- Jacobi sum
In
mathematics , a Jacobi sum is a type ofcharacter sum formed with one or moreDirichlet character s. 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 ofcyclotomy . In modern language, Jacobi sums are also certain multiplicative expressions involving powers ofGauss sum s that are chosen to lie in smallercyclotomic field s. Here "J" for example contains no "p"-throot of unity , just the values of χ which lie in the field of ("p" − 1)st roots of unity. "Cyclotomy" implies that theprime ideal factorisation of "J" is to be investigated. Some general answers to that can be obtained fromStickelberger'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 thelocal zeta-function s ofdiagonal form s. The result on the Legendre symbol amounts to the formula "p" + 1 for the number of points on aconic section that is aprojective line . A paper ofAndré Weil from 1949 very much revived the subject. In fact through theDavenport-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 character s. This was to become important once thecomplex multiplication of abelian varieties became established. The Hecke characters in question were exactly those one needs to express theHasse-Weil L-function s of theFermat curve s, for example. The exact conductors of these characters, a question Weil had left open, were determined in later work.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).
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