb R)$, we prove that the notions of exceptional pole of type $1$ and type $2$ coincide at every level. When both representations are in general position, we use this identification to express the Rankin--Selberg $L$-function $L(s,π_1\timesπ_2)$ in terms of the exceptional $L$-factors attached to the irreducible constituents of their derivatives.

math.NT↗