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SOME OBSERVATIONS

ON THE SIEGEL FORMULA*

by Jιm-ichi Igusa

This is a summary of some of our results in a certain area of mathematics
that has been created by Siegel (and, among others, by Minkowski). It
concerns the theory of quadratic forms and, in particular, an identity which
involves on one hand an integral of a theta-series and on the other hand
an Eisenstein series. This theory has been reconsidered and, in some
aspects, completed by Weil from the adelic viewpoint; he also gave the
name “Siegel formula” to the above-mentioned identity. It has turned
out that, for a classical group G defined over an algebraic number field k,
one has a Siegel formula. Meanwhile, Mars obtained a similar identity
for a special к-form of the exceptional simple group of type E₆.

Our first observation was to recognize the possibility that the Siegel
formula may depend on which “realization” one takes for G as a matrix
group (defined over k). This suggested considering not only the group G
but also its rational representation defined over k and thus freeing ourselves
from the time-honored doctrine of taking a “semisimple algebra with an
involution” as a starting point. Our second observation was to recognize
clearly the fact that a quadratic form is the invariant of the corresponding
orthogonal group. Putting these observations together, we were able to
formulate a conjectural Siegel formula of sufficient generality and to offer
a new example of such involving an invariant of arbitrarily high degree.
It appears that this area, the exact scope of which is still unclear, might be
called the “arithmetic of invariants.”

1. Throughout this paper, we shall denote by Ω a universal domain
of characteristic O and by k an algebraic number field. We shall use the
adelic language and denote by the subscript A the “adelization functor”
relative to k. If no ambiguity is expected, we shall sometimes drop the
subscript A . Also, if K is a finite algebraic extension of k, we shall denote

* This work was partially supported by the National Science Foundation.

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