Proceedings of Symposia in Pure Mathematics
Volume 58.2 (1995)
O n t h e W i t t R i n g o f E l l i p t i c C u r v e s
Very little is known about the classification of symmetric bilinear spaces
over an algebraic variety X over a field K. In fact, little computation has
been done in explicit cases. This is not surprising because so little is known
about the classification of vector bundles on X even if K is algebraically
closed. Hence most of the results so far are results about the Witt ring W(X)
of X. But even these are scarce. The most notable results are about affine
space, projective space, and hyperelliptic curves.
In our previous papers [AEJ2] and [AEJ3], we have, in some sense, re-
duced the problem of the classification of symmetric bilinear spaces over X
to the corresponding problem over X, the variety obtained from X by ex-
tending the base field to an algebraic closure K of K, when X is proper
over K (and K is perfect of characteristic not two). Of course, the explicit
classification of symmetric bilinear spaces over X is, in general, impossible
because it would involve an explicit classification of vector bundles on X.
However, Atiyah in [At] determined all vector bundles over an elliptic curve
over an algebraically closed field. This was extended by Tillman to the case
that the field is not algebraically closed (cf. [Ti]). In [AEJ3] we then used
these results to classify symmetric bilinear spaces over an elliptic curve.
In this paper, we produce a presentation of the Witt ring of an elliptic curve
that is quite explicit. Of course, our description of the Witt ring is closely
related to the descriptions given in [PS, Sh] in this case. However, our ap-
proach exhibits an interesting interplay between the geometry and arithmetic
of the curve that gives rise to our presentation. It also shows the complexity
of the classification problem even when X is an elliptic curve. In a further
paper [AEJ4], we shall show this even more explicitly by computing the Witt
ring of elliptic curves defined over specific fields, especially local fields. This
will indicate the richness of the theory even in this very simple case.
1991 Mathematics Subject Classification. Primary 11E81, 14F05, 14G27, 14H52, 14H60.
The second and third authors were supported by the National Science Foundation.
This paper is infinalform and no version of it will be submitted for publication elsewhere.
©1995 American Mathematical Society
0082-0717/95 $1.00+ $.25 per page
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