Integration in algebraically closed valued fields

Yin, Yimu

doi:10.48550/arxiv.0809.0473

Cited by 10 publications

(36 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…From here on, our discussion will be of an increasingly formal nature. Many statements are exact copies of those in [19,20,22] and often the same proofs work, provided that the auxiliary results are replaced by the corresponding ones obtained above. For the reader's convenience, we will write down all the details.…”

Section: Generalized Euler Characteristicmentioning

confidence: 95%

“…Obviously rv-affine implies val-affine. With the extra structure afforded by the total ordering, we can reproduce (an analogue of) [20,Lemma 3.18] with a somewhat simpler proof:…”

Section: Definition 327 (Contractions) a Functionmentioning

confidence: 99%

“…For a description of the ideas and the main results of the Hrushovski-Kazhdan style integration theory, we refer the reader to the original introduction in [10] and also the introductions in [20,22]. There is also a quite comprehensive introduction to the same materials in [11] and, more importantly, a specialized version that relates the Hrushovski-Kazhdan style integration to the geometry and topology of Milnor fibers over the complex field.…”

Section: Introductionmentioning

confidence: 99%

“…The method expounded there will be featured in a sequel to this paper as well. In fact, since much of the work below is closely modeled on that in [10,19,20,11], the reader may simply substitute the term "theory of power-bounded T -convex valued fields" for "theory of algebraically closed valued fields" or more generally "V -minimal theories" in those introductions and thereby acquire a quite good grip on what the results of this paper look like. For the reader's convenience, however, we shall repeat some of the key points, perhaps with minor changes here and there.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Generalized Euler characteristic in power-bounded T-convex valued fields

Yin¹

2017

Compositio Math.

Self Cite

View full text Add to dashboard Cite

We lay the groundwork in this first installment of a series of papers aimed at developing a theory of Hrushovski-Kazhdan style motivic integration for certain type of nonarchimedean o-minimal fields, namely power-bounded T -convex valued fields, and closely related structures. The main result of the present paper is a canonical homomorphism between the Grothendieck semirings of certain categories of definable sets that are associated with the VF-sort and the RV-sort of the language L T RV . Many aspects of this homomorphism can be described explicitly. Since these categories do not carry volume forms, the formal groupification of the said homomorphism is understood as a universal additive invariant or a generalized Euler characteristic. It admits, not just one, but two specializations to Z. The overall structure of the construction is modeled on that of the original Hrushovski-Kazhdan construction.

show abstract

Section: Generalized Euler Characteristicmentioning

confidence: 95%

“…Obviously rv-affine implies val-affine. With the extra structure afforded by the total ordering, we can reproduce (an analogue of) [20,Lemma 3.18] with a somewhat simpler proof:…”

Section: Definition 327 (Contractions) a Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Generalized Euler characteristic in power-bounded T-convex valued fields

Yin¹

2017

Compositio Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…One advantage of having a diagonal cross-section in our formalism is that, unlike in, say, [6,8,11,9], specialization of the universal additive invariant to Z (generalized Euler characteristic) does not require a lengthy structural analysis of definable sets in the RV-sort, since all the relevant prerequisites are now fullfilled by the o-minimality of K. The dimensional part is lost in the groupification…”

Section: Universal Additive Invariantmentioning

confidence: 99%

$T$-convex valued fields with tempered exponentiation

Yin¹

2020

Preprint

Self Cite

View full text Add to dashboard Cite

We continue the effort of grokking the structure of power-bounded T -convex valued fields, whose theory is in general referred to as TCVF. In the present paper our focus is on certain expansion of it that is equipped with a tempered exponential function beyond the valuation ring. In order to construct such a tempered exponential function, the signed value group Γ is also converted into a model of T plus exponentiation and is in fact identified with (a section of) the residue field via the composition of a diagonal cross-section and an angular component map. In a sense, the resulting universal theory TKVF is a halfway point between power-bounded TCVF and exponential TCVF. This theory is reasonably well-behaved. In particular, we show that it admits quantifier elimination in a natural language, a notion of dimension, a generalized Euler characteristic, etc.

show abstract

Fourier transform of the additive group in algebraically closed valued fields

Yin

2014

Sel. Math. New Ser.

Self Cite

View full text Add to dashboard Cite

We continue the study of the Hrushovski-Kazhdan integration theory and consider exponential integrals. The Grothendieck ring is enlarged via a tautological additive character and hence can receive such integrals. We then define the Fourier transform in our integration theory and establish some fundamental properties of it. Thereafter a basic theory of distributions is also developed. We construct the Weil representations in the end as an application. The results are completely parallel to the classical ones.1991 Mathematics Subject Classification. Primary 03C60; Secondary 11S80, 20C08.

show abstract

Integration in algebraically closed valued fields

Cited by 10 publications

References 7 publications

Generalized Euler characteristic in power-bounded T-convex valued fields

Generalized Euler characteristic in power-bounded T-convex valued fields

$T$-convex valued fields with tempered exponentiation

Fourier transform of the additive group in algebraically closed valued fields

Contact Info

Product

Resources

About