Algorithms for Drinfeld Modules
Loading...
Date
2024-04-23
Authors
Musleh, Yossef
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
Abstract
Drinfeld modules play an important role in number theory over function fields, analogizing that of elliptic curves for the number field setting. The broad success in translating results over from number fields to function fields has motivated the study of Drinfeld modules, particularly in light of the enormous theoretical and practical weight of their elliptic curve counterparts. More recently, this has taken a more computational focus, with the potential for applications to polynomial factorization over finite fields. The main focus of this work will be to study the computation of the characteristic polynomial of the Frobenius endomorphism of a Drinfeld module. This problem has its roots in point counting problems over elliptic curves.
This thesis introduces several new algorithms for computing the characteristic polynomial of the Frobenius endomorphism. In particular, we give three new algorithms for computing the characteristic polynomial for Drinfeld modules of any rank, as well as two modifications to existing algorithms for the rank two case only. We also prove their correctness and analyze their bit complexity. In addition, we analyze algorithms for computing basis representations for the space of morphisms between Drinfeld modules derived from pre-existing approaches described in the literature.
Description
Keywords
Drinfeld module, cohomology, skew polynomials, Frobenius, endomorphism