Sums of Kloosterman sums for real quadratic number fields Original Research Article

We estimate sums of Kloosterman sums for a real quadratic number field F of the typeimagewhere c runs through the integers of F that satisfy Cless-than-or-equals, slantN(c)<2C, Aless-than-or-equals, slantc/c′0. An estimate not taking cancellation between Kloosterman sums into account would yield image. The exponent image is less sharp than occurs in the bound image, obtained in our paper in J. reine angew. Math. 535 (2001) 103–164 for sums of Kloosterman sums where c runs over integers satisfying image, image. The proof is based on the Kloosterman-spectral sum formula for the corresponding Hilbert modular group. The Bessel transform in this formula has a product structure corresponding to the infinite places of F. This does not fit well to the bounds depending on N(c) and c/c′. Nevertheless, we do obtain non-trivial bounds for S.