# Deligne and the four Weil statements about polynomials over finite fields?

This article mentions he recently won 3 million of some money. Does anyone have a link to the original Deligne paper that proves this, and could anyone give a basic rundown of how "counting points on related shapes, using a zeta function" can help to find zeroes for polynomials over finite fields, and count the solutions?

All the objects mentioned so far I understand and these seem fairly straightforward, is there a way to begin to talk about this method of counting solutions and why it works using elementary number theory, a little group theory, and a lot of hand waving? If so, please contribute an answer. Otherwise, downvote with gleeful fervor.

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What does "solve polynomials over finite fields" mean? – Greg Martin Mar 24 '13 at 19:59
You could start by reading Gowers's description of Deligne's work on the official page of the Abel prize or some of the other information on the page. Here's a direct link to the pdf. – Martin Mar 24 '13 at 20:04
@GregMartin Solve polynomials over finite fields . . . Obtain values, in some finite field, of the variables of that polynomial, whose coefficients in same finite field, such that the polynomial evaluates to zero. 1 set of such variables is a 'solve' or a solution. The find the number of these is to count the solutions. I believe that Deligne's work had something to do with counting the number of solves. – Cris Stringfellow Mar 24 '13 at 20:15
@Martin Okay, thanks. That pdf looks to be pitched at the right level. At a cursory first inspection. Thanks again. – Cris Stringfellow Mar 24 '13 at 20:17
Okay, I think a better phrase would be "find roots of polynomials over finite fields", or "solve the equation $p(x_1,x_2,\dots)=0$ over finite fields". Problems and equations have solutions; functions and polynomials have zeros or roots. – Greg Martin Mar 24 '13 at 22:13