Numerical computation of a certain Dirichlet series attached to Siegel modular forms of degree two

Ryan, Nathan; Skoruppa, Nils-Peter; Strömberg, Fredrik

doi:10.1090/S0025-5718-2012-02584-1

Numerical computation of a certain Dirichlet series attached to Siegel modular forms of degree two
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by Nathan C. Ryan, Nils-Peter Skoruppa and Fredrik Strömberg PDF

Math. Comp. 81 (2012), 2361-2376 Request permission

Abstract:

The Rankin convolution type Dirichlet series $D_{F,G}(s)$ of Siegel modular forms $F$ and $G$ of degree two, which was introduced by Kohnen and the second author, is computed numerically for various $F$ and $G$. In particular, we prove that the series $D_{F,G}(s)$, which shares the same functional equation and analytic behavior with the spinor $L$-functions of eigenforms of the same weight are not linear combinations of those. In order to conduct these experiments a numerical method to compute the Petersson scalar products of Jacobi Forms is developed and discussed in detail.

References

Götz Alefeld and Jürgen Herzberger, Introduction to interval computations, Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Translated from the German by Jon Rokne. MR 733988
A. N. Andrianov, Euler products that correspond to Siegel’s modular forms of genus $2$, Uspehi Mat. Nauk 29 (1974), no. 3 (177), 43–110 (Russian). MR 0432552
Jan H. Bruinier, Borcherds products on O(2, $l$) and Chern classes of Heegner divisors, Lecture Notes in Mathematics, vol. 1780, Springer-Verlag, Berlin, 2002. MR 1903920, DOI 10.1007/b83278
Martin Eichler and Don Zagier, The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781735, DOI 10.1007/978-1-4684-9162-3
S. A. Evdokimov, Characterization of the Maass space of Siegel modular cusp forms of genus $2$, Mat. Sb. (N.S.) 112(154) (1980), no. 1(5), 133–142, 144 (Russian). MR 575936
David W. Farmer, Nathan C. Ryan, and Ralf Schmidt, Finding and verifying the functional equation for high degree Euler-products, Pac. J. Math. (To appear).
Carl-Friedrich Gauss, Summatio quarundam serierum singularium, Com Soc. Rec. Sci. Göttingensis Rec. 1 (1811).
B. Gross, W. Kohnen, and D. Zagier, Heegner points and derivatives of $L$-series. II, Math. Ann. 278 (1987), no. 1-4, 497–562. MR 909238, DOI 10.1007/BF01458081
Özlem Imamoḡlu and Yves Martin, On a Rankin-Selberg convolution of two variables for Siegel modular forms, Forum Math. 15 (2003), no. 4, 565–589. MR 1978335, DOI 10.1515/form.2003.031
W. Kohnen and N.-P. Skoruppa, A certain Dirichlet series attached to Siegel modular forms of degree two, Invent. Math. 95 (1989), no. 3, 541–558. MR 979364, DOI 10.1007/BF01393889
Nathan C. Ryan and N.-P. Skoruppa, Sage-add-ons: A Sage package for computing Siegel modular forms of degree 2, 2009, http://hg.countnumber.de.
Nathan C. Ryan, Nils-Peter Skoruppa, and Fredrik Strömberg, Data on the Rankin convolution of two Siegel modular forms of degree 2, August 2009, http://data.countnumber.de/Lfg.
The SAGE Group, SAGE Mathematics Software (Version 4.1), 2009.
Takuro Shintani, On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975), 83–126. MR 389772, DOI 10.1017/S0027763000016706
Nils-Peter Skoruppa, Über den Zusammenhang zwischen Jacobiformen und Modulformen halbganzen Gewichts, Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 159, Universität Bonn, Mathematisches Institut, Bonn, 1985 (German). Dissertation, Rheinische Friedrich-Wilhelms-Universität, Bonn, 1984. MR 806354
Nils-Peter Skoruppa, Computations of Siegel modular forms of genus two, Math. Comp. 58 (1992), no. 197, 381–398. MR 1106982, DOI 10.1090/S0025-5718-1992-1106982-0
Nils-Peter Skoruppa and Don Zagier, Jacobi forms and a certain space of modular forms, Invent. Math. 94 (1988), no. 1, 113–146. MR 958592, DOI 10.1007/BF01394347
Nils-Peter Skoruppa and Don Zagier, A trace formula for Jacobi forms, J. Reine Angew. Math. 393 (1989), 168–198. MR 972365, DOI 10.1515/crll.1989.393.168
Fredrik Strömberg, Weil representation associated to finite quadratic modules, preprint (2011), arXiv:1108.0202.