Review by Choice Review
The theory of modular forms concerns analytic functions that satisfy a kind of symmetry so strong that coefficients of the functions' series expansions must carry profound number-theoretic information. Many books introduce the one-dimensional theory, the simplest case, but already so rich that the theory has produced, say, the proof of Fermat's Last Theorem. At the other extreme, many very advanced books speak in generality about automorphic forms on reductive Lie groups and the associated Langlands philosophy, one of modern mathematics' most ambitious programs. This counts as the rare book that falls squarely in between. Andrianov (Russian Academy of Sciences), a leading authority whose career spans four decades, offers a comparatively concrete, lowbrow treatment of Siegel modular forms, a large and important but still special class of higher-dimensional modular forms. Readers familiar with standard accounts of the one-dimensional story will find the flow and organization here comfortingly familiar; indeed, this exposition subsumes the one-dimensional story, so does not presume it. Though one might expect to find Abelian varieties at the foundation of the theory, Andrianov has suppressed entirely the vantage of algebraic geometry, presumably for simplicity. He likewise omits representative applications and unexpected connections to other fields, such as Kac-Moody Lie algebras. Summing Up: Highly recommended. Upper-division undergraduate through professional collections. D. V. Feldman University of New Hampshire
Copyright American Library Association, used with permission.
Review by Choice Review
