Sums of Kloosterman sums for real quadratic number fields

AbstractWe estimate sums of Kloosterman sums for a real quadratic number field F of the typeS=∑c|N(c)|−1/2SF(r,r1;c)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 bySF(r,r1;c)=∑d|c∗e2πiTrF/Q(ra+r1d)/c,with ad≡1|(c).In the absence of exceptional eigenvalues for the corresponding Hilbert modular forms, our estimate implies thatS=OF,ε,r,r1,A,BC5/6+εfor each ε>0. An estimate not taking cancellation between Kloosterman sums into account would yield OC. The exponent 56+ε is less sharp than occurs in the bound OF,r,r1,εC3/4+ε, obtained in our paper in J. reine angew. Math. 535 (2001) 103–164 for sums of Kloosterman sums where c runs over integers satisfying C⩽|c|⩽2C, C⩽|c′|⩽2C. 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