In mathematics, the Weil conjectures were some highly influential proposals by André Weil on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields.

Quotes

• There can be no doubt that proving the Weil conjectures was the ultimate goal that Grothendieck had in mind with his reconstruction of algebraic geometry.
  • Chapter 10. The Weil Conjectures. 

• Like Moses, André Weil caught sight of the Promised Land, but unlike Moses, he was unable to cross the Red Sea on dry land, nor did he have an adequate vessel.
  • Cartier, Pierre (24 December 2004), "Un pays dont on ne connaîtrait que le nom (Grothendieck et les " motifs ")", in Cartier, Pierre; Charraud, Nathalie (in French), Réel en mathématiques-psychanalyse et mathématiques, Editions Agalma, English translation: A country of which nothing is known but the name Grothendieck and "motives". 

• Just as Weil's conjectures were about counting solutions to equations in a situation where the number of solutions is known to be finite, the BSD conjecture concerns the simplest class of polynomial equations—elliptic curves—for which there is no simple way to decide whether the number of solutions is finite or infinite.
  • Michael Harris (30 May 2017). Mathematics without Apologies: Portrait of a Problematic Vocation. Princeton University Press. p. 27. ISBN 978-1-4008-8552-7. 

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