DOCUMENTA MATHEMATICA, Vol. 13 (2008), 739-774

Goro Shimura

Arithmetic of Hermitian Forms

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 $\x\Phi\cdot\tr \x^{\r}=\Psi$, where $\Phi$ and $\Psi$ represent hermitian forms. We connect the number of such $\x$ modulo a group of units with the class number and mass of the unitary group of a form $\Th$ such that $\Phi\approx\Psi\oplus\Th$.

2000 Mathematics Subject Classification: 11E39 (primary), 11E41, 11D09 (secondary)

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