Kelly O'Connor, Colorado State University
Authors: Kelly O'Connor
2023 AWM Research Symposium
Recent Advances in Curves and Abelian Varieties [Organized by Renee Bell, Padmavathi Srinivasan, and Isabel Vogt]

In 2019 Zemková defined relative oriented class groups associated to quadratic extensions of number fields L/K, extending work of Bhargava concerning composition laws for binary quadratic forms over number fields of higher degree. Indeed, this work generalized the classical correspondence between the ideal class group of a quadratic number field and classes of binary quadratic forms to any base number field of narrow class number one. Zemková explicitly computed these relative oriented class groups for totally real quadratic extensions of the rationals. We extend Zemková's result for quadratic extensions L/K of a totally real field of narrow class number one, under certain conditions on the absolute Galois group of L/Q.

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