Algorithms for computing norms and characteristic polynomials on general Drinfeld modules

TL;DR

This paper introduces two algorithms for computing characteristic polynomials and norms of isogenies in Drinfeld modules, applicable to any rank and base curve, with complexity analysis showing high efficiency in many cases.

Contribution

The paper presents novel algorithms for Drinfeld modules that generalize previous methods and improve computational efficiency, especially for the Frobenius endomorphism.

Findings

Algorithms are effective for any rank and base curve.

Complexity analysis shows high asymptotic performance.

Specialized algorithm for Frobenius endomorphism based on reduced norm.

Abstract

We provide two families of algorithms to compute characteristic polynomials of endomorphisms and norms of isogenies of Drinfeld modules. Our algorithms work for Drinfeld modules of any rank, defined over any base curve. When the base curve is $\mathbb P^1_{\mathbb F_q}$, we do a thorough study of the complexity, demonstrating that our algorithms are, in many cases, the most asymptotically performant. The first family of algorithms relies on the correspondence between Drinfeld modules and Anderson motives, reducing the computation to linear algebra over a polynomial ring. The second family, available only for the Frobenius endomorphism, is based on a formula expressing the characteristic polynomial of the Frobenius as a reduced norm in a central simple algebra.