This is a characteristic $3$ analog of part of my earlier question, "Are these two subspaces of $\mathbb{Z}/2[[x]]$ the same?"
Notation
Fix a prime $N$ other than $3$. Let $F,G \in \mathbb{Z}/3[[x]]$ be the reductions of the expansions at infinity of the modular forms $\delta(z)$ and $\delta(Nz)$. (If you don't like modular forms, $F$ is $\sum a_nx^n$, where $a_n$ is $0$ if $3\mid n$ and is the mod $3$ reduction of the number of ideals of norm $n$ in the Eisenstein integers otherwise.)
There is a degree $N+1$ irreducible polynomial relation between $F$ and $G$ over $\mathbb{Z}/3$. The relation is monic in $F$ and is symmetric. (For example, when $N=5$, $(F^3+G^3-FG)^2=FG$). The reason for this is that in characteristics $2$ and $3$, the reduction of the Eisenstein series $E_4$ is $1$. So in these characteristics the expansions of $j(z)$ and of $\delta(z)$ are reciprocals, and the modular equation linking $j(z)$ and $j(Nz)$ goes over to our relation).
Let $M$ be the integral closure of $\mathbb{Z}/3[G]$ in the extension field of $\mathbb{Z}/3(G)$ generated by $F$. View $M$ as a subring of $\mathbb{Z}/3[[x]]$. There are formal Hecke operators $T_n: \mathbb{Z}/3[[x]]\to \mathbb{Z}/3[[x]]$ when $(n,3)=1$. I'll denote the algebra spanned by the $T_n$ with $(n,3N)=1$ by $HE$; it is the "shallow Hecke algebra". Just as in characteristic $2$ there is a modular interpretation of $M$ that shows that it is stabilized by $HE$.
Question
Is the subspace of $M$, consisting of elements whose trace from $\mathbb{Z}/3(F,G)$ to $\mathbb{Z}/3(G)$ is $0$, stable under $HE$?
I've examined one case in detail - $N=5$. All the computer evidence is that the answer is yes. But I haven't a clue as how to prove anything. In characteristic $2$, at least for small $N$, there was an alternative characterization of the space of trace $0$ elements that gave the desired conclusion, but nothing like that seems to apply here.
EDIT___I've passed over the case N=2, but this is perhaps the most interesting case of all. One has the relation (F+G)^3=FG, and the integral closure, M, of Z/3[G] in Z/3(F,G) is Z/3[A] where A=G/(F+G). Note that A*(1-A)^2 is F, while (1-A)A^2 is G. The trace 0 elements of M form a Z/3[G]-module of rank 2 generated by 1 and F. For i=1 or 2 let V_i consist of the trace 0 elements of M in which all the exponents that appear are i mod 3. Let pr be the map Z/3[[x]]-->Z/3[[x]] which removes from each h all terms with even exponent. Let D=pr(F)=F(x)-F(x^4)=x+2(x^7)+2*(x^13)+2*(x^19)+x^25+(higher degree terms). One can show that D^2=G.
Now V_1 is generated as Z/3[G^3]-module by F and G^2, while V_2 is generated by FG^2 and G. It follows that pr(V_1) is spanned by the pr(FG^3n). But this last power series is just DG^(3n)=D^(6n+1). Similarly we find that pr(V_2) is spanned by the D^(6n+5). Let HE# be the subalgebra of HE spanned by the T_n with n=1 mod 6. Then if my conjecture that the space of trace 0 elements of M is HE-stable holds, it will follow that:
(a)___The space spanned by the D^k with k=1 mod 6 is stable under HE#
(b)___The space spanned by the D^k with (k,6)=1 is stable under HE.
There's a great deal of computer evidence supporting (a) and (b). These should be compared with similar (proved!) results in characteristic 2---look under the CONJECTURAL ANALOGS head of my earlier question 135902--Higher level analogs of Nicolas-Serre theory, where I give a proved result 1* and conjectures 2*, 3* and 4* that (once proved) should extend the characteristic 2 Nicolas-Serre theory to level 3. I now think that I have empirical generalizations of all this that (once proved) should extend Bellaiche's characteristic 3 version of Nicolas-Serre theory to level 2. I'll report on these findings later. But I suspect it will take someone well-versed in representation theory and deformation theory to furnish any proofs.
