The following questions arise from modular form theory. But this theory isn't needed to formulate or understand them, and I'm not using the modular-forms tag.

NOTATION

Fix an odd prime N. Let F in Z/2[[x]] be sum (x^n) where n runs over the odd squares. Set G=F(x^N). There is a degree N+1 irreducible polynomial relation between F and G over Z/2, (The relation has the form (F+G)^(N+1)+(lower degree terms) =0, and is symmetric).

Examples: Let S=F+G and P=FG. When N=3, S^4=P. When N=5, S^6=P. When N=7, S^8=P^2+P. When N=11, (S^4+P)^3=P.

Now let M be the integral closure of Z/2[G] in the degree N+1 extension field of Z/2(G) generated by F. View M as a subring of Z/2[[x]], and let M(odd) consist of those g in M for which each exponent n appearing in g is odd. The trace map Z/2(F,G)-->Z/2(G) maps the Z/2[G^2] module M(odd) into the cyclic module generated by G. Let M_0 consist of all elements of M(odd) of trace 0. M_0 is a free rank N module over Z/2[G^2].

SOME REMARKABLE FACTS

For small N there are very nice bases of M(odd) and M_0 over Z/2[G^2].

N=3... Let C_1=F, C_3=G, C_5=(F^2)(G). Then the C_j are a basis of M_0. There is an element of M(odd) whose trace is G; it follows that this element and the C_j form a basis of M(odd). Furthermore for each exponent n appearing in C_j, n=j mod 8 and the Legendre symbol (n/3) is either (j/3) or 0.

N=5... Let C_1=F, C_3=(F^3)+(G^2)(F), C_5=G, C_7=(F^2)(G), C_9=(F^4)(G). Then the above results continue to hold, with 3 replaced by 5 in the final sentence.

When N=7 or 11 one can write down C_j where j is odd and <2N, and prove the corresponding facts. But when N=11 it's not possible for all the C_j to be in Z/2[F,G]; nevertheless one can arrange that the product of each C_j by (1+G^8) is in Z/2[F,G].

QUESTIONS

To what extent do the above results generalize to larger N? More precisely, let C(plus), (resp. C(minus)), consist of those g in M(odd) in which each exponent n that appears has (n/N) equal to 0 or 1 (resp. -1). Let C be C(plus)+C(minus).

Question 1--- Does M_0=C?

Remark 1... M is indeed the Serre Swinnerton-Dyer ring of characteristic 2 modular power series for Gamma_0 (N), though this isn't obvious. This allows one to introduce Hecke operators.

Remark 2... By using these operators one can show that C has rank N or N+1. The bases that I've exhibited when N=3,5,7 or 11 show that in each of these cases M_0 is contained in C. When N=3,5 or 7, I can show that M_0=C. This is surely also true when N=11, though I haven't proved it.

Remark 3... For each odd n prime to N there is a formal Hecke operator Z/2[[x]]-->Z/2[[x]]. As I've indicated, Remark 1 can be used to show that the T_n stabilize M, M(odd) and C. So one may ask the perhaps more accessible weakening of Question 1:

Question 2... Do the T_n stabilize M_0?