# Arithmetic of Hermitian forms

### Goro Shimura

Department of Mathematics Princeton University Princeton NJ 08544-1000 USA

## Abstract

We investigate the following two problems on a hermitian form $Φ$ over an algebraic number field: (1) classification of $Φ$ 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 $ξΦ⋅_{t}ξ_{ρ}=Ψ$, where $Φ$ and $Ψ$ represent hermitian forms. We connect the number of such $ξ$ modulo a group of units with the class number and mass of the unitary group of a form $Θ$ such that $Φ≈Ψ⊕Θ$.

## Cite this article

Goro Shimura, Arithmetic of Hermitian forms. Doc. Math. 13 (2008), pp. 739–774

DOI 10.4171/DM/258