Abstract
We show that a Siegel modular form with integral Fourier
coefficients in a number field K, for which
all but finitely many coefficients (up to equivalence) are divisible by a prime ideal
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: Grant-in-Aid for Young Scientists (B) 26800026
Funding statement: The second author is supported by JSPS Grant-in-Aid for Young Scientists (B) 26800026.
Acknowledgements
This work was done when the first author held a guest professorship at the Graduate School of Mathematical Sciences at the University of Tokyo. He wishes to thank Professor T. Oda for arranging this stay and supporting this collaboration. The authors thank Professor S. Nagaoka for helpful discussions. The authors would also like to thank the referee, whose advice was helpful in improving the presentation of this paper.
References
[1] Andrianov A. N. and Zhuravlev V. G., Modular Forms and Hecke Operators, Transl. Math. Monogr. 145, American Mathematical Society, Providence, 1995. Search in Google Scholar
[2] Bayer-Fluckiger E., Definite unimodular lattices having an automorphism with given characteristic polynomial, Comment. Math. Helv. 54 (1984), 509–538. 10.1007/BF02566364Search in Google Scholar
[3] Böcherer S., Über gewisse Siegelsche Modulformen zweiten Grades, Math. Ann. 261 (1982), 23–41. 10.1007/BF01456406Search in Google Scholar
[4] Böcherer S., Über die Fourier–Jacobi-Entwicklung Siegelscher Eisensteinreihen, Math. Z. 183 (1983), 21–46. 10.1007/BF01187213Search in Google Scholar
[5] Böcherer S., Über die Fourierkoeffizienten der Siegelschen Eisensteinreihen, Manuscripta Math. 45 (1985), 273–288. 10.1007/BF01158040Search in Google Scholar
[6] Böcherer S. and Nagaoka S., On mod p properties of Siegel modular forms, Math. Ann. 338 (2007), 421–433. 10.1007/s00208-007-0081-7Search in Google Scholar
[7] Böcherer S. and Nagaoka S., Congruences for Siegel modular forms and their weights, Abh. Math. Semin. Univ. Hambg. 80 (2010), 227–231. 10.1007/s12188-010-0042-zSearch in Google Scholar
[8] Böcherer S. and Nagaoka S., On p-adic properties of Siegel modular forms, Automorphic Forms: Research in Number Theory from Oman (Muscat 2012), Springer Proc. Math. Stat. 115, Springer, Cham (2014), 47–66. 10.1007/978-3-319-11352-4_4Search in Google Scholar
[9] Böcherer S. and Raghavan S., On Fourier coefficients of Siegel modular forms, J. Reine Angew. Math. 384 (1988), 80–101. 10.1017/nmj.2016.41Search in Google Scholar
[10] Choi D., Choie Y. and Richter O., Congruences for Siegel modular forms, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 4, 1455–1466. 10.5802/aif.2646Search in Google Scholar
[11] Diamond F. and Shurman J., A First Course in Modular Forms, Grad. Texts in Math. 228, Springer, Cham, 2005. Search in Google Scholar
[12] Eichler M. and Zagier D., The Theory of Jacobi forms, Progr. Math. 55, Birkhäuser, Boston, 1985. 10.1007/978-1-4684-9162-3Search in Google Scholar
[13] Freitag E., Siegelsche Modulfunktionen, Grundlehren Math. Wiss. 254, Springer, Berlin, 1983. 10.1007/978-3-642-68649-8Search in Google Scholar
[14] Haruki A., Explicit formulae of Siegel Eisenstein series, Manuscripta Math. 92 (1997), 107–134. . 10.1007/BF02678184Search in Google Scholar
[15] Katz N., p-adic properties of modular schemes and modular forms, Modular Functions of One Variable III (Antwerp 1972), Lecture Notes in Math. 350, Springer, Berlin (1973), 69–190. 10.1007/978-3-540-37802-0_3Search in Google Scholar
[16] Klingen H., Introductory Lectures on Siegel Modular Forms, Cambridge University Press, Cambridge, 1990. 10.1017/CBO9780511619878Search in Google Scholar
[17] Koblitz N., Introduction to Elliptic Curves and Modular Forms, Grad. Texts in Math. 97, Springer, New York, 1993. 10.1007/978-1-4612-0909-6Search in Google Scholar
[18] Mizumoto S., Fourier coefficients of generalized Eisenstein series of degree two. I, Invent. Math. 65 (1981), 115–135. 10.1007/BF01389298Search in Google Scholar
[19] Mizumoto S., On Integrality of Eisenstein liftings, Manuscripta Math. 89 (1996), 203–235. 10.1007/BF02567514Search in Google Scholar
[20] Nagaoka S., p-adic properties of Siegel modular forms of degree 2, Nagoya Math. J. 71 (1978), 43–60. 10.1017/S0027763000021620Search in Google Scholar
[21] Rasmussen J., Higher congruences between modular forms, PhD. thesis, University of Copenhagen, Department of Mathematical Sciences, Copenhagen, 2009. Search in Google Scholar
[22] Serre J.-P., Formes modulaires et fonctions zêta p-adiques, Modular Functions of One Variable III, Lec. Notes in Math. 350, Springer, Berlin (1973), 191–268. 10.1007/978-3-540-37802-0_4Search in Google Scholar
[23] Serre J.-P., Divisibilité de certaines fonctions arithmétiques, Enseign. Math. (2) 22 (1976), 227–260. 10.1007/978-3-642-39816-2_108Search in Google Scholar
[24] Shimura G., On Eisenstein series, Duke Math. J. 50 (1983), 417–476. 10.1215/S0012-7094-83-05019-6Search in Google Scholar
[25] Sturm J., On the congruence of modular forms, Number Theory, Lecture Notes in Math. 1240, Springer, Berlin (1987), 275–280. 10.1007/BFb0072985Search in Google Scholar
[26] Swinnerton-Dyer H. P. F., On l-adic representations and congruences for coefficients of modular forms, Modular Functions of One Variable III, Lec. Notes in Math. 350, Springer, Berlin (1973), 1–55. 10.1007/978-3-540-37802-0_1Search in Google Scholar
[27] Weissauer R., Stabile Modulformen und Eisensteinreihen, Lec. Notes in Math. 2119, Springer, Berlin, 1986. 10.1007/BFb0072235Search in Google Scholar
[28] Ziegler C., Jacobi forms of higher degree, Abh. Math. Semin. Univ. Hambg. 59 (1989), 191–224. 10.1007/BF02942329Search in Google Scholar
© 2016 by De Gruyter
