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Publisher:

Cambridge University Press

Publication Date:

2008

Number of Pages:

350

Format:

Hardcover

Price:

110.00

ISBN:

9780521493543

Category:

Anthology

[Reviewed by , on ]

Fernando Q. Gouvêa

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.

Preface

Contributors

1. *Modular forms*

Bas Edixhoven, Gerard van der Geer and Ben Moonen

*2. On the basis problem for Siegel modular forms with level*

Siegfried Böcherer, Hidenori Katsurada and Rainer Shulze-Pillot

*3. Mock theta functions, weak Maass forms, and applications*

Kathrin Bringmann

*4. Sign changes of coefficients of half integral weight modular forms*

Jan Hendrik Bruinier and Winfried Kohnen

*5. Gauss map on the theta divisor and Green’s functions*

Robin de Jong

*6. A control theorem for the images of Galois actions on certain infinite families of modular forms*

Luis Dieulefait

*7. Galois realizations of families of Projective Linear Groups via cusp forms*

Luis Dieulefait

*8. A strong symmetry property of Eisenstein series*

Bernhard Heim

*9. A conjecture on a Shimura type correspondence for Siegel modular forms, and Harder's conjecture on congruences*

Tomoyoshi Ibukiyama

*10. Petersson's trace formula and the Hecke eigenvalues of Hilbert modular forms*

Andrew Knightly and Charles Li

*11. Modular shadows and the Lévy-Mellin ∞-adic transform*

Yuri I. Manin and Matilde Marcolli

*12. Jacobi forms of critical weight and Weil representations*

Nils-Peter Skoruppa

*13. Tannakian categories attached to abelian varieties*

Rainer Weissauer

*14. Torelli's theorem from the topological point of view*

Rainer Weissauer

*15. Existence of Whittaker models related to four dimensional symplectic Galois representations*

Rainer Weissauer

*16. Multiplying modular forms*

Martin H. Weissman

*17. On projective linear groups over finite fields*

Gabor Wiese.

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