# Background material and notations#

In this section, we define some notations and give some definitions. We will that assume the reader possesses some knowledge of Drinfeld modules and their analytic theory. A classical reference is chapter 3 and 4 of [Gos1998].

Function field setting

Let $$q$$ be a prime power and let $$A$$ be the ring of functions of $$\mathbb{P}^1/\mathbb{F}_q$$ which are regular outside $$\infty$$. This ring is the polynomial ring $$\mathbb{F}_q[T]$$. We denote by $$K := \mathbb{F}_q(T)$$ its fraction field and let $$K_{\infty} := \mathbb{F}_q((1/T))$$. We define $$\mathbb{C}_{\infty}$$ to be the completion of an algebraic closure of $$K_{\infty}$$. Lastly, We denote by $$\tau : x\mapsto x^q$$ the $$q$$-Frobenius.

Drinfeld period domain and group action

The Drinfeld period domain of rank $$r > 1$$ is defined by

$\Omega^r(\mathbb{C}_{\infty}) := \mathbb{P}^{r-1}(\mathbb{C}_{\infty}) \setminus \{K_{\infty}\text{-rational hyperplanes}\}.$

This space is a rigid analytic space and plays the role of the complex upper half plane. We identify the elements of $$\Omega^r(\mathbb{C}_{\infty})$$ with the set of column vectors $$(w_1,\ldots, w_{r-1}, w_{r})^{\mathrm{T}}$$ in $$\mathbb{C}_{\infty}^r$$ such that the $$w_i$$ are $$K_{\infty}$$-linearly independant and $$w_r = \xi$$, a nonzero constant in $$\mathbb{C}_{\infty}$$.

We define a left action of $$\mathrm{GL}_r(K_{\infty})$$ on $$\Omega^r(\mathbb{C}_{\infty})$$ by setting

$\gamma(w) := j(\gamma, w)^{-1}\gamma w$

where $$j(\gamma, w) := \xi^{-1} \cdot (\text{last entry of }\gamma w)$$.

Universal Drinfeld module over $$\Omega^r(\mathbb{C}_{\infty})$$ and modular forms

For any $$w = (w_1, \ldots, w_{r-1}, \xi)$$ in $$\Omega^r(\mathbb{C}_{\infty})$$ we have a corresponding free $$A$$-lattice of rank $$r$$:

$\Lambda_w := Aw_1 \oplus \cdots \oplus Aw_{r-1} \oplus A\xi.$

By analytic uniformization, there exists a corresponding Drinfeld module

$\phi_w : T \mapsto T + g_1(w)\tau + \cdots + g_{r - 1}(w)\tau^{r-1} + g_{r}(w)\tau^{r}$

where the coefficients $$g_i : \Omega^r(\mathbb{C}_{\infty}) \rightarrow \mathbb{C}_{\infty}$$ are rigid analytic functions satisfying the invariance property:

$g_i(\gamma(w)) = j(\gamma, w)^{1 - q^i} g_i(w), ~\forall \gamma\in \mathrm{GL}_r(A)$

where $$g_{r}(w)$$ never vanishes. Moreover, these coefficients $$g_i$$ admits an expansion at infinity, analogous to the classical theory. In the rank two case, this expansion at infinity is of the form

$g_i(w) = \sum_{i = 0}^{\infty} a_n(g_i)u(w)^i$

where $$u(w) := e(w)^{-1}$$ and $$e$$ is the exponential of the Carlitz module. Any rigid analytic function $$f : \Omega^r(\mathbb{C}_{\infty}) \rightarrow \mathbb{C}_{\infty}$$ that satisfies the invariance property and the expansion at infinity is called a Drinfeld modular form of weight $$k$$. The forms $$g_i$$ are called the coefficients forms.

This Drinfeld module is the universal Drinfeld $$\mathbb{F}_q[T]$$-module over $$\Omega^r(\mathbb{C}_{\infty})$$. The coefficients $$g_i : \Omega^r(\mathbb{C}_{\infty}) \rightarrow \mathbb{C}_{\infty}$$ are rigid analytic function which satisfies a modular invariance properties under the action of the group $$\mathrm{GL}_r(A)$$. These function are examples of modular forms.

Definition.

A Drinfeld modular form of rank $$r$$, weight $$k$$, type $$m$$ for $$\mathrm{GL}_r(A)$$ is a rigid analytic function $$f:\Omega^r(\mathbb{C}_{\infty}) \rightarrow \mathbb{C}_{\infty}$$ such that

• $$f(\gamma(w)) = \mathrm{det}(\gamma)^m j(\gamma, w)^k f(w)$$ for all $$\gamma$$ in $$\mathrm{GL}_r(A)$$ and $$w\in \Omega^r(\mathbb{C}_{\infty})$$;

• $$f$$ is holomorphic at infinity.

The second condition is similar to the classical case. In the rank two situation, this expansion is simply given by $$f = \sum_{n\geq 0} a_n(f) u^n$$ where $$a_n(f)\in \mathbb{C}_{\infty}$$.

The reader is refered to part I of [BRP2018] for more information about the analytic theory of Drinfeld modular form of arbitrary rank.

Ring of Drinfeld modular forms

Letting $$M_k^{r, m}(\mathrm{GL}_r(A))$$ denote the space of rank $$r$$, weight $$k\in (q - 1)\mathbb{Z}$$ and type $$m~(\mathrm{mod}~q-1)$$ Drinfeld modular forms, we define

$M^{r, 0}(\mathrm{GL}_r(A)) := \bigoplus_{k\in ZZ} M_k^{r}(\mathrm{GL}_r(A))$

to be the graded ring of all Drinfeld modular forms. By theorem 17.5 in part III of [BRP2018], we have

$M^{r, 0}(\mathrm{GL}_r(A)) = \mathbb{C}_{\infty}[g_1,\ldots, g_{r-1}, g_{r}].$

Furthermore, in the rank two case, we also have

$M^r(\mathrm{GL}_r(A)) := \bigoplus_{k, m} M_k^{r, m}(\mathrm{GL}_r(A)) = \mathbb{C}_{\infty}[g_1, h]$

where $$h$$ is a modular form of weight $$q+1$$, type 1 with expansion $$u + O(u^2)$$ (see (5.13) of [Gek1988]).

Rank two examples

• Drinfeld Eisenstein series

For $$k \equiv 0$$ modulo $$q - 1$$. The Drinfeld Eisenstein series of weight $$k$$ and rank 2 is defined by

$\begin{split}E_{k}(w) := \sum_{\substack{ (c, c)\in A^{2} \\ (c, c) \neq (0, 0) }} \frac{1}{(cw + d)^k}.\end{split}$

This series is absolutely and uniformly convergent and admits the expansion

$\begin{split}E_k(w) = \tilde{\pi}^k\delta_k - \tilde{\pi}^k \sum_{\substack{a\in A \\a\text{ monic}}} G_k(u(aw))\end{split}$

where $$\tilde{\pi}$$ is the Carlitz period (analogue of $$\pi$$) and $$G_k$$ is the $$k$$-th Goss polynomial and $$\delta_k \in K$$ is some constant depending on $$k$$. See section 6 of [Gek1988] for the proof of this fact. We will denote by

$g_k := \tilde{\pi}^{q^k - 1}\delta_{q^k - 1} E_{q^k - 1}$

the normalized Eisenstein series. For $$k = 1,\ldots r-1$$, these forms corresponds to the coefficients forms defined above.

• Modular discriminant

The modular discriminant $$\Delta : \Omega^2(\mathbb{C}_{\infty}) \rightarrow \mathbb{C}$$ is the leading coefficient form of the rank 2 universal Drinfeld module over $$\Omega^2(\mathbb{C}_{\infty})$$:

$\phi^w : T \mapsto T + g_1(w)\tau + \Delta(w)\tau^2.$

By the work of López in [Lop2010], the discriminant function admits an expansion of the form

$\begin{split}-\tilde{\pi}^{1 - q^2}\Delta(w) = \sum_{\substack{a\in A\\a \text{ monic}}} a^{q(q-1)} u(aw)^{q-1}.\end{split}$
• Petrov $$A$$-expansions

We say that a Drinfeld modular forms of weight $$k$$ admits a Petrov expansion or an $$A$$-expansion if there exists an integer $$n$$ and elements $$c_{a}(f)\in \mathbb{C}_{\infty}$$ such that

$\begin{split}f = \sum_{\substack{a\in \mathbb{F}_q[T] \\ a\text{ monic}}} c_a(f)G_n(u(az)).\end{split}$

In [Pet2013], Petrov showed that

$\begin{split}f_{k, n} := \sum_{\substack{a\in \mathbb{F}_q[T] \\ a\text{ monic}}} a^{k - n}G_n(u(az))\end{split}$

defines an infinite family of Drinfeld modular forms of weight $$k$$ provided that $$k - 2n \equiv 0$$ modulo $$q - 1$$ and $$n \leq p^{v_p(k - n)}$$. See theorem 1.3 of loc. cit. for more details.

References

[BRP2018] (1,2)

Basson D., Breuer F., Pink R., Drinfeld modular forms of arbitrary rank: Part I: arxiv:1805.12335, Part II: arxiv:1805.12337, Part III: arxiv:1805.12339, 2018.

[Gek1988] (1,2)

Gekeler, EU. On the coefficients of Drinfeld modular forms. Invent Math 93, 667-700 (1988). doi.org/10.1007/BF01410204

[Gos1998]

Goss D. Basic structures of function field arithmetic. Springer, 1998. doi.org/10.1007/978-3-642-61480-4

[Lop2010]

López, B. A non-standard Fourier expansion for the Drinfeld discriminant function. Arch. Math. 95, 143-150 (2010). doi.org/10.1007/s00013-010-0148-7

[Pet2013]

Petrov A., A-expansions of Drinfeld modular forms, Journal of Number Theory, Volume 133, Issue 7, 2013, doi.org/10.1016/j.jnt.2012.12.012