Algebra & Number Theory

Bounds for traces of Hecke operators and applications to modular and elliptic curves over a finite field

Ian Petrow

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We give an upper bound for the trace of a Hecke operator acting on the space of holomorphic cusp forms with respect to certain congruence subgroups. Such an estimate has applications to the analytic theory of elliptic curves over a finite field, going beyond the Riemann hypothesis over finite fields. As the main tool to prove our bound on traces of Hecke operators, we develop a Petersson formula for newforms for general nebentype characters.

Article information

Source
Algebra Number Theory, Volume 12, Number 10 (2018), 2471-2498.

Dates
Received: 6 May 2018
Revised: 22 July 2018
Accepted: 23 August 2018
First available in Project Euclid: 14 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.ant/1550113229

Digital Object Identifier
doi:10.2140/ant.2018.12.2471

Mathematical Reviews number (MathSciNet)
MR3911137

Zentralblatt MATH identifier
07026824

Subjects
Primary: 11F25: Hecke-Petersson operators, differential operators (one variable)
Secondary: 11F11: Holomorphic modular forms of integral weight 11F72: Spectral theory; Selberg trace formula 11G20: Curves over finite and local fields [See also 14H25] 14G15: Finite ground fields

Keywords
traces of Hecke operators modular curves over a finite field elliptic curves over a finite field Petersson formula for newforms Tsfasman–Vlăduţ–Zink theorem

Citation

Petrow, Ian. Bounds for traces of Hecke operators and applications to modular and elliptic curves over a finite field. Algebra Number Theory 12 (2018), no. 10, 2471--2498. doi:10.2140/ant.2018.12.2471. https://projecteuclid.org/euclid.ant/1550113229


Export citation

References

  • A. O. L. Atkin and J. Lehner, “Hecke operators on $\Gamma \sb{0}(m)$”, Math. Ann. 185 (1970), 134–160.
  • W. D. Banks, “Twisted symmetric-square $L$-functions and the nonexistence of Siegel zeros on ${\rm GL}(3)$”, Duke Math. J. 87:2 (1997), 343–353.
  • O. Barrett, P. Burkhardt, J. DeWitt, R. Dorward, and S. J. Miller, “One-level density for holomorphic cusp forms of arbitrary level”, Res. Number Theory 3 (2017), art. id. 25.
  • V. Blomer and D. Milićević, “The second moment of twisted modular $L$-functions”, Geom. Funct. Anal. 25:2 (2015), 453–516.
  • V. Blomer, J. Buttcane, and P. Maga, “Applications of the Kuznetsov formula on $\rm GL(3$) II: the level aspect”, Math. Ann. 369:1-2 (2017), 723–759.
  • J. B. Conrey, W. Duke, and D. W. Farmer, “The distribution of the eigenvalues of Hecke operators”, Acta Arith. 78:4 (1997), 405–409.
  • H. Davenport, Multiplicative number theory, 3rd ed., Graduate Texts in Mathematics 74, Springer, 2000.
  • F. Diamond and J. Im, “Modular forms and modular curves”, pp. 39–133 in Seminar on Fermat's Last Theorem (Toronto, 1993–1994), CMS Conf. Proc. 17, Amer. Math. Soc., Providence, RI, 1995.
  • F. Diamond and J. Shurman, A first course in modular forms, Graduate Texts in Mathematics 228, Springer, 2005.
  • W. Duke and E. Kowalski, “A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations”, Invent. Math. 139:1 (2000), 1–39.
  • S. Gelbart and H. Jacquet, “A relation between automorphic representations of ${\rm GL}(2)$ and ${\rm GL}(3)$”, Ann. Sci. École Norm. Sup. $(4)$ 11:4 (1978), 471–542.
  • D. Goldfeld, J. Hoffstein, and D. Lieman, “Appendix: an effective zero-free region”, (1994). Appendix to J. Hoffstein and P. Lockhart, “Coefficients of Maass forms and the Siegel zero”, Ann. of Math. $(2)$ 140:1 (1994), 161–181.
  • D. R. Heath-Brown, “The fourth power mean of Dirichlet's $L$-functions”, Analysis 1:1 (1981), 25–32.
  • P. Humphries, “Density theorems for exceptional eigenvalues for congruence subgroups”, Algebra Number Theory 12:7 (2018), 1581–1610.
  • J. Igusa, “Kroneckerian model of fields of elliptic modular functions”, Amer. J. Math. 81 (1959), 561–577.
  • H. Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics 17, Amer. Math. Soc., Providence, RI, 1997.
  • H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications 53, Amer. Math. Soc., Providence, RI, 2004.
  • H. Iwaniec, W. Luo, and P. Sarnak, “Low lying zeros of families of $L$-functions”, Inst. Hautes Études Sci. Publ. Math. 91 (2000), 55–131.
  • N. Kaplan and I. Petrow, “Elliptic curves over a finite field and the trace formula”, Proc. Lond. Math. Soc. $(3)$ 115:6 (2017), 1317–1372.
  • A. Knightly and C. Li, Kuznetsov's trace formula and the Hecke eigenvalues of Maass forms, Mem. Amer. Math. Soc. 1055, Amer. Math. Soc., Providence, RI, 2013.
  • E. Kowalski and P. Michel, “The analytic rank of $J_0(q)$ and zeros of automorphic $L$-functions”, Duke Math. J. 100:3 (1999), 503–542.
  • W. C. W. Li, “Newforms and functional equations”, Math. Ann. 212 (1975), 285–315.
  • W. C. W. Li, “$L$-series of Rankin type and their functional equations”, Math. Ann. 244:2 (1979), 135–166.
  • J. S. Milne, “Modular functions and modular forms”, course notes, 2017, https://www.jmilne.org/math/CourseNotes/mf.html. version 1.31.
  • C. Moreno, Algebraic curves over finite fields, Cambridge Tracts in Mathematics 97, Cambridge University Press, 1991.
  • M. R. Murty and K. Sinha, “Effective equidistribution of eigenvalues of Hecke operators”, J. Number Theory 129:3 (2009), 681–714.
  • P. D. Nelson, “Analytic isolation of newforms of given level”, Arch. Math. $($Basel$)$ 108:6 (2017), 555–568.
  • M. Ng, The basis for space of cusp forms and Petersson trace formula, master's thesis, University of Hong Kong, 2012.
  • A. P. Ogg, “On the eigenvalues of Hecke operators”, Math. Ann. 179 (1969), 101–108.
  • I. Petrow and M. P. Young, “A generalized cubic moment and the Petersson formula for newforms”, Math. Ann. (online publication August 2018).
  • M. Ram Murty, “The analytic rank of $J_0(N)(\mathbb{Q})$”, pp. 263–277 in Number theory (Halifax, 1994), edited by K. Dilcher, CMS Conf. Proc. 15, Amer. Math. Soc., Providence, RI, 1995.
  • S. L. Ross, II, “A simplified trace formula for Hecke operators for $\Gamma_0(N)$”, Trans. Amer. Math. Soc. 331:1 (1992), 425–447.
  • D. Rouymi, “Formules de trace et non-annulation de fonctions $L$ automorphes au niveau $\mathfrak p^\nu$”, Acta Arith. 147:1 (2011), 1–32.
  • R. Schulze-Pillot and A. Yenirce, “Petersson products of bases of spaces of cusp forms and estimates for Fourier coefficients”, Int. J. Number Theory 14:8 (2018), 2277–2290.
  • J.-P. Serre, “Répartition asymptotique des valeurs propres de l'opérateur de Hecke $T_p$”, J. Amer. Math. Soc. 10:1 (1997), 75–102.
  • M. A. Tsfasman, S. G. Vlădu\commaaccentt, and T. Zink, “Modular curves, Shimura curves, and Goppa codes, better than Varshamov–Gilbert bound”, Math. Nachr. 109 (1982), 21–28.
  • A. Venkatesh, “Large sieve inequalities for ${\rm GL}(n)$-forms in the conductor aspect”, Adv. Math. 200:2 (2006), 336–356.
  • M. P. Young, “Explicit calculations with Eisenstein series”, J. Number Theory (online publication December 2018).