A geometric model for representations of Z

Lubna Shaheen

Frank Adams 1,

The aim of this project is to attach a geometric structure to the  ring of integers. 
It is generally assumed that the spectrum of Z defined by Grothendieck serves this purpose.
However, it is still not clear what geometry this object carries. E.g. Y I. Manin discusses what the dimension of Spec$(Z) could be,
speculating that it may be 1,3 or infinity. A.Connes and C.Consani published
recently an important paper which introduces a much more complicated structure
called the arithmetic site on the basis of Spec(Z). Our approach is based on the generalisation of constructions applied by B.Zilber
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 $\Z[\alpha]$ 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 multi-sorted 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. Note that the model-theoretic analysis of the structure establishes that the Morley rank
of $Spec($\mathbb{Z}$) is infinity while the u-rank is 1, thus identifying
formally two of the three Manin's dimensions.
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