Integer-valued quadratic forms and quadratic Diophantine equations.
Documenta mathematica, Tome 11 (2006), pp. 333-367
We investigate several topics on a quadratic form Φ over an algebraic number field including the following three: (A) an equation ξΦ⋅tξ=Ψ for another form Ψ of a smaller size; (B) classification of Φ over the ring of algebraic integers; (C) ternary forms. In (A) we show that the “class” of such a ξ determines a “class” in the orthogonal group of a form Θ such that Φ≈Ψ⊕Θ. Such was done in [S3] when Ψ is a scalar. We will treat the case of nonscalar Ψ, and prove a class number formula and a mass formula, both of new types. In [S5] we classified all genera of Z-valued Φ. We generalize this to the case of an arbitrary number field, which is topic (B). Topic (C) concerns some explicit forms of the formulas in (A) when Φ is of size 3 and Ψ is a scalar.
@article{10_4171_dm_213,
author = {Goro Shimura},
title = {Integer-valued quadratic forms and quadratic {Diophantine} equations.},
journal = {Documenta mathematica},
pages = {333--367},
year = {2006},
volume = {11},
doi = {10.4171/dm/213},
url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/213/}
}
Goro Shimura. Integer-valued quadratic forms and quadratic Diophantine equations.. Documenta mathematica, Tome 11 (2006), pp. 333-367. doi: 10.4171/dm/213
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