Generators and relations for the shallow mod 2 Hecke algebra in levels $\Gamma_{0}(3)$ and $\Gamma_{0}(5)$

TL;DR

This paper characterizes the structure of the shallow mod 2 Hecke algebra at levels b1 3 and 5, showing it is a power series ring over a specific local ring with explicit generators and relations.

Contribution

It provides explicit generators, relations, and structural description of the shallow mod 2 Hecke algebra at levels b1 3 and 5, extending understanding of its algebraic properties.

Findings

Hecke operators act as multiplication by elements in the maximal ideal.

The kernel of the action is generated by specific quadratic and linear relations.

The Hecke algebra is a 1-variable power series ring over a local ring with known relations.

Abstract

Let $M(\mathit{odd})\subset Z/2[[x]]$ be the space of odd mod~2 modular forms of level $\Gamma_{0}(3)$. It is known that the formal Hecke operators $T_{p}:Z/2[[x]]\rightarrow Z/2[[x]]$, $p$ an odd prime other than $3$, stabilize $M(\mathit{odd})$ and act locally nilpotently on it. So $M(\mathit{odd})$ is an $\mathcal{O} = Z/2[[t_{5},t_{7}, t_{11}, t_{13}]]$-module with $t_{p}$ acting by $T_{p}$, $p\in \{5,7,11,13\}$. We show: (1) Each $T_{p}:M(\mathit{odd})\rightarrow M(\mathit{odd})$, $p\ne 3$, is multiplication by some $u$ in the maximal ideal, $m$, of $\mathcal{O}$. (2) The kernel, $I$, of the action of $\mathcal{O}$ on $M(\mathit{odd})$ is $(A^{2},AC,BC)$ where $A,B,C$ have leading forms $t_{5}+t_{7}+t_{13},\, t_{7},\, t_{11}$. We prove analogous results in level $\Gamma_{0}(5)$. Now $\mathcal{O}$ is $Z/2[[t_{3},t_{7},t_{11},t_{13}]]$, and the leading forms of $A,B,C$ are $t_{3}+t_{7}+t_{11},\, t_{7},\, t_{13}$. Let $\mathit{HE}$, "the shallow mod~2 Hecke algebra (of level $\Gamma_{0}(3)$ or $\Gamma_{0}(5)$)" be $\mathcal{O}/I$. (1) and (2) above show that $\mathit{HE}$ is a 1 variable power series ring over the 1-dimensional local ring $Z/2[[A,B,C]]/(A^{2},AC,BC)$. For another approach to all these results, based on deformation theory, see Deo and Medvedovsky, "Explicit old components of mod-2 Hecke algebras with trivial $\bar{\rho}$."

Full text

1. (1)

Each $T_{p}:M(\mathit{odd})\rightarrow M(\mathit{odd})$, $p\neq 3$, is multiplication by some $u$ in the maximal ideal, $m$, of $\mathcal{O}$. 2. (2)

The kernel, $I$, of the action of $\mathcal{O}$ on $M(\mathit{odd})$ is $(A^{2},AC,BC)$ where $A,B,C$ have leading forms $t_{5}+t_{7}+t_{13},t_{7},t_{11}$.

We prove analogous results in level $\Gamma_{0}(5)$. Now $\mathcal{O}$ is $Z/2[[t_{3},t_{7},t_{11},t_{13}]]$, and the leading forms of $A,B,C$ are $t_{3}+t_{7}+t_{11},t_{7},t_{13}$.

Let $\mathit{HE}$, “the shallow mod 2 Hecke algebra (of level $\Gamma_{0}(3)$ or $\Gamma_{0}(5)$)” be $\mathcal{O}/I$. (1) and (2) above show that $\mathit{HE}$ is a 1 variable power series ring over the 1-dimensional local ring $Z/2[[A,B,C]]/(A^{2},AC,BC)$. For another approach to all these results, based on deformation theory, see Deo and Medvedovsky [4].

1 Introduction

For $p$ an odd prime, $T_{p}:Z/2[[x]]\rightarrow Z/2[[x]]$ is the formal Hecke operator $\sum c_{n}x^{n}\rightarrow\sum c_{pn}x^{n}+\sum c_{n}x^{pn}$; the $T_{p}$ commute. We’ll be concerned with certain subspaces of $Z/2[[x]]$, coming from modular forms of level $\Gamma_{0}(N)$, and stabilized by the $T_{p}$, $p\nmid N$. On the spaces we’re looking at, each $T_{p}$ acts locally nilpotently. Let $S$ be a finite set of odd primes $p$ not dividing $N$, and $\mathcal{O}$ be a power series ring over $Z/2$ in variables $t_{p}$, $p\in S$. Then our subspace is an $\mathcal{O}$-module with $t_{p}$ acting by $T_{p}$. We’ll show in some cases that each $T_{p}$ acting on the subspace is multiplication by an element of $\mathcal{O}$ (which lies in the maximal ideal since $T_{p}$ acts locally nilpotently). And we’ll describe the kernel, $I$, of the action of $\mathcal{O}$ on the subspace.

The motivating level 1 example appears in [3]. Let $F$ in $Z/2[[x]]$ be $x+x^{9}+x^{25}+\cdots$, the exponents being the odd squares. The subspace $V$ is spanned by the $F^{k}$, $k$ odd (and positive). $F$ is the mod 2 reduction of the weight 12 cusp form $\Delta$, and a modular forms interpretation of $V$ shows that the $T_{p}$ stabilize it; with more work one may show that they act locally nilpotently. Take $S=\{3,5\}$. Nicolas and Serre show:

1. (1)

Each $T_{p}:V\rightarrow V$ is multiplication by an element of $Z/2[[t_{3},t_{5}]]$. 2. (2)

$\mathcal{O}=Z/2[[t_{3},t_{5}]]$ acts faithfully on $V$.

Here is a level $\Gamma_{0}(3)$ example whose study was begun in [1]. Let $G=F(x^{3})$, and $M(\mathit{odd})$ be spanned by the $F^{i}G^{j}$, where $i,j$ are $\geq 0$ and $i+j$ is odd. Here’s a modular forms interpretation; $M(\mathit{odd})$ consists of all odd power series that are mod 2 reductions of elements of $Z[[x]]$ arising as expansions at infinity of holomorphic modular forms of level $\Gamma_{0}(3)$ (and any weight). (We write $x$ in place of the customary $q$ for the expansion variable throughout.) This interpretation shows that the $T_{p}$, $p\neq 3$, stabilize $M(\mathit{odd})$. Using the local nilpotence of the $T_{p}$ acting on $V$, and on a certain subquotient $W$ of $M(\mathit{odd})$ introduced in [1], we show that the $T_{p}$, $p\neq 3$, act locally nilpotently on $M(\mathit{odd})$.

If we take $G$ to be $F(x^{5})$ instead of $F(x^{3})$ we get another subspace, which we’ll also call $M(\mathit{odd})$; it has a similar interpretation with $\Gamma_{0}(3)$ replaced by $\Gamma_{0}(5)$. This $M(\mathit{odd})$ is stabilized by the $T_{p}$, $p\neq 5$, and we’ll use results from [2] to show that they act locally nilpotently on it.

We take $S$ to be $\{5,7,11,13\}$ in the level 3 example and to be $\{3,7,11,13\}$ in level 5. We will show that the $T_{p}$, $p\neq 3$, are multiplication by elements of $\mathcal{O}$ in the first case, while the $T_{p}$, $p\neq 5$, are multiplication by elements of $\mathcal{O}$ in the second. In each case we’ll determine the kernel, $I$, of the action. It is an ideal $(A^{2},AC,BC)$ where the degree 1 parts of $A$, $B$ and $C$ are linearly independent in the 4-dimensional vector space $m/m^{2}$. Apart from results from [1], [2], [3] there are 2 simple new ideas. One is making use of the fact that a certain $\mathcal{O}$-submodule of $V$ imbeds in the subquotient, $W$, of $M(\mathit{odd})$. The other is showing that there are no non-zero $\mathcal{O}$-linear maps $W\rightarrow V$.

Shaunak Deo and Anna Medvedovsky, [4], have derived the same results simultaneously with us. They use techniques from deformation theory in place of our arguments, which are more related to ideas from [3]. Communications in both directions were helpful in understanding precisely what the kernel, $I$, of the action of $\mathcal{O}$ on $M(\mathit{odd})$ should be, and in completing the proofs, both for us in our arguments and for them in theirs. It would be interesting to understand how the proofs are related.

2 $\bm{M(\mathit{odd})}$ in level $\bm{\Gamma}_{0}(3)$

Throughout this section $\mathit{pr}:Z/2[[x]]\rightarrow Z/2[[x]]$ is the map $\sum c_{n}x^{n}\rightarrow\sum_{(n,3)=1}c_{n}x^{n}$, $G$ is $F(x^{3})$ and $D=\mathit{pr}(F)$. (We’ll use a related but different notation in the next section.)

Definition 2.1**.**

$U_{3}:Z/2[[x]]\rightarrow Z/2[[x]]$* is the map $\sum c_{n}x^{n}\rightarrow\sum c_{3n}x^{n}$. $M(\mathit{odd})\subset Z/2[[x]]$ is spanned by the $F^{i}G^{j}$, $i,j\geq 0$, $i+j$ odd.*

As was shown in [1], there is an interpretation of $M(\mathit{odd})$ in terms of modular forms of level $\Gamma_{0}(3)$ that shows that the $T_{p}$, $p\neq 3$, stabilize it. It’s also stabilized by $U_{3}$ (by a similar argument) but we’ll only need the obvious fact that $U_{3}$ maps the space spanned by the $G^{k}$, $k$ odd, bijectively to $V$, and that this map commutes with $T_{p}$, $p\neq 3$. The following are proved in [1]:

Theorem 2.2**.**

1. (1)

$F$* has degree 4 over $Z/2(G)$ and $(F+G)^{4}=FG$.* 2. (2)

As $Z/2[G^{2}]$-module, $M(\mathit{odd})$ has basis $\{G,F,F^{2}G,F^{3}\}$. 3. (3)

The trace map $Z/2(F,G)\rightarrow Z/2(G)$ takes $G,F,F^{2}G,F^{3}$ to $0,0,0,G$. So it gives a $Z/2[G^{2}]$-linear map $\mathit{Tr}:M(\mathit{odd})\rightarrow M(\mathit{odd})$. The kernel $N2$ and image $N1$ of $\mathit{Tr}$ have $Z/2[G^{2}]$-bases $\{G,F,F^{2}G\}$ and $\{G\}$. 4. (4)

The filtration $M(\mathit{odd})\supset N2\supset N1\supset(0)$ of $M(\mathit{odd})$ is “Hecke-stable.” I.e., the $T_{p}$, $p\neq 3$, stabilize $N2$ and $N1$. 5. (5)

The image, $W$, of $N2$ under $\mathit{pr}$ has $Z/2[G^{2}]$-basis $\{D,D^{2}G\}$. $N1$ is the kernel of $\mathit{pr}:N2\rightarrow W$, and so the $T_{p}$, $p\neq 3$, stabilize $W$, and the isomorphism $N2/N1\rightarrow W$ commutes with $T_{p}$, $p\neq 3$.

Remarks* 

Besides the above bijection $N2/N1\rightarrow W$ commuting with $T_{p}$, $p\neq 3$, we have the bijection $U_{3}:N1\rightarrow V$ commuting with these $T_{p}$. Finally there is a composite bijection $M(\mathit{odd})/N2\stackrel{{\scriptstyle\mathit{Tr}}}{{\rightarrow}}N1\stackrel{{\scriptstyle U_{3}}}{{\rightarrow}}V$. We’ll show that this too commutes with $T_{p}$. *

Lemma 2.3**.**

$T_{3}:V\rightarrow V$* is onto.*

Proof.

By [3], $V$ has a $Z/2$-basis $\{m_{i,j}\}$ with $m_{0,0}=F$, “adapted to $T_{3}$ and $T_{5}$.” But then $T_{3}$ takes $m_{i+1,j}$ to $m_{i,j}$. ∎∎

Observation* 

Here’s an “injective modules are divisible” generalization of the above. Suppose $M$ is a $k[[X,Y]]$-module that admits a $k$-basis $m_{i,j}$ adapted to $X$ and $Y$. Then if $u$ in $k[[X,Y]]$ is non-zero, $uM=M$. (And in particular, the action is faithful.) To see this, totally order $N\times N$, taking $(0,0)<(1,0)<(0,1)<(2,0)<(1,1)<(0,2)<(3,0)<\cdots$. Let $cX^{a}Y^{b}$ be the monomial appearing in the leading form of $u$ with largest $a$. Then $u(m_{a+i,b+j})=cm_{i,j}+$ a $k$-linear combination of $m_{r,s}$ with $(r,s)<(i,j)$, and an inductive argument using the total ordering gives the result. *

Lemma 2.4**.**

The composite map $V\stackrel{{\scriptstyle\mathit{Tr}}}{{\rightarrow}}N1\stackrel{{\scriptstyle U_{3}}}{{\rightarrow}}V$ is $T_{3}$, and so, by Lemma 2.3, is onto.

Proof.

Let $U(X,Y)$ be $(X+Y)^{4}+XY$. Then $U(F(x),F(x^{3}))=0$. Replacing $x$ by $x^{3}$ we find that $U(G,F(x^{9}))=0$. So $U(F(x^{9}),G)=0$, and $F(x^{9})$ is a conjugate of $F$ over $Z/2(G)$. Similarly, replacing $x$ by $rx$ where $r$ is in the field of 4 elements, $r^{3}=1$, we find that the other 3 conjugates of $F$ are the $F(rx)$. So the conjugates of $F^{k}$ are $F^{k}(x^{9})$ and $F^{k}(rx)$. Writing $F^{k}$ as $\sum c_{n}x^{n}$, adding together the conjugates, and applying $U_{3}$ we get $T_{3}(F^{k})$. ∎∎

Theorem 2.5**.**

The composite bijection $M(\mathit{odd})/N2\stackrel{{\scriptstyle\mathit{Tr}}}{{\rightarrow}}N1\stackrel{{\scriptstyle U_{3}}}{{\rightarrow}}V$ commutes with $T_{p}$, $p\neq 3$. We conclude that $M(\mathit{odd})/N2$, $N2/N1$ and $N1$ identify with $V$, $W$ and $V$ as Hecke-modules.

Proof.

By Lemma 2.4 the restriction of our map to $V$ is onto So the elements of $V$ span $M(\mathit{odd})/N2$. And on $V$ our map is $T_{3}:V\rightarrow V$ which commutes with the $T_{p}:V\rightarrow V$. ∎∎

Theorem 2.6**.**

The $T_{p}$, $p\neq 3$, act locally nilpotently on $M(\mathit{odd})$. In other words, if $f\in M(\mathit{odd})$ and $p\neq 3$, some power of $T_{p}$ annihilates $f$.

Proof.

[3] and [1] show that $T_{p}$ acts locally nilpotently on $V$ and $W$. And the quotients in the filtration of $M(\mathit{odd})$ are Hecke-isomorphic to $V$, $W$ and $V$. ∎∎

For the rest of this section, unless otherwise noted, $S=\{5,7,11,13\}$ and $\mathcal{O}$ is the 4-variable power series ring over $Z/2$ in the $t_{p}$, $p\in S$. Then $V$, $W$ and $M(\mathit{odd})$ are all $\mathcal{O}$-modules with $t_{p}$ acting by $T_{p}$, $p\in S$. Let $I(V)$, $I(W)$ and $I$ be the kernels of the respective actions.

$I(V)$ is easily described. As we noted in section 1, when $V$ is viewed as a $Z/2[[t_{3},t_{5}]]$-module, the action is faithful, and each $T_{p}:V\rightarrow V$ is multiplication by some $u$ in $Z/2[[t_{3},t_{5}]]$. In [3] it’s shown that:

when $p=11$ $u=\mathrm{unit}(t_{3})$

when $p=13$ $u=\mathrm{unit}(t_{5})$

when $p=7$ $u=\mathrm{unit}(t_{3})(t_{5})$

It follows from the above that when $V$ is viewed as a $Z/2[[t_{11},t_{13}]]$-module the action is faithful, and each $T_{p}:V\rightarrow V$ is multiplication by some $u$ in $Z/2[[t_{11},t_{13}]]$. Furthermore when $p=5$, $u=\mathrm{unit}(t_{13})$, while when $p=7$, $u=\mathrm{unit}(t_{11})(t_{13})$. So $I(V)$ is generated by 2 elements, $t_{5}+\mathrm{unit}(t_{13})$ and $t_{7}+\mathrm{unit}(t_{11})(t_{13})$. This gives:

Theorem 2.7**.**

$I(V)$* is generated by 2 elements $A$ and $B$ whose leading forms can be taken to be $t_{5}+t_{7}+t_{13}$ and $t_{7}$.*

To describe $I(W)$ we use the following results from [1]:

Theorem 2.8**.**

Let $W1$ and $W5$ be the $Z/2[G^{2}]$-submodules of $W$ generated by $D$ and $D^{2}G$ respectively. (In fact $G=D^{3}$, so that a $Z/2$-basis of $W1$ consists of the $D^{k}$, $k\equiv 1\hskip 5.0pt(6)$, while a $Z/2$-basis of $W5$ consists of the $D^{k}$, $k\equiv 5\hskip 5.0pt(6)$.)

1. (1)

The $T_{p}$, $p\equiv 1\hskip 5.0pt(6)$, stabilize $W1$ and $W5$. The $T_{p}$, $p\equiv 5\hskip 5.0pt(6)$, map $W1$ to $W5$, and $W5$ to $W1$. 2. (2)

$W1$* has a basis $\{m_{i,j}\}$ with $m_{0,0}=D$, adapted to $T_{7}$ and $T_{13}$. The same holds for $W5$ with $m_{0,0}=D^{5}=D^{2}G$. It follows that $T_{7}$ and $T_{13}$ act locally nilpotently on $W1$, $W5$ and $W$.* 3. (3)

Taking $S=\{7,13\}$ and making $W$ into a $Z/2[[t_{7},t_{13}]]$-module, we find that each $T_{p}:W\rightarrow W$, $p\equiv 1\hskip 5.0pt(6)$, is multiplication by some $u$ in the maximal ideal of $Z/2[[t_{7},t_{13}]]$. 4. (4)

And each $T_{p}:W\rightarrow W$, $p\equiv 5\hskip 5.0pt(6)$, is the composition of $T_{5}$ with multiplication by some $u$ in $Z/2[[t_{7},t_{13}]]$. 5. (5)

$T_{5}^{2}:W\rightarrow W$* is multiplication by $\lambda^{2}$ for some $\lambda$ in $Z/2[[t_{7},t_{13}]]$ with leading form $t_{7}+t_{13}$.*

Remarks* 

(3), (4) and (5) show that each $T_{p}$, $p\neq 3$, acts locally nilpotently on $W$. And if we take $S=\{5,7,13\}$, each $T_{p}:W\rightarrow W$, $p\neq 3$, is multiplication by an element of $Z/2[[t_{5},t_{7},t_{13}]]$. Furthermore if we set $\varepsilon=t_{5}+\lambda$, then $\varepsilon^{2}$ kills $W$. Note that the leading form of $\varepsilon$ is $t_{5}+t_{7}+t_{13}$. *

Theorem 2.9**.**

$I(W)$* is generated by 2 elements $\varepsilon^{2}$ and $C$ where $\varepsilon$ is congruent to the $A$ of Theorem 2.7 mod $m^{2}$, and the leading form of $C$ is $t_{11}$.*

Proof.

First we determine the kernel of the action of $Z/2[[t_{5},t_{7},t_{13}]]$ on $W$. The remark above shows that the kernel contains $(\varepsilon^{2})=(t_{5}^{2}+\lambda^{2})$. If $R$ is in the kernel, we may modify $R$ by an element of this ideal, and assume that $R=u+u^{\prime}t_{5}$ with $u$ and $u^{\prime}$ in $Z/2[[t_{7},t_{13}]]$. Since $u$ stabilizes $W1$ and $W5$ while $u^{\prime}t_{5}$ takes $W1$ to $W5$ and $W5$ to $W1$, $u$ and $\lambda^{2}u^{\prime}$ are elements of $Z/2[[t_{7},t_{13}]]$ annihilating $W$. Since $Z/2[[t_{7},t_{13}]]$ acts faithfully on $W$ (see for example the observation following Lemma 2.3), $u=u^{\prime}=0$, and the kernel of the action is $(\varepsilon^{2})$; note that $\varepsilon$ has the same leading form as $A$. Finally by (4) of Theorem 2.8, $I(W)$ contains an element of the form $t_{11}+vt_{5}$ with $v$ in $\mathcal{O}$. It remains to show that $v$ is not a unit. But an easy calculation shows that $T_{11}(D^{5})=0$ while $T_{5}(D^{5})=D$. ∎∎

Definition 2.10**.**

$V(0,\star)$* is the kernel of $T_{3}:V\rightarrow V$.*

If $\{m_{i,j}\}$ is a basis of $V$ adapted to $T_{3}$ and $T_{5}$, the $m_{0,j}$ form a $Z/2$-basis of $V(0,\star)$. $Z/2[[t_{5}]]$ acts faithfully and locally nilpotently on $V(0,\star)$. $V(0,\star)$ is an $\mathcal{O}$-submodule of $V$, and the kernel of the action is a height 3 prime ideal $P$ generated by $t_{7}$, $t_{11}$ and an element with leading form $t_{13}+t_{5}$.

Lemma 2.11**.**

$W$, as well as $V$, contains a “Hecke-submodule” isomorphic to $V(0,\star)$.

Proof.

If $f\in V(0,\star)$, $T_{3}(f)=U_{3}(\mathit{Tr}(f))=0$. Since $U_{3}:N1\rightarrow V$ is bijective, $\mathit{Tr}(f)=0$ and $f$ is in $N2$. So we have a composite map $V(0,\star)\subset N2\stackrel{{\scriptstyle\mathit{pr}}}{{\rightarrow}}W$ which commutes with $T_{p}$, $p\neq 3$, and takes $m_{0,0}=F$ to $D$. Since $m_{0,0}$ is not in the kernel of this map, the kernel is $(0)$, giving the result. ∎∎

Theorem 2.12**.**

Let $A,B,C$ be as in Theorems 2.7 and 2.9. Then $(A,B,C)=I(V)+I(W)=P$, the kernel of the action of $\mathcal{O}$ on $V(0,\star)$.

Proof.

Evidently $(A,B,C)\subset I(V)+I(W)$. Since $V$ and $W$ each have an $\mathcal{O}$-submodule isomorphic to $V(0,\star)$, $I(V)+I(W)\subset P$. Finally $(A,B,C)$ and $P$ are height 3 primes of $\mathcal{O}$. ∎∎

Theorem 2.13**.**

The only $\mathcal{O}$-linear map $\alpha:W\rightarrow V$ is the zero-map.

Proof.

$\alpha(W)$ is annihilated both by $I(W)$ and $I(V)$. So by Theorem 2.12 it is annihilated by $P$, and thus by $t_{7}$. Then $\alpha(t_{7}W)=t_{7}\alpha(W)=(0)$. But since $W1$ and $W5$ have bases adapted to $T_{7}$ and $T_{13}$, $t_{7}(W)=W$, and $\alpha(W)=(0)$. ∎∎

Theorem 2.14**.**

If $p\neq 3$, $T_{p}:M(\mathit{odd})\rightarrow M(\mathit{odd})$ is multiplication by some $u$ in $\mathcal{O}$.

Proof.

We know that $T_{p}:V\rightarrow V$ and $W\rightarrow W$ are multiplication by some $u$ and $u^{\prime}$ in $\mathcal{O}$; for $W$ see the remarks following Theorem 2.8. Then $f\rightarrow T_{p}(f)-uf$ and $f\rightarrow T_{p}(f)-u^{\prime}f$ both annihilate $V(0,\star)$ by Lemma 2.11. So $u-u^{\prime}$ is in $P$, and by Theorem 2.12 it is in $(A,B,C)$. Modifying $u$ by an element of $(A,B)$, and $u^{\prime}$ by an element of $(C)$ we may assume $u-u^{\prime}=0$. Let $\alpha:M(\mathit{odd})\rightarrow M(\mathit{odd})$ be the map $f\rightarrow T_{p}(f)-uf$. We’ll show that $\alpha$ is the zero-map.

$\alpha$ annihilates $V$, and since $u=u^{\prime}$, it annihilates $W$. Since $\alpha$ commutes with $U_{3}$ and $\mathit{pr}$ it annihilates $N1$ and $N2/N1$. So $\alpha(N2)\subset N1$, and $\alpha$ induces an $\mathcal{O}$-linear map $N2/N1\rightarrow N1$. By Theorem 2.13 this is the zero-map, and $\alpha(N2)=(0)$. But the proof of Theorem 2.5 shows that the elements of $V$ span $M(\mathit{odd})/N2$. Since $\alpha(V)=0$, $\alpha=0$. ∎∎

Theorem 2.15**.**

There are $A,B,C$ in $\mathcal{O}$ with leading forms $t_{5}+t_{7}+t_{13}$, $t_{7}$ and $t_{11}$ such that $I(V)=(A,B)$ and $I(W)=(A^{2},C)$. The kernel, $I$, of the action of $\mathcal{O}$ on $M(\mathit{odd})$ is $I(V)\cap I(W)=(A^{2},AC,BC)$.

Proof.

Let $A,B,C,\varepsilon$ be as in Theorems 2.7 and 2.9; $I(V)=(A,B)$, $I(W)=(\varepsilon^{2},C)$ and $A-\varepsilon$ is in $m^{2}$. Then $A^{2}-\varepsilon^{2}$ is in $I(V)+I(W)$ which is $(A,B,C)$ by Theorem 2.12. Since $(A,B,C)$ is prime, $A-\varepsilon$ is in $(A,B,C)\cap m^{2}=mA+mB+mC$. Modifying $A$ by an element of $mA+mB$, and $\varepsilon$ by an element of $mC$ we may assume $A-\varepsilon=0$. Then $I(V)=(A,B)$, $I(W)=(A^{2},C)$ and it follows that $I(V)\cap I(W)=(A^{2},AC,BC)$. Suppose $u$ is in $I(V)\cap I(W)$. Let $\alpha:M(\mathit{odd})\rightarrow M(\mathit{odd})$ be multiplication by $u$. Then $\alpha(N1)=(0)$, $\alpha(N2)\subset N1$, and arguing as in the final paragraph of the proof of Theorem 2.14 we get the result; $u$ is in $I$. ∎∎

3 $\bm{M(\mathit{odd})}$ in level $\bm{\Gamma}_{0}(5)$

Throughout this section $\mathit{pr}:Z/2[[x]]\rightarrow Z/2[[x]]$ is the map $\sum c_{n}x^{n}\rightarrow\sum_{(n,5)=1}c_{n}x^{n}$, $G$ is $F(x^{5})$ and $D=\mathit{pr}(F)$.

Definition 3.1**.**

$U_{5}:Z/2[[x]]\rightarrow Z/2[[x]]$* is the map $\sum c_{n}x^{n}\rightarrow\sum c_{5n}x^{n}$. $M(\mathit{odd})\subset Z/2[[x]]$ is spanned by the $F^{i}G^{j}$, $i,j\geq 0$, $i+j$ odd.*

As was shown in [2], there is an interpretation of $M(\mathit{odd})$ in terms of modular forms of level $\Gamma_{0}(5)$ that shows that the $T_{p}$, $p\neq 5$, stabilize it. It is also stabilized by $U_{5}$, but we’ll only need the obvious fact that $U_{5}$ maps the space spanned by the $G^{k}$, $k$ odd, bijectively to $V$, and that this map commutes with $T_{p}$, $p\neq 5$. The following are proved in [2]:

Theorem 3.2**.**

1. (1)

$F$* has degree 6 over $Z/2(G)$ and $(F+G)^{6}=FG$.* 2. (2)

As $Z/2[G^{2}]$-module, $M(\mathit{odd})$ has basis $\{G,F,F^{2}G,F^{3},F^{4}G,F^{5}\}$. 3. (3)

The trace map $Z/2(F,G)\rightarrow Z/2(G)$ takes $G,F,F^{2}G,F^{3},F^{4}G,F^{5}$ to $0,0,0,0,0,G$. So it gives a $Z/2[G^{2}]$-linear map $\mathit{Tr}:M(\mathit{odd})\rightarrow M(\mathit{odd})$. The kernel $N2$ and image $N1$ of $\mathit{Tr}$ have $Z/2[G^{2}]$-bases $\{G,F,F^{2}G,F^{3},F^{4}G\}$ and $\{G\}$. 4. (4)

The filtration $M(\mathit{odd})\supset N2\supset N1\supset(0)$ of $M(\mathit{odd})$ is “Hecke-stable.” I.e., the $T_{p}$, $p\neq 5$, stabilize $N2$ and $N1$. 5. (5)

The image, $W$, of $N2$ under $\mathit{pr}$ has $Z/2[G^{2}]$-basis $\{D,D^{8}/G,D^{2}G,D^{4}G\}$. $N1$ is the kernel of $\mathit{pr}:N2\rightarrow W$, and so the $T_{p}$, $p\neq 5$, stabilize $W$, and the isomorphism $N2/N1\rightarrow W$ commutes with $T_{p}$, $p\neq 5$.

Remarks* 

Note that $\mathit{pr}$ takes the elements $F$, $F^{2}G$ and $F^{4}G$ of $N2$ to $D$, $D^{2}G$ and $D^{4}G$. Also $F(F+G)^{2}=F(F+G)^{8}/FG=(F^{8}/G)+G^{7}$; $\mathit{pr}$ takes this to $D^{8}/G$. Besides the bijection $N2/N1\rightarrow W$ commuting with $T_{p}$, $p\neq 5$, we have the bijection $U_{5}:N1\rightarrow V$ commuting with these $T_{p}$. Finally there is a composite bijection $M(\mathit{odd})/N2\stackrel{{\scriptstyle\mathit{Tr}}}{{\rightarrow}}N1\stackrel{{\scriptstyle U_{5}}}{{\rightarrow}}V$; we’ll show that this too commutes with $T_{p}$. *

Lemma 3.3**.**

$T_{5}:V\rightarrow V$* is onto.*

Proof.

If $m_{i,j}$ are as in Lemma 2.3, $T_{5}$ takes $m_{i,j+1}$ to $m_{i,j}$. ∎∎

Lemma 3.4**.**

The composite map $V\stackrel{{\scriptstyle\mathit{Tr}}}{{\rightarrow}}N1\stackrel{{\scriptstyle U_{5}}}{{\rightarrow}}V$ is $T_{5}$, and so by Lemma 3.3, is onto.

Proof.

We argue as in Lemma 2.4. Now, however, $U$ is $(X+Y)^{6}+XY$, and the conjugates of $F$ over $Z/2(G)$ are $F(x^{25})$ and the $F(rx)$ where $r^{5}=1$. The argument is otherwise unchanged. ∎∎

Theorem 3.5**.**

The composite bijection $M(\mathit{odd})/N2\stackrel{{\scriptstyle\mathit{Tr}}}{{\rightarrow}}N1\stackrel{{\scriptstyle U_{5}}}{{\rightarrow}}V$ commutes with $T_{p}$, $p\neq 5$. Together with the remarks following Theorem 3.2, this shows that $M(\mathit{odd})/N2$, $N2/N1$ and $N1$ identify with $V$, $W$ and $V$ as Hecke-modules.

Proof.

See the proof of Theorem 2.5. ∎∎

Theorem 3.6**.**

The $T_{p}$, $p\neq 5$, act locally nilpotently on $M(\mathit{odd})$. In other words, if $f\in M(\mathit{odd})$ and $p\neq 5$, some power of $T_{p}$ annihilates $f$.

Proof.

[3] and [2] show that $T_{p}$ acts locally nilpotently on $V$ and $W$. And the quotients in the filtration of $M(\mathit{odd})$ are Hecke-isomorphic to $V$, $W$ and $V$. ∎∎

For the rest of this section, unless otherwise noted, $S=\{3,7,11,13\}$ and $\mathcal{O}$ is the 4-variable power series ring over $Z/2$ in the $t_{p}$, $p\in S$. Then $V$, $W$ and $M(\mathit{odd})$ are all $\mathcal{O}$-modules with $t_{p}$ acting by $T_{p}$, $p\in S$. Let $I(V)$, $I(W)$ and $I$ be the kernels of the respective actions.

$I(V)$ is easily described. As in the paragraph before Theorem 2.7 we view $V$ as a $Z/2[[t_{11},t_{13}]]$-module. When $p=3$, $T_{p}:V\rightarrow V$ is multiplication by $\mathrm{unit}(t_{11})$, while when $p=7$, $T_{p}$ is multiplication by $\mathrm{unit}(t_{11})(t_{13})$. So $I(V)$ is generated by two elements $t_{3}+\mathrm{unit}(t_{11})$ and $t_{7}+\mathrm{unit}(t_{11})(t_{13})$, and:

Theorem 3.7**.**

$I(V)$* is generated by 2 elements $A$ and $B$ whose leading forms can be taken to be $t_{3}+t_{7}+t_{11}$ and $t_{7}$.*

To describe $I(W)$ we use the following results from [2]:

Theorem 3.8**.**

Let $D_{1},D_{3},D_{7},D_{9}$ be $D,D^{8}/G,D^{2}G$ and $D^{4}G$. Define $D_{k}$ for $k$ positive and prime to $10$ by $D_{k+10}=G^{2}D_{k}$. Let $W_{a}$ be spanned by $D_{k}$, $k\equiv 1,3,7,9\hskip 5.0pt(20)$, and $W_{b}$ be spanned by $D_{k}$, $k\equiv 11,13,17,19\hskip 5.0pt(20)$. Then $W=W_{a}\oplus W_{b}$ and:

1. (1)

The $T_{p}$, $p\equiv 1,3,7,9\hskip 5.0pt(20)$, stabilize $W_{a}$ and $W_{b}$. The $T_{p}$, $p\equiv 11,13,17,19\hskip 5.0pt(20)$, map $W_{a}$ to $W_{b}$ and $W_{b}$ to $W_{a}$. 2. (2)

$W_{a}$* has a basis $\{m_{i,j}\}$ with $m_{0,0}=D$ adapted to $T_{3}$ and $T_{7}$. The same holds for $W_{b}$ with $m_{0,0}=D_{11}$. It follows that $T_{3}$ and $T_{7}$ act locally nilpotently on $W_{a}$, $W_{b}$ and $W$.* 3. (3)

Taking $S=\{3,7\}$ and making $W$ into a $Z/2[[t_{3},t_{7}]]$-module, we find that each $T_{p}:W\rightarrow W$, $p\equiv 1,3,7,9\hskip 5.0pt(20)$, is multiplication by some $u$ in the maximal ideal of $Z/2[[t_{3},t_{7}]]$. 4. (4)

And each $T_{p}:W\rightarrow W$, $p\equiv 11,13,17,19\hskip 5.0pt(20)$ is the composition of $T_{11}$ with multiplication by some $u$ in $Z/2[[t_{3},t_{7}]]$. 5. (5)

$T_{11}^{2}:W\rightarrow W$* is multiplication by $\lambda^{2}$ for some $\lambda$ in $Z/2[[t_{3},t_{7}]]$ with leading form $t_{3}+t_{7}$.*

Remarks* 

Now each $T_{p}:W\rightarrow W$, $p\neq 5$, is locally nilpotent on $W$. And if $S=\{3,7,11\}$, each $T_{p}$ is multiplication by an element of $Z/2[[t_{3},t_{7},t_{11}]]$. And if we set $\varepsilon=t_{11}+\lambda$, then $\varepsilon^{2}$ kills $W$. Note that the leading form of $\varepsilon$ is $t_{3}+t_{7}+t_{11}$. *

Theorem 3.9**.**

$I(W)$* is generated by 2 elements $\varepsilon^{2}$ and $C$, where $\varepsilon$ is congruent mod $m^{2}$ to the $A$ of Theorem 3.7 and the leading form of $C$ is $t_{13}$.*

Proof.

The proof is essentially the same as that of Theorem 2.9. At the very end we use (4) of Theorem 3.8 to see that $I(W)$ contains an element of the form $t_{13}+vt_{11}$ with $v$ in $\mathcal{O}$. But an easy calculation shows that $T_{13}(D_{11})=T_{13}(DG^{2})=0$ while $T_{11}(D_{11})=T_{11}(x^{11}+\cdots)=x+\cdots\neq 0$. So $v$ is not a unit, and our element has leading form $t_{13}$. ∎∎

Definition 3.10**.**

$V(\star,0)$* is the kernel of $T_{5}:V\rightarrow V$.*

If $\{m_{i,j}\}$ is a basis of $V$ adapted to $T_{3}$ and $T_{5}$, the $m_{i,0}$ form a $Z/2$-basis of $V(\star,0)$. $Z/2[[t_{3}]]$ acts faithfully and locally nilpotently on $V(\star,0)$. $V(\star,0)$ is an $\mathcal{O}$-submodule of $V$, and the kernel of the action is a height 3 prime ideal, $P$, generated by $t_{7}$, $t_{13}$ and an element with leading form $t_{11}+t_{3}$.

Lemma 3.11**.**

$W$, as well as $V$, contains a “Hecke-submodule” isomorphic to $V(\star,0)$.

Proof.

If $f\in V(\star,0)$, $T_{5}(f)=U_{5}(\mathit{Tr}(f))=0$. As in the proof of Lemma 2.11 we conclude that $f$ is in $N2$ and we get a composite map $V(\star,0)\subset N2\stackrel{{\scriptstyle\mathit{pr}}}{{\rightarrow}}W$ taking $m_{0,0}$ to $D$, which is the desired imbedding. ∎∎

Theorem 3.12**.**

Let $A,B,C$ be as in Theorems 3.7 and 3.9. Then $(A,B,C)=I(V)+I(W)=P$, the kernel of the action of $\mathcal{O}$ on $V(\star,0)$.

The proof mimics that of Theorem 2.12.

Theorem 3.13**.**

The only $\mathcal{O}$-linear map $\alpha:W\rightarrow V$ is the zero-map.

The proof is like that of Theorem 2.13, but this time we use the fact that $W_{a}$ and $W_{b}$ have bases adapted to $T_{3}$ and $T_{7}$ to see that $T_{7}(W)=W$.

Theorem 3.14**.**

If $p\neq 5$, $T_{p}:M(\mathit{odd})\rightarrow M(\mathit{odd})$ is multiplication by some $u$ in $\mathcal{O}$.

We argue as in the proof of Theorem 2.14, using $V(\star,0)$ in place of $V(0,\star)$.

Theorem 3.15**.**

There are $A,B,C$ in $\mathcal{O}$ with leading forms $t_{3}+t_{7}+t_{11}$, $t_{7}$ and $t_{13}$ such that $I(V)=(A,B)$ and $I(W)=(A^{2},C)$. The kernel, $I$, of the action of $\mathcal{O}$ on $M(\mathit{odd})$ is $I(V)\cap I(W)=(A^{2},AC,BC)$.

The proof mimics that of Theorem 2.15.