MOTIVATION

Nicolas and Serre have analyzed the structure of the space of mod $2$ modular forms of level $1$, viewed as a "Hecke-module". They show that for each $p>2$, the operator $T_p$ acting on this space can be written uniquely as a power series with zero constant term in $T_3$ and $T_5$.

I've been working on generalizing their results to higher levels and have been largely successful in levels $3$ and $9$. But there remains one delicate and interesting question in level $9$. There is a Hecke-stable subspace $P_1$ of the space of level $9$ forms, and a larger Hecke-stable subspace $P_2$, such that $P_1$ and $P_2/P_1$ seem to be Hecke-isomorphic. Indeed there is a bijection of $P_1$ with $P_2/P_1$ that experimentally preserves the Hecke action.In many ways the situation parallels that of level $1$. For example if $p$ is $1$ mod $6$ then $T_p$ in its action on either $P_1$ or $P_2/P_1$ is uniquely a power series in $T_7$ and $T_{13}$ with zero constant term. And for any particular $p$ one can decide whether these $2$ power series are the same. The evidence is that they always are. Similarly if $p$ is $5$ mod $6$, then $T_p$ in its action on either $P_1$ or $P_2/P_1$ is uniquely the composition of $T_5$ with a power series in $T_7$ and $T_{13}$. These two power series are surely the same, but a proof for all $p$ eludes me.

NOTATION

$D$ is the element $x+x^{25}+x^{49}+\ldots$ of $\mathbb{Z}/2\mathbb{Z}[[x]]$, the exponents being the squares prime to $6$. $P_1$ is the space spanned by the $D^k$ with $k$ prime to $6$; $P_1$ is evidently the free rank $2$ $\mathbb{Z}/2\mathbb{Z}[D^6]$-module generated by $D$ and $D^5$.

$E$ is $x+x^4+x^{16}+x^{25}+\ldots$, the exponents being the squares prime to $3$. $P_2$ is the $\mathbb{Z}/2\mathbb{Z}[D^6]$ module having $D$, $D^5$, $D \cdot E^{16}$ and $D^5 \cdot E^8$ as a basis. The theory of level $9$ forms can be used to show that the $T_p$ with $p>3$ stabilize $P_1$ and $P_2$.

QUESTION

Does the $\mathbb{Z}/2\mathbb{Z}[D^6]$-linear bijection $P_1 \to P_2/P_1$ taking $D$ to $D \cdot E^{16}$ and $D^5$ to $D^5 \cdot E^8$ preserve the action of the $T_p$, $p>3$?

EDIT

I'll sketch a proof that the T_p, p>3, stabilize P1 and P2. If f is in Z/2[[x]] then f is f_0+f_1+f_2 where all the exponents appearing in f_i are i mod 3. Let F=x+x^9+x^25+... be the reduction of the normalized weight 12 level 1 cusp form; let G and H be F(x^3) and F(x^9). F_1 is D while F_2 is 0. One can show that D^3=G.

Let P consist of all level 9 modular forms annihilated by U_2 and by U_3. Since D=F+H is of level 9 and D^6=G^2 is a power series in x^6, P is a module over Z/2[D^6]; its rank is 8. P1 and P2 are submodules of P of ranks 2 and 4. One can show:

(A)___If M is the Z/2[G^2]-module consisting of level 3 forms that are annihilated by U_2 and whose trace from Z/2(F,G) to Z/2(G) is 0, then P1 is spanned by the f_1 and f_2 with f in M.

(B)___P2 consists of those f in P for which the traces from Z/2(F,G,H) to Z/2(F,G) of f_1 and f_2 are both 0.

The desired Hecke-stability results follow from the above characterizations of P1 and P2. That of P1 is straightforward; since F^4+F*G+G^4=0, M is generated as Z/2[G^2]-module by F,(F^2)*G, and G. And if f=(F^2)*G, then f_1 and f_2 are 0 and D^5, while if f=G, f_1 and f_2 are 0. The proof of (B) is messier.

EDIT-COMPUTER CALCULATIONS

I can show that the map P1-->P2/P1 preserves the actions of T_7, T_13 and T_5. And as I should have realized earlier, once a certain question that the experts must know the answer to has been resolved (see the end of this edit) it will be possible to handle all T_p. But this still doesn't explain in a satisfactory way WHY the conjecture holds. (And there is a similar conjecture in level 25, presumably provable, whose truth is still more mysterious). At any rate, here's the argument for T_7.

For k=1 mod 6 let Dk and Xk be D^k and(D^k)(E^16). For k=5 mod 6 let Dk and Xk be D^k and (D^k)(E^8). The Dk are a basis for P1, while the Dk and the Xk are a basis for P2. I'll describe the action of T_7 on P1 and on P2. The images of D1,D7,D13,D19,D25,D31,D37, and D43 under T_7 are:

___0, D1, 0, D13, D7, D25, D19, and D37

while the images of X1,X7,X13,X19,X25,X31,X37 and X43 are:

___0, X1, 0, X13+D5, X7, X25, X19+D11 and X37+D29

The above patterns continue. For as in Nicolas-Serre one has recursions coming from the level 7 modular equation--if A_n is either T_7(Dn) or T_7(Xn) then:

A_(n+48)=(D^48)(A_n)+(D^6)(A_(n+6))+(D^12)(A_(n+12))

So for example T_7(D55)=D49+D25 while T_7(X55)=X49+X25+D17. In like manner, T_7 takes D5, D11,D17,D23,D29,D35,D41, and D47 to:

___0, D5, 0, D17, D11, D29+D5, D23 and D41+D17

while the images of X5,X11,X17,X23,X29,X35,X41, and X47 are:

___0, X5, 0, X17, X11, X29+X5, X23+D7, and X41+X17

The recursion shows that the pattern continues, so that the map P1-->P2/P1 taking Dk to the image of Xk preserves the action of T_7. Similar arguments work for T_13 and T_5. Suppose now that we can find a finite set of primes p* such that the corresponding T_p* generate the maximal ideal in the completed Hecke algebra for mod 2 level 9 modular forms. (The algebra is known to be local and Noetherian.) Then any T_p, in its action on this full space of forms will be a power series in the T_p*, and will be this same power series in its actions on P1 and on P2/P1. So it will suffice to show that the map P1-->P2/P1 preserves the action of each of these T_p*, and this can be done as with T_7. So all I need is a list of p*--can someone provide it?

EDIT--AN ANSWER

I believe the following argument, along the lines of my last edit, proves the conjecture. Make P2 into a module over a 3-variable power series ring (O,m) over Z/2, with the variables acting by T_7, T_13, and T_5. The calculations of my last edit show that the isomorphism P1-->P2/P1 is O-linear. In the answer to question 135902, higher-level-analogs-of-nicolas-serre-theory, I sketched a proof that the only elements of P1 killed by m are 0 and D1.

I claim that the subspace of P2 annihilated by m is spanned by D1 and X1. For suppose that m*f =0 with f in P2. Then the image of f in P2/P1 is killed by m, and since P2/P1 identifies with P1, this image is 0 or X1. So f is either in P1 or is X1+(an element of P1). Since T_7, T_13 and T_5 all kill X1, f is 0 or D1 in the first case and X1 or D1+X1 in the second.

Now make P2 into a module over a 4 variable power series ring (O_1,m_1) over Z/2 with the variables acting by T_7, T_13, T_5, and T_17. Each element of P2 is annihilated by some power of m_1. Since T_17(X1)=D1, the paragraph above shows that the only elements of P2 killed by m_1 are 0 and D1.

So P2 is an Artinian O_1 module with 1 dimensional socle, and its (Noetherian) dual is a cyclic O_1 module. It follows that every O_1 linear map P2-->P2 is multiplication by an element of O_1, so that each T_p, acting on P2, is a power series in T_13,T_7,T_5,and T_17. So it's enough to show that P1-->P2/P1 preserves the action of T_17; this is a straightforward calculation like that carried out for T_7 in my last edit.

REMARK

The above result can be used to show that P, viewed as Hecke-module, has a curious structure. Namely there is a filtration of P by Hecke-stable Z/2[D^6]-modules, such that 3 quotients in the filtration identify with P1, while the other 2 quotients identify with the level 1 space of Nicolas and Serre. I'd be interested in understanding this conceptually, and seeing if there are similar connections between forms of levels p and p^2 in a more general setting.