Rankin–Selberg L-functions and “beyond endoscopy”


Let f and g be two holomorphic cuspidal Hecke eigenforms on the full modular group \( \text {SL}_{2}({\mathbb {Z}}). \) We show that the Rankin–Selberg L-function \(L(s, f \times g)\) has no pole at \(s=1\) unless \( f=g \), in which case it has a pole with residue \( \frac{3}{\pi }\frac{(4\pi )^{k}}{\Gamma (k)} \Vert f \Vert ^2 \), where \( \Vert f\Vert \) is the Petersson norm of f. Our proof uses the Petersson trace formula and avoids the Rankin–Selberg method.

This is a preview of subscription content, log in to check access.


  1. 1.

    Ali Altuǧ, S.: Beyond endoscopy via the trace formula: 1. Poisson summation and isolation of special representations. Compos. Math. 151(10), 1791–1820 (2015)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Ali Altuǧ, S.: Beyond endoscopy via the trace formula, II: Asymptotic expansions of Fourier transforms and bounds towards the Ramanujan conjecture. Am. J. Math. 139, 4 (2017)

    MathSciNet  Google Scholar 

  3. 3.

    Duke, W., Iwaniec, H.: Convolution \(L\)-series. Compos. Math. 91, 145–158 (1994)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Edward Herman, P.: Beyond endoscopy for the Rankin–Selberg L-function. J. Number Theory 131(9), 1691–1722 (2011)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Edward Herman, P.: Quadratic base change and the analytic continuation of the Asai L-function: a new trace formula approach. Am. J. Math. 138(6), 1669–1729 (2016). (English summary)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Iwaniec, H.: Topics in classical automorphic forms. American Mathematical Society Graduate Studies in Mathematics 17, Providence, RI, 1997. xii+259 pp

  7. 7.

    Iwaniec, H.: Spectral methods of automorphic forms. Second Edition. Graduate Studies in Mathematics, 53. American Mathematical Society, Providence; Revista Matemtica Iberoamericana, Madrid, 2002. xii+220 pp

  8. 8.

    Iwaniec, H., Kowalski, E.: Analytic number theory. American Mathematical Society Colloquium Publications 53, American Mathematical Society, Providence (2004)

  9. 9.

    Iwaniec, H., Michel, P.: The second moment of the symmetric square L-functions. Ann. Acad. Sci. Fenn. Math. 26, 465–482 (2001)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Kloosterman, H.D.: On the representation of numbers in the form \(ax^2+by^2+cz^2+dt^2\). Acta Math. 49(3–4), 407–464 (1927)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Endoscopy, Beyond: In contributions to automorphic forms, geometry, and number theory, pp. 611–697. Johns Hopkins University Press, Baltimore (2004)

    Google Scholar 

  12. 12.

    Sarnak, P.: Comments on Robert Langlands Lecture: Endoscopy and Beyond, https://publications.ias.edu/sites/default/files/SarnakLectureNotes-1.pdf

  13. 13.

    Venkatesh, A.: “Beyond endoscopy” and special forms on GL(2). J. Reine Angew. Math. 577, 23–80 (2004)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Watson, G.N.: A treatise on the theory of Bessel functions. Reprint of the second (1944) edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1995)

  15. 15.

    White, P.J.: The base change \(L\)-function for modular forms and beyond endoscopy. J. Number Theory 140, 13–37 (2014)

    MathSciNet  Article  Google Scholar 

Download references


The authors thank Farrell Brumley, Subhajit Jana, M. Ram Murty, V. Kumar Murty, Dipendra Prasad, Olivier Ramar and Peter Sarnak for their interest in this paper and their valuable comments. We especially thank the anonymous referee for helpful remarks which have improved the scope of the result in this paper.

Author information



Corresponding author

Correspondence to Ramdin Mawia.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ganguly, S., Mawia, R. Rankin–Selberg L-functions and “beyond endoscopy”. Math. Z. 296, 175–184 (2020). https://doi.org/10.1007/s00209-019-02431-5

Download citation

Mathematics Subject Classification

  • 11F12
  • 11F30
  • 11F66