## A geometric model for representations of Z

#### Lubna Shaheen

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$(Z) could be, speculating that it may be 1,3 or infinity. A.Connes and C.Consani published recently an important paper which introduces a much more complicated structure called the arithmetic site on the basis of Spec(Z). Our approach is based on the generalisation of constructions applied by B.Zilber for similar purposes in non-commutative (and commutative) algebraic geometry. The current version is quite basic. We describe a category of certain representationsof integral extensions$\Z[\alpha]$of$\Z$and establish its tight connection withthe space of elementary theories of pseudo-finite fields. From model-theoretic pointof view the category of representations is a multi-sorted structure which we prove to be superstable with pregeometry of trivial type. It comes as some surprise that a structure like this can code a rich mathematics of pseudo-finite fields. Note that the model-theoretic analysis of the structure establishes that the Morley rank of$Spec($\mathbb{Z}$) is infinity while the u-rank is 1, thus identifying formally two of the three Manin's dimensions.