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Emiliano Ambrosi

Publications and source records attributed to Emiliano Ambrosi.

13 recordsLinked to original sources
arXiv2026

Topological groupoids with involution and real algebraic stacks

To a topological groupoid endowed with an involution, we associate a topological groupoid of fixed points, generalizing the fixed-point subspace of a topological space with involution. We prove that when the topological groupoid with involution arises from a Deligne-Mumford stack over $\mathbb{R}$, this fixed locus coincides with the real locus of the stack. This provides a topological framework to study real algebraic stacks, and in particular real moduli spaces. Finally, we propose a Smith-Thom type conjecture in this setting, generalizing the Smith-Thom inequality for topological spaces endowed with an involution.

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arXiv2025

Wild Brauer classes via prismatic cohomology

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $X$ a smooth proper $K$-variety with good reduction. Under a mild assumption on the behaviour of Hodge numbers under reduction modulo $p$, we prove that the existence of a non-zero global 2-form on $X$ implies, after a finite extension of $K$, the existence of $p$-torsion Brauer classes with surjective evaluation map. This implies that any smooth proper variety over a number field which satisfies weak approximation over all finite extensions has no non-zero global 2-form. The proof is based on a prismatic interpretation of Brauer classes with eventually constant evaluation, and a Newton-above-Hodge result for the mod $p$ reduction of prismatic cohomology. This generalises work of Bright and the second-named author beyond the ordinary reduction case.

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arXiv2025

On the topology of real algebraic stacks

Motivated by questions arising in the theory of moduli spaces in real algebraic geometry, we develop a range of methods to study the topology of the real locus of a Deligne-Mumford stack over the real numbers. As an application, we verify in several cases the Smith-Thom type inequality for stacks that we conjectured in an earlier work. This requires combining techniques from group theory, algebraic geometry, and topology.

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arXiv2025

Geometrically simple counterexamples to a local-global principle for quadratic twists

Two abelian varieties $A$ and $B$ over a number field $K$ are said to be strongly locally quadratic twists if they are quadratic twists at every completion of $K$. While it was known that this does not imply that $A$ and $B$ are quadratic twists over $K$, the only known counterexamples (necessarily of dimension $\geq 4$) are not geometrically simple. We show that, for every prime $p\equiv 13 \pmod{24}$, there exists a pair of geometrically simple abelian varieties of dimension $p-1$ over $\mathbb{Q}$ that are strongly locally quadratic twists but not quadratic twists. The proof is based on Galois cohomology computations and class field theory.

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arXiv2025

An Artin--Mumford criterion for conic bundles in characteristic two

We prove a characteristic two version of the famous criterion of Artin and Mumford for irrationality of conic bundles. On the one hand, combined with the pathological behaviour of conic bundles in characteristic two, this allows us to construct easier and more explicit examples of irrational conic bundles. On the other hand, degeneration techniques à la Voisin allow to deduce irrationality results in characteristic zero.

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arXiv2025

Reduction modulo $p$ of the Noether problem

Let $R$ be a complete valuation ring of mixed characteristic $(0,p)$ with algebraically closed fraction field $K$ and residue field $k$. Let $X/R$ be a smooth projective morphism. We show that if $X_k$ is stably rational, then $H^3(X_K, \mathbb Z_p)$ is torsion-free. The proof uses integral $p$-adic Hodge theory of Bhatt-Morrow-Scholze and the study of differential forms in positive characteristic. We then apply this result to study the Noether problem for finite $p$-groups.

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arXiv2023

Perfect points of abelian varieties

Let $k$ be an algebraic extension of $\mathbb F_p$ and $K/k$ a regular extension of fields (e.g. $\mathbb F_p(T)/\mathbb F_p$). Let $A$ be a $K$-abelian variety such that all the isogeny factors are neither isotrivial nor of $p$-rank zero. We give a necessary and sufficient condition for the finite generation of $A(K^{perf})$ in terms of the action of $End(A)\otimes \mathbb Q_p$ on the $p$-divisible group $A[p^{\infty}]$ of $A$. In particular we prove that if $End(A)\otimes \mathbb Q_p$ is a division algebra then $A(K^{perf})$ is finitely generated. This implies the "full" Mordell-Lang conjecture for these abelian varieties. In addition we prove that all the infinitely $p$-divisible elements in $A(K^{perf})$ are torsion. These reprove and extend previous results to the non ordinary case. One of the main technical intermediate result is an overconvergence theorem for the Dieudonné module of certain semiabelian schemes over smooth varieties.

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arXiv2022

Betti numbers of real semistable degenerations via real logarithmic geometry

Let $X\rightarrow C$ be a totally real semistable degeneration over a smooth real curve $C$ with degenerate fiber $X_0$. Assuming that the irreducible components of $X_0$ are simple from a cohomological point of view, we give a bound for the individual Betti numbers of a real smooth fiber near $0$ in terms of the complex geometry of the degeneration. This generalizes previous work of Renaudineau-Shaw, obtained via combinatorial techniques, for tropical degenerations of hypersurfaces in smooth toric varieties. The main new ingredient is the use of real logarithmic geometry, which allows to work with not necessarily toric degenerations.

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arXiv2022

Maximal tori of monodromy groups of $F$-isocrystals and an application to abelian varieties

Let $X_0$ be a smooth geometrically connected variety defined over a finite field $\mathbb F_q$ and let $\mathcal E_0^{\dagger}$ be an irreducible overconvergent $F$-isocrystal on $X_0$. We show that if a subobject of minimal slope of the associated convergent $F$-isocrystal $\mathcal E_0$ admits a non-zero morphism to $\mathcal O_{X_0}$ as a convergent isocrystal, then $\mathcal E_0^{\dagger}$ is isomorphic to $\mathcal O^{\dagger}_{X_0}$ as an overconvergent isocrystal. This proves a special case of a conjecture of Kedlaya. The key ingredient in the proof is the study of the monodromy group of $\mathcal E_0^{\dagger}$ and the subgroup defined by $\mathcal E_0$. The new input in this setting is that the subgroup contains a maximal torus of the entire monodromy group. This is a consequence of the existence of a Frobenius torus of maximal dimension. As an application, we prove a finiteness result for the torsion points of abelian varieties, which extends the previous theorem of Lang--Néron and answers positively a question of Esnault.

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arXiv2021

Specialization of Néron-Severi groups in positive characteristic

Let $k$ be an infinite finitely generated field of characteristic $p>0$. Fix a separated scheme $X$ smooth, geometrically connected, and of finite type over $k$ and a smooth proper morphism $f:Y\rightarrow X$. The main result of this paper is that there are ``lots of" closed points $x\in X$ such that the fibre of $f$ at $x$ has the same geometric Picard rank as the generic fibre. If $X$ is a curve we show, under a minimal technical assumption, that this is true for all but finitely many $k$-rational points. In characteristic zero, these results have been proved by André (existence) and Cadoret-Tamagawa (finiteness) using Hodge theoretic methods. To extend the argument in positive characteristic we use the variational Tate conjecture in crystalline cohomology, the comparison between various $p$-adic cohomology theories and independence techniques. The result has applications to the Tate conjecture for divisors, uniform boundedness of Brauer groups, proper families of projective varieties and to the study of families of hyperplane sections of smooth projective varieties.

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arXiv2019

A uniform open image theorem for l-adic representations in positive characteristic

Let $k$ be a finitely generated field of characteristic $p > 0$ and $\ell$ a prime. Let $X$ be a smooth, separated, geometrically connected curve of finite type over $k$ and $ρ: π_1(X)\rightarrow GL_r(\mathbb Z_{\ell})$ a continuous representation of the \etale fundamental group of $X$ with image $G$. Any $k$-rational point $x:Spec(k)\rightarrow X$ induces a local representation $ρ_x: π_1(Spec(k)) \rightarrow π_1(X) \rightarrow GL_r(\mathbb Z_{\ell})$ with image $G_x$. The goal of this paper is to study how $G_x$ varies with $x\in X(k)$. In particular we prove that if $\ell\neq p$ and every open subgroup of $ρ(π_1(X_{\overline k}))$ has finite abelianization, then the set $X_ρ^{ex}(k)$ of $k$-rational points such that $G_x$ is not open in $G$ is finite and there exists a constant $C\geq 0$ such that $[G:G_x]\leq C$ for all $x\in X(k)-X_ρ^{ex}(k)$. This result can be applied to obtain uniform bounds for the $\ell$-primary torsion of groups theoretic invariants in one dimensional families of varieties. For example, torsion of abelian varieties and the Galois invariants of the geometric Brauer group. This extends to positive characteristic previous results of Anna Cadoret and Akio Tamagawa in characteristic 0.

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arXiv2019

Uniform boundedness for Brauer groups of forms in positive characteristic

Let $k$ be a finitely generated field of characteristic $p>0$ and $X$ a smooth and proper scheme over $k$. Recent works of Cadoret, Hui and Tamagawa show that, if $X$ satisfies the $\ell$-adic Tate conjecture for divisors for every prime $\ell\neq p$, the Galois invariant subgroup $Br(X_{\overline k})[p']^{π_1(k)}$ of the prime-to-$p$ torsion of the geometric Brauer group of $X$ is finite. The main result of this note is that, for every integer $d\geq 1$, there exists a constant $C:=C(X,d)$ such that for every finite field extension $k \subseteq k'$ with $[k':k]\leq d$ and every $(\overline k/k')$-form $Y$ of $X$ one has $|(Br(Y\times_{k'}\overline k)[p']^{π_1(k')}|\leq C$. The theorem is a consequence of general results on forms of compatible systems of $π_1(k)$-representations and it extends to positive characteristic a recent result of Orr and Skorobogatov in characteristic zero.

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