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The Stationary-Phase Method for Exponential Sums with Multiplicative Characters

✍ Scribed by Benji Fisher


Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
250 KB
Volume
96
Category
Article
ISSN
0022-314X

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✦ Synopsis


Let f ðxÞ be a polynomial with p-adic coefficients, and define

where w is a character of Γ°Z=p m ZÞ Γ‚ : For example, f Γ°xÞ ΒΌ x leads to the Gauss sum of w: An explicit formula for SΓ°f ; w; Z=p m ZÞ is given if f 0 Γ°xÞ has distinct roots in Z=p m Z and m52; or if f 0 Γ°xÞ has distinct roots p-adically and m is sufficiently large. In either case, the formula involves one term for each root of f 0 Γ°xÞ: If the derivative does not have distinct roots, there is a less explicit formula, requiring one term for each approximate root. Analogous results are proved for sums involving several variables. As applications of these results, Mauclaire's evaluation of the Gauss sum is rederived, and new results on n-variable Kloosterman sums are obtained. # 2002

Elsevier Science (USA)


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