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Ambrus Pál

Publications and source records attributed to Ambrus Pál.

At least 19 recordsLinked to original sources
arXiv2026

Weak and genuine A_n-formality for number fields

We introduce weak $A_n$-formality for Galois cochain algebras and, for profinite groups whose mod-$p$ cohomology is concentrated in degrees at most two, characterise weak and genuine $A_n$-formality through finite embedding problems and the projective system they form. We then study $A_n$-formality for number fields. Let $K$ be a number field and let $p$ be an odd prime with $μ_p\not\subset K$. For $p\geq 5$, and for $p=3$ under a natural local cyclotomic condition, we prove that every associated finite $A_3$-embedding problem is solvable. In contrast, for every odd $p$ with $μ_p\not\subset K$, we show that no compatible family of finite-stage solutions exists, and hence $C^\bullet(G_K,\mathbb F_p)$ is not genuinely $A_3$-formal. The proof uses Chebotarev density, governing fields, and the Gras-Munnier criterion to obstruct the compatibility of finite-stage solutions, and it builds on previous work of Maire-Mináč-Ramakrishna-Tân. This shows that genuine $A_3$-formality detects a global compatibility obstruction for number fields invisible to every individual finite embedding problem and invisible to obstructions arising from individual strong Massey vanishing problems.

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arXiv2026

A homotopy exact sequence for overconvergent isocrystals

In this article we prove exactness of the homotopy sequence of overconvergent $p$-adic fundamental groups for a smooth and projective morphism in characteristic $p$. We do so by first proving a corresponding result for rigid analytic varieties in characteristic $0$, following dos Santos in the algebraic case. In characteristic $p$, we then proceed by a series of reductions to the case of a liftable family of curves, where we can apply the rigid analytic result.

math.AG↗
arXiv2026

$A_3$-formality for pro-2 Demushkin groups

We study a weak form of formality for differential graded algebras, called $A_3$-formality, and show that the differential graded $\mathbb{F}_2$-algebras of continuous cochains of all pro-$2$ Demushkin groups are $A_3$-formal. We prove this result by an explicit computation of the Benson--Krause--Schwede canonical class using the classification of pro-$2$ Demushkin groups by Demushkin, Serre, and Labute. Compared to the case of odd primes, the new idea is to interpret the data of the canonical class as defining systems of higher Massey products.

math.GR↗
arXiv2026

The minimal width of universal $p$-adic ReLU neural networks

We determine the minimal width of $p$-adic neural networks with the universal approximation property for continuous $\mathbb Q_p$-valued functions on compact open subsets with respect to the $L_q$ norms and the $C_1$ norm, where the activation function is a natural $p$-adic analogue of the ReLU function.

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arXiv2026

$A_3$-formality for Demushkin groups at odd primes

We study a weak form of formality for differential graded algebras, called $A_3$-formality, for the cohomology of pro-p Demushkin groups at odd primes p. We show that the differential graded $\mathbb{F}_p$-algebras of continuous cochains of Demushkin groups with q-invariant not equal 3 are $A_3$-formal, whereas Demushkin groups with q-invariant 3 are not $A_3$-formal. We prove these results by an explicit computation of the Benson-Krause-Schwede canonical class in Hochschild cohomology.

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arXiv2025

An arithmetic Yau-Zaslow formula

We prove an arithmetic refinement of the Yau-Zaslow formula by replacing the classical Euler characteristic in Beauville's argument by a "motivic Euler characteristic", related to the work of Levine. Our result implies similar formulas for other related invariants, including a generalisation of a formula of Kharlamov and Rasdeaconu on counting real rational curves on real K3 surfaces, and Saito's determinant of cohomology.

math.AG↗
arXiv2025

Quasi-Boolean groups

We give several equivalent characterisations of the maximal pro-2 quotients of real projective groups. In particular, for pro-2 real projective groups we provide a presentation in terms of generators and relations, and a purely cohomological characterisation. As a consequence we explicitly reconstruct such groups from their mod 2 cohomology rings.

math.GR↗
arXiv2025

Real projective groups are formal

We prove that the mod 2-cohomology algebras of real projective groups are formal. As a consequence we derive the Hopkins-Wickelgren formality and the strong Massey vanishing conjecture for a large class of fields.

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arXiv2022

The strong Massey vanishing conjecture for fields with virtual cohomological dimension at most $1$

We show that a strong vanishing conjecture for $n$-fold Massey products holds for fields of virtual cohomological dimension at most $1$ using a theorem of Haran. We also prove the same for PpC fields, using results of Haran--Jarden. Finally we construct a pro-$2$ group which satisfies the weak Massey vanishing property for every $n\geq3$, but does not satisfy the strong Massey vanishing property for $n=4$.

math.AG↗
arXiv2022

Simplicial homotopy theory of algebraic varieties over real closed fields, Part 1

We study the homotopy type of the simplicial set of continuous semi-algebraic simplexes of an algebraic variety defined over a real closed field, which we will call the real homotopy type. We prove an analogue of the theorem of Artin-Mazur comparing the real homotopy type with the étale homotopy type. This paper is part one of a sequence of papers on this topic.

math.AG↗
arXiv2022

Rank 2 local systems and abelian varieties II

Let $X/\mathbb{F}_{q}$ be a smooth, geometrically connected, quasiprojective variety. Let $\mathcal{E}$ be a semisimple overconvergent $F$-isocrystal on $X$. Suppose that irreducible summands $\mathcal{E}_i$ of $\mathcal E$ have rank 2, determinant $\bar{\mathbb{Q}}_p(-1)$, and infinite monodromy at $\infty$. Suppose further that for each closed point $x$ of $X$, the characteristic polynomial of $\mathcal{E}$ at $x$ is in $\mathbb{Q}[t]\subset \mathbb Q_p[t]$. Then there exists a non-trivial open set $U\subset X$ such that $\mathcal{E}|_U$ comes from a family of abelian varieties on $U$. As an application, let $L_1$ be an irreducible lisse $\bar{\mathbb{Q}}_l$ sheaf on $X$ that has rank 2, determinant $\bar{\mathbb{Q}}_l(-1)$, and infinite monodromy at $\infty$. Then all crystalline companions to $L_1$ exist (as predicted by Deligne's crystalline companions conjecture) if and only if there exists a non-trivial open set $U\subset X$ and an abelian scheme $π_U\colon A_U\rightarrow U$ such that $L_1|_U$ is a summand of $R^1(π_U)_*\bar{\mathbb{Q}}_l$.

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arXiv2021

Rank 2 local systems and abelian varieties

Let $X/\mathbb{F}_{q}$ be a smooth geometrically connected variety. Inspired by work of Corlette-Simpson over $\mathbb{C}$, we formulate a conjecture that absolutely irreducible rank 2 local systems with infinite monodromy on $X$ come from families of abelian varieties. When $X$ is a projective variety, we prove a Lefschetz-style theorem for abelian schemes of $\text{GL}_2$-type on $X$, modeled after a theorem of Simpson. If one assumes a strong form of Deligne's ($p$-adic) \emph{companions conjecture} from Weil II, this implies that our conjecture for projective varieties also reduces to the case of projective curves. We also answer affirmitavely a question of Grothendieck on extending abelian schemes via their $p$-divisible groups.

math.AG↗
arXiv2019

Cycle classes in overconvergent rigid cohomology and a semistable Lefschetz $(1,1)$ theorem

In this article we prove a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, stating that a rational (logarithmic) line bundle on the special fibre of a semistable scheme over $k [\![ t ]\!]$ lifts to the total space if and only if its first Chern class does. The proof is elementary, using standard properties of the logarithmic de Rham-Witt complex. As a corollary, we deduce similar algebraicity lifting results for cohomology classes on varieties over global function fields. Finally, we give a counter example to show that the variational Tate conjecture for divisors cannot hold with $\mathbb{Q}_p$-coefficients.

math.AG↗
arXiv2017

Iterated line integrals over Laurent series fields of characteristic p

Inspired by Besser's work on Coleman integration, we use $\nabla$-modules to define iterated line integrals over Laurent series fields of characteristic $p$ taking values in double cosets of unipotent $n\times n$ matrices with coefficients in the Robba ring divided out by unipotent $n\times n$ matrices with coefficients in the bounded Robba ring on the left and by unipotent $n\times n$ matrices with coefficients in the constant field on the right. We reach our definition by looking at the analogous theory for Laurent series fields of characteristic $0$ first, and reinterpreting the classical formal logarithm in terms of $\nabla$-modules on formal schemes. To illustrate that the new $p$-adic theory is non-trivial, we show that it includes the $p$-adic formal logarithm as a special case.

math.AG↗
arXiv2015

Pathologies of the Brauer-Manin obstruction

We construct a conic bundle over an elliptic curve over a real quadratic field that is a counterexample to the Hasse principle not explained by the étale Brauer-Manin obstruction. We also give simple examples of threefolds with the same property that are families of 2-dimensional quadrics, and discuss some other examples and general properties of the Brauer-Manin obstruction.

math.AG↗
arXiv2015

Rigid cohomology over Laurent series field I: First definitions and basic properties

This is the first in a series of papers in which we construct and study a new $p$-adic cohomology theory for varieties over Laurent series fields $k(\!(t)\!)$ in characteristic $p$. This will be a version of rigid cohomology, taking values in the bounded Robba ring $\mathcal{E}_K^\dagger$, and in this paper, we give the basic definitions and constructions. The cohomology theory we define can be viewed as a relative version of Berthelot's rigid cohomology, and is constructed by compactifying $k(\!(t)\!)$-varieties as schemes over $k[\![ t]\!]$ rather than over $k(\!(t)\!)$. We reprove the foundational results necessary in our new context to show that the theory is well defined and functorial, and we also introduce a category of `twisted' coefficients. In latter papers we will show some basic structural properties of this theory, as well as discussing some arithmetic applications including the weight monodromy conjecture and independence of $\ell$ results for equicharacteristic local fields.

math.NT↗
arXiv2015

Rigid cohomology over Laurent series fields II: Finiteness and Poincaré duality for smooth curves

In this paper we prove that the $\mathcal{E}^\dagger_K$-valued cohomology, introduced in [9] is finite dimensional for smooth curves over Laurent series fields $k((t))$ in positive characteristic, and forms an $\mathcal{E}^\dagger_K$-lattice inside `classical' $\mathcal{E}_K$-valued rigid cohomology. We do so by proving a suitable version of the p-adic local monodromy theory over $\mathcal{E}^\dagger_K$, and then using an étale pushforward for smooth curves to reduce to the case of $\mathbb{A}^1$. We then introduce $\mathcal{E}^\dagger_K$-valued cohomology with compact supports, and again prove that for smooth curves, this is finite dimensional and forms an $\mathcal{E}^\dagger_K$-lattice in $\mathcal{E}_K$-valued cohomology with compact supports. Finally, we prove Poincaré duality for smooth curves, but with restrictions on the coefficients.

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