Jacobi sum

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In mathematics, a Jacobi sum is a type of character sum formed with one or more Dirichlet characters. The simplest example would be for a Dirichlet character χ modulo a prime number p. Then take

J(χ) = Σ χ(a)χ(1 − a)

where the summation runs over all residues a = 2, 3, ..., p − 1 mod p for which neither a nor 1 − a is 0. Such sums were introduced by C. G. J. Jacobi early in the nineteenth century in connection with the theory of cyclotomy. In modern language, Jacobi sums are also certain multiplicative expressions involving powers of Gauss sums that are chosen to lie in smaller cyclotomic fields. Here J for example contains no p-th root of unity, just the values of χ which lie in the field of (p − 1)st roots of unity. Cyclotomy implies that the prime ideal factorisation of J is to be investigated. Some general answers to that can be obtained from Stickelberger's theorem.

When χ is the Legendre symbol, J can be evaluated, as −χ(−1) = (−1) (p+1)/2 (here p is the defining prime modulus). In general the values of Jacobi sums occur in relation with the local zeta-functions of diagonal forms. The result on the Legendre symbol amounts to the formula p + 1 for the number of points on a conic section that is a projective line. A paper of André Weil from 1949 very much revived the subject. In fact through the Davenport-Hasse theorem of the previous decade the formal properties of powers of Gauss sums had become current once more.

Weil pointed out, not only the possibility of writing down local zeta-functions for diagonal hypersurfaces by means of general Jacobi sums, but their properties as Hecke characters. This was to become important once the complex multiplication of abelian varieties became established. The Hecke characters in question were exactly those one needs to express the Hasse-Weil L-functions of the Fermat curves, for example. The exact conductors of these characters, a question Weil had left open, were determined in later work.

[edit] References

  • B. C. Berndt, R. J. Evans, K. S. Williams, Gauss and Jacobi Sums (1998)
  • S. Lang, Cyclotomic fields, Graduate texts in mathematics vol. 59, Springer Verlag 1978. ISBN 0-387-90307-0. See in particular chapter 1 (Character Sums).