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Publications (23)

math.AG2020

On the rationality problem for forms of moduli spaces of stable marked curves of positive genus

Mathieu Florence, Norbert Hoffmann, Zinovy Reichstein

Let $M_{g, n}$ (respectively, $M_{g, n}$ ) be the moduli space of smooth (respectively stable) curves of genus $g$ with $n$ marked points. Over the field of complex numbe…

math.AG2021

Residues on Affine Grassmannians

Mathieu Florence, Philippe Gille

Given a linear group G over a field k, we define a notion of index and residue of an element g of G(k((t)). This provides an alternative proof of Gabber's theorem stating that G ha…

math.AG2025

Algebraic Groups with Torsors That Are Versal for All Affine Varieties

Uriya A. First, Mathieu Florence, Zev Rosengarten

Let $k$ be a field and let $G$ be an affine algebraic group over $k$ . Call a $G$ -torsor weakly versal for a class of $k$ -schemes $C$ if it specializes to every $G$ -torsor over…

math.AG2025

Smooth profinite groups, II: the Uplifting Pattern

Mathieu Florence

This text presents a scheme-theoretic enhancement of the theory of smooth profinite groups and cyclotomic pairs, introduced in the paper `Smooth profinite groups, I'. To do so, our…

math.AG2013

The Lie algebra of type G_2 is rational over its quotient by the adjoint action

Dave Anderson, Mathieu Florence, Zinovy Reichstein

Let G be a split simple group of type G_2 over a field k, and let g be its Lie algebra. Answering a question of Colliot-ThÃ©lÃ¨ne, KunyavskiÄ, Popov, and Reichstein, we show that…

math.NT2026

Lifting Galois representations via Kummer flags

Andrea Conti, Cyril Demarche, Mathieu Florence

Let $I^$ be either i) the absolute Galois group of a local field $F$ , or ii) the topological fundamental group of a closed connected orientable surface of genus $g$ . In case i), as…

math.AG2025

Lifting Galois representations via Galois cohomology

Mathieu Florence

This is a survey paper, on recent progress around lifting Galois representations. I focused on approaches that solely rely on Galois cohomology. Emphasis is laid on material I talk…

math.RA2022

Common Splitting Fields of Symbol Algebras

Adam Chapman, Mathieu Florence, Kelly McKinnie

We study the common splitting fields of symbol algebras of degree $p^{m}$ over fields $F$ of $char (F) = p$ . We first show that if any finite number of such algebras shar…

math.AG2017

The rationality problem for forms of $M_{0, n}$

Mathieu Florence, Zinovy Reichstein

Let $X$ be a del Pezzo surface of degree $5$ defined over a field $F$ . A theorem of Yu. I. Manin and P. Swinnerton-Dyer asserts that every Del Pezzo surface of degree $5$ is ration…

math.AG2025

Smooth profinite groups, III: the Smoothness Theorem

Charles De Clercq, Mathieu Florence

Let $p$ be a prime. In this article, we prove the Smoothness Theorem, which asserts that a $(1, 1)$ -cyclotomic pair is $(n, 1)$ -cyclotomic, for all $n \geq 1$ . In the particular case…

math.AG2023

Realisation of linear algebraic groups as automorphism groups

Mathieu Florence

Let $G$ be a linear algebraic group, over a field $F$ . We show that $G$ is isomorphic to the automorphism group scheme of a smooth projective $F$ -variety, defined as the blow-up of…

math.AG2022

Realisation of abelian varieties as automorphism groups

Mathieu Florence

Let $A / F$ be an abelian variety over a field. Does there exist a smooth projective $F$ -variety $X$ , such that $A$ is isomorphic to the automorphism group scheme of $X / F$ ? We show t…

math.AG2017

A constructive approach to a conjecture by Voskresenskii

Mathieu Florence, Michel van Garrel

Voskresenskii conjectured that stably rational tori are rational. Klyachko proved this assertion for a wide class of tori by general principles. We re-prove Klyachko's result by pr…

math.AG2010

GÃ©omÃ©trie birationnelle Ã©quivariante des grassmanniennes

Mathieu Florence

Let k be a field, and A a finite-dimensional k-algebra. Let d be an integer. Denote by Gr(d,A) the Grassmannian of d-subspaces of A (viewed as a k-vector space), and by GL_1(A) the…

math.AG2020

On extensions of algebraic groups

Mathieu Florence, Giancarlo Lucchini Arteche

We extend to the context of algebraic groups a classic result on extensions of abstract groups relating the set of isomorphism classes of extensions of $G$ by $H$ with that of exte…

math.AG2023

On composition of torsors

Mathieu Florence, Diego Izquierdo, Giancarlo Lucchini Arteche

Let $K$ be a field, let $X$ be a connected smooth $K$ -scheme and let $G, H$ be two smooth connected $K$ -group schemes. Given $Y \to X$ a $G$ -torsor and $Z \to Y$ an $H$ -torsor, we s…

math.AG2025

Smooth profinite groups, I: geometrizing Kummer theory

Charles De Clercq, Mathieu Florence

In this series of three papers, we introduce and study cyclotomic pairs and smooth profinite groups. They are a geometric axiomatisation of Kummer theory for fields, with coefficie…

math.AG2021

Lifting low-dimensional local systems

Charles De Clercq, Mathieu Florence

Let $k$ be a field of characteristic $p > 0$ . Denote by $W_{r} (k)$ the ring of truntacted Witt vectors of length $r \geq 2$ , built out of $k$ . In this text, we consider the following q…

math.AG2023

Lifting vector bundles to Witt vector bundles

Charles De Clercq, Mathieu Florence, Giancarlo Lucchini Arteche

Let $X$ be a scheme. Let $r \geq 2$ be an integer. Denote by $W_{r} (X)$ the scheme of Witt vectors of length $r$ , built out of $X$ . We are concerned with the question of extending (=…

math.AG2018

Splitting families in Galois cohomology

Cyril Demarche, Mathieu Florence

Let $k$ be a field, with absolute Galois group $I^$ . Let $A / k$ be a finite Ã©tale group scheme of multiplicative type, i.e. a discrete $I^$ -module. Let $n \geq 2$ be an integer, an…

math.NT2010

The valuation criterion for normal basis generators

Bart de Smit, Mathieu Florence, Lara Thomas

If $L / K$ is a finite Galois extension of local fields, we say that the valuation criterion $V C (L / K)$ holds if there is an integer $d$ such that every element $x \in L$ with valuati…

math.AG2026

Mathieu Florence, Akinari Hoshi, Aiichi Yamasaki

Let $k$ be a field. Let $A = \prod_{i = 1}^{r} K_{i}$ and $B = \prod_{j = 1}^{s} E_{j}$ be Ã©tale $k$ -algebras where $K_{i}$ and $E_{j}$ are finite separable field extensions of $k$ with $[K_{i} : k] = m_{i}$ …

math.CO2009

Completely symmetric configurations for sigma-games on grid graphs

The paper deals with sigma-games on grid graphs (in dimension 2 and more) and conditions under which any completely symmetric configuration of lit vertices can be reached -- in par…

Publications (23)

On the rationality problem for forms of moduli spaces of stable marked curves of positive genus

Residues on Affine Grassmannians

Algebraic Groups with Torsors That Are Versal for All Affine Varieties

Smooth profinite groups, II: the Uplifting Pattern

The Lie algebra of type G_2 is rational over its quotient by the adjoint action

Lifting Galois representations via Kummer flags

Lifting Galois representations via Galois cohomology

Common Splitting Fields of Symbol Algebras

The rationality problem for forms of M0,n​​

Smooth profinite groups, III: the Smoothness Theorem

Realisation of linear algebraic groups as automorphism groups

Realisation of abelian varieties as automorphism groups

A constructive approach to a conjecture by Voskresenskii

GÃ©omÃ©trie birationnelle Ã©quivariante des grassmanniennes

On extensions of algebraic groups

On composition of torsors

Smooth profinite groups, I: geometrizing Kummer theory

Lifting low-dimensional local systems

Lifting vector bundles to Witt vector bundles

Splitting families in Galois cohomology

The valuation criterion for normal basis generators

Rationality problem for norm one tori of tensor products of Ã©tale algebras and Hasse norm principle

Completely symmetric configurations for sigma-games on grid graphs

The rationality problem for forms of $M_{0, n}$