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Preprint Number 985

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985. Boris Zilber and Lubna Shaheen
Model theory and geometry of representations of rings of integers

Submission date: 12 January 2016


The aim of this project is to attach a geometric structure to the ring of integers. It is generally assumed that the spectrum Spec Z defined by Grothendieck serves this purpose. However, it is still not clear what geometry this object carries. A.Connes and C.Consani published recently an important paper which introduces a much more complex structure called the arithmetic site which includes Spec Z.
Our approach is based on the generalisation of constructions applied by the first author for similar purposes in non-commutative (and commutative) algebraic geometry.
The current version is quite basic. We describe a category of certain representations of integral extensions of Z and establish its tight connection with the space of elementary theories of pseudo-finite fields. From model-theoretic point of view the category of representations is a multisorted structure which we prove to be superstable with pregeometry of trivial type. It comes as some surprise that a structure like this can code a rich mathematics of pseudo-finite fields.

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Full text arXiv 1601.02957: pdf, ps.

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