Correlation sums in the wild

In my last post concerning my joint work with É. Fouvry and Ph. Michel, I reported a few weeks ago how happy we were to have found in the literature a specific case of the general correlation sums that we introduced in our paper to deal with “algebraic twists” of modular forms. The example, we are happy to report, turns out not to be isolated: we have found three more in the last few days. I only list them in the briefest way below, since some of them are rather complicated looking, but precise statements and references are found in a short note we just finished typing. It is rather nice to see how some order emerges from these sums (the last one has no less than 8 parameters, in addition to the three variables of summation) once they are considered from the point of view of general correlation sums.

• In a paper from 1990 concerning small eigenvalues of the hyperbolic Laplace operator in special situations, H. Iwaniec considers correlation sums related to the weight $K(x)=e(2\bar{x}/p)S(\bar{x},\bar{x};p)$, where $\bar{x}$ is the inverse of $x$ modulo $p$, and $S(m,n;p)$ is the usual Kloosterman sum; the matrices $\gamma$ in the correlation sums are here all upper-triangular, the difficult case being when $\gamma$ is not diagonal.  It turns out that the general machinery we develop proves the desired estimates for these sums.
• In a paper of 1995, N. Pitt considers correlation sums related to the weight $K(x)=e(\bar{x}/p)$, which was also the one involved in the sums of Friedlander and Iwaniec that we had earlier identified as examples of correlation sums. However, whereas the matrices in that first case were lower-triangular, there is no particular restriction on the sums of Pitt (which are somewhat involved, with 4 parameters), except that they are not upper-triangular.
• Finally, in a recent preprint, R. Munshi also considers correlations sums related to $K(x)=e(\bar{x}/p)$, also without any particular restriction on the matrices involved except that they are not upper-triangular. These sums differ from those of Pitt by the number and configuration of parameters (there are 8 here…)

We have not yet fully updated the text of the paper to mention these examples, but this will be done soon…