Searcharxiv⌕ Search

arXiv subjects

Magnus Carlson

Publications and source records attributed to Magnus Carlson.

11 recordsLinked to original sources
arXiv2026

Elementary anabelian varieties are anabelian

We show that isomorphisms of fundamental groups of elementary anabelian varieties -- varieties obtained as iterated fibrations of hyperbolic curves -- over sub-$p$-adic fields correspond bijectively to isomorphisms of varieties. Moreover, dominant maps between proper elementary anabelian varieties are in bijection with ``stably cohomologically injective'' maps of fundamental groups: open maps whose pullbacks to all open subgroups induce injections on cohomology rings with $\ell$-adic coefficients, for any prime $\ell$. This verifies conjectures of Grothendieck from his letter to Faltings. Finally, we establish \'etale homotopical generalizations of these results.

math.NT↗
arXiv2024

The \'etale topos reconstructs varieties over sub-p-adic fields

Let $K$ be a sub-$p$-adic field. We show that the functor sending a finite type $K$-scheme to its \'etale topos is fully faithful after localizing at the class of universal homeomorphisms. This generalizes a result of Voevodsky, who proved the analogous theorem for fields finitely generated over $\mathbb{Q}$. Our proof relies on Mochizuki's Hom-theorem in anabelian geometry, and a study of point-theoretic morphisms of fundamental groups of curves.

math.AG↗
arXiv2024

Reconstruction of schemes from their \'{e}tale topoi

Let $k$ be a field that is finitely generated over its prime field. In Grothendieck's anabelian letter to Faltings, he conjectured that sending a $k$-scheme to its \'{e}tale topos defines a fully faithful functor from the localization of the category of finite type $k$-schemes at the universal homeomorphisms to a category of topoi. We prove Grothendieck's conjecture for infinite fields of arbitrary characteristic. In characteristic $0$, this shows that seminormal finite type $k$-schemes can be reconstructed from their \'{e}tale topoi, generalizing work of Voevodsky. In positive characteristic, this shows that perfections of finite type $k$-schemes can be reconstructed from their \'{e}tale topoi.

math.AG↗
arXiv2022

Massey products in the \'etale cohomology of number fields

We give formulas for 3-fold Massey products in the \'etale cohomology of the ring of integers of a number field and use these to find the first known examples of imaginary quadratic fields with class group of $p$-rank two possessing an infinite $p$-class field tower, where $p$ is an odd prime. Furthermore, a necessary and sufficient condition, in terms of class groups of $p$-extensions, for the vanishing of 3-fold Massey products is given. As a consequence, we give an elementary and sufficient condition for the infinitude of class field towers of imaginary quadratic fields. We also disprove McLeman's $(3,3)$-conjecture. Lastly, we relate the vanishing of Massey products to the existence of Galois representations of $G_{\mathbb{Q},S}$ which realize an unexpectedly large class group for certain extensions of a quadratic imaginary number field.

math.NT↗
arXiv2022

Path Integrals and p-adic L-functions

We prove an arithmetic path integral formula for the inverse p-adic absolute values of the Kubota-Leopoldt p-adic L-functions at roots of unity.

math.NT↗
arXiv2021

The \'etale cohomology ring of a punctured arithmetic curve

We compute the cohomology ring $H^*(U,\mathbb{Z}/n\mathbb{Z})$ for $U=X\setminus S$ where $X$ is the spectrum of the ring of integers of a number field $K$ and $S$ is a finite set of finite primes. As a consequence, we obtain an efficient way to compute presentations of $Q_2(G_S)$, where $G_S$ is Galois group of the maximal extension of $K$ unramified outside of a finite set of primes $S$, for varying $K$. This includes the following cases (for $p$ any prime dividing $n$): $\mu_p(\overline{K}) \not\subseteq K$; $S$ does not contain the primes above $p$; and $p=2$ with $K$ admitting real archimedean places. We also show how to recover the classical reciprocity law of the Legendre symbol from the graded commutativity of the cup product.

math.NT↗
arXiv2019

A note on abelian arithmetic BF-theory

We compute some arithmetic path integrals for BF-theory over the ring of integers of a totally imaginary field, which evaluate to natural arithmetic invariants associated to $\mathbb{G}_m$ and abelian varieties.

math.NT↗
arXiv2018

Tensor products of affine and formal abelian groups

In this paper we study tensor products of affine abelian group schemes over a perfect field $k.$ We first prove that the tensor product $G_1 \otimes G_2$ of two affine abelian group schemes $G_1,G_2$ over a perfect field $k$ exists. We then describe the multiplicative and unipotent part of the group scheme $G_1 \otimes G_2$. The multiplicative part is described in terms of Galois modules over the absolute Galois group of $k.$ We describe the unipotent part of $G_1 \otimes G_2$ explicitly, using Dieudonn\'e theory in positive characteristic. We relate these constructions to previously studied tensor products of formal group schemes.

math.AG↗
arXiv2017

Classification of plethories in characteristic zero

We classify plethories over fields of characteristic zero, thus answering a question of Borger-Wieland and Bergman-Hausknecht. All plethories over characteristic zero fields are linear, in the sense that they are free plethories on a bialgebra. For the proof we need some facts from the theory of ring schemes where we extend previously known results. We also classify plethories with trivial Verschiebung over a perfect field of non-zero characteristic and indicate future work.

math.AC↗
arXiv2016

The unramified inverse Galois problem and cohomology rings of totally imaginary number fields

We employ methods from homotopy theory to define new obstructions to solutions of embedding problems. By using these novel obstructions we study embedding problems with non-solvable kernel. We apply these obstructions to study the unramified inverse Galois problem. That is, we show that our methods can be used to determine that certain groups cannot be realized as the Galois groups of unramified extensions of certain number fields. To demonstrate the power of our methods, we give an infinite family of totally imaginary quadratic number fields such that $\text{Aut}(\text{PSL}(2,q^2))$ for $q$ an odd prime power, cannot be realized as an unramified Galois group over $K,$ but its maximal solvable quotient can. To prove this result, we determine the ring structure of the \'etale cohomology ring $H^*(\text{Spec }\mathcal{O}_K;\mathbb{Z}/ 2\mathbb{Z})$ where $\mathcal{O}_K$ is the ring of integers of an arbitrary totally imaginary number field $K.$

math.NT↗