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The zeta function of a cubic surface over a finite field

Published online by Cambridge University Press:  24 October 2008

H. P. F. Swinnerton-Dyer
H. P. F. Swinnerton-Dyer
Affiliation:
Trinity College, Cambridge
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Extract

Introduction. The object of this paper is to obtain an explicit formula for the zeta function of an arbitrary non-singular cubic surface over a finite field. Let k denote the finite field of q elements, and kn the field of qn elements which is the unique algebraic extension of k of degree n. Let be a non-singular variety defined over k, and for each n > 0 let be the number of points defined over kn which lie on . The zeta function of is given by

Dwork has shown in (3) that for any this is a rational function of q−s; and in particular it follows from the results he proves in (4) that if is a non-singular cubic surface then

and hence also

Here the numbers w o depend only on q and on .

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 63 , Issue 1 , January 1967 , pp. 55 - 71
Copyright
Copyright © Cambridge Philosophical Society 1967

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References

REFERENCES

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