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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.

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  • $\begingroup$ Would this imply that there are no new mod $2$ modular two-dimensional Galois representations of conductor 3 or 9? $\endgroup$ – Will Sawin Apr 25 '15 at 15:06
  • $\begingroup$ @Will--I can show by methods similar to this that the T_p are locally nilpotent on the space of level 9 mod 2 modular forms--the conjectured isomorphism isn't needed. I guess this gives a new proof of the result you state, though the result isn't new. $\endgroup$ – paul Monsky Apr 25 '15 at 15:42
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It's much easier than I thought (granting a result about the Fricke involution in level Gamma_0 (9)). Let M(odd) consist of those odd power series lying in Z/2[E]; it is the space of odd mod 2 modular forms of level Gamma_0 (9). Suppose f in Z[1/3][[x]] is a modular form of weight w and the above level. Then the same is true of the image of f under Fricke--this is the slightly delicate point. So Fricke induces an automorphism f-->f* of the space Z/2[E] that commutes with the formal characteristic 2 Hecke operators T_p; one sees that this is the involution E--> E+1. Note that it is Z/2[G]-linear. Let J be the kernel of U_3: M(odd)--> M(odd). J is a rank 8 Z/2[G^2]-module with basis D, D^5, G* E^8, G* E^16, D* E^16, D^5* E^8, D* E^24, D^5* E^24.

Now consider the map s: J-->J taking f to pr(f+f*), where pr throws away all terms x^j in a power series where 3 divides j. This s is Z/2[G^2]-linear and commutes with the T_p. It sends the first 4 elements in the above basis to 0 and the last 4 to D, D^5, D* (1+E^16), D^5* (1+E^8). So s(J) has D, D^5, D* E^16, D^5* E^8 as Z/2[G^2] basis and is just P_2. Thus the T_p stabilize P2. And s, which as we've said is Z/2[G^2}-linear and preserves the action of the T_p, maps P2 onto P1 with kernel P1.

EDIT (3/17/17)

  1. Both in my answers to this question and to the companion question 204722, the essential tool is the map f-->f* on mod 2 modular forms of level Gamma_0 (N^2), N=3 or 5, induced by the Fricke involution of level N^2. Because this map is a ring automorphism fixing the reduction G of delta(Nz), and preserving the Hecke action, one can use the kernels and images of maps attached to it to get Z/2[G^2]-submodules of M(odd) stabilized by the T_p, and maps between such modules preserving the Hecke action. This leads to the results I had observed in a fairly natural way. And it should give results of the same flavor for larger primes N.

  2. But specific details depend on N, as the cases N=3 and N=5 show. When N=3, E*=E+1, while when N=5, E*=E (because 1+E^8 = D^5/G; see my answer to question 204722). So the technique used here won't work for the other question. Conversely one may define J, J1, and J2 as in question 204722. J1 and J2 now have ranks 3 and 6 and one gets a map J2-->J1 preserving the Hecke action and killing J1, as in my other answer. But the kernel is larger than J1; the map kills E^8*G which is not in J1 precisely because E*=E+1.

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