Contemporary Mathematics
Volume 249, 1999
Arithmetic of Indefinite Quadratic Forms
J. S. Hsia
To the memory of my friend Dennis R. Estes
ABSTRACT. The arithmetic of indefinite quadratic forms admits a field-theoretic
perspective which may be referred to as the "class field theory of integral qua-
dratic forms" . We demonstrate how the fundamental arithmetic questions of
classification and representation can be successfully treated from this new ap-
proach. This supplements the adelic or group-theoretic aspects of Eichler and
Kneser in the 1950s.
The basic message is simple and is the following. The spinor genus theory
from the 1950s admits an adelic or
formulation. In recent times
there emerges a "class field theory" of integral quadratic forms allowing a
perspective which we address here. This talk is on some reflections of the
arithmetic of indefinite integral quadratic forms, a subject which I had periodically
contemplated in the past years. The present account is largely based on some joint
works which I have had the pleasure to work with a number of colleagues: Earnest,
Estes, Shao, Xu. Over the years this subject has evolved to a more mature state
and the landscape has become clearer to the point where a reassessment here of the
theory is perhaps not unjustified.
The arithmetics of indefinite integral quadratic forms, or more generally that
of spinor genera, has been known (or at least suspected) since the 1950s to have
strong abelian characteristics. As such its studies should be particularly susceptible
to methods and ideas from class field theory and algebraic groups. We shall see
below that the modern theory of spinor genera is not merely the correct abelian
component of the subject, it is perhaps the only "algebraic" theory of integral
quadratic forms.
occupies an analogous positon in the general theory of integral
quadratic forms as the family of elliptic curves with complex multiplication does in
the general theory of elliptic curves. On the other hand, it should be noted that
1991 Mathematics Subject Classification. Primary 11E12, 11D85; Secondary 11E20, 11E2,
11R20, 11R23, 11R37, 11R56.
Research supported in part by N.S.F. DMS9401015
N.S.A. MDA904-98-1-0031.
1999 American Mathematical Society
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