Setup and Classification Rule

Let $N$ be a prime and let $$V_N = S_2(\Gamma_0(N))\otimes \mathbf Q,\qquad M_N= V_N\otimes_{\mathbf Q}\mathbf F_3.$$ Let $T_2$ be the Hecke operator $T_2$ and let $$Q_N:=\ker\left(T_2^2\colon M_N\to M_N\right),\qquad q_N=\dim_{\mathbf F_3}Q_N,\qquad f_N=\text{minimal polynomial of }T_2|_{Q_N}\text{ over }\mathbf F_3.$$ For $q_N>0$ we define the syndrome algebra $$\mathcal A_N:=\mathrm{span}_{\mathbf F_3}\{T_\ell|_{Q_N}\mid \ell\in \{3,5,7,11,13,17,19,23\},\ \ell\neq N\} \subset \mathrm{End}_{\mathbf F_3}(Q_N),$$ as in the computation script below.

We assign one of four categories to level $N$ as follows:

• smooth iff $q_N=0$;

• split node iff $q_N>0$ and $\mathcal A_N$ contains a nontrivial idempotent that lifts to $\mathbf Z_3$;

• cusp iff $q_N>0$, $f_N=x$, and no split idempotent lift exists;

• non-split node iff $q_N>0$, $f_N=x^2$, and no split idempotent lift exists.

The split/non-split distinction is implemented by the same Hensel lifting test used in src/hecke/hecke_prime_node_census.py.

Computation

The classification is computed by

• computing a basis of $M_N$ from ModularSymbols(N,2).cuspidal_subspace(),

• forming $Q_N$,

• computing $\mathcal A_N$ on $Q_N$ from a small prime Hecke set,

• searching for a nontrivial mod-$3$ idempotent and lifting it to $\mathbf Z_3$.

[thm:census500] The scan returns exactly $$\#\text{smooth}=56,\quad \#\text{cusp}=20,\quad \#\text{split node}=8,\quad \#\text{non-split node}=7.$$ Smooth levels are the complement of the 44 non-smooth levels.

[thm:census1000] The scan over the 164 prime levels in this range gives $$\#\text{smooth}=95,\quad \#\text{cusp}=39,\quad \#\text{split node}=17,\quad \#\text{non-split node}=13.$$

Observed Split Nodes

The split-node primes in the range $N\le 1000$ are $$127,163,199,307,337,449,487,499,541,577,631,773,811,829,857,883,919.$$ The non-split-node primes in this range are $$71,269,271,359,421,439,443,509,523,571,601,947,997.$$ The cusp levels are $$19,37,73,101,109,113,131,149,151,167,181,223,293,379,383,397,431,433,463,467,503,521,557,587,593,599,613,619,659,727,739,751,757,761,827,937,941,953,991.$$

qdim-2+ Block

Across $11\le N\le 1000$, exactly $35$ levels have $q_N\ge 2$. The values of $\dim_{\mathbf F_3}Q_N$ occurring in each category are: $$\text{cusp: }1,2;\quad \text{split node: }2,3,4,5,6,7;\quad \text{non-split node: }2,4;\quad \text{smooth: }0.$$

Asymptotics

The split-node counts up to these points are $$N\le 500:\ 8,\qquad N\le 1000:\ 17.$$ At these values this is close to linear growth in the number of primes scanned: $$\frac{8}{91}\approx 0.088,\qquad \frac{17}{164}\approx 0.104.$$ This motivates the conjecture that split-node primes have positive density among prime levels (with respect to prime counting in this family), and that non-split nodes also persist with positive lower density.

The quantities $$\frac{\#\{\text{split node }N\le X\}}{\#\{p\le X:\ p\ \text{prime},\ p\ne 3\}} \quad\text{and}\quad \frac{\#\{\text{non-split node }N\le X\}}{\#\{p\le X:\ p\ \text{prime},\ p\ne 3\}}$$ have nonzero limits as $X\to\infty$.

Reproducibility

All classification data are generated by src/hecke/hecke_prime_node_census.py, with outputs src/hecke/hecke_prime_node_census.json, src/hecke/hecke_prime_node_census.md, src/hecke/hecke_prime_node_census_upto_1000.json, and src/hecke/hecke_prime_node_census_upto_1000.md. Exact matrices and syndrome-algebra consistency checks are in src/hecke/hecke_syndrome_lift.py, src/heegner/heegner_syndrome_algebra.py.

9 The Sage Developers, SageMath, https://www.sagemath.org.