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