Hybrid subconvexity for class group 𝐿-functions and uniform sup norm bounds of Eisenstein series

Asbjørn Christian Nordentoft 1
  • 1 Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark
Asbjørn Christian Nordentoft
  • Corresponding author
  • Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark
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Abstract

In this paper, we study hybrid subconvexity bounds for class group 𝐿-functions associated to quadratic extensions K/Q (real or imaginary). Our proof relies on relating the class group 𝐿-functions to Eisenstein series evaluated at Heegner points using formulas due to Hecke. The main technical contribution is the uniform sup norm bound for Eisenstein series E(z,1/2+it)εy1/2(|t|+1)1/3+ε, y1, extending work of Blomer and Titchmarsh. Finally, we propose a uniform version of the sup norm conjecture for Eisenstein series.

  • [1]

    E. Assing, On sup-norm bounds part II: GL ( 2 ) \mathrm{GL}(2) Eisenstein series, Forum Math. 31 (2019), no. 4, 971–1006.

    • Crossref
    • Export Citation
  • [2]

    V. Blomer, Non-vanishing of class group 𝐿-functions at the central point, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 4, 831–847.

    • Crossref
    • Export Citation
  • [3]

    V. Blomer, Epstein zeta-functions, subconvexity, and the purity conjecture, J. Inst. Math. Jussieu 19 (2020), no. 2, 581–596.

    • Crossref
    • Export Citation
  • [4]

    V. Blomer, G. Harcos and P. Michel, Bounds for modular 𝐿-functions in the level aspect, Ann. Sci. Éc. Norm. Supér. (4) 40 (2007), no. 5, 697–740.

    • Crossref
    • Export Citation
  • [5]

    S. Dittmer, M. Proulx and S. Seybert, Some arithmetic problems related to class group 𝐿-functions, Ramanujan J. 37 (2015), no. 2, 257–268.

    • Crossref
    • Export Citation
  • [6]

    W. Duke, J. Friedlander and H. Iwaniec, Class group 𝐿-functions, Duke Math. J. 79 (1995), no. 1, 1–56.

    • Crossref
    • Export Citation
  • [7]

    W. Duke, J. B. Friedlander and H. Iwaniec, The subconvexity problem for Artin 𝐿-functions, Invent. Math. 149 (2002), no. 3, 489–577.

    • Crossref
    • Export Citation
  • [8]

    W. Duke, O. Imamoḡlu and A. Tóth, Cycle integrals of the 𝑗-function and mock modular forms, Ann. of Math. (2) 173 (2011), no. 2, 947–981.

    • Crossref
    • Export Citation
  • [9]

    W. Duke, O. Imamoḡlu and A. Tóth, Geometric invariants for real quadratic fields, Ann. of Math. (2) 184 (2016), no. 3, 949–990.

    • Crossref
    • Export Citation
  • [10]

    Y. Hu and A. Saha, Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, preprint (2019), https://arxiv.org/abs/1905.06295.

  • [11]

    B. Huang and Z. Xu, Sup-norm bounds for Eisenstein series, Forum Math. 29 (2017), no. 6, 1355–1369.

  • [12]

    H. Iwaniec, Spectral Methods of Automorphic Forms, 2nd ed., Grad. Stud. Math. 53, American Mathematical Society, Providence, 2002.

  • [13]

    H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ. 53, American Mathematical Society, Providence, 2004.

  • [14]

    H. Iwaniec and P. Sarnak, L L^{\infty} norms of eigenfunctions of arithmetic surfaces, Ann. of Math. (2) 141 (1995), no. 2, 301–320.

    • Crossref
    • Export Citation
  • [15]

    P. Michel and A. Venkatesh, Heegner points and non-vanishing of Rankin/Selberg 𝐿-functions, Analytic Number Theory, Clay Math. Proc. 7, American Mathematical Society, Providence (2007), 169–183.

  • [16]

    P. Michel and A. Venkatesh, The subconvexity problem for GL 2 \mathrm{GL}_{2}, Publ. Math. Inst. Hautes Études Sci. (2010), no. 111, 171–271.

  • [17]

    P. Sarnak, Arithmetic quantum chaos, The Schur Lectures (1992), Israel Math. Conf. Proc. 8, Bar-Ilan University, Ramat Gan (1995), 183–236.

  • [18]

    P. C. Sarnak, Class numbers of indefinite binary quadratic forms. II, J. Number Theory 21 (1985), no. 3, 333–346.

    • Crossref
    • Export Citation
  • [19]

    P. Söhne, An upper bound for Hecke zeta-functions with Groessencharacters, J. Number Theory 66 (1997), no. 2, 225–250.

    • Crossref
    • Export Citation
  • [20]

    N. Templier, Heegner points and Eisenstein series, Forum Math. 23 (2011), no. 6, 1135–1158.

  • [21]

    E. C. Titchmarsh, On Epstein’s Zeta-Function, Proc. London Math. Soc. (2) 36 (1934), 485–500.

  • [22]

    E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed., Oxford University, New York, 1986.

  • [23]

    H. Wu, Burgess-like subconvexity for GL 1 \mathrm{GL}_{1}, Compos. Math. 155 (2019), no. 8, 1457–1499.

    • Crossref
    • Export Citation
  • [24]

    H. Wu and N. Andersen, Explicit subconvexity for GL 2 \mathrm{GL}_{2} and some applications, preprint (2018), https://arxiv.org/abs/1812.04391.

  • [25]

    M. P. Young, Weyl-type hybrid subconvexity bounds for twisted 𝐿-functions and Heegner points on shrinking sets, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 5, 1545–1576.

    • Crossref
    • Export Citation
  • [26]

    M. P. Young, A note on the sup norm of Eisenstein series, Q. J. Math. 69 (2018), no. 4, 1151–1161.

  • [27]

    S.-W. Zhang, Gross–Zagier formula for GL 2 \mathrm{GL}_{2}, Asian J. Math. 5 (2001), no. 2, 183–290.

    • Crossref
    • Export Citation
  • [28]

    S.-W. Zhang, Gross-Zagier formula for GL ( 2 ) \mathrm{GL}(2). II, Heegner Points and Rankin 𝐿-Series, Math. Sci. Res. Inst. Publ. 49, Cambridge University, Cambridge (2004), 191–214.

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