Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T02:11:07.748Z Has data issue: false hasContentIssue false
  • English
  • Français

Special values of L-functions for Saito–Kurokawa lifts

Published online by Cambridge University Press:  02 May 2013

JIM BROWN  and
AMEYA PITALE
JIM BROWN
Affiliation:
Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, U.S.A. e-mail: [email protected]
AMEYA PITALE
Affiliation:
Department of Mathematics, University of Oklahoma, Norman, OK 73019, U.S.A. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we obtain special value results for L-functions associated to classical and paramodular Saito–Kurokawa lifts. In particular, we consider standard L-functions associated to Saito–Kurokawa lifts as well as degree eight L-functions obtained by twisting with an automorphic form defined on GL(2). The results are obtained by combining classical and representation theoretic arguments.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 2 , September 2013 , pp. 237 - 255
Copyright
Copyright © Cambridge Philosophical Society 2013 

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]Agarwal, M. and Brown, J. Saito–Kurokawa lifts of square-free level and lower bounds on Selmer groups. preprint (2011).Google Scholar
[2]Andrianov, A.On functional equations satisfied by Spinor Euler products for Siegel modular forms of genus 2 with characters. Abh. Math. Sem. Univ. Hamburg 71 (2001), 123–142.CrossRefGoogle Scholar
[3]Asgari, M. and Schmidt, R.Siegel modular forms and representations. Manuscripta Math 104 (2001), 173–200.CrossRefGoogle Scholar
[4]Böcherer, S.Über die Fourier–Jacobi–Entwicklung der Siegelschen Eisensteinreihen II. Math. Z. 189 (1985), 81–100.CrossRefGoogle Scholar
[5]Böcherer, S. and Heim, B.Critical values of L-functions on GSp2 × GL2. Math. Z. 254 (2006), 485–503.CrossRefGoogle Scholar
[6]Deligne, P.Valeurs de fonctions L et periodes d'integrales. Proc. Symp. Pure Math 33, part 2 (1979), 313–346.CrossRefGoogle Scholar
[7]Diamond, F. and Shurman, J.A first course in modular forms. Graduate Text in Mathematics 228 (Springer-Verlag, New York, 2005).Google Scholar
[8]Eichler, M. and Zagier, D.The theory of Jacobi forms. Progr. Math. 55 (Boston-Birkhauer 1985).CrossRefGoogle Scholar
[9]Gan, W. T. and Takeda, S.The local Langlands conjecture for GSp(4). Ann. Math. 173 no 3 (2011), 1841–1882.CrossRefGoogle Scholar
[10]Gritsenko, V.Irrationality of the moduli spaces of polarized abelian surfaces. In: Abelian varieties, Proccedings of the international conference held in Egloffstein, Germany (October 3–8, 1993), 63–81 (Walter de Gruyter, Berlin, 1995).Google Scholar
[11]Gross, B., Kohnen, W. and Zagier, D.Heegner points and derivatives of L-Series II. Math. Ann. 278 (1987), 497–562.CrossRefGoogle Scholar
[12]Harris, M.Special values of zeta functions attached to Siegel modular forms. Ann. Sci. École. Norm. Super 14 (1981), 77–120.CrossRefGoogle Scholar
[13]Ibukiyama, T.Saito–Kurokawa liftings of level N and practical construction of Jacobi forms. Kyoto J. Math 52 (1) (2012), 141–178.CrossRefGoogle Scholar
[14]Kohnen, W.A remark on the Shimura correspondence. Glasgow Math. J. 30 (1988), 285–291.CrossRefGoogle Scholar
[15]Kohnen, W. and Skoruppa, N.-P.A certain Dirichlet series attached to Siegel modular forms of degree two. Invent. Math 95 (1989), 541–558.CrossRefGoogle Scholar
[16]Kramer, J.Jacobiformen und Thetareihen. Manuscripta Math 54 (1986), 279–322.CrossRefGoogle Scholar
[17]Miyake, T.Modular forms. Springer Monographs in Mathematics (Springer-Verlag, Berlin, 2006).Google Scholar
[18]Mizumoto, S.Poles and residues of standard L-functions attached to Siegel modular forms. Math. Ann. 289 (1991), 589–612.CrossRefGoogle Scholar
[19]Neukirsch, J.Algebraic Number Theory (Springer Verlag, Berlin, 1999).CrossRefGoogle Scholar
[20]Pitale, A. Steinberg representation of GSp(4): Bessel models and integral representation of L-functions. Pacific J. Math., to appear (2010).CrossRefGoogle Scholar
[21]Pitale, A. and Schmidt, R.Integral representation for L-functions for GSp4 × GL2. J. Number Theory 129 (2009), 1272–1324.CrossRefGoogle Scholar
[22]Pitale, A. and Schmidt, R. Integral representation for L-functions for GSp(4) × GL(2), II. Preprint.Google Scholar
[23]Raghuram, A.On the special values of certain Rankin–Selberg L-functions and applications to odd symmetric power L-functions of modular forms. Int. Math. Res. Not. vol. 2010, no. 2 (2010), 334–372.CrossRefGoogle Scholar
[24]Roberts, B. and Schmidt, R.On modular forms for the paramodular groups. Automorphic forms and zeta functions (World Sci. Publ., 2006), 334–364.Google Scholar
[25]Roberts, B. and Schmidt, R.Local newforms for GSp(4). Lecture Notes in Mathematics 1918 (Springer Verlag 2007).CrossRefGoogle Scholar
[26]Saha, A. Pullbacks of Eisenstein series from GU(3,3) and critical L-values for GSp4 × GL2. Pacific J. Math., to appear.Google Scholar
[27]Schmidt, R.The Saito–Kurokawa lifting and functoriality. Amer. J. Math 127 (2005), 209–240.CrossRefGoogle Scholar
[28]Schmidt, R.On classical Saito–Kurokawa liftings. J. Reine Angew. Math 604 (2007), 211–236.Google Scholar
[29]Shimura, G.The special values of zeta functions associated with cusp forms. Comm. Pure Appl. Math. 29 (1976), 783–804.CrossRefGoogle Scholar
[30]Shimura, G.On the periods of modular forms. Math. Ann 229 (1977), 211–221.CrossRefGoogle Scholar
[31]Skoruppa, N.-P. and Zagier, D.Jacobi forms and a certain space of modular forms. Invent. Math. 94 (1988), 113–146CrossRefGoogle Scholar
[32]Stevens, G.Λ-adic modular forms of half-integral weight and a Λ-adic Shintani lifting. Contemp. Math 174 (1994), 129–146.CrossRefGoogle Scholar
[33]Sturm, J.The critical values of zeta-functions associated to the symplectic group. Duke Math. J. 48 (1981), 327–350. CrossRefGoogle Scholar