Arithmetic of Hermitian forms

Abstract

We investigate the following two problems on a hermitian form $\Phi$ over an algebraic number field: (1) classification of $\Phi$ over the ring of algebraic integers; (2) hermitian Diophantine equations. The same types of problems for quadratic forms were treated in the author's previous articles. Here we discuss the hermitian case. Problem (2) concerns an equation $\xi \Phi \cdot {}^t{\xi }^{\rho }=\Psi$, where $\Phi$ and $\Psi$ represent hermitian forms. We connect the number of such $\xi$ modulo a group of units with the class number and mass of the unitary group of a form $\Theta$ such that $\Phi \approx \Psi \oplus \Theta$.