Trace formulas for Hecke operators, Gaussian hypergeometric functions, and the modularity of a threefold. (English) Zbl 1277.11037
Summary: We present simple trace formulas for Hecke operators \(T_k(p)\) for all \(p>3\) on \(S_k(\varGamma _{0}(3))\) and \(S_k(\varGamma _{0}(9))\), the spaces of cusp forms of weight \(k\) and levels 3 and 9. These formulas can be expressed in terms of special values of Gaussian hypergeometric series and lend themselves to recursive expressions in terms of traces of Hecke operators on spaces of lower weight. Along the way, we show how to express the traces of Frobenius of a family of elliptic curves equipped with a 3-torsion point as special values of a Gaussian hypergeometric series over \(\mathbb F_q\), when \(q \equiv 1\). As an application, we use these formulas to provide a simple expression for the Fourier coefficients of \(\eta (3z)^8\), the unique normalized cusp form of weight 4 and level 9, and then show that the number of points on a certain threefold is expressible in terms of these coefficients.
MSC:
| 11F25 | Hecke-Petersson operators, differential operators (one variable) |
| 11G20 | Curves over finite and local fields |
| 11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |
| 11T24 | Other character sums and Gauss sums |
| 33E50 | Special functions in characteristic \(p\) (gamma functions, etc.) |
Keywords:
traces of Hecke operators; Gaussian hypergeometric series; Fourier coefficients of modular formsReferences:
| [1] | Ahlgren, S., The points of a certain fivefold over finite fields and the twelfth power of the eta function, Finite Fields Appl., 8, 1, 18-33 (2002) · Zbl 1026.11053 |
| [2] | Ahlgren, S.; Ono, K., Modularity of a certain Calabi-Yau threefold, Monatsh. Math., 129, 3, 177-190 (2000) · Zbl 0999.11031 |
| [3] | Clemens, C. H., A Scrapbook of Complex Curve Theory, Grad. Stud. Math., vol. 55 (1980), Plenum Press: Plenum Press New York · Zbl 0456.14016 |
| [4] | Cox, D. A., Primes of the Form \(x^2 + n y^2\). Fermat, Class Field Theory and Complex Multiplication, Wiley-Interscience Publ. (1989), John Wiley and Sons: John Wiley and Sons New York · Zbl 0701.11001 |
| [5] | Frechette, S.; Ono, K.; Papanikolas, M., Gaussian hypergeometric functions and traces of Hecke operators, Int. Math. Res. Not., 60, 3233-3262 (2004) · Zbl 1088.11029 |
| [6] | Fuselier, J., Hypergeometric functions over \(F_p\) and relations to elliptic curves and modular forms, Proc. Amer. Math. Soc., 138, 109-123 (2010) · Zbl 1222.11058 |
| [7] | Greene, J., Hypergeometric functions over finite fields, Trans. Amer. Math. Soc., 301, 1, 77-101 (1987) · Zbl 0629.12017 |
| [8] | Hijikata, H.; Pizer, A. K.; Shemanske, T. R., The basis problem for modular forms on \(\Gamma_0(N)\), Mem. Amer. Math. Soc., 82, 418 (1989), vi+159 pp · Zbl 0689.10034 |
| [9] | Husemöller, D., Elliptic Curves, Grad. Texts in Math., vol. 111 (2004), Springer-Verlag: Springer-Verlag New York · Zbl 1040.11043 |
| [10] | Igusa, J., Class number of a definite quaternion with prime discriminant, Proc. Natl. Acad. Sci., 44, 312-314 (1958) · Zbl 0081.03601 |
| [11] | Ireland, K.; Rosen, M., A Classical Introduction to Modern Number Theory, Grad. Texts in Math., vol. 84 (1990), Springer-Verlag: Springer-Verlag New York · Zbl 0712.11001 |
| [12] | Katz, N., Exponential Sums and Differential Equations, Ann. of Math. Stud., vol. 124 (1990), Princeton University Press: Princeton University Press New Jersey · Zbl 0731.14008 |
| [13] | Koike, M., Orthogonal matrices obtained from hypergeometric series over finite fields and elliptic curves over finite fields, Hiroshima Math. J., 25, 43-52 (1995) · Zbl 0944.11021 |
| [14] | Lang, S., Cyclotomic Fields I and II, Grad. Texts in Math., vol. 121 (1990), Springer-Verlag: Springer-Verlag New York · Zbl 0704.11038 |
| [15] | Lennon, C., Gaussian hypergeometric evaluations of traces of Frobenius for elliptic curves, Proc. Amer. Math. Soc., 139, 1931-1938 (2011) · Zbl 1281.11104 |
| [16] | Manin, J. I., The Hasse-Witt matrix of an algebraic curve, Trans. Amer. Math. Soc., 45, 245-264 (1965) · Zbl 0148.28002 |
| [17] | Meyer, C., Modular Calabi-Yau Threefolds, Fields Inst. Monogr., vol. 22 (2005), American Mathematical Society: American Mathematical Society Providence · Zbl 1096.14032 |
| [18] | Miret, J.; Moreno, R.; Rio, A.; Valls, M., Computing the \(ℓ\)-power torsion of an elliptic curve over a finite field, Math. Comp., 78, 1767-1786 (2009) · Zbl 1215.11062 |
| [19] | Ono, K., Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc., 350, 1205-1223 (1998) · Zbl 0910.11054 |
| [20] | Schoof, R., Counting points on elliptic curves over finite fields, J. Theor. Nombres Bordeaux, 7, 219-254 (1995) · Zbl 0852.11073 |
| [21] | Schoof, R., Nonsingular plane cubic curves over finite fields, J. Combin. Theory Ser. A, 46, 2, 183-211 (1987) · Zbl 0632.14021 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
