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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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