In this question Joel Bellaiche constructed an algebra, M, of modular forms for gamma_0 (N) in finite characteristic (which he called p, but I'll call ell) and asked to know its structure. Matt Emerton gave a "comment-answer" that showed, among other things, that M is integrally closed.
I've made some explicit calculations (particularly when ell=2 and 3; see my question for some of these when ell=2). The empirical "discoveries" unveiled by these seem likely to be theorems. I'll present a couple of these here, saving the remaining ones for edits. I'll assume that the level N is a prime, p. M contains f1 and fp; the mod ell reductions of the Fourier expansions of the cusp forms delta(z) and delta(pz).
Discovery 1---M is integral over Z/ell[f1,fp]
Discovery 2---When ell=2 or 3, then M is the integral closure of Z/ell[f1,fp] in its field of fractions.
Are these really true? I'll also hazard the following guess when l>3. The mod ell reductions of the Eisenstein series E_4 and E_6 generate an extension of Z/ell(f1,fp) of degree (l-1)/2, and M is the integral closure of Z/ell[f1,fp] in this extension.
EDIT: To describe further observations I'll introduce some notation. If k is even and non-negative, M[k] will be the Z/ell subspace of M consisting of the mod ell reductions of modular power series in Z[[x]] corresponding to weight k forms for gamma_0 (p). C will be a non-singular projective curve over Z/ell with function field the field of fractions of M, and D will be the divisor of poles of the element fp of M[12].
For example, when ell=2 and p=11, then in the notation of my question referenced above, M[k] has dimension k, while M2 is spanned by 1 and t, and M[4] is spanned by 1,t,t^2 and r. We have the relation r^2+r=t^3+t, and C is the curve r^2+r=t^3+t with the point O at infinity adjoined. f11=r^3+r^4+t^3 has zeros of orders 11 and 1 at (t,r)=(0,0) and (0,1) and D=12(O). Furthermore M[12] is spanned by 1,t,t^2,t^3,t^4,t^5,t^6,r,t*r,t^2*r,t^3*r and t^4*r and is the complete linear series attached to the divisor D.
Here's what I think is true in general. Suppose ell=2. Then:
1---fp has one zero of order p, and one of order 1 on C.
2a--When p is 11 mod 12, fp has (p+1)/12 poles of order 12.
2b--When p is 5 mod 12, fp has 1 pole of order 6 and (p-5)/12 of order 12.
2c--When p is 7 mod 12, fp has 2 poles of order 4 and (p-7)/12 of order 12.
2d--When p is 13 mod 12, fp has 1 pole of order 6, 2 of order 4 and (p-13)/12 of order 12.
3---For each k there is a divisor D, easily describable in terms of D and k such that M[k] is the complete linear series attached to this divisor; in particular D<12m>=m(D).
I think that entirely similar results hold when ell=3. My belief is that all of this is known, but I'd appreciate proofs and/or references.
