Indices of nilpotency in certain spaces of modular forms

We study the index of nilpotency relative to certain Hecke operators in spaces of modular forms with integer weight and level $N$ with integer coefficients modulo primes $p$ for $(p, N) \in {(3, 1), (5, 1), (7, 1), (3, 4)}$. In these settings, we prove upper bounds on certain indices of nilpotency. As an application of our bounds, we prove infinite families of congruences for $p^t$-core partition functions modulo $p$ for $p\in {3, 5, 7}$ and $t\geq 1$, and we prove an infinite family of congruences modulo $3$ for the $r$th power partition function, $p_r(n)$, when $r = 12k$ with $\gcd(k,6) = 1$. We also include conjectures on a function which quantifies degree lowering on powers of the Delta function by the relevant Hecke operators in these settings, and on the index of nilpotency relative to a modification of this degree-lowering function.

💡 Research Summary

The paper investigates the nilpotency indices of certain Hecke operators acting on spaces of modular forms with integer weight and level N, reduced modulo a prime p. The authors focus on the four pairs (p,N) = (3,1), (5,1), (7,1), (3,4) and study the modified Hecke operators (T_{1,\ell}) defined by
(T_{1,\ell}=T_\ell) if (\ell\equiv1\pmod p) and (T_{1,\ell}=T_\ell^2) if (\ell\equiv-1\pmod p).
For a non‑zero reduced form (\tilde f) in the space (\widetilde M_p(\Gamma_0(N))) they define the nilpotency index
(N_{\ell}^{p}(\tilde f)=\min{t\ge1:\tilde f|T_{1,\ell}^t=0}).
The main contributions are:

1. Upper bounds for nilpotency indices.

   • For (p=3) and any prime (\ell) with (\ell\equiv\pm1\pmod3) they prove
     (N_{\ell}^{3}(\Delta^k)\le 1+\lfloor k/3\rfloor) when (\ell\equiv1) and
     (N_{\ell}^{3}(\Delta^k)\le 1+\lfloor 2k/3\rfloor) when (\ell\equiv-1).
   • For (p=5) they obtain
     (N_{\ell}^{5}(\Delta^k)\le 1+\lfloor 2k/5\rfloor) for (\ell\equiv1) and
     (N_{\ell}^{5}(\Delta^k)\le 1+\lfloor 3k/5\rfloor) for (\ell\equiv-1).
   • For (p=7) they obtain analogous linear bounds involving (\lfloor 3k/7\rfloor) and (\lfloor 2k/7\rfloor).
   • For the weight‑12 cusp form (D_2(z)=\eta(2z)^{12}) (which equals (\Delta(z)) modulo 3) they prove the same type of bounds for all primes (\ell=2,3).

2. Filtered sequences and recurrence relations.
   The authors connect nilpotency to the notion of an “i‑filtered” sequence: if the degrees of (\Delta^k|T_{1,\ell}^t) satisfy (\deg_{\Delta}(\Delta^k|T_{1,\ell}^t)=k-t) for (t\le i) and then drop faster, the sequence is i‑filtered and one expects (N_{\ell}^{p}(\Delta^k)=O(k^{1/i})). Their theorems show that for the considered primes the sequences are at least 1‑filtered, and for many (\ell) they are 2‑filtered, giving the linear bounds above. Computational evidence suggests that for (p=5) and (\ell=11) the sequence is 4‑filtered, which would improve the bound to (1+\lfloor k/4\rfloor).

3. Applications to partition congruences.

   • p‑core partitions. Let (a_t(n)) denote the number of (t)-core partitions of (n). For (p=3,5,7) they define a shift (k_{p,t}) (e.g. (k_{3,t}=\lfloor 2k/3\rfloor) when (p=3)) and prove infinite families of congruences:
     (a_t(p^{r}n-k_{p,t})\equiv a_t(p^{r+2}n-k_{p,t})\pmod p) for all (r) large enough and all (n) coprime to the chosen prime (\ell).
     When (\ell\equiv-1\pmod p) they obtain a similar congruence with a factor of (p^2) in the exponent.
   • 12r‑colored partitions. For (r) coprime to 6 they consider the partition function (p_{12r}(n)) (the coefficient of (\eta(z)^{-12r})). They prove congruences modulo 3 of the shape
     (p_{12r}(\ell^{u}n - r^2/2)\equiv p_{12r}(\ell^{u}n - r^2/2)\pmod3)
     for all odd primes (\ell\equiv\pm1\pmod3) and appropriate exponents (u).

4. Degree‑lowering function and conjectures.
   For each prime (\ell) they define
   (D_{\ell}^{p}(k)=\deg_{\Delta}(\Delta^k|T_{1,\ell}))
   and study its iterates (D_{\ell}^{p,(t)}(k)). They also define a nilpotency index for this degree‑lowering process:
   (S_{\ell}^{p}(k)=\min{t\ge1: D_{\ell}^{p,(t)}(k)=0}).
   Extensive computer experiments lead to detailed conjectural formulas for (D_{19}^{5}(k)) (when (p=5,\ell=19)) expressed in terms of the base‑5 expansion of (k). They conjecture that (S_{19}^{5}(k)=O(k^{\alpha})) with (\alpha\approx0.7892) (the exponent comes from the dominant root of the characteristic polynomial (x^2-3x-2)). Similar conjectures are made for (p=7,\ell=29) with (\alpha\approx0.864). Moreover, they observe experimentally that the nilpotency index of the original Hecke operator satisfies (N_{\ell}^{p}(\Delta^k)\le S_{\ell}^{p}(k)).

5. Methodology and computations.
   The proofs rely on classical modular form theory (Eisenstein series, the Dedekind eta function, and the discriminant (\Delta)), the structure of Hecke algebras modulo p, and the theory of “mod‑p” modular forms developed by Serre, Nicolas, and others. The authors use the fact that (\widetilde M_{0,p}(\Gamma_0(p))) is generated by powers of (\Delta) (or by (\Delta) and (\eta(2z)^{12}) for (p=3)). They then analyze how (T_{1,\ell}) acts on these generators, obtaining explicit degree‑lowering formulas. All numerical data were produced with PARI/GP and SageMath for (k) up to about (10^5); tables of constants (c) and (\alpha) for various ((p,\ell)) are provided.

6. Conclusion and outlook.
   The paper extends the theory of nilpotent Hecke operators from the classical case (p=2) to the primes 3, 5, 7, providing linear upper bounds for nilpotency indices and deriving new infinite families of partition congruences. The conjectures on the degree‑lowering function suggest a deeper combinatorial structure linking the base‑(p) expansion of exponents to the behavior of Hecke operators modulo p. Future work may include proving these conjectures, extending the analysis to higher levels (N>1), and exploring connections with Galois representations and (p)-adic modular forms.