Curvature and Topology of Riemannian Manifolds: Proceedings by Werner Ballmann (auth.), Katsuhiro Shiohama, Takashi Sakai,

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2 /i - 2 ) establishing projection Then 0) that V = E U E -I, v° <0 1i) where Z/p). number. Let us fix as generators Then, for 28 For a and b relatively prime to p, and × (that is, x(ab) = ×(a) ×(b)), Sx(a,b,p ) = where ~ = e 2~i/p, a multiplicative character (mod p the Kloosterman sum Sx(a,b,p) is defined by: ~ ~ ( a y + by -l ) y~0(mod p) and y-i is the multiplicative inverse of y (mod p). One has the deep estimate of Weil [18]: (***) ISx(a,b,p) I ~ 2 /p sharpening the estimate of Davenport [7] that ISx(a,b,p)I < 0(p2/3).

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