Monnet, Sebastian;
(2025)
Low-degree extensions of number fields and local fields.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
We present new results in arithmetic statistics, particularly in the statistics of number fields and p-adic fields. For n ≤ 5, and conjecturally for n ≥ 6, the asymptotic counting of so-called “Sn-n-ic” extensions of number fields amounts to computing the “masses” of certain sets of étale extensions. This notion of mass was studied by Serre in his famous “Serre’s mass formula”, and subsequently by Bhargava and others. By counting various families of étale extensions of p-adic fields, we obtain novel refinements of Serre’s formula and apply them to prove results about counting Sn-n-ic extensions of number fields with certain prescribed norm elements. Our results are divided into two categories: a “pure” study of masses for wildly ramified extensions of 2-adic fields, and a more “applied” study of Sn-n-ic extensions, where we use our techniques to deduce results about counting number fields. The upshot of our pure study of masses is as follows. Given a 2-adic field F, a finite group G, and a positive integer m, we obtain a formula for the number of isomorphism classes of totally ramified quartic field extensions L/F with Galois closure group G and discriminant valuation m. As a corollary, we then use these counts to deduce our refinements of Serre’s mass formula. As for the applied study of Sn-n-ic extensions, let k be a number field and let A ⊆ k × be a finitely generated subgroup. For a positive integer n and a real number X, let Nk,n(X; A) be the number of Sn-n-ic extensions K/k with A ⊆ NK/kK× and Nm(disc(K/k)) ≤ X. For n ≤ 5, and conjecturally for n ≥ 6, we express the limit lim X→∞ Nk,n(X; A) X as an Euler product, whose term at each prime p of k is the mass of a certain set of étale algebras over the completion kp. Using, among other things, the techniques from our pure study of masses, we evaluate almost all of these local factors explicitly and give an efficient algorithm for computing the rest.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Low-degree extensions of number fields and local fields |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2025. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10211878 |
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