Distribution of moments of Hurwitz class numbers in arithmetic progressions and holomorphic projection

Kane, Ben; Pujahari, Sudhir

doi:10.1090/tran/8885

Distribution of moments of Hurwitz class numbers in arithmetic progressions and holomorphic projection
HTML articles powered by AMS MathViewer

by Ben Kane and Sudhir Pujahari

Trans. Amer. Math. Soc. 376 (2023), 5503-5519

DOI: https://doi.org/10.1090/tran/8885

Published electronically: May 23, 2023

HTML | PDF | Request permission

Abstract:

In this paper, we study moments of Hurwitz class numbers associated to imaginary quadratic orders restricted into fixed arithmetic progressions. In particular, we fix $t$ in an arithmetic progression $t\equiv m\ \, \left ( \operatorname {mod} \, M \right )$ and consider the ratio of the $2k$-th moment to the zeroeth moment for $H(4n-t^2)$ as one varies $n$. The special case $n=p^r$ yields as a consequence asymptotic formulas for moments of the trace $t\equiv m\ \, \left ( \operatorname {mod} \, M \right )$ of Frobenius on elliptic curves over finite fields with $p^r$ elements.

References

O. Ahmadi and I. E. Shparlinski, On the distribution of the number of points on algebraic curves in extentions of finite fields, arXiv:0907.3664, 2009.
E. Artin, Quadratische Körper im Gebiete der höheren Kongruenzen. II. Analytischer Teil, Math. Z. 19 (1924), 207–246.
M. B. Barban and G. Gordover, Moments of the number of classes of purely radical quadratic forms with negative determinant, Sov. Math. 7 (1966), 356–358.
T. Barnet-Lamb, D. Geraghty, M. Harris, and R. Taylor, A family of Calabi–Yau varieties and potential automorphy II, Publ. Res. Inst. Math. Sci. 47 (2011), 29–98.
A. Bhand and M. R. Murty, Class number of quadratic fields, Hardy-Ramanujan J. 42 (2019), 17–25.
B. Birch, How the number of points of an elliptic curve over a fixed prime field varies, J. London Math. Soc. 43 (1968), 57–60.
K. Bringmann and B. Kane, Sums of class numbers and mixed mock modular forms, Math. Proc. Cambridge Philos. Soc. 167 (2019), 321–333.
B. Brock and A. Granville, More points than expected on curves over finite field extensions, Finite Fields Appl. 7 (2001), 70–91.
Brittany Brown, Neil J. Calkin, Timothy B. Flowers, Kevin James, Ethan Smith, and Amy Stout, Elliptic curves, modular forms, and sums of Hurwitz class numbers, J. Number Theory 128 (2008), no. 6, 1847–1863. MR 2419197, DOI 10.1016/j.jnt.2007.10.008
W. Castryck and H. Hubrechts, The distribution of the number of points modulo an integer on elliptic curves over finite fields, Ramanujan J. 30 (2013), 223–242.
L. Clozel, M. Harris, and R. Taylor, Automorphy for some $\ell$-adic lifts of automorphic mod $\ell$ Galois representations, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181.
H. Cohen, Sums involving the values at negative integers of $L$-functions of quadratic characters, Math. Ann. 217 (1975), 217–285.
C. David, D. Koukoulopoulos, and E. Smith, Sums of Euler products and statistics of elliptic curves, Math. Ann. 368 (2017), 685–752.
P. Deligne, La conjecture de Weil I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307.
M. Deuring, Die Typen der Multiplikatorenringe elliptische Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197–272.
Digital Library of Mathematical Functions, National Institute of Standards and Technology, http://dlmf.nist.gov/.
R. Donepudi, J. Li, and A. Zaharescu, Exact evaluation of second moments associated with some families of curves over a finite field, Finite Fields Appl. 48 (2017), 331–355.
M. Eichler, On the class of imaginary quadratic fields and sums of divisors of natural numbers, J. Indian Math. Soc. 19 (1956), 153–180.
Martin Eichler and Don Zagier, The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781735, DOI 10.1007/978-1-4684-9162-3
S. Galbraith and J. McKee, The probability that the number of points on an elliptic curve over a finite field is prime, J. London Math. Soc. 62 (2000), 671–684.
E. Gekeler, Frobenius distributions of elliptic curves over finite prime fields, Int. Math. Res. Not. 2003, no. 37, (2003), 1999–2018.
B. Gross and D. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), 225–320.
Michael Harris, Galois representations, automorphic forms, and the Sato-Tate conjecture, Indian J. Pure Appl. Math. 45 (2014), no. 5, 707–746. MR 3286083, DOI 10.1007/s13226-014-0085-4
M. Harris, N. Shepherd-Barron, and R. Taylor, A family of Calabi–Yau varieties and potential automorphy, Ann. Math. 171 (2010), 779–813.
H. Hasse, Zur Theorie der abstrakten elliptischen Funktionenkörper. I, II, and III, J. Reine Angew. Math. 175 (1936), 55–62, 69–88, 193–208.
F. Hirzebruch and D. Zagier, Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, Invent. Math. 36 (1976), 57–113.
Y. Ihara, Hecke polynomials as congruence $\zeta$ functions in elliptic modular case, Ann. Math. 85 (1967), 267–295.
N. Kaplan and I. Petrow, Elliptic curves over a finite field and the trace formula, Proc. London Math. Soc. 115 (2017), 1317–1372.
N. Kaplan and I. Petrow, Traces of Hecke operators and refined weight enumerators of Reed–Solomon codes, Trans. Amer. Math. Soc. 370 (2018), 2537–2561.
Nicholas M. Katz and Peter Sarnak, Random matrices, Frobenius eigenvalues, and monodromy, American Mathematical Society Colloquium Publications, vol. 45, American Mathematical Society, Providence, RI, 1999. MR 1659828, DOI 10.1090/coll/045
N. Koblitz, Introduction to elliptic curves and modular forms, Graduate texts in Math. 97, Springer-Verlag, 1993.
E. Kowalski, Analytic problems for elliptic curves, J. Ramanujan Math. Soc. 21 (2006), 19–114.
A. F. Lavrik, On the problem of distribution of the values of class number of properly primitive quadratic forms with negative determinant, Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. 1 (1959), 81–90.
J. Littlewood, On the class number of the corpus $P(\sqrt {-k})$, Proc. London Math. Soc. 27 (1928), 358–372.
J. McKee, Subtleties in the distribution of the numbers of points on elliptic curves over a finite prime field, J. Lond. Math. Soc. 59 (1999), 448–460.
Michael H. Mertens, Eichler-Selberg type identities for mixed mock modular forms, Adv. Math. 301 (2016), 359–382. MR 3539378, DOI 10.1016/j.aim.2016.06.016
Michael H. Mertens, Ken Ono, and Larry Rolen, Mock modular Eisenstein series with Nebentypus, Int. J. Number Theory 17 (2021), no. 3, 683–697. MR 4254772, DOI 10.1142/S179304212040028X
Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and $q$-series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR 2020489
R. Schoof, Nonsingular plane cubic curves over finite fields, J. Comb. Theory Ser. A 46 (1987), 183–211.
J.-P. Serre and H. Stark, Modular forms of weight $\frac {1}{2}$, in Modular functions of one variable VI, Lecture Notes in Math. 627 (1977), Springer, Berlin, pp. 27–67.
C. Siegel, Über die Classenzahl quadratischer Zahlkörper, Acta. Arith. 1 (1935), 83–86.
Joseph H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094, DOI 10.1007/978-0-387-09494-6
I. E. Shparlinski and L. Zhao, Elliptic curves in isogeny classes, J. Number Theory 191 (2018), 194–212.
J. Sturm, Projections of $C^{\infty }$ automorphic forms, Bull. Amer. Math. Soc. 2 (1980), 435–439.
D. Wolke, Moments of the number of classes of primitive quadratic forms with negative discriminant, J. Number Theory 1 (1969), 502–511.
D. Wolke, Momente der klassenzahlen, II, Arch. Math. 22 (1971), 65–69.
D. Wolke, Moments of class numbers, III, J. Number Theory 4 (1972), 523–531.