1,720,980 research outputs found
Grothendieck rings of Laurent series fields
No full textAbstractWe study Grothendieck rings (in the sense of model theory) of fields, extending previous work of Haskell and the author in [R. Cluckers, D. Haskell, Bull. Symbolic Logic 7 (2) (2001) 262–269]. We construct definable bijections from the line to the line minus one point in the language of rings for valued fields like fields of formal Laurent series over p-adic numbers and fields of formal Laurent series over local fields of strictly positive characteristic. It follows that the Grothendieck rings of these fields are trivial
ORBITAL INTEGRALS FOR LINEAR GROUPS
No full textAbstract. For a linear group G acting on an absolutely irreducible variety X over Q, we describe the orbits of X(Qp) under G(Qp) and of X(Fp((t))) under G(Fp((t))) for p big enough. This allows us to show that the degree of a wide class of orbital integrals over Qp or Fp((t)) is ≤ 0 for p big enough, and similarly for all finite field extensions of Qp and Fp((t))
Grothendieck rings of Z-valued fields
No full textWe prove the triviality of the Grothendieck ring of a Z-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K² to itself minus a point. When we specialize to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point
Analytic van der Corput Lemma for p-adic and Fq((t)) oscillatory integrals, singular Fourier transforms, and restriction theorems
No full textAbstractWe give a non-archimedean analogue of the van der Corput Lemma on oscillating integrals, where the condition of sufficient smoothness for the phase in the real case is replaced by the condition that the phase is a convergent power series. This result allows us, in analogy to the real situation, to study singular Fourier transforms on suitably curved (p-adic analytic) manifolds. As an application we give a restriction theorem for Fourier transforms of Lq functions to suitably curved analytic manifolds over non-archimedean local fields, similar to a real restriction result by E.M. Stein. Several analogues of the van der Corput Lemma were already known when the phase is a polynomial
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