Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-05T05:11:39.279Z Has data issue: false hasContentIssue false
  • English
  • Français

Period polynomials and explicit formulas for Hecke operators on Γ0(2)

Published online by Cambridge University Press:  01 March 2009

SHINJI FUKUHARA  and
YIFAN YANG
SHINJI FUKUHARA
Affiliation:
Department of Mathematics, Tsuda College, Tsuda-machi 2-1-1, Kodaira-shi, Tokyo 187-8577, Japan. e-mail: [email protected]
YIFAN YANG
Affiliation:
Department of Applied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan300. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let Sw+2(Γ0(N)) be the vector space of cusp forms of weight w + 2 on the congruence subgroup Γ0(N). We first determine explicit formulas for period polynomials of elements in Sw+2(Γ0(N)) by means of Bernoulli polynomials. When N = 2, from these explicit formulas we obtain new bases for Sw+2(Γ0(2)), and extend the Eichler–Shimura–Manin isomorphism theorem to Γ0(2). This implies that there are natural correspondences between the spaces of cusp forms on Γ0(2) and the spaces of period polynomials. Based on these results, we will find explicit form of Hecke operators on Sw+2(Γ0(2)). As an application of main theorems, we will also give an affirmative answer to a speculation of Imamoglu and Kohnen on a basis of Sw+2(Γ0(2)).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 2 , March 2009 , pp. 321 - 350
Copyright
Copyright © Cambridge Philosophical Society 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Antoniadis, J. A.Modulformen auf Γ0(N) mit rationalen Perioden. Manuscripta Math. 74 (4) (1992), 359–384.CrossRefGoogle Scholar
[2]Chan, H. H. and Chua, K. S.Representations of integers as sums of 32 squares. Ramanujan J. 7 (1–3) (2003), 79–89. Rankin memorial issues.CrossRefGoogle Scholar
[3]Cohen, H.Sums involving the values at negative integers of L-functions of quadratic characters. Math. Ann. 217 (3) (1975), 271–285.CrossRefGoogle Scholar
[4]Cohen, H. Sur certaines sommes de séries liées aux périodes de formed modulaires (1981).Google Scholar
[5]Diamantis, N.Hecke operators and derivatives of L-functions. Compositio Math. 125 (1) (2001), 39–54.CrossRefGoogle Scholar
[6]Diamond, F. and Shurman, J. A first course in modular forms, Graduate Texts in Mathematics, vol. 228 (Springer–Verlag, 2005).Google Scholar
[7]Eichler, M.Eine Verallgemeinerung der Abelschen Integrale. Math. Z. 67 (1957), 267–298.CrossRefGoogle Scholar
[8]Fukuhara, S.Explicit formulas for Hecke operators on cusp forms, Dedekind symbols and period polynomials. J. Reine Angew. Math. 607 (2007), 163–216, arXiv:math.NT/0506373.Google Scholar
[9]Fukuhara, S.Hecke operators on weighted Dedekind symbols. J. Reine Angew. Math. 593 (2006), 1–29.CrossRefGoogle Scholar
[10]Imamoglu, O. and Kohnen, W.Representations of integers as sums of an even number of squares. Math. Ann. 333 (4) (2005), 815–829.CrossRefGoogle Scholar
[11]Jones, W. B. and Thron, W. J. Continued fractions, Encyclopedia of Mathematics and its Applications, vol. 11 (Addison-Wesley Publishing Co., 1980). Analytic theory and applications, with a foreword by Felix E. Browder, and an introduction by Peter Henrici.Google Scholar
[12]Kohnen, W. and Zagier, D. Modular forms with rational periods. In: Modular forms (Durham, 1983) Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., pp. 197–249. (Horwood, Chichester, 1984).Google Scholar
[13]Krattenthaler, C.Advanced determinant calculus: a complement. Linear Algebra Appl. 411 (2005), 68–166.CrossRefGoogle Scholar
[14]Manin, Y. I.Explicit formulas for the eigenvalues of Hecke operators. Acta Arith. 24 (1973), 239–249. Collection of articles dedicated to Carl Ludwig Siegel on the occasion of his seventy-fifth birthday. Ill.CrossRefGoogle Scholar
[15]Manin, Y. I.Periods of cusp forms, and p-adic Hecke series. Mat. Sb. (N.S.) 92 (134) (1973), 378–401, 503.Google Scholar
[16]Rankin, R. A.The construction of automorphic forms from the derivatives of a given form. J. Indian Math. Soc. (N.S.) 20 (1956), 103–116.Google Scholar
[17]Shimura, G.Sur les intégrates attachées aux formes automorphes. J. Math. Soc. Japan 11 (1959), 291–311.Google Scholar
[18]Skoruppa, N. P.Binary quadratic forms and the Fourier coefficients of elliptic and Jacobi modular forms. J. Reine Angew. Math. 411 (1990), 66–95.Google Scholar
[19]Zagier, D.Hecke operators and periods of modular forms. In: Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part II (Ramat Aviv, 1989). Israel Math. Conf. Proc., vol. 3 pp. 321–336 (Weizmann, Jerusalem 1990).Google Scholar
[20]Zagier, D.Periods of modular forms and Jacobi theta functions. Invent. Math. 104 (3) (1991), 449–465.CrossRefGoogle Scholar