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. ...

Get The Princeton Companion to Mathematics now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.