History
Weil's estimate can now be studied in W. M. Schmidt, Equations over finite fields: an elementary approach, 2nd. edn. (Kendrick Press, 2004). The underlying ideas here are due to S. Stepanov and draw inspiration from Axel Thue's work in Diophantine approximation.
There are many connections between Kloosterman sums and modular forms. In fact the sums first appeared (minus the name) in a 1912 paper of Henri Poincaré on modular forms. Hans Salié introduced a form of Kloosterman sum that is twisted by a Dirichlet character: such Salié sums have an elementary evaluation.
After the discovery of important formulae connecting Kloosterman sums with non-holomorphic modular forms by Kuznetsov in 1979, which contained some 'savings on average' over the square root estimate, there were further developments by Iwaniec and Deshouillers in a seminal paper in Inventiones Mathematicae (1982). Subsequent applications to analytic number theory were worked out by a number of authors, particularly Bombieri, Fouvry, Friedlander and Iwaniec.
The field remains somewhat inaccessible. A detailed introduction to the spectral theory needed to understand the Kuznetsov formulae is given in R. C. Baker, Kloosterman Sums and Maass Forms, vol. I (Kendrick press, 2003). Also relevant for students and researchers interested in the field is Iwaniec & Kowalski (2004).
Read more about this topic: Kloosterman Sum
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