Definition of the ring of weakly modular forms over $\mathbb{Z}_{(p)}$ I am an undergraduate in a small mathematics course focused on a single project. Right now we are in the preliminary stage, the part where our advisor has given us a high-level reference, referred us to sections, and asks we do our best to make left from right of it.
In doing this, I need some help. Early into Behrens' paper here, specifically on page 8, he defines the following:

$MF_*$ is the ring of weakly modular forms over $\mathbb{Z}_{(p)}$.
$$MF_* = \mathbb{Z}_{(p)}[c_4,c_6,\Delta^{\pm 1}]/(c_4^3-c_6^2 = 1728\Delta)$$
We must invert $\Delta$ because we are taking sections over the complement of the singular locus of the moduli space of generalized elliptic curves.

I do not understand where this definition of $MF_*$ comes from. This is what I would like clarification of. I would guess it is not a straightforward definition to motivate, so I would be fine with references that would help me answer my question (in particular, references that lead nothing to presumption.)
If it is useful, here is some additional context:

*

*I am interested in the $p=3$ case.

*I have very light background in algebraic geometry.

*I have very light background in elliptic curves/modular forms (having only introduced these  to myself about two weeks ago.)

 A: This was going to be a comment but got way too long. I'm not familiar with weakly modular forms but there are a couple potential motivations that come to mind. Over $\mathbb{C}$ there is a ring of modular forms (of all weights), and this is generated as a $\mathbb{C}$-algebra by forms $E_4, E_6$ of weights 4 and 6 respectively. There is also a cusp form commonly called $\Delta$ of weight 12 that satisfies the relation $E_4^3 - E_6^2 = 1728 \Delta$, or something similar depending on your normalization. This can be found in any reasonable book about modular forms and doesn't require knowledge of algebraic geometry. I recommend Diamond and Shurman.
Another motivation behind the definition is the idea of a moduli space. You may be aware that modular forms can have many equivalent definitions. One is as sections of a sheaf over some moduli space (be that a scheme, stack, or something else) of elliptic curves. A detailed explanation of this can be found in Arithmetic Moduli by Katz and Mazur, but this may be impenetrable until you learn more algebraic geometry. The rough idea is that points of your space correspond to elliptic curves. Each elliptic curve $E$ has a discriminant $\Delta(E)$, and this discriminant is invertible since $E$ is smooth (by definition). I recommend taking a look at Silverman's book on elliptic curves if you're not comfortable with this.
We can then think of $\Delta$ as a function on our moduli space of elliptic curves, that takes a point $x$ (necessarily associated to some elliptic curve $E_x$) to the value $\Delta(E_x)$ in our base ring/field. If we call our sheaf $\omega$ and pretend its the structure sheaf (which it is locally) then sections of $\omega$ are just functions on our space. So $\Delta$ is an element of the ring of global sections on our moduli space. Similarly, we can define certain quantities $c_4, c_6$ for every elliptic curve, and view these as functions as well. I think the idea should be that every generalized elliptic curve $E$ (up to iso) is determined by the values $c_4(E), c_6(E), \Delta(E)$, with the caveat that we always know
$$
c_4(e)^3 - c_6(E)^2 = 1728 \Delta(E).
$$
A good example to keep in mind is that the (coarse) moduli space of elliptic curves over $\mathbb{C}$ is isomorphic to $\mathbb{A}^1_{\mathbb{C}}$ via the $j$-invariant; the moduli space is just $\operatorname{Spec}\mathbb{C}[j]$. In the paper you linked, they consider a moduli space of generalized elliptic curves (ref for this is the paper by Deligne-Rapoport, but it may not be accessible to you yet), so not all points will be smooth curves. Some points will be singular, and these correspond to where $\Delta$ vanishes. Inverting $\Delta$ corresponds to taking the complement of the singular locus, i.e., only considering the points corresponding to smooth curves. Finally, I would also check out the paper referenced as [8] in the link you gave (P. Deligne, Courbes elliptiques: formulaire d’après J. Tate), as this appears immediately after the definition you copied and will probably be quite helpful.
