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

2025/11/26 by Chen, Chien-Hua
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2511.21329

Abstract

In this paper, we develop a view of self-isogenous modular polynomials and the \mathfrakl-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,\mathfraka(X,X) for Drinfeld modules of rank r\geqslant 3 with \mathfraka a prime ideal of \mathbbF_q[T]. When \mathfraka=(T), we provide an algorithm to compute Φ_J,\mathfraka(X,X); when \mathfraka=(T²+T+1), we give an explicit degree bound on Φ_J,\mathfraka(X,X). On the structural side, we formulate a generalized \mathfrakl-cyclic volcano structure and prove that the generalized volcano appears in a component of the full \mathfrakl-cyclic isogeny graph for rank-r Drinfeld modules with complex multiplication.

Citations

• Isogeny graphs of abelian varieties and singular ideals in orders
• Isogeny graphs of ordinary abelian varieties
• Drinfeld modular polynomials in higher rank II: Kronecker congruences

Related

• Drinfeld Modular Polynomials in Higher Rank
• Drinfeld modular polynomials of level T
• Drinfeld modular polynomials in higher rank II: Kronecker congruences
• Explicit description of isogeny and isomorphism classes of Drinfeld\n modules over finite field
• Heights and isogenies of Drinfeld modules