The Lifting Of An Exponential Sum To A Cyclic Algebraic Number Field Of A Prime Degree

. Let E be a cyclic algebraic number field of a prime degree. We prove an identity which lifts an exponential sum similar to the Kloosterman sum to an exponential sum taken over certain algebraic integers in E. 1. Introduction. Based on the relative trace formula for GL(2) (Ye [13] and Jacquet and Ye [8]) and inspired by early work of Zagier [15], Ye [14] established a lifting identity of Kloosterman sums. Let E = Q( p ø ) be a quadratic field with a square-free integer ø , and let c be a positive odd integer with (c; ø) = 1. Assume that for any prime factor p of c we have i ø p j = \Gamma1. Then for any nonzero integers m and n with (m; c) = (n; c) = 1 we have X 1xc; (x;c)=1 e 2ßi(xm+¯xn)=c = (\Gamma1) c) X 1ac; 1bc; a 2 \Gammaø b 2 jmn(mod c) e 4ßia=c (1) where ¯ x is the inverse of x modulo c. Here\Omega\Gamma c) is the number of prime factors of c counting multiplicity. Note that the exponential sum on the right side of (1) is actually a sum taken over so..