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
Famous quotes containing the word history:
“The history of all Magazines shows plainly that those which have attained celebrity were indebted for it to articles similar in natureto Berenice—although, I grant you, far superior in style and execution. I say similar in nature. You ask me in what does this nature consist? In the ludicrous heightened into the grotesque: the fearful coloured into the horrible: the witty exaggerated into the burlesque: the singular wrought out into the strange and mystical.”
—Edgar Allan Poe (1809–1849)
“The history of modern art is also the history of the progressive loss of art’s audience. Art has increasingly become the concern of the artist and the bafflement of the public.”
—Henry Geldzahler (1935–1994)
“My good friends, this is the second time in our history that there has come back from Germany to Downing Street peace with honour. I believe it is peace for our time. We thank you from the bottom of our hearts. And now I recommend you to go home and sleep quietly in your beds.”
—Neville Chamberlain (1869–1940)