Ph.D Thesis
Ph.D Student Cohen Itai Some Theorems on the Arithmetic Plane Department of Mathematics Professor Shai Haran

Abstract

We consider an analogue of the category of commutative rings, where the initial object is "the field with one element" F. We show how to relate the categories  and by adjusting the commutativity and constructing an idempotent endofunctor in . The pushout generalized ring

is defined under  in  different ways, which gives  different (but isomorphic) ways to go back to commutative rings. It has

as its symmetry group, by permuting additions .

We explain how  injects into
, as an analogue of the  injection.

The research stems from Weil's approach of the Riemann's Hypothesis for projective smooth curves \$\Gamma\$ over finite fields , whose 1946 proof [Weil2] uses intersection theory on  for the Frobenius graph intersected with the diagonal  The proof uses the Castelnuovo- Severi inequality for any divisor  on a product surface , for  projective smooth curves, with injections

The theory of schemes over  constructs  as a pro- object in the geometric category of affine schemes parallel to the generalized rings category together with an open dense embedding , and a non-reduced plane:

The research shows the suggested plane is an integral domain and cancelable as a multiplicative monoid.