On the kernel of the theta operator mod p
Article
First Online:
Received:
Accepted:
- 33 Downloads
Abstract
We construct many examples of level one Siegel modular forms in the kernel of theta operators mod p by using theta series attached to positive definite quadratic forms.
Mathematics Subject Classification
11F33 11F30Preview
Unable to display preview. Download preview PDF.
References
- 1.Andrianov, A.N.: Quadratic Forms and Hecke operators. In: (ed.) Grundlehren Math. Wiss, vol. 286. Springer, Berlin (1987)Google Scholar
- 2.Aoki, H., Ibukiyama, T.: Simple graded rings of Siegel modular forms, differential operators and Borcherds products. Int. J. Math. 16, 249–279 (2005)MathSciNetCrossRefMATHGoogle Scholar
- 3.Böcherer, S.: Über gewisse Siegelsche Modulformen zweiten Grades. Math. Ann. 261, 23–41 (1982)MathSciNetCrossRefMATHGoogle Scholar
- 4.Böcherer, S., Funke, J., Schulze-Pillot, R.: Trace operator and theta series. J. Number Theory 78, 119–139 (1999)MathSciNetCrossRefMATHGoogle Scholar
- 5.Böcherer, S., Kikuta, T.: On mod \(p\) singular modular forms. Forum Math. 28, 1051–1065 (2016)MathSciNetCrossRefMATHGoogle Scholar
- 6.Böcherer, S., Nagaoka, S.: On mod \(p\) properties of Siegel modular forms. Math. Ann. 338, 421–433 (2007)MathSciNetCrossRefMATHGoogle Scholar
- 7.Böcherer, S., Nagaoka, S.: On Siegel modular forms of level \(p\) and their properties mod \(p\). Manuscr. math. 132, 501–515 (2010)MathSciNetCrossRefMATHGoogle Scholar
- 8.Böcherer, S., Nagaoka, S.: On \(p\)-adic properties of Siegel modular forms. In: Heim, B., et al. (eds.) Automorphic Forms, Springer Proceedings in Mathematics and Statistics, vol. 115, pp. 47–66. Springer, Berlin (2014)Google Scholar
- 9.Böcherer, S., Nebe, G.: On theta series attached to maximal lattices and their adjoints. J. Ramanujan Math. Soc. 25(3), 265–284 (2000)MathSciNetMATHGoogle Scholar
- 10.Böcherer, S., Kikuta, T., Takemori, S.: Weights of mod \(p\) kernels of theta operators. arXiv: 1606.06390v1 [math.NT] (to appear in Canadian J. Math.)
- 11.Böcherer, S., Schulze-Pillot, R.: Siegel modular forms and theta series attached to quaternion algebras. Nagoya Math. J. 121, 35–96 (1991)MathSciNetCrossRefMATHGoogle Scholar
- 12.Choi, D., Choie, Y., Kikuta, T.: Sturm type theorem for Siegel modular forms of genus 2 modulo \(p\). Acta Arith. 158, 129139 (2013)MathSciNetCrossRefMATHGoogle Scholar
- 13.Freitag, E.: Siegelsche Modulfunktionen Grundlehren der mathematischen Wissenschaften, vol. 254. Springer, Berlin (1983)MATHGoogle Scholar
- 14.Ichikawa, T.: Vector-valued \(p\)-adic Siegel modular forms. J. Rein. Angew. Math. 690, 35–49 (2014)MathSciNetMATHGoogle Scholar
- 15.Igusa, J.-I.: On the ring of modular forms of degree two over \(\varvec {Z}\). Am. J. Math. 101, 149–183 (1979)MathSciNetCrossRefMATHGoogle Scholar
- 16.Kikuta, T., Kodama, H., Nagaoka, S.: Note on Igusa’s cusp form of weight 35. Rocky Mt. J. Math. 45, 963–972 (2015)MathSciNetCrossRefMATHGoogle Scholar
- 17.Klingen, H.: Bemerkungen über Kongruenzuntergruppen der Modulgruppe \(n\)-ten Grades. Archiv der Math. 10, 113–122 (1959)CrossRefMATHGoogle Scholar
- 18.Klingen, H.: Introductory Lectures on Siegel Modular Forms. Cambridge University Press, Cambridge (1990)CrossRefMATHGoogle Scholar
- 19.Kodama, H., Nagaoka, S.: A congruence relation satisfied by Siegel cusp form of odd weight. J. School Sci. Eng. Kinki Univ. 49, 9–15 (2013)Google Scholar
- 20.Krieg, A.: Hecke Algebras, Memoirs of AMS, 435, (1990)Google Scholar
- 21.Maaß, H.: Konstruktion von Spitzenformen beliebigen Grades mit Hilfe von Thetareihen. Math. Ann. 226, 275–284 (1977)MathSciNetCrossRefMATHGoogle Scholar
- 22.Maaß, H.: Siegel’s modular forms and Dirichlet series, Lecture Notes in Mathematics 216, Springer, Berlin (1971)Google Scholar
- 23.Mizumoto, S.: On integrality of certain algebraic numbers associated with modular forms. Math. Ann. 265, 119–135 (1983)MathSciNetCrossRefMATHGoogle Scholar
- 24.Nagaoka, S.: On the mod \(p\) kernel of the theta operator. Proc. Am. Math. Soc. 143, 4237–4244 (2015)MathSciNetCrossRefMATHGoogle Scholar
- 25.Nagaoka, S.: Note on mod \(p\) Siegel modular forms. Math. Z. 235, 405–420 (2000)MathSciNetCrossRefMATHGoogle Scholar
- 26.Peters, M.: Ternäre und quaternäre quadratische Formen und Quaternionenalgebren. Acta Arithm. 15, 329–365 (1969)CrossRefMATHGoogle Scholar
- 27.Scharlau, W.: Quadratic and Hermitian Forms, Grundlehren der mathematischen Wissenschaften, vol. 270. Springer, Berlin (1985)CrossRefGoogle Scholar
- 28.Shimura, G.: On the Fourier coefficients of modular forms in several variables. Nachrichten der Akademie der Wissenschaften in Göttingen, Mathematisch–Physikalische Klasse 1975, 261-268 (=[75d] in Collected Papers II)Google Scholar
- 29.Waldspurger, J.-L.: Engendrement par des séries thêta de certains espaces de formes modulaires. Invent. Math. 50, 135–168 (1978)MathSciNetCrossRefMATHGoogle Scholar
- 30.Zagier, D.B.: Zetafunktionen und quadratische Körper. Springer, Berlin (1981)CrossRefMATHGoogle Scholar
Copyright information
© Springer-Verlag GmbH Germany 2017