Philippe Michel, Will Sawin and I have just finished the first draft of a paper where we prove estimates for general bilinear forms of the type
where (1) is a (normalized) hyper-Kloosterman sum (for , this is a classical Kloosterman sum) modulo a prime ; and (2) the ranges and are such that we have non-trivial bounds even if is a bit smaller than in logarithmic scale. In other words, we obtain non-trivial results below the “Pólya-Vinogradov range”.
The basic strategy to get this result is not new: it was devised by Fouvry and Michel a number of years ago (inspired at least in part by earlier work of Friedlander-Iwaniec and by the Vinogradov-Karatsuba-style “shift” method to estimate certain short exponential sums). What was missing (despite the strong motivation provided by applications that were known to follow from such a result, one of which is described in a recent preprint of Blomer, Fouvry, Milicevic, Michel and myself) was a way to prove certain estimates for (complete) sums over finite fields, of the type
unless the parameters are in some “diagonal” positions. And we cannot afford too many diagonal cases…
The main contribution of our paper (much of which comes from the ideas of Will!) is to find a relatively robust approach to such estimates.
This relies, as one can expect, from extensive algebraic-geometric arguments to apply the Riemann Hypothesis over finite fields. In fact, from this point of view, this paper is by far the most complicated I’ve ever been involved in. We use, among other things:
- The Riemann Hypothesis over finite fields, in its most general version of Deligne — indeed, we use it multiple times;
- The interpretation of the sum over (in the sum above) as itself a trace function of sbome sheaf on the space of parameters ; this follows from the formalism of étale cohomology, which is also used in many other ways (e.g., to detect irreducibility of sheaves by properties of the top-degree cohomology);
- A very general version of the Euler-Poincaré characteristic formula in étale cohomology – this comes from SGA5;
- The formalism and properties of vanishing and nearby cycles in étale cohomology, and in particular their relations with local monodromy representations of sheaves on curves;
- The global -adic Fourier transform of Deligne as well as the local Fourier transform of Laumon;
- A special case of the homogeneous Fourier transform of Laumon (which we might be able to avoid, although with an argument involving perverse sheaves);
- Katz’s theory of Kloosterman and hypergeometric sheaves, in particular with respect to the computation of their geometric monodromy groups (and its implication through the Goursat-Kolchin-Ribet Criterion), but also (and equally importantly) with respect to their local monodromy properties;
- The diophantine criterion for geometric irreducibility (which is again a case of the Riemann Hypothesis)…
Many of these are results and ideas that I was aware of but had never actually used before, and I learnt a lot by seeing how Will exploited and combined them. I will try to write a few more posts later to (attempt to) explain and motivate them (and how we use them) from an analytic nunber theorist’s viewpoint. The theory of vanishing cycles, in particular, should have many more applications in extending the range of applicability of Deligne’s Riemann Hypothesis to problems in analytic number theory.
The paper is dedicated to Henryk Iwaniec, who has been over the years the most eloquent and powerful advocate for a deeper use of the work of Deligne (and Katz and others) in applications to analytic number theory.