Integer-valued quadratic forms and quadratic Diophantine equations.
Goro ShimuraDepartment of Mathematics Princeton University Princeton, New Jersey 08544-1000 U.S.A
We investigate several topics on a quadratic form over an algebraic number field including the following three: (A) an equation for another form of a smaller size; (B) classification of over the ring of algebraic integers; (C) ternary forms. In (A) we show that the "class" of such a determines a "class" in the orthogonal group of a form such that Such was done in [S3] when is a scalar. We will treat the case of nonscalar and prove a class number formula and a mass formula, both of new types. In [S5] we classified all genera of -valued 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 is of size 3 and is a scalar.
Cite this article
Goro Shimura, Integer-valued quadratic forms and quadratic Diophantine equations.. Doc. Math. 11 (2006), pp. 333–367DOI 10.4171/DM/213