Ph.D Thesis | |

Ph.D Student | Cohen Itai |
---|---|

Subject | Some Theorems on the Arithmetic Plane |

Department | Department of Mathematics |

Supervisor | ASSOCIATE PROF. Shai Haran |

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.