In mathematics, the Weil conjectures were some highly-influential proposals by André Weil (1949) on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields.
A variety V over a finite field with q elements has a finite number of rational points, as well as points over every finite field with qk elements containing that field. The generating function has coefficients derived from the numbers Nk of points over the (essentially unique) field with qk elements.
Weil conjectured that such zeta-functions should be rational functions, should satisfy a form of functional equation, and should have their zeroes in restricted places. The last two parts were quite consciously modeled on the Riemann zeta function and Riemann hypothesis. The rationality was proved by Dwork (1960), the functional equation by Grothendieck (1965), and the analogue of the Riemann hypothesis was proved by Deligne (1974)
Read more about Weil Conjectures: Background and History, Statement of The Weil Conjectures, Weil Cohomology, Grothendieck's Formula For The Zeta Function, Deligne's First Proof, Deligne's Second Proof, Applications
Famous quotes containing the words weil and/or conjectures:
“With no matter what human being, taken individually, I always find reasons for concluding that sorrow and misfortune do not suit him; either because he seems too mediocre for anything so great, or, on the contrary, too precious to be destroyed.”
—Simone Weil (1909–1943)
“Our conjectures pass upon us for truths; we will know what we do not know, and often, what we cannot know: so mortifying to our pride is the base suspicion of ignorance.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)