CM Drinfeld Modules, Self-isogenous Modular Polynomials, and Volcano Structure

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 $\Phi_{J,\mathfrak{a}}(X,X)$ 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 $\Phi_{J,\mathfrak{a}}(X,X)$; when $\mathfrak{a}=(T^2+T+1)$, we give an explicit degree bound on $\Phi_{J,\mathfrak{a}}(X,X)$. 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.