Sums of Kloosterman sums for real quadratic number fields

Abstract

We estimate sums of Kloosterman sums for a real quadratic number field F of the type where c runs through the integers of F that satisfy C⩽|N(c)|<2C, A⩽|c/c′|<B, with A<B fixed and C→∞. By x↦x′ we indicate the non-trivial automorphism of F. The Kloosterman sums are given by with . In the absence of exceptional eigenvalues for the corresponding Hilbert modular forms, our estimate implies that for each ε>0. An estimate not taking cancellation between Kloosterman sums into account would yield . The exponent is less sharp than occurs in the bound , obtained in our paper in J. reine angew. Math. 535 (2001) 103–164 for sums of Kloosterman sums where c runs over integers satisfying , . 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.