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On the coefficients of Drinfeld modular forms. (English) Zbl 0653.14012
Let \(\mathbb A=\mathbb F_ r[T]\) and \(k=\mathbb F_ r(T)\). Let \(K=k_{\infty}\) be the completion of \(k\) at the prime \(\infty\), and let \(K^{\text{ac}}\) be an algebraic closure of \(K\). We denote the space \(K^{\text{ac}}-K\) by \(\Omega\). One knows that \(\Omega\) inherits a rigid-analytic structure as a subspace of the affine line and that it is connected as a rigid space. The group \(\text{GL}_ 2(\mathbb A)]\), and its congruence subgroups, act discretely on \(\Omega\) and the quotient spaces can also be given a rigid structure. In fact, they are the underlying rigid spaces of certain algebraic curves which are the moduli spaces of Drinfeld modules [of rank two and with, possibly, “level structure”]. A “Drinfeld module” is obtained by looking at the \(\mathbb A\)-module structure that is obtained on \(K^{\text{ac}}\) when one divides \(K^{\text{ac}}\) by a discrete finitely generated \(\mathbb A\)-submodule (i.e., a “lattice”). Drinfeld modules thus are the correct \(\mathbb A\)-analogs of elliptic curves.
On \(\Omega\) one can talk about “modular forms” with the straightforward definition. Through the use of rigid analysis one can show that such forms have “\(q\)-expansions” (called “\(t\)-expansions” by the author). Such forms have been around for a number of years and much of the recent work on them has been carried out by the author. The paper under review contains very interesting results in a number of directions:
(1) Given a lattice \(\Lambda\) one forms the rigid-meromorphic function \(S_{k,\Lambda}\) defined by \(S_{k,\Lambda}=\sum (Z+\lambda)^{- k}\), where one sums over all \(\lambda\) in \(\Lambda\). One knows that \(S_{k,\Lambda}\) is a polynomial of degree \(k\), \(G_{k,\Lambda}\), in the function \(S_{1,\Omega}\). The author presents a lovely formula for the generating function of the \(G_{k,\Lambda}\). This allows him to e.g., make some progress in understanding the action of the Hecke operators on modular forms.
(2) By defining an analog of the false Eisenstein series of weight one, the author defines a differential operator of weight two on the graded ring of modular forms. This allows him to find various relations between the modular forms \(g\), \(\Delta\) and \(h\) (the first two forms generate the ring of all modular forms for \(\text{GL}_ 2(\mathbb A)\) and the first and third generate the ring of generalized modular forms).
(3) The author presents a version of the classical theorem of Swinnerton-Dyer which allows him to describe the ring of modular forms modulo \(\mathfrak p\), where \(\mathfrak p\) is a prime of \(\mathbb A\).
There are still many, many questions remaining about modular forms on \(\Omega\). Among them are the deeper understanding of the information contained in the action of the Hecke operators, the construction of the “correct” (if any exist) versions of theta functions, and, the generalization of all of this to the case of general \(\mathbb A\).

MSC:
11F52 Modular forms associated to Drinfel’d modules
11G09 Drinfel’d modules; higher-dimensional motives, etc.
11G20 Curves over finite and local fields
11R58 Arithmetic theory of algebraic function fields
11F11 Holomorphic modular forms of integral weight
11F25 Hecke-Petersson operators, differential operators (one variable)
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References:
[1] Carlitz, L.: An analogue of the Staudt-Clausen theorem. Duke Math. J.3, 503-517 (1937) and7, 62-67 (1940) · JFM 63.0879.03
[2] Drinfeld, V.G.: Elliptic Modules (Russian). Math. Sbornik94, 594-627 (1974) English Translation: Math. USSR-Sbornik23, 561-592 (1976)
[3] Fresnel, J., Put, M. van der: Géométrie Analytique Rigide et Applications. Progr. Math.18, Basel Boston: Birkhäuser, 1981 · Zbl 0479.14015
[4] Gekeler, E.-U.: Zur Arithmetik von Drinfeld-Moduln. Math. Ann.262, 167-182 (1983) · Zbl 0536.14028
[5] Gekeler, E.-U.: A Product Expansion for the Discriminant Function of Drinfeld Modules of Rank Two. J. Number Theory21, 135-140 (1985) · Zbl 0572.10021
[6] Gekeler, E.-U.: Modulare Einheiten für Funktionenkörper. J. Reine Angew. Math.348, 94-115 (1984) · Zbl 0523.14021
[7] Gekeler, E.-U.: Drinfeld Modular Curves (Lect. Notes Math., vol. 1231) Berlin Heidelberg New York: Springer, 1986 · Zbl 0657.14012
[8] Gerritzen, L., Put, H. van der: Schottky Groups and Mumford Curves (Lect. Notes Math., vol. 817) Berlin Heidelberg New York: Springer, 1980 · Zbl 0442.14008
[9] Goss, D.: Von Staudt forF q [T]. Duke Math. J.45, 885-910 (1978) · Zbl 0404.12013
[10] Goss, D.: ?-adic Eisenstein Series for Function Fields. Comp. Math.41, 3-38 (1980) · Zbl 0388.10020
[11] Goss, D.: Modular Forms forF r [T]. J. Reine Angew. Math.317, 16-39 (1980) · Zbl 0422.10021
[12] Goss, D.: The Algebraist’s Upper Half-Plane. Bull. Am. Math. Soc. NS2, 391-415 (1980) · Zbl 0433.14017
[13] Hayes, D.: Explicit class field theory for rational function fields. Trans. Am. Math. Soc.189, 77-91 (1974) · Zbl 0292.12018
[14] Igusa, J.: On the algebraic theory of elliptic modular functions. J. Math. Soc. Jpn20, 96-106 (1968) · Zbl 0164.21101
[15] Katz, N.: Higher congruences between modular forms. Ann. Math.101, 332-367 (1975) · Zbl 0356.10020
[16] Lang, S.: Introduction to Modular Forms. Berlin Heidelberg New York: Springer, 1976 · Zbl 0344.10011
[17] Radtke, W.: Diskontinuierliche arithmetische Gruppen im Funktionenkörperfall. J. Reine Angew. Math.363, 191-200 (1985) · Zbl 0572.14018
[18] Swinnerton-Dyer, H.P.F.: Onl-adic representations and congruences for coefficients of modular forms (Lect. Notes Math., vol. 350, pp. 1-55) Berlin Heidelberg New York: Springer, 1973 · Zbl 0267.10032
[19] Yu, J.: Transcendence and Drinfeld Modules. Invent. Math.83, 507-517 (1986) · Zbl 0586.12010
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