The lifting of Kloosterman sums
To prove the relative trace formula for \(\text{GL}(2)\) [see \textit{H. Jacquet} and the author, Bull. Soc. Math. Fr. 120, 263-295 (1992; Zbl 0785.11032), the author, J. Reine Angew. Math. 400, 57-121 (1989; Zbl 0665.10020)]\ Jacquet and the author have shown that there are certain identities for local Kloosterman sums. On basis of those local results the author proves here identities for global Kloosterman sums. These identities are related to the relative trace formula associated to the field extension \(\mathbb{Q} (\sqrt {\tau})/ \mathbb{Q}\), where \(\tau\not\equiv 5\bmod 8\) is a negative square-free integer. An example is \[ \sum_{x\bmod c, (x,c)=1} e^{2\pi i(xN+ \bar x M)/C}= (-1)^{\Omega (C)} \sum_{a,b} e^{4\pi ia/C}, \] where \(C\geq 1\), \((N,C)= (M,C)=1\), and \(({\tau\over p}) =-1\) for each odd prime \(p\) dividing \(C\). The bar on \(x\) denotes the inverse modulo \(C\), and \(\Omega (C)\) is the number of prime factors of \(C\) counting multiplicities. The sum on the right is over \(a,b\bmod c\) such that \(a^ 2- \tau b^ 2\equiv MN\bmod C\). This equality is a special case of similar results for generalized Kloosterman sums of the type \[ \sum_{x\bmod C, (x,C)=1} e^{2\pi i(xN+ \bar xM)/C} \eta(x), \] where the character \(\eta\) is determined by \(C\) and the imaginary quadratic field extension \(\mathbb{Q} (\sqrt {\tau})/ \mathbb{Q}\). Results of the same type are given for (generalized) Kloosterman sums for \(\text{GL} (3)\).
scientific article; zbMATH DE number 751315