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 S = ∑ c |N(c)| −1/2 S F(r,r 1;c) where c runs through the integers of F that satisfy C⩽| N( c)|<2 C, 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 S F(r,r 1;c)= ∑ d|c ∗e 2πi Tr F/ Q (ra+r 1d)/c , with ad≡1| (c) . In the absence of exceptional eigenvalues for the corresponding Hilbert modular forms, our estimate implies that S =O F,ε,r,r 1,A,B C 5/6+ε for each ε>0. An estimate not taking cancellation between Kloosterman sums into account would yield O C . The exponent 5 6 +ε is less sharp than occurs in the bound O F,r,r 1,ε C 3/4+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|⩽2 C , C ⩽|c′|⩽2 C . 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.