Author

Erik Medina

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)

Vassilev, Janet

Second Committee Member

Nakamaye, Michael

Third Committee Member

Borger, James

Language

English

Keywords

lifts of Frobenius, arithmetic geometry, vector fields

Document Type

Dissertation

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