The Weil Conjectures

The Weil conjectures constitute one of the central landmarks of 20th century algebraic geometry: not only was their proof a dramatic triumph, but they served as a 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. 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 such as the Birch-Swinnerton-Dyer conjecture establish strong, albeit subtle, relationships between the two cases. Moreover, there are some basic questions that have non-obvious connections to the Weil conjectures. The most famous of these is the Ramanujan conjecture, which treats the coefficients of ∆(q), one of the most fundamental examples of a modular form. We obtain the function τ(n) from the formula for ∆(q) as follows: