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Let C be a hyperelliptic curve of genus 3, with potentially good reduction at all primes not dividing 3, defined over a number field K. A function defined on the Jacobian J of C will be constructed which, when evaluated at the 3-torsion points J[3] of J, will give S-units in K(J[3]); at most, the set S will consist of primes dividing 3.

Recently Bost and Connes considered a Hecke C -algebra arising from the ring inclusion of Z in Q, and a C -dynamical system involving this algebra. Laca and Raeburn realized this algebra as a semigroup crossed product, and studied it using techniques they had previously developed for studying Toeplitz algebras. Here we associate Hecke algebras to general number elds, realize them as semigroup crossed products, and analyze their representations.

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