This paper concerns bilinear sums involving generalized Kloosterman sums (hyper-Kloosterman sums) to prime modulus \(p\), given by \[ \mathrm{Kl}_k(a;p)=p^{-(k-1)/2}\sum_{1\le x_1,\ldots,x_{k-1}<p}e_p(x_1+\ldots+x_{k-1}+a \overline{x_1\ldots x_{k-1}}), \] where \(p\nmid a\). The sums considered take the form \[ \Sigma=\sum_{m\in\mathcal{M},\,n\in\mathcal{N}}\alpha_m\beta_n\mathrm{Kl}_k(amn;p), \] where \(\alpha_m\) and \(\beta_n\) are complex coefficients and \(\mathcal{M}=\{1,2,\ldots,M\}\) and \(\mathcal{N}=\{N_0+1,N_0+2,\ldots,N_0+N\}\) for positive integers \(M,N<p\).
The first, most general, result states that if \(p\) is prime and \(M\le Np^{1/4}\) and \(p^{1/4}<MN<p^{5/4}\) then for any fixed \(\varepsilon>0\) one has \[ \Sigma\ll_{\varepsilon,k}p^{\varepsilon}||\alpha||_2 ||\beta||_2(MN)^{1/2}\left(M^{-1/2}+(MN)^{-3/16}p^{11/64}\right),\tag{\(\ast\)} \] where \[ ||\alpha||_k=\left(\sum|\alpha_m|^k\right)^{1/k} \] is the \(\ell^k\)-norm. For comparison, the trivial bound would be \(||\alpha||_2||\beta||_2(MN)^{1/2}\), while a relatively straightforward argument yields \[ \Sigma\ll_{k}||\alpha||_2 ||\beta||_2(MN)^{1/2}\left(p^{-1/4}+M^{-1/2}+N^{-1/2}p^{1/4}\right). \] This latter bound is trivial when \(N\ll p^{1/2}\), and the significance of the new result (*) is that it is non-trivial for a range including the case \(M=N=p^{1/2}\).
The result above handles a Type-II bilinear sum, that is to say, with unrestricted coefficients. The second main result handles a Type-I sum, in which we suppose that \(\beta_n=1\) for all \(n\in\mathcal{N}\). Then, if we replace the conditions on \(M\) and \(N\) by \(M\le N^2\) and \(MN<p^{3/2}\), it is shown that \[ \Sigma\ll_{\varepsilon,k}p^{\varepsilon}||\alpha||_1^{1/2}||\alpha||_2^{1/2}M^{1/4}N\left(\frac{M^2N^5}{p^3}\right)^{-1/12}.\tag{\(\ast\ast\)} \] For comparison, there is a trivial bound \(||\alpha||_1^{1/2}||\alpha||_2^{1/2}M^{1/4}N\). In the special case \(k=2\), Blomer et. al. [to appear] give a slightly stronger estimate than (**), building on work of É. Fouvry and P. Michel [Ann. Sci. Éc. Norm. Supér. (4) 31, No. 1, 93–130 (1998; Zbl 0915.11045)]. The latter paper is also an important starting point for the present work.
One interesting application is to moments of twisted cuspidal \(L\)-functions. Suppose that \(f\) and \(g\) are Hecke eigenforms (holomorphic, or Maass forms), of level 1, with the same root number. Then one has an asymptotic formula, with a power saving, for \[ \sum_{\chi(\mathrm{mod}\; p)}L(f\otimes\chi,1/2)\overline{L(g\otimes\chi,1/2)}. \] The proof of the Type-II bound employs the “shift by \(ab\)” method used by Vinogradov, Burgess and Karatsuba. This requires square-root estimates for some complicated averages involving \(\mathrm{Kl}_k(a;p)\), and the proofs of these need detailed knowledge of the ramification properties of the associated sheaves.