Bibilography: Riemann's hypothesis and positive semi-definite billinear forms

Question

This is a bibliography request: I remember browsing through a book, some years ago, in a library, in which Riemann's hypothesis was proved over some type of fields (I cannot remember what type), the main tool being some billinear form and $p$-adic cohomology (if I recall correctly). It was proven that Riemann's hypothesis in this context was equivalent to proving the positive semi-definiteness of that billinear form, I think. The book was physically small, quite thin too (maybe less than 100 pages?), was a succession of numbered paragraphs and was not recently published. I was under the impression that it was by Weil, but I seem unable to find it among Weil's published works (judging by their titles). It was in English, its style was concise and very clear.

Unfortunately, not being an algebraist, I cannot be more precise (and possibly some of the memories above are distorted by the passage of time). I would like to meet this book again and spend my holidays together with it. Can you help me, please?

Answer 1

Almost 8 years after having stumbled upon this mysterious book in a moment of boredom in a library, I have found it, yay! It is "An Introduction to the Theory of the Riemann Zeta-Function" by S. J. Patterson. (My memories about the book not being recently published were wrong, since its first edition was printed in 1988.) Google and a judicious choice of search terms did the job. Huh, I feel relieved!

Answer 2

The Weil Conjectures contain statements about the location of zeroes of local zeta function $\zeta(X, s)$ associated to an algebraic variety $X$ over the finite field $\mathbb{F}_q$. In particular, he conjectured that $\zeta(X,s)$ is actually a rational function (his first conjecture) whose zeroes lie on the $1/2$ line (his third conjecture, with the correct identification of the $1/2$ line). This can be called the Riemann Hypothesis for the algebraic variety.

He suspected his conjectures would be proved through (what's now called) Weil cohomology.

In 1974, Deligne proved this Riemann Hypothesis.

It seems likely to me that you read an expository account of Deligne's proof. Many such accounts exist, but I must admit that I haven't read a good one. (I first came across this result in Rosen's Number Theory in Function Fields which has an expository account of a slightly dumbed-down version of Deligne's result).

Perhaps this will help you find your book.