On the Frobenius Endomorphisms of Fermat and Artin-schreier Curves

The following article offers an explanation of the relationship of Jacobi and Gauss sums to Fermât and Artin-Schreier curves which is an analogue of the proof of Stickleberger's theorem. 1. Correspondences. The connection between cubic Fermât curves and cubic Jacobi sums was first observed by Gauss [G], who used it to study such sums. That one can compute the number of points on a Fermât curve over a finite field using Jacobi sums has long been known. The same is true for Artin-Schreier curves and Gauss sums and were applied by Davenport and Hasse [D-H] to compute Artin ¿-functions associated with such curves. This computation, in turn, yielded the Hasse-Davenport identity. Ultimately, Weil [W] interpreted these computations via a Lifshetz fixed point formula by showing that the eigenvalues of Frobenius acting on what is now known as the étale cohomology groups of Fermât curves are Jacobi sums and on Artin-Schreier curves are Gauss sums. This can be interpreted as an identity which says, in a suitable sense, that Frobenius is a Gauss or Jacobi sum (see below). One can now use this to produce annihilators of the divisor class groups of these curves which are also given by the Brumer-Stark conjecture for function fields proven in [T]. Below, we will give an elementary proof of the aforementioned identity which is analogous to Stickelberger's proof of Stickelberger's theorem. We just write down especially simple functions with appropriate divisors. Let p be a rational prime and m an integer prime to p. Let K be a field of characteristic p. Let Am and Fm denote the complete nonsingular curves over K with affine equations: Am:yp-y = xm, Fm:um + vm = l. Suppose q — p3* = 1 mod m and suppose K DFq. Define the homomorphisms ip: F, -♦ Aut(Am), x: F,-► Aut(Am), Xo,Xi : Fg -> Aut(Fm) by <P(a)(x,y) = (x,y + TFg/Fp(a)), X(b)(x,y) = (b^'mx,y), Xo(b)(u,v) = (b^-^mu,v), xi(b)(u,v) = (uM9~1)/mv) for o G F+ and bG F*. Now recall that a correspondence between a curve A" to a curve Y is a divisor Z on X x Y with no vertical or horizontal components. Among other things, Z gives Received by the editors April 15, 1986 and, in revised form, December 9, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 11G20. ©1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page