## Bilinear forms with Kloosterman sums and applications.(English)Zbl 1441.11194

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.

### MSC:

 11L07 Estimates on exponential sums 11T23 Exponential sums 11L05 Gauss and Kloosterman sums; generalizations 11L26 Sums over arbitrary intervals 11M41 Other Dirichlet series and zeta functions 11N37 Asymptotic results on arithmetic functions 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 14D05 Structure of families (Picard-Lefschetz, monodromy, etc.) 14F20 Étale and other Grothendieck topologies and (co)homologies

Zbl 0915.11045
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