Computational Aspects of Mixed Characteristic Witt Vectors and Denominators in Canonical Liftings of Elliptic Curves

Author

Jacob Dennerlein, University of Tennessee, KnoxvilleFollow

Date of Award

5-2023

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Luís R. A. Finotti

Committee Members

Marie Jameson, Shashikant Mulay, Michael W. Berry

Abstract

Given an ordinary elliptic curve E over a field 𝕜 of characteristic p, there is an elliptic curve E over the Witt vectors W(𝕜) for which we can lift the Frobenius morphism, called the canonical lifting of E. The Weierstrass coefficients and the elliptic Teichmüller lift of E are given by rational functions over 𝔽_p that depend only on the coefficients and points of E. Finotti studied the properties of these rational functions over fields of characteristic p ≥ 5. We investigate the same properties for fields of characteristic 2 and 3, make progress on some conjectures of Finotti, and introduce some conjectures of our own. We also investigate the structure of rings of Witt vectors over arbitrary commutative rings and give a computationally useful isomorphism for Witt vectors over ℤ/p^αℤ [alpha].

Recommended Citation

Dennerlein, Jacob, "Computational Aspects of Mixed Characteristic Witt Vectors and Denominators in Canonical Liftings of Elliptic Curves. " PhD diss., University of Tennessee, 2023.
https://trace.tennessee.edu/utk_graddiss/8145