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 *q**k* elements containing that field. The generating function has coefficients derived from the numbers *N*_{k} of points over the (essentially unique) field with *q**k* 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

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### Famous quotes containing the words conjectures and/or weil:

“After all, it is putting a very high price on one’s *conjectures* to have a man roasted alive because of them.”

—Michel de Montaigne (1533–1592)

“A test of what is real is that it is hard and rough. Joys are found in it, not pleasure. What is pleasant belongs to dreams.”

—Simone *Weil* (1909–1943)