# 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

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

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 \(r\):

By analytic uniformization, there exists a corresponding Drinfeld module

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

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

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

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

Furthermore, in the rank two case, we also have

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

This series is absolutely and uniformly convergent and admits the expansion

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

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
\(\Omega^2(\mathbb{C}_{\infty})\):

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

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

In [Pet2013], Petrov showed that

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

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.

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

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

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

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