43 research outputs found
Trees of definable sets over the p-adics
To a definable subset of Z_p^n (or to a scheme of finite type over Z_p) one
can associate a tree in a natural way. It is known that the corresponding
Poincare series P(X) = \sum_i N_i X^i is rational, where N_i is the number of
nodes of the tree at depth i. This suggests that the trees themselves are far
from arbitrary. We state a conjectural, purely combinatorial description of the
class of possible trees and provide some evidence for it. We verify that any
tree in our class indeed arises from a definable set, and we prove that the
tree of a definable set (or of a scheme) lies in our class in three special
cases: under weak smoothness assumptions, for definable subsets of Z_p^2, and
for one-dimensional sets.Comment: 33 pages, 1 figur
A definable henselian valuation with high quantifier complexity
We give an example of a parameter-free definable henselian valuation ring
which is neither definable by a parameter-free -formula nor by
a parameter-free -formula in the language of rings. This
answers a question of Prestel.Comment: 6 page
Integrability of oscillatory functions on local fields: transfer principles
For oscillatory functions on local fields coming from motivic exponential
functions, we show that integrability over implies integrability over
for large , and vice versa. More generally, the integrability
only depends on the isomorphism class of the residue field of the local field,
once the characteristic of the residue field is large enough. This principle
yields general local integrability results for Harish-Chandra characters in
positive characteristic as we show in other work. Transfer principles for
related conditions such as boundedness and local integrability are also
obtained. The proofs rely on a thorough study of loci of integrability, to
which we give a geometric meaning by relating them to zero loci of functions of
a specific kind.Comment: 44 page
Local integrability results in harmonic analysis on reductive groups in large positive characteristic
Let be a connected reductive algebraic group over a non-Archimedean local
field , and let be its Lie algebra. By a theorem of Harish-Chandra, if
has characteristic zero, the Fourier transforms of orbital integrals are
represented on the set of regular elements in by locally constant
functions, which, extended by zero to all of , are locally integrable. In
this paper, we prove that these functions are in fact specializations of
constructible motivic exponential functions. Combining this with the Transfer
Principle for integrability [R. Cluckers, J. Gordon, I. Halupczok, "Transfer
principles for integrability and boundedness conditions for motivic exponential
functions", preprint arXiv:1111.4405], we obtain that Harish-Chandra's theorem
holds also when is a non-Archimedean local field of sufficiently large
positive characteristic. Under the hypothesis on the existence of the mock
exponential map, this also implies local integrability of Harish-Chandra
characters of admissible representations of , where is an
equicharacteristic field of sufficiently large (depending on the root datum of
) characteristic.Comment: Compared to v2/v3: some proofs simplified, the main statement
generalized; slightly reorganized. Regarding the automatically generated text
overlap note: it overlaps with the Appendix B (which is part of
arXiv:1208.1945) written by us; the appendix and this article cross-reference
each other, and since the set-up is very similar, some overlap is unavoidabl
Achieving snaky
We prove that the polyomino generally known as snaky is a three-dimensional winner, that it loses on an 8 × 8 board, and that its handicap number is at most one