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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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V.35 The Weil Conjectures

Brian Osserman

The Weil conjectures constitute one of the central landmarks of twentieth-century ALGEBRAIC GEOMETRY [IV.4]: not only was their proof a dramatic triumph, but they were the driving force behind a striking number of fundamental advances in the field. The conjectures treat a very elementary problem: how to count the number of solutions to systems of polynomial equations over finite FIELDS [I.3 §2.2]. While one might ultimately be more interested in solutions over, say, the field of rational numbers, the problem is far more tractable Over finite fields, and LOCAL–GLOBAL PRINCIPLES [III.51] such as THE BIRCH–SWINNERTON-DYER CONJECTURE [V.4] establish strong, albeit subtle, relationships between the two cases. ...

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