44 research outputs found
Nonstandard Mathematics and New Zeta and L-Functions
This Ph.D. thesis, prepared under the supervision of Professor Ivan Fesenko,
defines new zeta functions in a nonstandard setting and their analytical
properties are developed. Further, p-adic interpolation is presented within a
nonstandard setting which enables the concept of interpolating with respect to
two, or more, distinct primes to be defined.
The final part of the dissertation examines the work of M. J. Shai Haran and
makes initial attempts of viewing it from a nonstandard perspective.Comment: Ph.D. Thesis, University of Nottingham, 2007, 163 page
Real closed fields with nonstandard and standard analytic structure
We consider the ordered field which is the completion of the Puiseux series
field over \bR equipped with a ring of analytic functions on [-1,1]^n which
contains the standard subanalytic functions as well as functions given by
t-adically convergent power series, thus combining the analytic structures from
[DD] and [LR3]. We prove quantifier elimination and o-minimality in the
corresponding language. We extend these constructions and results to rank n
ordered fields \bR_n (the maximal completions of iterated Puiseux series
fields). We generalize the example of Hrushovski and Peterzil [HP] of a
sentence which is not true in any o-minimal expansion of \bR (shown in [LR3] to
be true in an o-minimal expansion of the Puiseux series field) to a tower of
examples of sentences \sigma_n, true in \bR_n, but not true in any o-minimal
expansion of any of the fields \bR,\bR_1,...,\bR_{n-1}.Comment: 15 pages, no figure
Panorama of p-adic model theory
ABSTRACT. We survey the literature in the model theory of p-adic numbers since\ud
Denef’s work on the rationality of Poincaré series. / RÉSUMÉ. Nous donnons un aperçu des développements de la théorie des modèles\ud
des nombres p-adiques depuis les travaux de Denef sur la rationalité de séries de Poincaré,\ud
par une revue de la bibliographie
Operations on integral lifts of K(n)
This very rough sketch is a sequel to arXiv:1808.08587; it presents evidence
that operations on lifts of the functors K(n) to cohomology theories with
values in modules over valuation rings of local number fields, indexed by
Lubin-Tate groups of such fields, are extensions of the groups of automorphisms
of the indexing group laws, by the exterior algebras on the normal bundle to
the orbits of the group laws in the space of lifts.Comment: \S 2.0 hopefully less cryptic. To appear in the proceedings of the
2015 Nagoya conference honoring T Ohkawa. Comments very welcome
Analytic Nullstellens\"atze and the model theory of valued fields
We present a uniform framework for establishing Nullstellens\"atze for power
series rings using quantifier elimination results for valued fields. As an
application we obtain Nullstellens\"atze for -adic power series (both formal
and convergent) analogous to R\"uckert's complex and Risler's real
Nullstellensatz, as well as a -adic analytic version of Hilbert's 17th
Problem. Analogous statements for restricted power series, both real and
-adic, are also considered.Comment: 49 p
Relative p-adic Hodge theory and Rapoport-Zink period domains
As an example of relative p-adic Hodge theory, we sketch the construction of
the universal admissible filtration of an isocrystal (\phi$-module) over the
completion of the maximal unramified extension of Q_p, together with the
associated universal crystalline local system.Comment: 20 page
The perfectoid Tate algebra has uncountable Krull dimension
Let be a perfectoid field with pseudo-uniformizer . We adapt an
argument of Du to show that the perfectoid Tate algebra has an uncountable chain of distinct prime ideals. First,
we conceptualize Du's argument, defining the notion of a 'Newton polygon
formalism' on a ring. We prove a version of Du's theorem in the prescence of a
sufficiently nondiscrete Newton polygon formalism. Then, we apply our framework
to the perfectoid Tate algebra via a "nonstandard" Newton polygon formalism
(roughly, the roles of the series variable and the pseudo-uniformizer
are switched). We conclude a similar statement for multivatiate perfectoid Tate
algebras using the one-variable case. We also answer a question of Heitmann,
showing that if is a complete local noetherian domain of mixed
characteristic , the -adic completion of it's absolute integral
closure has uncountable Krull dimension.Comment: 15 pages, 2 figures, comments welcom