#### Publication Date

6-9-2016

#### Abstract

Lifts of Frobenius on formal schemes X over the p-adic completion of the maximal unramified extension of the p-adic integers may be viewed as arithmetic analogues of vector fields on manifolds. In particular, vector fields on the tangent bundle of a manifold, appearing for instance in Hamiltonian mechanics, have as arithmetic analogues lifts of Frobenius on arithmetic jet spaces J^1(X) of schemes. In this thesis, we first consider the projective space P^m and prove that lifts of Frobenius do not exist on its arithmetic jet spaces J^n(P^m_R) for n, m >= 1. Exhibiting a contrast in the case n=m=1 between the arithmetic and geometric frameworks, we show on the other hand that the space of vector fields on the tangent bundle T(P^1_k) lifting vector fields on P^1_k, where k is an algebraically closed field, has dimension 6 over k. Nevertheless, "normalized" vector fields, which play a role in Hamiltonian mechanics, do not exist on T(P^1_k). We proceed to prove a stronger result for the case n=m=1, that there are no effective Cartier divisors on J^2(P^1) that are finite-to-one over J^1(P^1), and discover that an analogous result holds in geometry. As a final result, we prove the nonexistence of lifts of Frobenius on the first jet space of any smooth quadric hypersurface in projective space.

#### Degree Name

Mathematics

#### Level of Degree

Doctoral

#### Department Name

Mathematics & Statistics

#### First Advisor

Buium, Alexandr

#### First Committee Member (Chair)

Alexandru Buium

#### Second Committee Member

Janet Vassilev

#### Third Committee Member

Michael Nakamaye

#### Fourth Committee Member

James Borger

#### Language

English

#### Keywords

lifts of Frobenius, arithmetic geometry, vector fields

#### Document Type

Dissertation

#### Recommended Citation

Medina, Erik. "Lifts of Frobenius on Arithmetic Jet Spaces of Schemes." (2016). https://digitalrepository.unm.edu/math_etds/28