User:Charles Matthews/Weildraft

This is a timeline of the Weil conjectures, proved completely by Pierre Deligne in 1973. According to Jean-Pierre Serre, they played an essential role in the development of algebraic geometry over a period of 25 years.

• c.1800 Example: a special cubic curve (Carl Friedrich Gauss).
• Galois introduces finite fields.
• Riemann Hypothesis enunciated.
• c.1900 Points counted on the general linear groups, implying counting also for numerous flag varieties including projective spaces and Grassmannians. These examples were folklore but remained dormant until Weil's work.
• 1923 Emil Artin introduces local zeta functions, for hyperelliptic curves over finite fields in hia dissertation. He had verified some cases of RH already in 1921 (letters to Gustav Herglotz), but at the time the emphasis was elsewhere, on the function field analogue of class field theory.
• 1920s Solomon Lefschetz develops his fixed-point theorem; also the theory of Lefschetz pencils.
• 1931 F. K. Schmidt proves the functional equation for local zeta functions of curves. "Artin-Schmidt zeta-function".
• 1933 Hasse's theorem proves the RH part of the conjectures for elliptic curves.
• 1940-1 Weil gives proofs of the RH part for curves, conditional on foundational work on the geometry of curves.
• 1949 Weil states his conjectures.
• Example: cubic surfaces
• Example: diagonal hypersurfaces
• 1958 Tate-Mattuck, On the inequality of Castelnuovo-Severi, dedicated to Artin on his 60th birthday; reproves the RH for curves.
• 1960 Serre on Kahler analogue.
• 1960 Bernard Dwork proves the rationality and functional equation parts of the conjectures.
• c. 1965 Grothendieck and Bombieri give a conditional proof of the conjectures based on the standard conjectures on algebraic cycles.
• 1967 Example: cubic threefolds (E. Bombieri and Peter Swinnerton-Dyer).
• 1968 Example: unirational projective threefolds (Yu. I. Manin).
• 1968-9 Deligne gives a conditional proof of the Ramanujan-Petersson conjecture based on the Weil conjectures, following work of Eichler, Shimura, Kuga, Ihara.
• 1969 Stepanov provides an elementary method for some cases of the RH for curves, developed by others.
• 1972 Examples: K3-surfaces and complete intersections (P. Deligne).
• 1973
• 1974 N. KATZ et W. MESSING - Some consequences of the Weil conjectures, Invent. Math., 23 (1974), p. 73 -77 .
• 1979 Weil II
• 1981-3 Work on sharpening the upper bound for point on curves over finite fields, and examples of curves with many points, begun by Ihara, Drinfelʹd, Vlăduţ, Serre and others.

References

• Peter Roquette, The history of the Riemann hypothesis in characteristic p: Report on the present state of our work.