Working on this topic for some time,I've come to understand the level 3 and 5 analogues to the characteristic 2 level 1 theory of modular forms developed by Nicolas and Serre. My results are presented in three arXiv articles: Variations on a Lemma of Nicolas and Serre (1604.02622 [math.NT]), A Hecke algebra attached to mod 2 modular forms of level 3 (1508.07523 [math. NT]), A Hecke algebra attached to mod 2 modular forms of level 5 (1610.07058 [math NT]). Along the way I give a simplification of the level 1 theory.

In each of levels 1,3,5 one has a subspace of Z/2[[x]] having as basis certain D_k. In level 1 (resp. 3,5), k runs over the positive integers prime to 2 (resp. 6,10). The formal Hecke operators T_p, p odd and not equal to the level stabilize our space; in fact T_p (D_k) is a sum of D_j with j < k. Furthermore the completed (weak) Hecke algebra attached to the space is a power series ring Z/2[[X,Y]] in level 1, while in levels 3 and 5 one needs to adjoin an element of square zero to this power series ring.

EDIT---I'll explain finally how the three spaces described above,spanned by various D_k, are related to modular forms of level Gamma_0 (N) where N is 1,3, and 5. Let M(odd,N) consist of those "odd" elements of Z/2[[x]] that are the mod 2 reductions of expansions at infinity lying in Z[[x]] of holomorphic modular forms of level Gamma_0 (N).(Any weight is allowed).

1.---The first space, that spanned by the D_k with (k,2)=1, is M(odd,1).

2.---Let W1 be the subspace of the second space spanned by the D_k with k=1 mod 6. The T_p with p=1 mod 6 stabilize W1. Let K be the subspace of M(odd,3) annihilated by the Hecke operator 1+U_3. Then the T_p with (p,6)=1 stabilize K and we get a shallow Hecke algebra attached to K. There is an identification of W1 with K, taking D to F+G, and preserving the action of the T_p with p=1 mod 6. It follows from 1508.07523[NT] that K admits a basis adapted to T_7 and T_13 with m_0,0 = F+G, and that the shallow Hecke algebra attached to K is a power series ring in T_7 and T_13.

3.---Let W_a be the subspace of the third space spanned by the D_k with k=1,3,7,9 mod 20. Then the T_p with p=1,3,7,9 mod 20 stabilize W_a. Now let K be the subspace of M(odd,5) annihilated by 1+U_5. We again have a shallow Hecke algebra attached to K. There is an identification of W_a with K taking D to F+G and preserving the action of the T_p with p=1,3,7,9 mod 20. It follows from 1610.07058[NT] that K admits a basis adapted to T_3 and T_7 and that the shallow Hecke algebra attached to K is a power series ring in T_3 and T_7.

Working on this topic for some time,I've come to understand the level 3 and 5 analogues to the characteristic 2 level 1 theory of modular forms developed by Nicolas and Serre. My first results were presented in three arXiv articles: Variations on a Lemma of Nicolas and Serre (1604.02622 [math.NT]), A Hecke algebra attached to mod 2 modular forms of level 3 (1508.07523 [math.NT]), A Hecke algebra attached to mod 2 modular forms of level 5 (1610.07058 [math.NT]). Along the way I gave a simplification of the level 1 theory. EDIT--More precise results relating the earlier findings to structure results for more naturally appearing Hecke algebras are found in three further arXiv postings; 1603.03910, 1612.01599 and 1703.04193. I'll start with a description of the earlier results.

In each of levels 1,3,5 one has a subspace of Z/2[[x]] having as basis certain D_k. In level 1 (resp. 3,5), k runs over the positive integers prime to 2 (resp. 6,10). The formal Hecke operators T_p, p odd and not equal to the level stabilize our space; in fact T_p (D_k) is a sum of D_j with j < k. Furthermore the completed (shallow) Hecke algebra attached to the space is a power series ring Z/2[[X,Y]] in level 1, while in levels 3 and 5 one needs to adjoin an element of square zero to this power series ring. (X and Y may be taken to be T_3 and T_5 for the first space, T_7 and T_13 for the second space, T_3 and T_7 for the third space).

I'll explain next the connection of the three spaces described above with modular forms of level Gamma_0 (N) where N is 1,3, and 5. Let M(odd,N) consist of those "odd" elements of Z/2[[x]] that are the mod 2 reductions of expansions at infinity lying in Z[[x]] of holomorphic modular forms of level Gamma_0 (N).(Any weight is allowed).

1.---The first space, that spanned by the D_k with (k,2)=1, is M(odd,1).

EDIT:

2.---The second and third spaces are subquotients of M(odd,3) and M(odd,5). Namely there is a Hecke-stable filtration M(odd,3)> N2 > N1 > (0) with the middle quotient N2/N1 identifying with the second of our spaces. (The two outer quotients identify with M(odd,1)). An identical result holds for M(odd,5) and the last of our spaces.

3.---Using the filtration of M(odd,3) together with my results about the Hecke algebra attached to the second space, and the Nicolas-Serre results about M(odd,1), one gets the precise structure of the shallow Hecke algebra attached to M(odd,3). Namely it is the quotient of Z/2[[t_5, t_7, t_11, t_13]] (with t_p acting by T_p) by an ideal (A^2, AC, BC) where the leading forms of A, B and C are t_5 + t_7 + t_13, t_11 and t_7. There is an entirely similar result for M(odd,5).

4.---For the results of 3. see: Generators and relations for the shallow mod 2 Hecke algebra in levels Gamma_0 (3) and Gamma_0 (5), (1703.04193 math.[NT]). These results have also been obtained by Shaunak Deo and Anna Medvedovsky using techniques from deformation theory.

5.---Let K be the subspace of M(odd,3) consisting of f fixed by U_3.Then the shallow Hecke algebra attached to K is just the reduced shallow Hecke algebra attached to our second space. So it is a power series ring in T_7 and T_13, Similarly if K is the subspace of M(odd,5) consisting of f fixed by U_5, then the shallow Hecke algebra attached to K is the reduced shallow Hecke algebra attached to our third space, and is a power series ring in T_3 and T_7.

Here's the definition of the various spaces. Let F be x+x^9+x^25+x^49+.... In level 1, D_k is F^k. In level 3, let G(x) be F(x^3) and D(x) be F(x)+F(x^9). Then D_1 = D , D_5 = (D^2)G and D_(k+6) = (G^2)D_k. In level 5 let G(x) be F(x^5) and D(x) be F(x)+F(x^25). Then D_1 = D, D_3 = (D^8)/G, D_7 = (D^2)G, D^9 = (D^4)G and D_(k+10) = (G^2)D_k.

Here's the definition of the various spaces. Let F be x+x^9+x^25+x^49+.... In level 1, D_k is F^k. In level 3, let G(x) be F(x^3) and D(x) be F(x)+F(x^9). Then D_1 = D , D_5 = (D^2)G and D_(k+6) = (G^2)D_k. In level 5 let G(x) be F(x^5) and D(x) be F(x)+F(x^25). Then D_1 = D, D_3 = (D^8)/G, D_7 = (D^2)G, D_9 = (D^4)G and D_(k+10) = (G^2)D_k.

EDIT---I'll explain finally how the three spaces described above,spanned by various D_k, are related to modular forms of level Gamma_0 (N) where N is 1,3, and 5. Let M(odd,N) consist of those "odd" elements of Z/2[[x]] that are the mod 2 reductions of expansions at infinity lying in Z[[x]] of holomorphic modular forms of level Gamma_0 (N).(Any weight is allowed).

1.---The first space, that spanned by the D_k with (k,2)=1, is M(odd,1).

2.---Let W1 be the subspace of the second space spanned by the D_k with k=1 mod 6. The T_p with p=1 mod 6 stabilize W1. Let K be the subspace of M(odd,3) annihilated by the Hecke operator 1+U_3. Then the T_p with (p,6)=1 stabilize K and we get a shallow Hecke algebra attached to K. There is an identification of W1 with K, taking D to F+G, and preserving the action of the T_p with p=1 mod 6. It follows from 1508.07523[NT] that K admits a basis adapted to T_7 and T_13 with m_0,0 = F+G, and that the shallow Hecke algebra attached to K is a power series ring in T_7 and T_13.

3.---Let W_a be the subspace of the third space spanned by the D_k with k=1,3,7,9 mod 20. Then the T_p with p=1,3,7,9 mod 20 stabilize W_a. Now let K be the subspace of M(odd,5) annihilated by 1+U_5. We again have a shallow Hecke algebra attached to K. There is an identification of W_a with K taking D to F+G and preserving the action of the T_p with p=1,3,7,9 mod 20. It follows from 1610.07058[NT] that K admits a basis adapted to T_3 and T_7 and that the shallow Hecke algebra attached to K is a power series ring in T_3 and T_7.