文档介绍：BULLETIN (New Series) OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 43, Number 3, July 2006, Pages 285–304
S 0273-0979(06)01107-4
Article electronically published on April 14, 2006
QUADRATIC DIOPHANTINE EQUATIONS, THE CLASS
NUMBER, AND THE MASS FORMULA
GORO SHIMURA
1. The basic setting and two ternary cases
We take a finite-dimensional vector space V over a field F and take also an
F -bilinear symmetric form ϕ: V × V → F . We then put ϕ[x]=ϕ(x, x)for
x ∈ V, thus using the same letter ϕ for the quadratic form and the corresponding
symmetric form. By a quadratic Diophantine equation we mean an equation of the
type
(1) ϕ[x]=q
with a given q ∈ F ×. In particular, in the classical case with F = Q and V = Qn,
we usually assume that ϕ is Z-valued on Zn and q ∈ Z. The purpose of the present
article is to present some new ideas on various arithmetical questions on such an
equation. We start with some of our basic symbols and terminology. For a set X
we denote by #X or #{X} the number (≤∞)ofelementsofX. For an associative
ring R with identity element, we denote by R× the group of invertible elements of
R and by Mn(R) the ring of all square matrices of size n with entries in
×
then put GLn(R)=Mn(R) and denote by 1n the identity element of Mn(R). For
two square matrices A and B of size m and n we denote by diag[A, B] the square
matrix of size m + n with A and B in diagonal blocks and zeros in the remaining
blocks.
Now, given (V, ϕ)asabove,wealwaysassumethat ϕ is nondegenerate. We also
put n =dim(V ) and define, as usual, the orthogonal group Oϕ(V ) and the special
orthogonal group SOϕ(V )by
Oϕ(V )= α∈ GL(V ) ϕ[xα]=ϕ[x] for every x ∈ V ,
SOϕ(V )=Oϕ(V ) ∩ SL(V ), written also SO(ϕ)andSO(V, ϕ).
We let GL(V )actonV on the right, so that xα is the image of x under α.
AsthebasefieldF we take, for the moment, an algebraic number field or its
completion at a nonarchimedean prime. We denote by g the ring of algebraic
integers in the f