In the frontispiece to Rigid Cohomology, by Bernard Le Stum, we find the following brief description of the book’s raison d’être : “It is a p-adic Weil cohomology suitable for computing Zeta and L-functions for algebraic varieties on finite fields… it is effective, in the sense that it gives algorithms to compute the number of rational points of such varieties.”
Thus, right off the bat, we find ourselves in the presence of, at least, loud echoes of the now all-but-legendary work of Alexandre Grothendieck surrounding the Weil conjectures, including Weil’s geometric Riemann Hypothesis, and Pierre Deligne’s papers “Weil I” and “Weil II” (to use Nicholas Katz’ well-known slang), all entailing the evolution of a cohomological method that is all but unparalleled.
A cohomology theory is called a Weil cohomology if it satisfies a set of axioms that make it tailor-made to handle questions concerning the abundance and distribution of rational points of varieties or schemes. Thus, we are dealing with Diophantine analysis in the most modern sense. And the centrality of zeta and L-functions, most generally Grothendieck L-functions (with “trâce de Frobenius” doing much of the heavy lifting), comes as no surprise nowadays: their ubiquity in arithmetic algebraic geometry is readily traced to the groundbreaking work of Bernard Dwork and André Weil, and then to Jean-Pierre Serre and, of course, Grothendieck.
Le Stum's rigid cohomology is a p-adic cohomology theory for algebraic schemes over a p-adic field. It arose from work by Pierre Berthelot in crystalline cohomology (which is in fact traced directly to Grothendieck), though it also owes some of its inspiration to another cohomology theory due to Washnitzer and Monsky.
The book under review presents itself as “an accessible tract… of interest to researchers working in arithmetic geometry, p-adic cohomology theory, and related cryptographic areas.” In the latter connection the recent work of Kiran Kedlaya figures prominently, and this added attraction of a cryptography connection augurs for a quantum jump in the book’s projected popularity.
In actuality this currently sexy topic appears explicitly only in the first and last (ninth) chapters of the book: in between, the attendant “calculus,” featuring the notion of overconvergence, is carefully and thoroughly dealt with, with the culmination coming in the eighth chapter, tilted simply “Rigid Cohomology.” This material is not meant for the novice, as the aforementioned designation of an audience of “researchers” conveys without equivocation.
Rigid Cohomology is a very nice book, poised to play a role of considerable importance in the literature. Its style is a bit terse but this should be no problem for the reader coming to this field, whose very interest in this fascinating subject bespeaks a commensurate maturity. This brand new book obviously fills an important niche.
Michael Berg is Professor of Mathematics at Loyola Marymount University.