358 research outputs found
A note on the axioms for Zilber's pseudo-exponential fields
We show that Zilber's conjecture that complex exponentiation is isomorphic to
his pseudo-exponentiation follows from the a priori simpler conjecture that
they are elementarily equivalent. An analysis of the first-order types in
pseudo-exponentiation leads to a description of the elementary embeddings, and
the result that pseudo-exponential fields are precisely the models of their
common first-order theory which are atomic over exponential transcendence
bases. We also show that the class of all pseudo-exponential fields is an
example of a non-finitary abstract elementary class, answering a question of
Kes\"al\"a and Baldwin.Comment: 10 pages, v2: substantial alteration
Model theory of special subvarieties and Schanuel-type conjectures
We use the language and tools available in model theory to redefine and
clarify the rather involved notion of a {\em special subvariety} known from the
theory of Shimura varieties (mixed and pure)
Integration of Modules II: Exponentials
We continue our exploration of various approaches to integration of
representations from a Lie algebra \mbox{Lie} (G) to an algebraic group
in positive characteristic. In the present paper we concentrate on an approach
exploiting exponentials. This approach works well for over-restricted
representations, introduced in this paper, and takes no note of -stability.Comment: Accepted by Transactions of the AMS. This paper is split off the
earlier versions (1, 2 and 3) of arXiv:1708.06620. Some of the statements in
these versions of arXiv:1708.06620 contain mistakes corrected here. Version 2
of this paper: close to the accepted version by the journal, minor
improvements, compared to Version
Natural models of theories of green points
We explicitly present expansions of the complex field which are models of the
theories of green points in the multiplicative group case and in the case of an
elliptic curve without complex multiplication defined over . In
fact, in both cases we give families of structures depending on parameters and
prove that they are all models of the theories, provided certain instances of
Schanuel's conjecture or an analogous conjecture for the exponential map of the
elliptic curve hold. In the multiplicative group case, however, the results are
unconditional for generic choices of the parameters
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