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Mathieu Florence

Publications and source records attributed to Mathieu Florence.

At least 19 recordsLinked to original sources
arXiv2026

Rationality problem for norm one tori of tensor products of \'etale algebras and Hasse norm principle

Let $k$ be a field. Let $A=\prod_{i=1}^r K_i$ and $B=\prod_{j=1}^s E_j$ be \'etale $k$-algebras where $K_i$ and $E_j$ are finite separable field extensions of $k$ with $[K_i:k]=m_i$ and $[E_j:k]=n_j$. Let $\mathcal{T}_A=R^{(1)}_{A/k}(\mathbb{G}_m)$ be the norm one torus of the \'etale $k$-algebra $A$. We prove that if $\gcd(m_i,n_j\mid 1\leq i\leq r, 1\leq j\leq s)=1$ and $\mathcal{T}_A$ and $\mathcal{T}_B$ are stably $($resp. retract$)$ $k$-rational, then the algebraic $k$-torus $\mathcal{T}_A\otimes \mathcal{T}_B$ and the norm one torus $\mathcal{T}_{A\otimes B}$ are stably $($resp. retract$)$ $k$-rational. In particular, if $k$ is a global field, then the Hasse norm principle holds for $(A\otimes B)/k$. We introduce a new invariant of $G$-lattices, the permutation order, whose triviality is equivalent to invertibility, and use it to study the rationality of tensor products $T_1\otimes T_2$ of algebraic $k$-tori. As an application, we obtain large families of field extensions $K/k$ for which the Hasse norm principle holds.

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arXiv2025

Lifting Galois representations via Galois cohomology

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 talked about, at the conference held in Ottawa in June 2024, to celebrate J\'an Min\'a\v{c}'s 71th birthday.

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arXiv2024

Lifting Galois representations via Kummer flags

Let $\Gamma$ 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), assume that $\mu_{p^2} \subset F$. We give an elementary and unified proof that every representation $\rho_1: \Gamma \to \mathbf{GL}_d(\mathbb{F}_p)$ lifts to a representation $\rho_2: \Gamma \to \mathbf{GL}_d(\mathbb{Z}/p^2)$. [In case i), it is understood these are continuous.] The actual statement is much stronger: for all $r \geq 1$, under "suitable" assumptions, triangular representations $\rho_r: \Gamma \to \mathbf{B}_d(\mathbb{Z}/p^r)$ lift to $\rho_{r+1}: \Gamma \to \mathbf{B}_d(\mathbb{Z}/p^{r+1})$, in the strongest possible step-by-step sense. Here "suitable" is made precise by the concept of $\textit{Kummer flag}$. An essential aspect of this work is to identify the common properties of groups i) and ii) that suffice to ensure the existence of such lifts.

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arXiv2024

Algebraic Groups with Torsors That Are Versal for All Affine Varieties

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 $\cal C$ if it specializes to every $G$-torsor over a scheme in $\cal C$. A recent result of the first author, Reichstein and Williams says that for any $d\geq 0$, there exists a $G$-torsor over a finite type $k$-scheme that is weakly versal for finite type affine $k$-schemes of dimension at most $d$. The first author also observed that if $G$ is unipotent, then $G$ admits a torsor over a finite type $k$-scheme that is weakly versal for all affine $k$-schemes, and that the converse holds if $\operatorname{char} k=0$. In this work, we extend this to all fields, showing that $G$ is unipotent if and only if it admits a $G$-torsor over a quasi-compact base that is weakly versal for all finite type regular affine $k$-schemes. Our proof is characteristic-free and it also gives rise to a quantitative statement: If $G$ is a non-unipotent subgroup of $\mathbf{GL}_n$, then a $G$-torsor over a quasi-projective $k$-scheme of dimension $d$ is not weakly versal for finite type regular affine $k$-schemes of dimension $n(d+1)+2$. This means in particular that every such $G$ admits a nontrivial torsor over a regular affine $(n+2)$-dimensional variety. When $G$ contains a nontrivial torus, we show that nontrivial torsors already exist over $3$-dimensional smooth affine varieties (even when $G$ is special), and this is optimal in general. In the course of the proof, we show that for every $m,\ell\in\mathbb{N}\cup\{0\}$ with $\ell\neq 1$, there exists a smooth affine $k$-scheme $X$ carrying an $\ell$-torsion line bundle that cannot be generated by $m$ global sections. We moreover study the minimal possible dimension of such an $X$ and show that it is $m$, $m+1$ or $m+2$.

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arXiv2022

On composition of torsors

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 study whether one can find an extension $E$ of $G$ by $H$ so that the composite $Z \to X$ is an $E$-torsor. We give both positive and negative results, depending on the nature of the groups $G$ and $H$.

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arXiv2021

Realisation of abelian varieties as automorphism groups

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 that the answer is positive, if and only if $A$ has only finitely many automorphisms, over an algebraic closure of $F$. When $F=\mathbb C$, this result is due to Lombardo and Maffei. When $F$ is algebraically closed, it was obtained independently by Blanc and Brion.

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arXiv2020

Smooth profinite groups, III: the Smoothness Theorem

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 of Galois cohomology, the Smoothness Theorem provides a new proof of the Norm Residue Isomorphism Theorem, entirely disjoint from motivic cohomology. A byproduct of this approach, is that the latter Theorem follows from mod $p^2$ Kummer theory for fields alone. We moreover extend it, from absolute Galois groups of fields, to algebraic fundamental groups of (not necessarily smooth, nor proper) curves over algebraically closed fields.

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arXiv2020

Common Splitting Fields of Symbol Algebras

We study the common splitting fields of symbol algebras of degree $p^m$ over fields $F$ of $\operatorname{char}(F)=p$. We first show that if any finite number of such algebras share a degree $p^m$ simple purely inseparable splitting field, then they share a cyclic splitting field of the same degree. As a consequence, we conclude that every finite number of symbol algebras of degrees $p^{m_0},\dots,p^{m_t}$ share a cyclic splitting field of degree $p^{m_0+\dots+m_t}$. This generalization recovers the known fact that every tensor product of symbol algebras is a symbol algebra. We apply a result of Tignol's to bound the symbol length of classes in $\operatorname{Br}_{p^m}(F)$ whose symbol length when embedded into $\operatorname{Br}_{p^{m+1}}(F)$ is 2 for $p\in \{2,3\}$. We also study similar situations in other Kato-Milne cohomology groups, where the necessary norm conditions for splitting exist.

math.RA↗
arXiv2020

Smooth profinite groups, I: geometrizing Kummer theory

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 coefficients $p$-primary roots of unity, for a prime $p$. These coefficients are enhanced, to $G$-linearized line bundles in Witt vectors, over $G$-schemes of characteristic $p$. In the second paper, this upgrade is pushed even further, to the scheme-theoretic setting. In this first article, we introduce cyclotomic pairs, smooth profinite groups and $(G,S)$-cohomology. We prove a first lifting theorem for $G$-linearized torsors under line bundles (Theorem A). With the help of the algebro-geometric tools developed in the second article, this formalism is applied in the third one, to prove the Smoothness Theorem, whose essence reads as follows. Let $G$ be profinite group. Assume that, for every open subgroup $H \subset G$, and for $n=1$, the natural arrow $H^n(H,\mathbb{Z}/p^2) \to H^n(H,\mathbb{Z}/p)$ is surjective. Then, it is also surjective for every such $H$, and every $n \geq 2$. Applied to absolute Galois groups, the Smoothness Theorem provides a new proof of the Norm Residue Isomorphism Theorem, entirely disjoint from motivic cohomology.

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arXiv2020

Smooth profinite groups, II: the Uplifting Pattern

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 main technical tools are Hochschild cohomology of affine group schemes and lifting frobenius of vector bundles. The main contribution of this work is the Uplifting Pattern. It is a natural process, to lift a given equivariant extension of vector bundles, to its $\mathbf W_2$-counterpart, upon a `reasonable' combination of base-change and group-change. This is the key ingredient to prove the Smoothness Theorem, in the paper `Smooth profinite groups, III'.

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arXiv2019

Residues on Affine Grassmannians

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 has no subgroups isomorphic to the additive or the commutative group iff G(k[[t]])= G(k((t))). In the case of a reductive group, we offer an explicit connection with the theory of affine grassmannians.

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arXiv2019

On extensions of algebraic groups

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 extensions of $G$ by the center $Z$ of $H$. The proof should be easily generalizable to other contexts. We also study the subset of classes of split extensions and give a quick application by proving a finiteness result on these sets over a finite field.

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arXiv2018

Lifting low-dimensional local systems

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 question, depending on a given profinite group $G$. $Q(G)$: Does every (continuous) representation $G\longrightarrow GL_d(k)$ lift to a representation $G\longrightarrow GL_d(W_r(k))$? We work in the class of cyclotomic pairs (Definition 4.3), first introduced in [DCF] under the name "smooth profinite groups". Using Grothendieck-Hilbert' theorem 90, we show that the algebraic fundamental groups of the following schemes are cyclotomic: spectra of semilocal rings over $\mathbb{Z}[\frac{1}{p}]$, smooth curves over algebraically closed fields, and affine schemes over $\mathbb{F}_p$. In particular, absolute Galois groups of fields fit into this class. We then give a positive partial answer to $Q(G)$, for a cyclotomic profinite group $G$: the answer is positive, when $d=2$ and $r=2$. When $d=2$ and $r=\infty$, we show that any $2$-dimensional representation of $G$ stably lifts to a representation over $W(k)$: see Theorem 6.1. \\When $p=2$ and $k=\mathbb{F}_2$, we prove the same results, up to dimension $d=4$. We then give a concrete application to algebraic geometry: we prove that local systems of low dimension lift Zariski-locally (Corollary 6.3).

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arXiv2018

Lifting vector bundles to Witt vector bundles

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 (=lifting) vector bundles on $X$, to vector bundles on $W_r(X)$-promoting a systematic use of Witt modules and Witt vector bundles. To begin with, we investigate two elementary but significant cases, in which the answer to this question is positive: line bundles, and the tautological vector bundle of a projective bundle over an affine base. We then offer a simple (re)formulation of classical results in deformation theory of smooth varieties over a field $k$ of characteristic $p>0$, and extend them to reduced $k$-schemes. Some of these results were recently recovered, in another form, by Stefan Schr\"oer. As an application, we prove that the tautological vector bundle of the Grassmannian $Gr_{\mathbb{F}_p}(m,n)$ does not extend to $W_2(Gr_{\mathbb{F}_p}(m,n))$, if $2 \leq m \leq n-2$. To conclude, we establish a connection to the work of Zdanowicz, on non-liftability of some projective bundles.

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arXiv2017

Splitting families in Galois cohomology

Let $k$ be a field, with absolute Galois group $\Gamma$. Let $A/k$ be a finite \'etale group scheme of multiplicative type, i.e. a discrete $\Gamma$-module. Let $n \geq 2$ be an integer, and let $x \in H^n(k,A)$ be a cohomology class. We show that there exists a countable set $I$, and a familiy $(X_i)_{i \in I}$ of (smooth, geometrically integral) $k$-varieties, such that the following holds. For any field extension $l/k$, the restriction of $x$ vanishes in $H^n(l,A)$ if and only if (at least) one of the $X_i$'s has an $l$-point. We moreover show that the $X_i$'s can be made into an ind-variety. In the case $n=2$, we note that one variety is enough.

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arXiv2017

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

Let $M_{g, n}$ (respectively, $\overline{M_{g, n}}$) be the moduli space of smooth (respectively stable) curves of genus $g$ with $n$ marked points. Over the field of complex numbers, it is a classical problem in algebraic geometry to determine whether or not $M_{g, n}$ (or equivalently, $\overline{M_{g, n}}$) is a rational variety. Theorems of J. Harris, D. Mumford, D. Eisenbud and G. Farkas assert that $M_{g, n}$ is not unirational for any $n \geqslant 0$ if $g \geqslant 22$. Moreover, P. Belorousski and A. Logan showed that $M_{g, n}$ is unirational for only finitely many pairs $(g, n)$ with $g \geqslant 1$. Finding the precise range of pairs $(g, n)$, where $M_{g, n}$ is rational, stably rational or unirational, is a problem of ongoing interest. In this paper we address the rationality problem for twisted forms of $\overline{M_{g, n}}$ defined over an arbitrary field $F$ of characteristic $\neq 2$. We show that all $F$-forms of $\overline{M_{g, n}}$ are stably rational for $g = 1$ and $3 \leqslant n \leqslant 4$, $g = 2$ and $2 \leqslant n \leqslant 3$, $g = 3$ and $1 \leqslant n \leqslant 14$, $g = 4$ and $1 \leqslant n \leqslant 9$, $g = 5$ and $1 \leqslant n \leqslant 12$.

math.AG↗
arXiv2017

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

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 rational. In this paper we generalize this result as follows. Recall that del Pezzo surfaces of degree $5$ over a field $F$ are precisely the twisted $F$-forms of the moduli space $\overline{M_{0, 5}}$ of stable curves of genus $0$ with $5$ marked points. Suppose $n \geq 5$ is an integer, and $F$ is an infinite field of characteristic $\neq 2$. It is easy to see that every twisted $F$-form of $\overline{M_{0, n}}$ is unirational over $F$. We show that (a) if $n$ is odd, then every twisted $F$-form of $\overline{M_{0, n}}$ is rational over $F$. (b) If $n$ is even, there exists a field extension $F/k$ and a twisted $F$-form $X$ of $\overline{M_{0, n}}$ such that $X$ is not retract rational over $F$.

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