Geodesic
Arithmetic of Hermitian forms
Documenta mathematica, Tome 13 (2008), pp. 739-774
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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⋅tξρ=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.
DOI : 10.4171/dm/258
Classification : 11D09, 11E39, 11E41 
@article{10_4171_dm_258,
     author = {Goro Shimura},
     title = {Arithmetic of {Hermitian} forms},
     journal = {Documenta mathematica},
     pages = {739--774},
     year = {2008},
     volume = {13},
     doi = {10.4171/dm/258},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/258/}
}
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Goro Shimura. Arithmetic of Hermitian forms. Documenta mathematica, Tome 13 (2008), pp. 739-774. doi: 10.4171/dm/258

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