Drinfeld modular forms of weight one. (English) Zbl 0894.11028
Let \(A=\mathbb{F}_q[T]\) be the polynomial ring over the finite field \(\mathbb{F}_q\), \(K= \mathbb{F}_q(T)\) its quotient field, \(K_\infty= \mathbb{F}_q((T^{-1}))\) the \(\infty\)-adic completion, \(C\) the completed algebraic closure of \(K_\infty\), and \(\Omega= C-K_\infty\) the Drinfeld upper half-plane. The modular group \(\Gamma(1)= GL(2,A)\) acts on \(\Omega\), and the associated \(C\)-algebras \(M(\Gamma(1))\) of Drinfeld modular forms is a polynomial ring \(C[g,\Delta]\) in two modular forms \(g\) and \(\Delta\) of respective weights \(q-1\) and \(q^2-1\).
It is a basic albeit unsolved problem to determine the structure of the algebra of modular forms for convergence subgroups \(\Gamma\) of \(\Gamma(1)\), especially for full congruence subgroups \(\Gamma(N)= \{\gamma\in\Gamma\mid \gamma\equiv 1\pmod N\}\) of \(\Gamma(1)\). Fix such a \(\Gamma=\Gamma(N)\) with \(N\in A-\mathbb{F}_q\), and let \(M(\Gamma)=\oplus M_k(\Gamma)\) be its algebra of modular forms (\(k\)= weight). \(M_k\) certainly contains the Eisenstein series \(E_u^{(k)}\) \((u\in(N^{-1}/ A)^2)\), where \[ E_u^{(k)}(z)= \mathop{{\sum}'}_{\substack{ (a,b)\in A^2\\ (a,b)\equiv u\pmod A}} \frac{1} {(az+b)^k}. \tag \(*\) \] Using an argument of Hecke, the author first shows that the \(E_u^{(k)}\) generate a vector space complement in \(M_k\) of the cusp forms of weight \(k\) (Prop. 1.12). Since there are no cusp forms of weight one, the result of Mumford may be applied to conclude that \(M(\Gamma)\) is generated by the \(E_u^{(1)}\) and the cusp forms of weight two (Prop. 1.15).
Two natural questions arise: (1) Do we already have \[ \begin{aligned} M(\Gamma) & = C\left[E_u^{(1)}\mid u \text{ as in (*)}\right]\\ & =: E(\Gamma)?\end{aligned} \] (2) What are the relations between the different \(E_u^{(1)}\)?
At least, \(M(\Gamma)\) is the integral closure of \(E(\Gamma)\) in their common field of fractions (Prop. 1.17). Now the main result of the paper is Theorem 2.2, too complicated to be stated here in detail, which relates \(E(\Gamma)\) with some ring \(R/J\) that admits a simple and explicit description through Drinfeld modules. Finally, some examples are given \((\Gamma= \Gamma(N)\) with \(N\in A\) linear, or \((q,\text{deg }N)= (2,2))\), where the above questions can be answered.
It is a basic albeit unsolved problem to determine the structure of the algebra of modular forms for convergence subgroups \(\Gamma\) of \(\Gamma(1)\), especially for full congruence subgroups \(\Gamma(N)= \{\gamma\in\Gamma\mid \gamma\equiv 1\pmod N\}\) of \(\Gamma(1)\). Fix such a \(\Gamma=\Gamma(N)\) with \(N\in A-\mathbb{F}_q\), and let \(M(\Gamma)=\oplus M_k(\Gamma)\) be its algebra of modular forms (\(k\)= weight). \(M_k\) certainly contains the Eisenstein series \(E_u^{(k)}\) \((u\in(N^{-1}/ A)^2)\), where \[ E_u^{(k)}(z)= \mathop{{\sum}'}_{\substack{ (a,b)\in A^2\\ (a,b)\equiv u\pmod A}} \frac{1} {(az+b)^k}. \tag \(*\) \] Using an argument of Hecke, the author first shows that the \(E_u^{(k)}\) generate a vector space complement in \(M_k\) of the cusp forms of weight \(k\) (Prop. 1.12). Since there are no cusp forms of weight one, the result of Mumford may be applied to conclude that \(M(\Gamma)\) is generated by the \(E_u^{(1)}\) and the cusp forms of weight two (Prop. 1.15).
Two natural questions arise: (1) Do we already have \[ \begin{aligned} M(\Gamma) & = C\left[E_u^{(1)}\mid u \text{ as in (*)}\right]\\ & =: E(\Gamma)?\end{aligned} \] (2) What are the relations between the different \(E_u^{(1)}\)?
At least, \(M(\Gamma)\) is the integral closure of \(E(\Gamma)\) in their common field of fractions (Prop. 1.17). Now the main result of the paper is Theorem 2.2, too complicated to be stated here in detail, which relates \(E(\Gamma)\) with some ring \(R/J\) that admits a simple and explicit description through Drinfeld modules. Finally, some examples are given \((\Gamma= \Gamma(N)\) with \(N\in A\) linear, or \((q,\text{deg }N)= (2,2))\), where the above questions can be answered.
Reviewer: E.-U.Gekeler (Saarbrücken)
MSC:
| 11G09 | Drinfel’d modules; higher-dimensional motives, etc. |
| 11F11 | Holomorphic modular forms of integral weight |
Keywords:
projective normality; modular forms for full convergence subgroups; Drinfeld upper half-plane; Drinfeld modular forms; Eisenstein series; cusp formsReferences:
| [1] | Arbarello, E.; Cornalba, M.; Griffiths, P. A.; Harris, J., Geometry of Algebraic Curves, Vol. 1. Geometry of Algebraic Curves, Vol. 1, Grundlehren der Math. Wiss., 267 (1985), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0559.14017 |
| [2] | Cornelissen, G., Sur les zéros des séries d’Eisenstein de poids \(q^k −1 pour GL_2F_q[T\), C. R. Acad. Sci. Paris Sér. I. Math., 321, 817-820 (1995) · Zbl 0847.11033 |
| [3] | Cornelissen, G., A survey of Drinfeld modular forms, (Van Geel, J., Drinfeld Modules, Drinfeld Modular Schemes and Applications (1997), World Scientific: World Scientific Singapore), 167-187 · Zbl 0926.11030 |
| [4] | Cornelissen, G., Drinfeld modular forms of level \(T\), (Van Geel, J., Drinfeld Modules, Drinfeld Modular Schemes and Applications (1997), World Scientific: World Scientific Singapore), 272-281 · Zbl 0926.11031 |
| [5] | G. Cornelissen, 1997, Geometric Properties of Modular Forms over Rational Function Fields, University of Ghent; G. Cornelissen, 1997, Geometric Properties of Modular Forms over Rational Function Fields, University of Ghent |
| [6] | Drinfeld, V. G., Elliptic modules, Math. USSR Sb., 23, 561-592 (1976) · Zbl 0321.14014 |
| [7] | Eisenbud, D., Commutative Algebra. Commutative Algebra, Graduate Texts in Math., 150 (1995), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0819.13001 |
| [8] | Fujita, T., Defining equations for certain types of polarized varieties, (Baily, W. L.; Shioda, T., Complex Analysis and Algebraic Geometry (1977), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 165-173 · Zbl 0353.14011 |
| [9] | Gekeler, E.-U., Modulare Einheiten für Funktionenkörper, J. Reine Angew. Math., 348, 94-115 (1984) · Zbl 0523.14021 |
| [10] | Gekeler, E.-U., Drinfeld Modular Curves. Drinfeld Modular Curves, Lecture Notes in Math., 1231 (1986), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0607.14020 |
| [11] | Gekeler, E.-U.; Reversat, M., Jacobians of Drinfeld modular curves, J. Reine Angew. Math., 476, 27-93 (1996) · Zbl 0848.11029 |
| [12] | Gianni, P.; Trager, B.; Zacharias, G., Gröbner bases and primary decomposition of polynomial ideals, (Robbiano, Computational Aspects of Commutative Algebra (1988), Academic Press: Academic Press London/New York), 15-33 |
| [13] | Goss, D., Modular forms for\(F_r}[T\), J. Reine Angew. Math., 317, 16-39 (1980) · Zbl 0422.10021 |
| [14] | Goss, D., \(π\), Compositio Math., 41, 3-38 (1980) · Zbl 0422.10020 |
| [15] | Hecke, E., Theorie der Eisensteinschen Reihen höherer Stufe and ihre Anwendung auf Funktionentheorie und Arithmetik, Abh. Math. Sem. Univ. Hamburg, 5, 199-224 (1927) · JFM 53.0345.02 |
| [16] | Igusa, J., Theta Functions. Theta Functions, Grundlehren der Math. Wiss., 194 (1972), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0251.14016 |
| [17] | Iitaka, S., Algebraic Geometry. Algebraic Geometry, Graduate Texts in Math., 76 (1982), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0491.14006 |
| [18] | Kapranov, M. M., On cuspidal divisors on the modular varieties of elliptic modules, Math. USSR Izv., 30, 533-547 (1988) · Zbl 0664.14025 |
| [19] | D. Mumford, 1969, Varieties defined by quadratic equations, Proceedings of the Centro Internationale Matematico Estivo “Questions on algebraic varieties,” 31, 100, Edizioni Cremonese, Roma; D. Mumford, 1969, Varieties defined by quadratic equations, Proceedings of the Centro Internationale Matematico Estivo “Questions on algebraic varieties,” 31, 100, Edizioni Cremonese, Roma · Zbl 0198.25801 |
| [20] | R. Pink, On compactification of Drinfeld moduli schemes, Moduli Spaces, Galois Representations and \(L\); R. Pink, On compactification of Drinfeld moduli schemes, Moduli Spaces, Galois Representations and \(L\) · Zbl 0900.11037 |
| [21] | Stanley, R., Hilbert functions of graded algebras, Adv. in Math., 28, 57-83 (1978) · Zbl 0384.13012 |
| [22] | Teitelbaum, J. T., The Poisson kernel for Drinfeld modular curves, J. Amer. Math. Soc., 4, 491-511 (1991) · Zbl 0735.11025 |
| [23] | W. Vasconcelos, 1993, Constructions in commutative algebra, Computational Algebraic Geometry and Commutative Algebra, D. EisenbudL. Robbiano, Symposia Mathematica, 34, 151, 197, Cambridge Univ. Press, Cambridge; W. Vasconcelos, 1993, Constructions in commutative algebra, Computational Algebraic Geometry and Commutative Algebra, D. EisenbudL. Robbiano, Symposia Mathematica, 34, 151, 197, Cambridge Univ. Press, Cambridge · Zbl 0831.13008 |
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