CM Drinfeld Modules, Self-isogenous Modular Polynomials, and Volcano Structure
Chien-Hua Chen
TL;DR
This work develops a theory for CM Drinfeld modules of arbitrary rank $r$ by introducing and algorithmically constructing single-variable self-isogenous modular polynomials $\Phi_{J,\mathfrak{a}}(X,X)$, including the case $\mathfrak{a}=(T)$ and a degree bound when $\mathfrak{a}=(T^2+T+1)$. It integrates CM uniformization and ideal-isogeny constructions to relate endomorphism rings, Picard-group actions, and Hilbert class polynomials through $\Phi_{J,a}(X,X)$, revealing a deep connection to Hilbert-class polynomials via multiplicities $\gamma(\mathcal{O},a)$. The paper also introduces a generalized $\mathfrak{l}$-cyclic volcano structure in the CM rank-$r$ Drinfeld isogeny graph, showing that a crater-and-levels architecture governs ascending edges and endomorphism types, mirroring the elliptic-case volcano. These results provide both algorithmic tools for computation of modular polynomials in higher rank and structural insights into isogeny graphs, with potential implications for CM theory and computational aspects in function-field arithmetic.
Abstract
In this paper, we develop a view of self-isogenous modular polynomials and the $\mathfrak{l}$-cyclic isogeny graph for CM Drinfeld modules of arbitrary rank $r$. On the computational side, we give an explicit procedure to construct the modular polynomial $Φ_{J,\mathfrak{a}}(X,X)$ for Drinfeld modules of rank $r\geqslant 3$ with $\mathfrak{a}$ a prime ideal of $\mathbb{F}_q[T]$. When $\mathfrak{a}=(T)$, we provide an algorithm to compute $Φ_{J,\mathfrak{a}}(X,X)$; when $\mathfrak{a}=(T^2+T+1)$, we give an explicit degree bound on $Φ_{J,\mathfrak{a}}(X,X)$. On the structural side, we formulate a generalized $\mathfrak{l}$-cyclic volcano structure and prove that the generalized volcano appears in a component of the full $\mathfrak{l}$-cyclic isogeny graph for rank-$r$ Drinfeld modules with complex multiplication.