arXiv subjects

Konstantinos Kartas

Publications and source records attributed to Konstantinos Kartas.

8 recordsLinked to original sources

arXiv2026

Transfer principles and the Kato-Kuzumaki conjecture

We show that for tame valued fields of equal characteristic with divisible value group, the $C_i$ property lifts from the residue field to the valued field under suitable hypotheses on the residue field. We apply this transfer principle to prove Kato-Kuzumaki's conjecture in full generality for several arithmetically significant fields, for instance the field $\mathbf{C}(x_1,\dots,x_m)(\!(t_1)\!)\dots(\!(t_n)\!)$, and the perfections of both $\overline{\mathbf{F}}_p(x_1,\dots,x_m)(\!(t_1)\!)\dots(\!(t_n)\!)$ and $\mathbf{F}_p(\!(t_1)\!)\dots(\!(t_n)\!)$. Finally, we prove that $\mathbf{Q}_p$ satisfies the strong $C_1^1$ property, thereby answering a question of Wittenberg.

math.NT↗

arXiv2026

Perfectoid $C_i$ transfer

We prove a perfectoid analogue of the Ax-Kochen theorem on zeros of $p$-adic forms: Given $d\in \mathbb{N}$, there is a finite totally ramified extension $E/\mathbb{Q}_p$ such that every untilt of $\mathbb{F}_p(\!(t^{1/p^{\infty}})\!)$ containing $E$ is $C_2(d)$. We also prove a similar result for the existence of rational points in rationally connected varieties over perfectoid field extensions of $\mathbb{Q}_p^{ur}$.

math.NT↗

arXiv2024

Beyond the Fontaine-Wintenberger theorem

Given a perfectoid field, we find an elementary extension and a henselian defectless valuation on it, whose value group is divisible and whose residue field is an elementary extension of the tilt. This specializes to the almost purity theorem over perfectoid valuation rings and Fontaine-Wintenberger. Along the way, we prove an Ax-Kochen/Ershov principle for certain deeply ramified fields, which also uncovers some new model-theoretic phenomena in positive characteristic. Notably, we get that the perfect hull of $\mathbb{F}_p(t)^h$ is an elementary substructure of the perfect hull of $\mathbb{F}_p(\!(t)\!)$.

math.AC↗

arXiv2024

On geometrically $C_1$ fields

A field $k$ is called geometrically $C_1$ if every smooth projective separably rationally connected $k$-variety has a $k$-rational point. Given a henselian valued field of equal characteristic $0$ with divisible value group, we show that the property of being geometrically $C_1$ lifts from the residue field to the valued field. We also prove that algebraically maximal valued fields with divisible value group and finite residue field are geometrically $C_1$. In particular, any maximal totally ramified extension of a local field is geometrically $C_1$.

math.AG↗

arXiv2023

Valued fields with a total residue map

When $k$ is a finite field, Becker-Denef-Lipschitz (1979) observed that the total residue map $\text{res}:k(\!(t)\!)\to k$, which picks out the constant term of the Laurent series, is definable in the language of rings with a parameter for $t$. Driven by this observation, we study the theory $\text{VF}_{\text{res},ι}$ of valued fields equipped with a linear form $\text{res}:K\to k$ which specializes to the residue map on the valuation ring. We prove that $\text{VF}_{\text{res},ι}$ does not admit a model companion. In addition, we show that the power series field $(k(\!(t)\!),\text{res})$, equipped with such a total residue map, is undecidable whenever $k$ is an infinite field. As a consequence, we get that $(\mathbb{C}(\!(t)\!), \text{Res}_0)$ is undecidable, where $\text{Res}_0:\mathbb{C}(\!(t)\!)\to \mathbb{C}:f\mapsto \text{Res}_0(f)$ maps $f$ to its complex residue at $0$.

math.LO↗

arXiv2022

Decidability via the tilting correspondence

We prove a relative decidability result for perfectoid fields. This applies to show that the fields $\mathbb{Q}_p(p^{1/p^{\infty}})$ and $\mathbb{Q}_p(ζ_{p^{\infty}})$ are (existentially) decidable relative to the perfect hull of $ \mathbb{F}_p(\!(t)\!)$ and $\mathbb{Q}_p^{ab}$ is (existentially) decidable relative to the perfect hull of $\overline{ \mathbb{F}}_p(\!(t)\!)$. We also prove some unconditional decidability results in mixed characteristic via reduction to characteristic $p$.

math.LO↗

arXiv2022

An undecidability result for the asymptotic theory of $p$-adic fields

Fix a prime $p$. We prove that the set of sentences true in all but finitely many finite extensions of $\mathbb{Q}_p$ is undecidable in the language of valued fields with a cross-section. The proof goes via reduction to characteristic $p$, adapting Pheidas' proof of the undecidability of $\mathbb{F}_p(\!(t)\!)$ with a predicate for powers of $t$. This answers a variant of a question of Derakhshan-Macintyre.

math.LO↗

arXiv2022

Diophantine problems over tamely ramified fields

Assuming a certain form of resolution of singularities, we prove a general existential Ax-Kochen/Ershov principle for tamely ramified fields in all characteristics. This specializes to well-known results in residue characteristic $0$ and unramified mixed characteristic. It also encompasses the conditional existential decidability results known for $\mathbb{F}_p(\!(t)\!)$ and its finite extensions, due to Denef-Schoutens. On the other hand, it also applies to the setting of infinite ramification, providing us with an abundance of infinitely ramified extensions of $\mathbb{Q}_p$ and $\mathbb{F}_p(\!(t)\!)$ that are existentially decidable.

math.AG↗