On some constancy of Hecke eigensystems for Drinfeld cuspforms of level $\Gamma_1(\mathfrak{n}\wp^r)$

Technical Summary: Constancy of Hecke Eigensystems for Drinfeld Cuspforms

Problem Statement
The paper investigates the behavior of Hecke eigensystems within the space of Drinfeld cuspforms, specifically focusing on how these systems vary as the level of the modular forms increases. In the classical theory of elliptic modular forms, Hecke eigensystems of finite $p$-slope are known to vary in $p$-adic analytic families as the weight and the $p$-adic valuation of the level change. The author contrasts this with the Drinfeld setting, where previous work (specifically [Hat2]) demonstrated an unexpected "triviality" of the Hecke action on the $t$-ordinary part of cuspforms of level $\Gamma_1(t^r)$. The central problem addressed here is to generalize this phenomenon to the finite $\wp$-slope part for more general levels of the form $\Gamma_1(n\wp^r)$, where $n$ is a nonzero element in the polynomial ring $A = \mathbb{F}_q[t]$, $\wp$ is a monic irreducible polynomial, and $r \geq 1$. The specific question is whether Hecke eigensystems of finite $\wp$-slope appearing in higher levels $\Gamma_1(n\wp^r)$ are merely repetitions of those found in the base level $\Gamma_1(n\wp)$.

Methodology
The author employs a combinatorial approach to Drinfeld modular forms, utilizing the description of cuspforms via harmonic cocycles on the Bruhat–Tits tree $T$ for $SL_2(K_\infty)$, following the work of Teitelbaum [Tei] and Böckle [Böc].

1. Combinatorial Framework: The space of Drinfeld cuspforms $S_k(\Gamma)$ is identified with the space of harmonic cocycles $\text{Char}_k(\Gamma)$, which in turn is isomorphic to a tensor product involving the Steinberg module and a space of homogeneous polynomials. This allows the problem to be translated into the language of group modules over the congruence subgroup $\Gamma$.
2. Diamond Operators and Group Rings: The paper analyzes the action of the group $\Theta_r = 1 + \wp(A/\wp^r A)$, a commutative $p$-group, on the space of cuspforms via diamond operators. The space $S_k(\Gamma_1(n\wp^r))$ is viewed as a module over the group ring $C_\infty[\Theta_r]$.
3. Freeness of Modules: A critical technical step involves proving that $S_k(\Gamma_1(n\wp^r))$ is a free module of finite rank over the local ring $C_\infty[\Theta_r]$. This relies on the fact that $C_\infty$ has characteristic $p$, making $C_\infty[\Theta_r]$ a local ring where projective modules are free. The proof utilizes the exact sequence relating the Steinberg module to the chain complex of the tree and the freeness of induced modules.
4. Isomorphism of Finite Slope Parts: The author establishes a Hecke-equivariant isomorphism between the finite $\wp$-slope parts of the cuspform spaces at levels $\Gamma_1(n\wp)$ and $\Gamma_1^\wp(n, \wp^r)$ (a specific intermediate level). This is achieved by constructing a map between the spaces and showing that the cokernel vanishes for non-zero eigenvalues of the $U_\wp$ operator.

Key Contributions and Results
The primary result is Theorem 1.1, which establishes a "constancy" of Hecke eigensystems for Drinfeld cuspforms as the level increases. Specifically:

• Multiplicity Formula: For any non-zero eigenvalue $\alpha$ of the operator $U_\wp$ and any slope $a$, the multiplicity of $\alpha$ (or the dimension of the slope $a$ subspace) in $S_k(\Gamma_1(n\wp^r))$ is exactly $q^{(r-1)\deg(\wp)}$ times the multiplicity in $S_k(\Gamma_1(n\wp))$.
  $$m(\Gamma_1(n\wp^r), k, \wp, \alpha) = q^{(r-1)\deg(\wp)} m(\Gamma_1(n\wp), k, \wp, \alpha)$$
• Constancy of Eigensystems: A Hecke eigensystem $\lambda = (\lambda_Q)_{Q \in \Pi}$ with $\lambda_\wp \neq 0$ appears in the space $S_k(\Gamma_1(n\wp^r))$ if and only if it appears in $S_k(\Gamma_1(n\wp))$.

This result implies that the set of non-zero $U_\wp$-eigenvalues and the set of finite slopes do not change as the exponent $r$ of the level increases; they merely scale in multiplicity.

Significance and Claims
The paper claims that this result highlights a "stark contrast" to the theory of elliptic modular forms. In the elliptic case, varying the level typically yields $p$-adic analytic families of eigenforms. In the Drinfeld case, as demonstrated by this theorem, varying the level (specifically the power of $\wp$) yields only "constant families" of Hecke eigensystems. While the author acknowledges that $\wp$-adic continuous families of Drinfeld eigenforms do exist when varying the weight (citing [Hat1]), the variation of the level in this context does not generate new eigensystems but rather replicates existing ones with increased multiplicity.

The work generalizes the author's previous findings on the $t$-ordinary part (where the action was trivial) to the broader context of finite slope, suggesting a fundamental rigidity in the arithmetic of Drinfeld modular forms regarding level variation. The proof relies heavily on the structural properties of the group ring $C_\infty[\Theta_r]$ and the combinatorial description of cuspforms, avoiding the need for $p$-adic analytic families in the level variable.