Explicit Kummer Theory for some 2-dimensional Formal Group Laws 144 views

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.

PHD (Doctor of Philosophy)

Explicit Kummer Theory; Formal Groups; Ramification Theory

English

2025-07-24