BACKGROUND

Let p be an odd prime. An element of Z/2[[x]] is "modular of level p" if it is the mod 2 reduction of a g in Z[[x]] with g the Fourier expansion of a modular form for gamma_0(p). In various cases when f is modular of level p then f(x^p) can be expressed as a polynomial of a special shape in "characteristic 2 theta series".

I've developed a connection between modular function theory and theta series. Namely let [i] in Z/2[[x]] be the sum of the x^(n^2), where n runs over the integers that are congruent to i mod p. Then the field of fractions of the ring generated over the algebraic closure of Z/2 by these theta series identifies with Igusa's field of modular functions for gamma(p). But computer calculation suggests further connections, involving modular forms for gamma_0(p). Here are some examples:

1.___ If f is x+x^9+x^25+x^49+..., the mod 2 reduction of the expansion of delta, then f(x^p) lies in the ring B generated over Z/2 by the [i]. Indeed if we let B' be the subring of B consisting of those elements that are power series in x^p, and are fixed by the automorphisms of B taking [i] to [ni] when n is in (Z/p)*, then f is in B'. (Proofs are in my MO questions and answers).

2.___ If the exponents appearing in the element r of Z/2[[x]] are just the products of the non-zero squares by 1,2,p and 2p, then r is the mod 2 reduction of the expansion of a weight 4 Eisenstein series, and once again r(x^p) is in B'. (Again, proofs are in my MO questions and answers).

3.___ If p=7 and s is the modular power series of shape x^2+... coming from a weight 4 modular form, then s(x^7)=[1][2][3], and so is in B'.

4.___ If p=11 and t=x+... comes from the weight 2 cusp form, then t(x^11) is the sum of [1][1][3]+[2][2][5]+[4][4][1]+[3][3][2]+[5][5][1] and [1][1][2][4]+[2][2][4][3]+[4][4][3][5]+[3][3][5][1]+[5][5][1][2], and so lies in B'.

QUESTION

I can show that when p is 3,5,7 or 11, then f is modular of level p if and only if f(x^p) lies in B'. To what extent does this generalize to larger p?

EDIT: It seems likely that the answer to my question is always yes--f is in the ring A of level p characteristic 2 modular power series if and only if f(x^p) is in the subring, B', of the ring B generated by the "theta series". In this edit I'll present further conjectures, some in part known, that would imply this. In a later edit I'll give explicit descriptions of A and B' for p<20. Fix an odd prime p.

THE RING A

A consists of the mod 2 reductions of all elements of Z[[x]] that are Fourier expansions of modular forms, of arbitrary weight, for gamma_0 (p). A is closed under multiplication. Using the fact that the expansions of the normalized Eisenstein series of weights 4 and 6 lie in 1+2*Z[[x]] we see that A is closed under addition as well, and is a ring. f1=x+x^9+x^25+..., and fp=f1(x^p) are, as I remarked, in A, and so Z/2[f1,fp] is a subring of A.

Conjecture 1: A is the integral closure of Z/2[f1,fp] in its field of fractions. (There are 3 separate questions here. Is A integrally closed? Is A integral over the subring? And is the field of fractions of A equal to Z/2(f1,fp)? I suspect that the first, at least, of these is known. My investigations for p<20 support the conjecture).

THE RING B'

Let L be the quotient of (Z/p)* by {1,-1}, and P be a polynomial ring over Z/2 in the variables x_i, with i running over L. There is a gradation of P by Z/p, with the "degree" of x_i being i^2. Also, L acts on P by permutation of variables with m taking x_i to x_mi, and the effect of m is to multiply degrees by m^2. Let P' be the subring of P consisting of L-stable elements of "degree" 0. For example when p=13, (x_1)(x_5)+(x_2)(x_3)+(x_4)(x_6) is in P', while when p=7 the same is true of (x_2)(x_1)(x_1)(x_1)+(x_3)(x_2)(x_2)(x_2)+(x_1)(x_3)(x_3)(x_3). Now if i is prime to p, let [i] in Z/2[[x]] be the sum of the x^(n^2) where n runs over the integers congruent to i mod p. [i] only depends on the image of i in L.

We define B' to be the image of P' under the ring homomorphism P-->B taking x_i to [i]. There's a simple compact notation for elements of B' that I'll use when presenting my results. For example the image [1][5]+[2][3]+[4][6] of the p=13 element of the last paragraph will be called C(1,5) (or C(2,3) or C(4,6)), while that of the p=7 element is called C(2,1,1,1) (or C(3,2,2,2) or C(1,3,3,3)). Now the answers I've given to other questions on this site show that f1(x^p) and fp(x^p) lie in B'. Evidently my conjecture would follow from Conjecture 1 combined with:

Conjecture 2: B' is the integral closure of the ring Z/2[f1(x^p),fp(x^p)] in its field of fractions.

I find myself on firmer ground here--I think my MO answers come close to establishing Conjecture 2, though there may be some separability questions when p is 1 mod 4. The key to showing that B' is integrally closed should lie in the facts I've established about the curve attached to the ring B and the action of PSL_2(Z/p) on this curve.