Une Introduction aux Motifs (Motifs Purs, Motifs Mixtes, Périodes)

When I was in graduate school, it sometimes seemed as if one couldn't refer to motives without using the phrase "Grothendieck's conjectural theory of motives." As Yves André's Une Introduction Aux Motifs shows, things have changed.

Grothendieck's vision, 40 years ago, was that the arithmetic properties of algebraic varieties suggested the existence of some sort of underlying object, which he termed a "motive" (or, perhaps, a "motif", as in music or literature). The study of cohomological properties of varieties suggested common themes, motifs that ran through the theory (for example, the notion of "weight"). At the same time, common properties of very different cohomological theories seemed to call for an explanation, a motive. At first this was only a conjecture, but over the last fifteen years the theory has taken shape and become an important part of Arithmetical Algebraic Geometry.

Grothendieck's punning choice of name seems to have inspired his followers. There are "pure motives" and "mixed motives". I understand that someone once wrote a paper with the sole purpose of defining "ulterior motives".

André's book is an introduction to the theory aimed at "non-specialists" (meaning roughly, I think, specialists in arithmetical algebraic geometry who don't yet know about motives). Its first two sections discuss pure motives and mixed motives, respectively, and the third section puts the theory to work in an interesting way by studying periods of motives and their connection to polyzeta functions. Each of the first two parts is preceded by a motivational chapter that, though still not easy, tries to show why the theory to be developed is necessary. All in all, a valuable contribution to the literature.

Fernando Q. Gouvêa is a number theorist and historian of mathematics; he teaches at Colby College in Waterville, ME.