Estimates for Kloosterman sums for totally real number fields (Q2720958)

We prove a sum formula of Kuznetsov type for \(SL_2\) over a totally real number field. This sum formula takes into account automorphic forms of all \(K\)-types. In this sense, it forms an extension of the sum formula in \textit{N. R. Wallach} and \textit{R. J. Miatello} [Isr. Math. Conf. Proc. 3, 305-320 (1990; Zbl 0719.11032)].NEWLINENEWLINENEWLINEThe presence of all \(K\)-types makes it possible to invert the Bessel transformation that occurs in the sum formula, and to take the function in the sum of Kloosterman sums as the independent test function. This enables us to obtain estimates of sums of Kloosterman sums of the following type NEWLINE\[NEWLINE\sum_{c\in I\atop 0< |c^{\sigma_j} |\leq X}{S(r,r';c) \over \bigl |N_{F/\mathbb{Q}} (c)\bigr|} \ll X^{(d-1)/2 +\varepsilon}\text{ as }X\to \infty,NEWLINE\]NEWLINE under the assumption that there are no exceptional eigenvalues. \(F\) is a totally real number field of degree \(d\) over \(\mathbb{Q}\), \(I\) a non-zero ideal in its ring of integers, \(\sigma_1, \dots, \sigma_d\) the embeddings of \(F\) into \(\mathbb{R}\). The Kloosterman sum is NEWLINE\[NEWLINES(r,r';c)= \sum_{d\bmod (c)}\exp \biggl(2\pi i\text{ Tr}_{F/ \mathbb{Q}}\bigl((rd+ r'a)/c\bigr) \biggr),NEWLINE\]NEWLINE where \(d\) runs over representatives of invertible classes modulo the ideal \((c)\), and \(a\) represents the inverse class. The non-zero numbers \(r\) and \(r'\) correspond to the order of Fourier terms of automorphic forms. We make explicit their influence on the estimate.NEWLINENEWLINENEWLINEIn the presence of exceptional eigenvalues, there are other terms in the estimate, that may be larger than \(X^{(d-1)/2+ \varepsilon}\). But if we input the best known bounds for exceptional eigenvalues, our result implies cancellation of Kloosterman sums for any totally real number field \(F\).