Online Archive of University of Virginia Scholarship
Explicit Kummer Theory for some 2-dimensional Formal Group Laws144 views
Author
Aik, Kian Cheong, Mathematics - Graduate School of Arts and Sciences, University of Virginia
Advisors
Gazaki, Evangelia, AS-Mathematics (MATH), University of Virginia
Abstract
The work of Kawachi studies Explicit Kummer Theory in the 1-dimensional case by using ramification theory and formal group properties. Let ϕ : E(K) → E′(K) be a degree p isogeny of elliptic curves; by computing the jumps in the ramification groups and applying it to different formal groups, she is able to obtain an expression for the image of E′(K)/ϕE(K) under the Kummer map. Our goal in this paper is to generalize her results to the case where we have 2-dimensional formal groups. The main difficulty in this case comes from the increase in variables and equations in higher dimensions, which means the methods in the 1-dimensional case do not carry over nicely. We resolve this by restricting our attention to two special cases: the product case and the symmetric case, and we will see that we can obtain analogues of the result in 1-dimensional case by further use of ramification theory.
Degree
PHD (Doctor of Philosophy)
Keywords
Explicit Kummer Theory; Formal Groups; Ramification Theory
Aik, Kian Cheong. Explicit Kummer Theory for some 2-dimensional Formal Group Laws. University of Virginia, Mathematics - Graduate School of Arts and Sciences, PHD (Doctor of Philosophy), 2025-07-24, https://doi.org/10.18130/0xn7-8x79.