arXiv subjects

Will Johnson

Publications and source records attributed to Will Johnson.

At least 37 records · Page 2Linked to original sources

arXiv2022

Topologizing interpretable groups in $p$-adically closed fields

We consider interpretable topological spaces and topological groups in a $p$-adically closed field $K$. We identify a special class of "admissible topologies" with topological tameness properties like generic continuity, similar to the topology on definable subsets of $K^n$. We show every interpretable set has at least one admissible topology, and every interpretable group has a unique admissible group topology. We then consider definable compactness (in the sense of Fornasiero) on interpretable groups. We show that an interpretable group is definably compact if and only if it has finitely satisfiable generics (fsg), generalizing an earlier result on definable groups. As a consequence, we see that fsg is a definable property in definable families of interpretable groups, and that any fsg interpretable group defined over $\mathbb{Q}_p$ is definably isomorphic to a definable group.

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arXiv2022

Abelian groups definable in $p$-adically closed fields

Recall that a group $G$ has finitely satisfiable generics ($fsg$) or definable $f$-generics ($dfg$) if there is a global type $p$ on $G$ and a small model $M_0$ such that every left translate of $p$ is finitely satisfiable in $M_0$ or definable over $M_0$, respectively. We show that any abelian group definable in a $p$-adically closed field is an extension of a definably compact $fsg$ definable group by a $dfg$ definable group. We discuss an approach which might prove a similar statement for interpretable abelian groups. In the case where $G$ is an abelian group definable in the standard model $\mathbb{Q}_p$, we show that $G^0 = G^{00}$, and that $G$ is an open subgroup of an algebraic group, up to finite factors. This latter result can be seen as a rough classification of abelian definable groups in $\mathbb{Q}_p$.

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arXiv2022

On non-compact $p$-adic definable groups

Peterzil and Steinhorn proved that if a group $G$ definable in an $o$-minimal structure is not definably compact, then $G$ contains a definable torsion-free subgroup of dimension one. We prove here a $p$-adic analogue of the Peterzil-Steinhorn theorem, in the special case of abelian groups. Let $G$ be an abelian group definable in a $p$-adically closed field $M$. If $G$ is not definably compact then there is a definable subgroup $H$ of dimension one which is not definably compact. In a future paper we will generalize this to non-abelian $G$.

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arXiv2022

A note on fsg groups in p-adically closed fields

Let $G$ be a definable group in a $p$-adically closed field $M$. We show that $G$ has finitely satisfiable generics (fsg) if and only if $G$ is definably compact. The case $M = \mathbb{Q}_p$ was previously proved by Onshuus and Pillay.

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arXiv2022

Type-definable NIP fields are Artin-Schreier closed

Let $K$ be a type-definable infinite field in an NIP theory. If $K$ has characteristic $p > 0$, then $K$ is Artin-Schreier closed (it has no Artin-Schreier extensions). As a consequence, $p$ does not divide the degree of any finite separable extension of $K$. This generalizes a theorem of Kaplan, Scanlon, and Wagner.

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arXiv2021

Henselianity in NIP $\mathbb{F}_p$-algebras

We prove an assortment of results on (commutative and unital) NIP rings, especially $\mathbb{F}_p$-algebras. Let $R$ be a NIP ring. Then every prime ideal or radical ideal of $R$ is externally definable, and every localization $S^{-1}R$ is NIP. Suppose $R$ is additionally an $\mathbb{F}_p$-algebra. Then $R$ is a finite product of Henselian local rings. Suppose in addition that $R$ is integral. Then $R$ is a Henselian local domain, whose prime ideals are linearly ordered by inclusion. Suppose in addition that the residue field $R/\mathfrak{m}$ is infinite. Then the Artin-Schreier map $R \to R$ is surjective (generalizing the theorem of Kaplan, Scanlon, and Wagner for fields).

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arXiv2021

The étale-open topology and the stable fields conjecture

For an arbitrary field $K$ and $K$-variety $V$, we introduce the étale-open topology on the set $V(K)$ of $K$-points of $V$. This topology agrees with the Zariski topology, Euclidean topology, or valuation topology when $K$ is separably closed, real closed, or $p$-adically closed, respectively. Topological properties of the étale-open topology corresponds to algebraic properties of $K$. For example, the étale-open topology on $\mathbb{A}^1(K)$ is not discrete if and only if $K$ is large. As an application, we show that a large stable field is separably closed.

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arXiv2020

Dp-finite fields VI: the dp-finite Shelah conjecture

We prove the dp-finite case of the Shelah conjecture on NIP fields. If K is a dp-finite field, then K admits a non-trivial definable henselian valuation ring, unless K is finite, real closed, or algebraically closed. As a consequence, the conjectural classification of dp-finite fields holds. Additionally, dp-finite valued fields are henselian. Lastly, if K is an unstable dp-finite expansion of a field, then K admits a unique definable V-topology.

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arXiv2020

Dp-finite fields V: topological fields of finite weight

We prove that unstable dp-finite fields admit definable V-topologies. As a consequence, the henselianity conjecture for dp-finite fields implies the Shelah conjecture for dp-finite fields. This gives a conceptually simpler proof of the classification of dp-finite fields of positive characteristic. For $n \ge 1$, we define a local class of "$W_n$-topological fields", generalizing V-topological fields. A $W_1$-topology is the same thing as a V-topology, and a $W_n$-topology is some higher-rank analogue. If $K$ is an unstable dp-finite field, then the canonical topology is a definable $W_n$-topology for $n = \operatorname{dp-rk}(K)$. Every $W_n$-topology has between 1 and $n$ coarsenings that are V-topologies. If the given $W_n$-topology is definable in some structure, then so are the V-topological coarsenings.

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arXiv2020

Dp-finite fields IV: the rank 2 picture

We investigate fields of characteristic 0 and dp-rank 2. While we do not obtain a classification, we prove that any unstable field of characteristic 0 and dp-rank 2 admits a unique definable V-topology. If this statement could be generalized to higher ranks, we would obtain the expected classification of fields of finite dp-rank. We obtain the unique definable V-topology by investigating the "canonical topology" defined in earlier work. Contrary to earlier expectations, the canonical topology need not be a V-topology. However, we are able to characterize the canonical topology (on fields of dp-rank 2 and characteristic 0) in terms of differential valued fields. This differential valued structure is obtained through a partial classification of "2-inflators," a sort of generalized valuation that arises naturally in fields of finite rank. Additionally, we give an example of a dp-rank 2 expansion of ACVF with a definable set of full rank and empty interior. This example interferes with certain strategies for proving the henselianity conjecture.

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arXiv2020

Dp-finite fields I: infinitesimals and positive characteristic

We prove that NIP valued fields of positive characteristic are henselian. Furthermore, we partially generalize the known results on dp-minimal fields to dp-finite fields. We prove a dichotomy: if K is a sufficiently saturated dp-finite expansion of a field, then either K has finite Morley rank or K has a non-trivial Aut(K/A)-invariant valuation ring for a small set A. In the positive characteristic case, we can even demand that the valuation ring is henselian. Using this, we classify the positive characteristic dp-finite pure fields.

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arXiv2020

Forking and Dividing in Fields with Several Orderings and Valuations

We consider existentially closed fields with several orderings, valuations, and $p$-valuations. We show that these structures are NTP$_2$ of finite burden, but usually have the independence property. Moreover, forking agrees with dividing, and forking can be characterized in terms of forking in ACVF, RCF, and $p$CF.

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arXiv2020

A criterion for uniform finiteness in the imaginary sorts

Let $T$ be a theory. If $T$ eliminates $\exists^\infty$, it need not follow that $T^{eq}$ eliminates $\exists^\infty$, as shown by the example of the $p$-adics. We give a criterion to determine whether $T^{eq}$ eliminates $\exists^\infty$. Specifically, we show that $T^{eq}$ eliminates $\exists^\infty$ if and only if $\exists^\infty$ is eliminated on all interpretable sets of "unary imaginaries." This criterion can be applied in cases where a full description of $T^{eq}$ is unknown. As an application, we show that $T^{eq}$ eliminates $\exists^\infty$ when $T$ is a C-minimal expansion of ACVF.

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arXiv2019

Counting mod n in pseudofinite fields

We show that in an ultraproduct of finite fields, the mod-$n$ nonstandard size of definable sets varies definably in families. Moreover, if $K$ is any pseudofinite field, then one can assign "nonstandard sizes mod $n$" to definable sets in $K$. As $n$ varies, these nonstandard sizes assemble into a definable strong Euler characteristic on $K$, taking values in the profinite completion $\hat{\mathbb{Z}}$ of the integers. The strong Euler characteristic is not canonical, but depends on the choice of a nonstandard Frobenius. When $\operatorname{Abs}(K)$ is finite, the Euler characteristic has some funny properties for two choices of the nonstandard Frobenius. Additionally, we show that the theory of finite fields remains decidable when first-order logic is expanded with parity quantifiers. However, the proof depends on a computational algebraic geometry statement whose proof is deferred to a later paper.

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arXiv2019

A pathological o-minimal quotient

We give an example of a definable quotient in an o-minimal structure which cannot be eliminated over any set of parameters, giving a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Equivalently, there is an o-minimal structure M whose elementary diagram does not eliminate imaginaries. We also give a positive answer to a related question, showing that any imaginary in an o-minimal structure is interdefinable over an independent set of parameters with a tuple of real elements. This can be interpreted as saying that interpretable sets look "locally" like definable sets, in a sense which can be made precise.

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arXiv2019

Interpretable sets in dense o-minimal structures

We give an example of a dense o-minimal structure in which there is a definable quotient that cannot be eliminated, even after naming parameters. Equivalently, there is an interpretable set which cannot be put in parametrically definable bijection with any definable set. This gives a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Additionally, we show that interpretable sets in dense o-minimal structures admit definable topologies which are "tame" in several ways: (a) they are Hausdorff, (b) every point has a neighborhood which is definably homeomorphic to a definable set, (c) definable functions are piecewise continuous, (d) definable subsets have finitely many definably connected components, and (e) the frontier of a definable subset has lower dimension than the subset itself.

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arXiv2019

Dp-finite fields III: inflators and directories

We develop some tools for analyzing dp-finite fields, including a notion of an ``inflator'' which generalizes the notion of a valuation/specialization on a field. For any field $K$, let $\operatorname{Sub}_K(K^n)$ denote the lattice of $K$-linear subspaces of $K^n$. An ordinary valuation on $K$ with residue field $k$ induces order-preserving dimension-preserving specialization maps from $\operatorname{Sub}_K(K^n)$ to $\operatorname{Sub}_k(k^n)$, satisfying certain compatibility across $n$. An $r$-inflator is a similar family of maps $\{\operatorname{Sub}_K(K^n) \to \operatorname{Sub}_k(k^{rn})\}_{n \in \mathbb{N}}$ scaling dimensions by $r$. We show that 1-inflators are equivalent to valuations, and that $r$-inflators naturally arise in fields of dp-rank $r$. This machinery was ``behind the scenes'' in §10 of [10]. We rework §10 of [10] using the machinery of $r$-inflators.

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arXiv2019

Dp-finite fields II: the canonical topology and its relation to henselianity

We continue our earlier investigation of dp-finite fields. We show that the "heavy sets" of [6] are exactly the sets of full dp-rank. As a consequence, full dp-rank is a definable property in definable families of sets. If $I$ is the group of infinitesimals, we show that $1 + I$ is the group of multiplicative infinitesimals. From this, we deduce that the canonical topology is a field topology. Lastly, we consider the (unlikely) conjecture that the canonical topology is a V-topology. Assuming this conjecture, we deduce the classification of dp-finite fields conjectured by Halevi, Hasson, and Jahnke.

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