CoCalc Shared Fileswww / mazur / katz / problem.tex
Author: William A. Stein
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6William,
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8   I wonder whether I could appeal to you to make a computation-- but
9only if it is really dead-easy and doesn't waste too much of your
10time. I'm supposed to give a lecture in Princeton shortly (in the Katz
11conference) and it is going to be a general overview of Katz's
12work. Now I would love to have some numerical displays which
13"illustrate" some of the problems that Katz worked on and contributed
14to.  For example, the distribution of angles of Kloosterman sums.
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17Specifically, let $a$ be an integer, and $p$ a prime number. Define
18 $$19{\rm Kl}(p,a):=\sum_{xy=a;\;x,y \ {\rm mod} p }\exp\left({2\pi 20i\over p}(x+y)\right).$$
21These are real numbers, and using the Weil bounds if we write
22$$23{\rm Kl}(p,a)= 2{\sqrt p}\ {\rm cos}\ \theta(p,a)$$
24we might try to study the variation of the angles $\theta(p,a)$, either as
25$p$ varies or as $a$
26varies, or both.  One expects these angles, viewed as points in $[0,\pi]$,
27for fixed nonzero $a$, varying over all primes $p$, to be
28equidistributed with respect to the Sato-Tate measure ${2\over \pi}{\rm 29sin}^2\theta d\theta$.  (This isn't yet proved, if I understand right, but
30things "close to it"-- e.g., various averagings over choices of $a$-- are
31what Katz proves
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33Would it be possible, for specific choices of $a$,  to make some sort of
34graph of these angles for $p \le 100$ or so, demonstrating approximations
35to Sato\_Tate distributions?  Needless to say, I'd be eternally grateful!
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