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

JON KR. ARASON, RICHARD ELMAN, AND BILL JACOB

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

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http://dx.doi.org/10.1090/pspum/058.2/1327286