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Modular Forms on Schiermonnikoog

Bas Edixhoven, Gerard van der Geer, and Ben Moonen
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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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.



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.