The Stationary-Phase Method for Exponential Sums with Multiplicative Characters
β Scribed by Benji Fisher
- Publisher
- Elsevier Science
- Year
- 2002
- Tongue
- English
- Weight
- 250 KB
- Volume
- 96
- Category
- Article
- ISSN
- 0022-314X
No coin nor oath required. For personal study only.
β¦ Synopsis
Let f Γ°xΓ be a polynomial with p-adic coefficients, and define
where w is a character of Γ°Z=p m ZΓ Γ : For example, f Γ°xΓ ΒΌ x leads to the Gauss sum of w: An explicit formula for SΓ°f ; w; Z=p m ZΓ is given if f 0 Γ°xΓ has distinct roots in Z=p m Z and m52; or if f 0 Γ°xΓ 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°xΓ: 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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