convert_syndrome-algebra

At level

N=163

, the cuspidal modular-symbols space

V = S_2(\Gamma_0(163))

has dimension

13

over

\mathbf Q

. The integral modular-symbols lattice

V_{\mathbf Z}\subset V

is preserved by all Hecke operators. Reducing modulo

3

gives a

13

-dimensional

\mathbf F_3

-vector space

V_{\mathbf F_3}=V_{\mathbf Z}/3V_{\mathbf Z}

with commuting Hecke operators.

Earlier computations in this project established two separate mod-

3

phenomena:

The second construction depends on a choice of primitive root modulo

163

and therefore does not give a canonical bridge between the Hecke module and the Hamming code. The purpose of this note is to record the strongest canonical bridge presently known: the action of the Heegner operators on the canonical

3

-dimensional summand.

The canonical Hecke-stable summand

Let $V_{\mathbf F_3}$ be the mod- $3$ reduction of the cuspidal modular-symbols space at level $163$ . Then $$\dim \kerop(T_2^2)=3,\qquad \dim \operatorname{im}(T_2^2)=10,$$ and $$V_{\mathbf F_3}=\kerop(T_2^2)\oplus \operatorname{im}(T_2^2).$$ Since every Hecke operator commutes with $T_2$ , both summands are Hecke-stable.

Proof. Modulo $3$ , the characteristic polynomial of $T_2$ factors as $\chi_{T_2}(x)=x^3(x^4+2x^3+2)(x^6+x^4+x^3+x+1).$ The three factors are pairwise coprime in $\mathbf F_3[x]$ , so primary decomposition gives $V_{\mathbf F_3}=V_{x^2}\oplus V_{x^4+2x^3+2}\oplus V_{x^6+x^4+x^3+x+1},$ with dimensions $3$ , $4$ , and $6$ respectively. Since $\kerop(T_2^2)=V_{x^2}$ has dimension $3$ and $\operatorname{im}(T_2^2)$ is the complementary summand of dimension $10$ , the result follows. ◻

Set $$Q:=\kerop(T_2^2)\subset V_{\mathbf F_3}.$$ Then $Q$ is a $3$ -dimensional Hecke-stable direct summand of $V_{\mathbf F_3}$ .

The summand $Q$ is canonical in the following sense: it is the $x^2$ -primary component of $T_2$ acting on $V_{\mathbf F_3}$ . The choice of $T_2$ as the splitting operator is natural since $2$ is the smallest prime not dividing the level. However, the same $3$ -dimensional summand arises as a Hecke-stable piece for any operator whose mod- $3$ reduction separates the three primary components.

The Heegner syndrome algebra

Let

\mathcal H = \{3,7,11,19,43,67,163\}

be the Heegner operator set at level

163

The restricted Heegner operators $T_d|_Q \in \operatorname{End}_{\mathbf F_3}(Q), \qquad d\in\mathcal H,$ span a $3$ -dimensional commutative algebra $A_Q\subset \operatorname{End}(Q)$ .

More precisely, with $I = \mathrm{Id}_Q,\qquad E = T_3|_Q,\qquad N = T_{11}|_Q,$ one has $E^2 = E,\qquad N^2=0,\qquad EN = NE = 0,$ and the Heegner operators satisfy $\begin{aligned} T_3|_Q &= E,\\ T_{19}|_Q &= -E,\\ T_{11}|_Q &= N,\\ T_7|_Q &= -I + N,\\ T_{43}|_Q &= T_{67}|_Q = I + E - N,\\ T_{163}|_Q &= -I - E. \end{aligned}$ In particular, $A_Q = \operatorname{Span}_{\mathbf F_3}\{I,E,N\},$ and $A_Q$ is generated by one idempotent and one orthogonal nilpotent.

Proof. The matrices were computed explicitly on $Q$ in an adapted modular-symbols basis. Direct multiplication gives the stated relations. Since all seven Heegner operators are linear combinations of $I,E,N$ , the span has dimension at most $3$ . Since $I,E,N$ are linearly independent, the dimension is exactly $3$ . ◻

Proof. Put $F=I-E$ . Then $E^2=E,\qquad F^2=F,\qquad EF=FE=0,\qquad FN=NF=N.$ Thus $E$ spans a copy of $\mathbf F_3$ , while $(F,N)$ spans the dual-number algebra $\mathbf F_3[\varepsilon]/(\varepsilon^2)$ with $\varepsilon$ corresponding to $N$ . ◻

The canonical collision $T_{43}|_Q = T_{67}|_Q$ is exact. This is the strongest base-free identification between Heegner labels currently visible on the mod- $3$ side.

The quotient trace form

Define the quotient trace form on $\mathcal H$ by $G_Q(d,d') = \operatorname{Tr}(T_d T_{d'}|_Q).$ Then $G_Q$ has rank $2$ .

Proof. In the commutative algebra $A_Q$ , multiplication by the nilpotent element $N$ is a nilpotent endomorphism: its only eigenvalue is $0$ , so $\operatorname{Tr}(NX)=0$ for all $X\in A_Q$ . Hence the $N$ -line lies in the radical of the trace form, and the trace form on $A_Q$ has rank at most $2$ . The explicit values $\operatorname{Tr}(I^2|_Q)=3\equiv 0$ , $\operatorname{Tr}(E^2|_Q)=1$ , $\operatorname{Tr}(IE|_Q)=1$ show the rank is exactly $2$ over $\mathbf F_3$ . ◻

Thus the quotient retains a $3$ -dimensional Hecke algebra but only a $2$ -dimensional trace geometry. The nilpotent direction survives algebraically but is invisible to trace.

The projective unit group

The unit group of $A_Q$ has $12$ elements, and its projectivization modulo scalar units is cyclic of order $6$ : $A_Q^\times/\mathbf F_3^\times \cong C_6.$ The invertible Heegner operators $T_7|_Q$ , $T_{43}|_Q$ , $T_{67}|_Q$ , and $T_{163}|_Q$ generate the full projective unit group.

Proof. From Corollary [cor:structure], $A_Q^\times \cong \mathbf F_3^\times \times \bigl(\mathbf F_3[\varepsilon]/(\varepsilon^2)\bigr)^\times.$ Now $\mathbf F_3^\times$ has order $2$ , while $\bigl(\mathbf F_3[\varepsilon]/(\varepsilon^2)\bigr)^\times \cong \mathbf F_3^\times \times (1+\varepsilon\mathbf F_3) \cong C_2\times C_3 \cong C_6.$ Hence $|A_Q^\times| = 2\cdot 6 = 12$ , and modding out by scalar units gives a group of order $6$ , necessarily isomorphic to $C_6$ .

The explicit projective permutations induced by the invertible Heegner operators have orders $|T_7|=3,\qquad |T_{43}|=|T_{67}|=6,\qquad |T_{163}|=2,$ and the subgroup they generate has order $6$ . Therefore it is the full projective unit group. ◻

The projective action of $A_Q^\times/\mathbf F_3^\times$ on the $13$ points of $\mathbf P(Q)$ has orbit sizes $1+1+2+3+6.$

Proof. Direct computation of the projective unit action on $\mathbf P(Q)$ . ◻

This explains the previously observed order- $6$ projective Hecke group exactly: it is the full projectivized unit group of the syndrome algebra, not an accidental subgroup of $\operatorname{PGL}(3,3)$ .

Family context

Among the class-number-one prime levels $19,\ 43,\ 67,\ 163,$ the dimensions of the canonical mod- $3$ quotient $Q_N=\kerop(T_2^2)$ are $1,\ 0,\ 0,\ 3$ respectively. Thus $163$ is the first class-number-one level with a genuinely $3$ -dimensional syndrome quotient.

Proof. Direct computation in SageMath. ◻

The phenomenon is not globally unique: other prime levels up to $199$ also have nontrivial $Q_N$ . What is special within the class-number-one family is the first appearance of the full $3$ -dimensional syndrome space.

Discussion

The canonical object on the mod-

3

side is not a seven-point configuration in

\mathbf P^2(\mathbf F_3)

. The seven Heegner primes do not canonically embed there. What is canonical is stronger in a different direction:

So the Hamming–Hecke bridge that survives canonically is not a point-set bridge but an algebra-and-group bridge.

Acknowledgements

T. Miyake, Modular Forms, Springer, 1989.

The Sage Developers, SageMath, the Sage Mathematics Software System, https://www.sagemath.org.

Introduction