Triviality of the Hecke action on ordinary Drinfeld cuspforms of level $Γ_1(t^n)$

Dimension variation of Gouvêa-Mazur type for Drinfeld cuspforms of level $Γ_1(t)$

Let $p$ be a rational prime and $q>1$ a $p$-power. Let $S_k(Γ_1(t))$ be the space of Drinfeld cuspforms of level $Γ_1(t)$ and weight $k$ for $\mathbb{F}_q[t]$. For any non-negative rational number $α$, we denote by $d(k,α)$ the dimension of the slope $α$ generalized eigenspace for the $U$-operator acting on $S_k(Γ_1(t))$. In this paper, we prove a function field analogue of the Gouvêa-Mazur conjecture for this setting. Namely, we show that for any $α\leq m$ and $k_1,k_2>α+1$, if $k_1\equiv k_2 \bmod p^m$, then $d(k_1,α)=d(k_2,α)$.

关于有限斜率 Drinfeld 尖形式的 Hecke 特征系统的一些恒定性

On some constancy of Hecke eigensystems for Drinfeld cuspforms of finite slope

本文证明在 Drinfeld 尖形式空间中，有限 $\wp$-斜率的 Hecke 特征系统的重数在水平 $\Gamma_1(\mathfrak{n}\wp^r)$ 中是水平 $\Gamma_1(\mathfrak{n}\wp)$ 中的 $q^{(r-1)\deg(\wp)}$ 倍，且特征系统出现与否等价。

On Drinfeld cusp forms of prime level

Let $(P_d)$ be any prime of $\mathbb{F}_q[t]$ of degree $d$ and consider the space of Drinfeld cusp forms of level $P_d$, i.e. for the modular group $Γ_0(P_d)$. We provide a definition for oldforms and newforms of level $P_d$. Moreover, when the dimension of the vector space of oldforms is one and $P_1=t$ we prove that the space of cuspforms of level $t$ is the direct sum of oldforms and newforms and that the Hecke operator $\mathbf{T}_t$ acting on Drinfeld cusp forms of level 1 is injective, thus providing more evidence for the conjectures presented and stated in [2] and [3].

Twisting eigensystems of Drinfeld Hecke eigenforms by characters

We address some questions posed by Goss related to the modularity of Drinfeld modules of rank 1 defined over the field of rational functions in one variable with coefficients in a finite field. For each positive characteristic valued Dirichlet character, we introduce certain projection operators on spaces of Drinfeld modular forms with character of a given weight and type such that when applied to a Hecke eigenform return a Hecke eigenform whose eigensystem has been twisted by the given Dirichlet character. Unlike the classical case, however, the effect on Goss' $u$-expansions for these eigenf…

$\wp$-adic continuous families of Drinfeld eigenforms of finite slope

Let $p$ be a rational prime, $v_p$ the normalized $p$-adic valuation on $\mathbb{Z}$, $q>1$ a $p$-power and $A=\mathbb{F}_q[t]$. Let $\wp\in A$ be an irreducible polynomial and $\mathfrak{n}\in A$ a non-zero element which is prime to $\wp$. Let $k\geq 2$ and $r\geq 1$ be integers. We denote by $S_k(Γ_1(\mathfrak{n}\wp^r))$ the space of Drinfeld cuspforms of level $Γ_1(\mathfrak{n}\wp^r)$ and weight $k$ for $A$. Let $n\geq 1$ be an integer and $a\geq 0$ a rational number. Suppose that $\mathfrak{n}\wp$ has a prime factor of degree one and the generalized eigenspace in $S_k(Γ_1(\mathfrak{n}\wp^r…