Abstract
We obtain a first moment formula for Rankin–Selberg convolution L-series of holomorphic modular forms or Maass forms of arbitrary level on \({{\,\mathrm{GL}\,}}(2)\), with an orthonormal basis of Maass forms. One consequence is the best result to date, uniform in level, spectral value and weight, for the equality of two Maass or holomorphic cusp forms if their Rankin–Selberg convolutions with the orthonormal basis of Maass forms \(u_j\) is equal at the center of the critical strip for sufficiently many \(u_j\). The main novelty of our approach is the new way the error terms are treated. They are brought into an exact form that provides optimal estimates for this first moment case, and also provide a basis for an extension to second moments, which will appear in another work.
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References
Andrews, G.E., Askey, R., Roy, R.: Special Functions, Encyclopedia of Mathematics and Its Applications, vol. 71. Cambridge University Press, Cambridge (1999)
Bettin, S., Bober, J.W., Booker, A.R., Conrey, B., Lee, M., Molteni, G., Oliver, T., Platt, D.J., Steiner, R.S.: A conjectural extension of Hecke’s converse theorem. Ramanujan J. 47(3), 659–684 (2018)
Blomer, V.: Shifted convolution sums and subconvexity bounds for automorphic $L$-functions. Int. Math. Res. Not. 73, 3905–3926 (2004)
Bykovskiĭ, V.A.: A trace formula for the scalar product of Hecke series and its applications. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. 226, 235–236 (1996)
Chinta, G., Diaconu, A.: Determination of a ${\rm GL}_3$ cuspform by twists of central $L$-values. Int. Math. Res. Not. 48, 2941–2967 (2005)
Diamantis, N., Hoffstein, J., Kiral, E.M., Lee, M.: Additive twists and a conjecture by Mazur, Rubin and Stei. J. Number Theory 209, 1–36 (2020)
Ganguly, S., Hoffstein, J., Sengupta, J.: Determining modular forms on ${\rm SL}_2(\mathbb{Z})$ by central values of convolution $L$-functions. Math. Ann. 345, 843–857 (2009)
Goldfeld, D., Zhang, S.: The holomorphic kernel of the Rankin–Selberg convolution. Asian J. Math. 3(4), 729–747 (1999)
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 6th edn. Academic Press Inc., San Diego (2000)
Hoffstein, J., Hulse, T.A.: Multiple Dirichlet series and shifted convolutions. J. Number Theory 161, 457–533 (2016)
Hoffstein, J., Lee, M.: Second moments of Rankin–Selberg convolutions and shifted Dirichlet series, in preparation (2020)
Holowinsky, R., Templier, N.: First moment of Rankin–Selberg central $L$-values and subconvexity in the level aspect. Ramanujan J. 33(1), 131–155 (2014)
Hu, Y.: The Petersson/Kuzenetsov trace formula with prescribed local ramifications. Preprint at http://arxiv.org/abs/2005.09959 (2020)
Humphries, P., Radziwiłł, M.: Optimal small scale equidistribution of lattice points on the sphere, Heegner points, and closed geodesics. Preprint at http://arxiv.org/abs/1910.01360 (2019)
Iwaniec, H.: Spectral Methods of Automorphic Forms. Graduate Studies in Mathema, vol. 53, 2nd edn. American Mathematical Society, Revista Matemática Iberoamericana, Providence, Madrid (2002)
Iwaniec, H., Kowalski, E.: Analytic Number Theory, American Mathematical Society Colloquium Publications, vol. 53. American Mathematical Society, Providence (2004)
Luo, W.: Special $L$-values of Rankin–Selberg convolutions. Math. Ann. 314(3), 591–600 (1999)
Luo, W., Ramakrishnan, D.: Determination of modular forms by twists of critical $L$-values. Invent. Math. 130(2), 371–398 (1997)
Munshi, R., Sengupta, J.: On effective determination of Maass forms from central values of Rankin–Selberg $L$-function. Forum Math. 27(1), 467–484 (2015)
Nelson, P.D.: Stable averages of central values of Rankin-Selberg $L$-functions: some new variants. J. Number Theory 133(8), 2588–2615 (2013)
Olver, F.W.J. et al. (eds): NIST Digital Library of Mathematical Functions, Release 1.0.26 of 2020-03-15. http://dlmf.nist.gov/
Saha, B., Sengupta, J.: Determination of ${ {GL}}(2)$ Maass forms from twists in the level aspect. Acta Arith. 189(2), 165–178 (2019)
Selberg, A.: On the estimation of Fourier coefficients of modular forms. In: Whiteman, A.L. (ed.) Proceedings of Symposia in Pure Mathematics, vol. 8, pp. 1–15. American Mathematical Society, Providence (1965)
Sengupta, J.: Distinguishing Hecke eigenvalues of primitive cusp forms. Acta Arith. 114(1), 23–34 (2004)
Young, M.P.: Explicit calculations with Eisenstein series. J. Number Theory 199, 1–48 (2019)
Zhang, Y.: Determining modular forms of general level by central values of convolution $L$-functions. Acta Arith. 150(1), 93–103 (2011)
Acknowledgements
The authors would like to thank Peter Humphries for some very helpful comments, and POSTECH for providing a welcoming working environment during part of the preparation of this paper.
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M. L. was supported by Royal Society University Research Fellowship “Automorphic forms, L-functions and trace formulas”. (AMS classifications 11M32, 11M36)
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Hoffstein, J., Lee, M. & Nastasescu, M. First moments of Rankin–Selberg convolutions of automorphic forms on \({{\,\mathrm{GL}\,}}(2)\). Res. number theory 7, 60 (2021). https://doi.org/10.1007/s40993-021-00281-x
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DOI: https://doi.org/10.1007/s40993-021-00281-x