Drinfeld modular polynomials in higher rank II: Kronecker congruences

TL;DR

This paper extends the algebraic construction of higher rank Drinfeld modular polynomials and generalizes Kronecker congruences, clarifying their behavior under reduction modulo primes.

Contribution

It provides a more general algebraic framework for Drinfeld modular polynomials and proves a broader version of Kronecker congruences, correcting previous errors.

Findings

Generalized algebraic construction of Drinfeld modular polynomials

Proved a generalized Kronecker congruences relation

Corrected an earlier error in the literature

Abstract

This is a sequel to the paper [F. Breuer, H.-G. R\"uck, Drinfeld modular polynomials in higher rank, J. Number Theory 129 (2009), 59-83.], in which we introduced Drinfeld modular polynomials of higher rank, using an analytic construction. These polynomials relate the isomorphism invariants of Drinfeld F_q[T]-modules of rank r\geq 2 linked by isogenies of a specified type. In the current paper, we give an algebraic construction of greater generality, and prove a generalization of the Kronecker congruences relations, which describe what happens when modular polynomials associated to P-isogenies are reduced modulo a prime P. We also correct an error in [loc. cit.].