MAA Reviews

Modular Forms on Schiermonnikoog

Bas Edixhoven, Gerard van der Geer, and Ben Moonen

Publisher: Cambridge University Press (2008)
Details: 350 pages, Hardcover
Price: $110.00
ISBN: 9780521493543
Category: Anthology
Topics: Modular Forms, Automorphic Forms, Arithmetic Algebraic Geometry

[Reviewed by Fernando Q. Gouvêa, on 02/19/2009]

This book is the proceedings volume for a conference on modular forms held in 2006 on the Dutch island of Schiermonikoog (no, I can't pronounce the name either). As is clear from the table of contents, the term "modular form" was taken in the most general sense, ranging all the way from the very classical holomorphic modular forms in one variable to automorphic representations and Gross's very general "algebraic modular forms." The authors include many leaders in the field, so that specialists are likely to find one or more articles here that will be of value to them.

Of particular interest is the editors' introduction, which tries to lay out in a few pages a panorama and historical sketch of the field. When Martin Eichler said (if he did say) that "There are five fundamental operations in mathematics: addition, subtraction, multiplication, division, and modular forms", he was one of a small band of devotés of the subject. As they point out, beginning with the work of Shimura, Langlands, and Weil in the 1960s, and even more so after Wiles' stunning breakthrough on the modularity conjecture for elliptic curves over Q, modular forms have been popping up everywhere, and new ideas have yielded far more theorems than anyone would have predicted. This article would serve as a nice outline and reading guide (it'd mean a lot of reading!) for anyone who wants to try to grasp the broad sweep of ideas that lead from the work of Gauss, Abel, and Jacobi to the Langlands program and beyond.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.