Integration in algebraically closed valued fields

Yin, Yimu

doi:10.1016/j.apal.2010.12.001

Cited by 7 publications

(9 citation statements)

References 13 publications

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“…Obviously -affine implies -affine. With the extra structure afforded by the total ordering, we can reproduce (an analogue of) [Yin11, Lemma 3.18] with a somewhat simpler proof.…”

Section: Definable Sets Inmentioning

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 [HK06] and also the introductions in [Yin11, Yin13b]. There is also a quite comprehensive introduction to the same material in [HL15] 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 [HK06, Yin10, Yin11, HL15], the reader may simply substitute the term ‘theory of power-bounded -convex valued fields’ for ‘theory of algebraically closed valued fields’ or more generally ‘ -minimal theories’ in those introductions and thereby acquire quite a 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%

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

Yin¹

2017

Compositio Math.

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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.

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“…Obviously -affine implies -affine. With the extra structure afforded by the total ordering, we can reproduce (an analogue of) [Yin11, Lemma 3.18] with a somewhat simpler proof.…”

Section: Definable Sets Inmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Yin¹

2017

Compositio Math.

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“…Though we do not currently have a general framework for "constructible motivic distributions", Corollary 3.2 allows us to handle many distributions in the context of motivic integration. [24,40,41] includes distributions; however, it is not yet known if this theory specializes to p-adic integration in the same way as the theory we are presently using. If this specialization were shown, it might yield an alternative approach to the prior discussion.…”

Section: 1mentioning

confidence: 99%

On the computability of some positive-depth supercuspidal characters near the identity

Cluckers¹,

Cunningham²,

Gordon³

et al. 2011

Represent. Theory

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Abstract. This paper is concerned with the values of Harish-Chandra characters of a class of positive-depth, toral, very supercuspidal representations of p-adic symplectic and special orthogonal groups, near the identity element. We declare two representations equivalent if their characters coincide on a specific neighbourhood of the identity (which is larger than the neighbourhood on which the Harish-Chandra local character expansion holds). We construct a parameter space B (that depends on the group and a real number r > 0) for the set of equivalence classes of the representations of minimal depth r satisfying some additional assumptions. This parameter space is essentially a geometric object defined over Q. Given a non-Archimedean local field K with sufficiently large residual characteristic, the part of the character table near the identity element for G(K) that comes from our class of representations is parameterized by the residue-field points of B. The character values themselves can be recovered by specialization from a constructible motivic exponential function, in the terminology of Cluckers and Loeser in a recent paper. The values of such functions are algorithmically computable. It is in this sense that we show that a large part of the character table of the group G(K) is computable.

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“…In this case we say that f ↓ is the contraction of f . The following technical result is a major tool for the Hrushovski-Kazhdan construction as presented in [15].…”

Section: Rv-pullbacks and Special Bijectionsmentioning

confidence: 99%

Special transformations in algebraically closed valued fields

Yin

2010

Annals of Pure and Applied Logic

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We present two of the three major steps in the construction of motivic integration, that is, a homomorphism between Grothendieck semigroups that are associated with a first-order theory of algebraically closed valued fields, in the fundamental work of Hrushovski and Kazhdan [8]. We limit our attention to a simple major subclass of V -minimal theories of the form ACVF S (0, 0), that is, the theory of algebraically closed valued fields of pure characteristic 0 expanded by a (VF, Γ)-generated substructure S in the language L RV . The main advantage of this subclass is the presence of syntax. It enables us to simplify the arguments with many different technical details while following the major steps of the Hrushovski-Kazhdan theory.

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Integration in algebraically closed valued fields

Cited by 7 publications

References 13 publications

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

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

On the computability of some positive-depth supercuspidal characters near the identity

Special transformations in algebraically closed valued fields

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