On Universal Binary Hermitian Forms

TL;DR

This paper provides a unified proof for the universality of binary Hermitian forms, leveraging Ramanujan's quadratic forms list and the Bhargava-Hanke 290-Theorem, simplifying previous ad hoc methods.

Contribution

It offers a streamlined, unified proof of the universality of binary Hermitian forms, improving upon prior ad hoc approaches.

Findings

Unified proof of universality for binary Hermitian forms

Relies on Ramanujan's quadratic forms and Bhargava-Hanke 290-Theorem

Simplifies previous classification methods

Abstract

Earnest and Khosravani, Iwabuchi, and Kim and Park recently gave a complete classification of the universal binary Hermitian forms. We give a unified proof of the universalities of these Hermitian forms, relying primarily on Ramanujan's list of universal quadratic forms and on the Bhargava-Hanke 290-Theorem. Our methods bypass nearly all of the ad hoc universality arguments required in the original classification.