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RANKIN’S METHOD AND JACOBI FORMS OF SEVERAL VARIABLES

  • B. RAMAKRISHNAN (a1) and BRUNDABAN SAHU (a2)
Abstract

Following R. A. Rankin’s method, D. Zagier computed the nth Rankin–Cohen bracket of a modular form g of weight k1 with the Eisenstein series of weight k2, computed the inner product of this Rankin–Cohen bracket with a cusp form f of weight k=k1+k2+2n and showed that this inner product gives, up to a constant, the special value of the Rankin–Selberg convolution of f and g. This result was generalized to Jacobi forms of degree 1 by Y. Choie and W. Kohnen. In this paper, we generalize this result to Jacobi forms defined over ℋ×ℂ(g,1).

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Copyright
Corresponding author
For correspondence; e-mail: brundaban.sahu@ucd.ie
References
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[1]Böcherer, S. and Kohnen, W., ‘Estimates for Fourier coefficients of Siegel cusp forms’, Math. Ann. 297 (1993), 499517.
[2]Choie, Y., ‘Jacobi forms and the heat operator’, Math. Z. 225(1) (1997), 95101.
[3]Choie, Y., ‘Jacobi forms and the heat operator II’, Illinois J. Math. 42 (1998), 179186.
[4]Choie, Y. and Kim, H., ‘Differential operators on Jacobi forms of several variables’, J. Number Theory 82 (2000), 140163.
[5]Choie, Y. and Kohnen, W., ‘Rankin’s method and Jacobi forms’, Abh. Math. Sem. Univ. Hamburg 67 (1997), 307314.
[6]Cohen, H., ‘Sums involving the values at negative integers of L-functions of quadratic characters’, Math. Ann. 217 (1977), 8194.
[7]Eichler, M. and Zagier, D., The Theory of Jacobi Forms, Progress in Mathematics, 55 (Birkhäuser, Boston, MA, 1985).
[8]Kohnen, W., ‘On the uniform convergence of Poincaré series of exponential type on Jacobi groups’, Abh. Math. Sem. Univ. Hamburg 66 (1996), 131134.
[9]Maass, H., ‘Über die gleichmässige Konvergenz der Poincaréschen Reihen n-ten Grades’, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 12 (1964), 137144.
[10]Rankin, R. A., ‘The construction of automorphic forms from the derivatives of a given form’, J. Indian Math. Soc. 20 (1956), 103116.
[11]Rankin, R. A., ‘The construction of automorphic forms from the derivatives of given forms’, Michigan Math. J. 4 (1957), 181186.
[12]Sahu, B., ‘Some problems in number theory’, PhD Thesis, Harish-Chandra Research Institute, University of Allahabad, 2008.
[13]Zagier, D., ‘Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields’, in: Modular Functions of One Variable, VI (Proc. 2nd Int. Conf., Univ. Bonn, Bonn, 1976), Lecture Notes in Mathematics, 627 (Springer, Berlin, 1977), pp. 105169.
[14]Zagier, D., ‘Modular forms and differential operators’, Proc. Indian Acad. Sci. Math. Sci. 104 (1994), 5775.
[15]Ziegler, C., ‘Jacobi forms of higher degree’, Abh. Math. Sem. Univ. Hamburg 59 (1989), 191224.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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