Book Review
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MathSciNet review:
2507285
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Book Information:
Authors:
J. H. Bruinier,
G. van der Geer,
G. Harder and
D. Zagier
Title:
The 1-2-3 of modular forms
Additional book information:
Universitext,
Springer-Verlag,
Berlin, Heidelberg,
2008,
x+266 pp.,
ISBN 978-3-540-74117-6,
US $69.95$,
softcover
Anatolij N. Andrianov, Quadratic forms and Hecke operators, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 286, Springer-Verlag, Berlin, 1987. MR 884891, DOI 10.1007/978-3-642-70341-6
A. N. Andrianov and V. G. Zhuravlëv, Modular forms and Hecke operators, Translations of Mathematical Monographs, vol. 145, American Mathematical Society, Providence, RI, 1995. Translated from the 1990 Russian original by Neal Koblitz. MR 1349824, DOI 10.1090/mmono/145
Richard E. Borcherds, Automorphic forms on $\textrm {O}_{s+2,2}(\textbf {R})$ and infinite products, Invent. Math. 120 (1995), no. 1, 161–213. MR 1323986, DOI 10.1007/BF01241126
Stefan Breulmann and Michael Kuss, On a conjecture of Duke-Imamoḡlu, Proc. Amer. Math. Soc. 128 (2000), no. 6, 1595–1604. MR 1707138, DOI 10.1090/S0002-9939-00-05586-6
Jan Hendrik Bruinier and Tonghai Yang, CM-values of Hilbert modular functions, Invent. Math. 163 (2006), no. 2, 229–288. MR 2207018, DOI 10.1007/s00222-005-0459-7
Pierre Deligne and Jean-Pierre Serre, Formes modulaires de poids $1$, Ann. Sci. École Norm. Sup. (4) 7 (1974), 507–530 (1975) (French). MR 379379
W. Duke and Ö. Imamoḡlu, A converse theorem and the Saito-Kurokawa lift, Internat. Math. Res. Notices 7 (1996), 347–355. MR 1389957, DOI 10.1155/S1073792896000220
E. Freitag, Siegelsche Modulfunktionen, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 254, Springer-Verlag, Berlin, 1983 (German). MR 871067, DOI 10.1007/978-3-642-68649-8
Benedict H. Gross and Don B. Zagier, On singular moduli, J. Reine Angew. Math. 355 (1985), 191–220. MR 772491
Tamotsu Ikeda, On the lifting of elliptic cusp forms to Siegel cusp forms of degree $2n$, Ann. of Math. (2) 154 (2001), no. 3, 641–681. MR 1884618, DOI 10.2307/3062143
Winfried Kohnen, Lifting modular forms of half-integral weight to Siegel modular forms of even genus, Math. Ann. 322 (2002), no. 4, 787–809. MR 1905104, DOI 10.1007/s002080100285
Isao Miyawaki, Numerical examples of Siegel cusp forms of degree $3$ and their zeta-functions, Mem. Fac. Sci. Kyushu Univ. Ser. A 46 (1992), no. 2, 307–339. MR 1195472, DOI 10.2206/kyushumfs.46.307
Jean-Pierre Serre, Valeurs propres des opérateurs de Hecke modulo $l$, Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, 1974), Astérisque, Nos. 24-25, Soc. Math. France, Paris, 1975, pp. 109–117 (French). MR 0382173
Goro Shimura, On modular correspondences for $Sp(n,\,Z)$ and their congruence relations, Proc. Nat. Acad. Sci. U.S.A. 49 (1963), 824–828. MR 157009, DOI 10.1073/pnas.49.6.824
H. P. F. Swinnerton-Dyer, On $l$-adic representations and congruences for coefficients of modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 1–55. MR 0406931
- 1.
- A. N. Andrianov, Quadratic forms and Hecke operators, Grund. der Math. 289, Springer-Verlag, Berlin (1987). MR 0884891
- 2.
- A. N. Andrianov and V. G. Zhuravlev, Modular forms and Hecke operators, translated from the Russian (1990) original by N. Koblitz. Translations of Mathematical Monographs 145, AMS, Providence, RI (1995). MR 1349824
- 3.
- R. E. Borcherds, Automorphic forms on
and infinite products, Invent. Math. 120 (1995), 161-213. MR 1323986
- 4.
- S. Breulmann and M. Kuss, On a conjecture of Duke-Omamoglu, Proc. AMS 128 (2000), 1595-1604. MR 1707138
- 5.
- J. H. Bruinier and T. Yang, CM values of Hilbert modular functions, Invent. Math. 163 (2006), 229-288. MR 2207018
- 6.
- P. Deligne and J. P. Serre, Formes modulaires de poids
. Ann. Sci. École Norm. Sup. (4) 7 (1974), 507-530 (1975). MR 0379379
- 7.
- W. Duke and O. Imamoglu, A converse theorem and the Saito-Kurokawa lift, Int. Math. Res. Not. 7 (1996), 347-355. MR 1389957
- 8.
- E. Freitag, Siegelsche Modulfunktionen, Grund. der Math. Wiss. 254, Springer-Verlag, Berlin, (1983). MR 0871067
- 9.
- B. Gross and D. Zagier, On singular moduli, J. Reine Agnew. Math. 355 (1985), 191-220. MR 0772491
- 10.
- T. Ikeda, On the lifting of elliptic cusp forms to Siegel cusp forms of degree
, Ann. of Math. 2 154 (2001), 641-681. MR 1884618
- 11.
- W. Kohnen, Lifting modular forms of half-integral weight to Siegel modular forms of even genus, Math. Ann 322 (2003), 787-809. MR 1905104
- 12.
- I. Miyawaki, Numerical examples of Siegel cusp forms of degree
and their zeta-functions, Mem. Fac. Sci. Kyushu Univ. Ser. A 46 (1992), no. 2, 307-339. MR 1195472
- 13.
- J. P. Serre, Valeurs propres des opérateurs de Hecke modulo
, Journés Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, 1974), 109-117. Asterisque, Nos. 24-25, Soc. Math. France, Paris, 1975. MR 0382173
- 14.
- G. Shimura, On modular correspondences for
and their congruence relations, Proc. Ac. Sci. USA 49 (1963), 824-828. MR 0157009
- 15.
- P. Swinnerton-Dyer, On
-adic representations and congruences for coefficients of modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), 1-55. Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973. MR 0406931
Review Information:
Reviewer:
Amanda Folsom
Affiliation:
University of Wisconsin, Madison
Email:
folsom@math.wisc.edu
Journal:
Bull. Amer. Math. Soc.
46 (2009), 527-533
DOI:
https://doi.org/10.1090/S0273-0979-09-01256-7
Published electronically:
March 23, 2009
Review copyright:
© Copyright 2009
American Mathematical Society
