Geodesic
Integer-valued quadratic forms and quadratic Diophantine equations.
Documenta mathematica, Tome 11 (2006), pp. 333-367.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We investigate several topics on a quadratic form $\Phi$ over an algebraic number field including the following three: (A) an equation $\,\xi\Phi\cdot\tr\xi=\Psi$ for another form $\Psi$ of a smaller size; (B) classification of $\Phi$ over the ring of algebraic integers; (C) ternary forms. In (A) we show that the "class" of such a $\,\xi\,$ determines a "class" in the orthogonal group of a form $\Th$ such that $\Phi \approx\Psi{\o}plus\Th.$ Such was done in [S3] when $\,\Psi$ is a scalar. We will treat the case of nonscalar $\Psi,$ and prove a class number formula and a mass formula, both of new types. In [S5] we classified all genera of $\Z$-valued $\Phi.$ 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 $\Phi$ is of size 3 and $\Psi$ is a scalar.
Classification : 11E12, 11D09, 11E41 
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     author = {Shimura, Goro},
     title = {Integer-valued quadratic forms and quadratic {Diophantine} equations.},
     journal = {Documenta mathematica},
     pages = {333--367},
     publisher = {mathdoc},
     volume = {11},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2006__11__a6/}
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Shimura, Goro. Integer-valued quadratic forms and quadratic Diophantine equations.. Documenta mathematica, Tome 11 (2006), pp. 333-367. http://geodesic.mathdoc.fr/item/DOCMA_2006__11__a6/