We present a new algorithm for computing the characteristic polynomial of an arbitrary endomorphism of a finite Drinfeld module using its associated crystalline cohomology. Our approach takes inspiration from Kedlaya’s p-adic algorithm for computing the characteristic polynomial of the Frobenius endomorphism on a hyperelliptic curve using Monsky-Washnitzer cohomology. The method is specialized using a baby-step giant-step algorithm for the particular case of the Frobenius endomorphism, and in this case we include a complexity analysis that demonstrates asymptotic gains over previously existing approaches.

Motivated by finding analogues of elliptic curve point counting techniques, we introduce one deterministic and two new Monte Carlo randomized algorithms to compute the characteristic polynomial of a finite rank-two Drinfeld module. We compare their asymptotic complexity to that of previous algorithms given by Gekeler, Narayanan and Garai-Papikian and discuss their practical behavior. In particular, we find that all three approaches represent either an improvement in complexity or an expansion of the parameter space over which the algorithm may be applied. Some experimental results are also presented.