English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this commentary the author disucsses the article “Numbers and solutions of equations in finite fields” by André Weil, which appeared in the Bulletin of the American Mathematical Society, 55 (1949), no. 5, 497–508. The paper “Numbers of solutions of equations in finite fields” of André Weil [W1] ranks among the most influential articles in the mathematics of the last hundred years. On the last page of this paper Weil proposes a collection of precise conjectures which initiated a vast development in mathematics. It would require a much longer article than these few paragraphs to describe the evolution of the “Weil conjectures”. The Wikipedia article [Wik] is a good starting point, perhaps followed by J. Milne’s comprehensive study [M]. See also [Di,H,O,K]. Weil’s article begins with a laborious computation of the number of solutions over a finite field to a certain equation ∑ arx r = 0. Initially, Weil offers a somewhat strained motivation for doing this calculation, and (other than in the title of the article) he gives no indication that he is interested in situations more general than this particular equation. However, at the end of the article the real point of the calculation is revealed: it is a nontrivial example which he uses to motivate the conjectures. Weil then states that “this, and many other examples that we cannot discuss here” support the conjectures. He had clearly been thinking about this circle of ideas for some time. Weil’s conjectures were revolutionary because they proposed a precise and deep relation between two completely different sorts of objects: the topology of a complex algebraic variety versus the number of points (mod p) in that variety. For most mathematicians this probably looked like a message from outer space, and Carlitz in his Mathematical Reviews article does little more than quote, without comment, Weil’s sentence that introduces the conjectures. In his 1954 ICM lecture [W2], perhaps as a result of daydreaming about these questions for the intervening five years, Weil points out that his conjectures could be conveniently rephrased in terms of the Lefschetz fixed point formula for the self-map that is defined by the Frobenius morphism, thus cementing the relation between topology and number theory that he had proposed in the Bulletin article. He does not discuss the obvious problem that a new cohomology theory (valid for Received by the editors March 16, 2018. 2010 Mathematics Subject Classification. Primary 11G25, 11M38, 14F20. c ©2018 American Mathematical Society

Commentary on “Numbers of solutions of equations in finite fields” by André Weil