# Are these empirical discoveries about the Serre Swinnerton-Dyer ring of prime level modular power series actual theorems?

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.

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@paul: I took the liberty of adding links to the MO questions you refer to in the body. I'd suggest you change the title to something more descriptive that does not include an MO question number. – Alberto García-Raboso Mar 11 '13 at 16:46
@Alberto: Thanks. I followed your suggestion. – paul Monsky Mar 11 '13 at 19:45