Cette rubrique regroupe mes travaux sur les modules de Drinfeld et leurs généralisations (e.g. motifs d'Anderson). J'étudie en particulier les aspects calculatoires de ces objets.

	X. Caruso, Q. Gazda, A. Lucas Wieferich primes for Drinfeld modules prépublication, 32 pages The aim of this paper is to discuss the notion of Wieferich primes in the context of Drinfeld modules. Our main result is a surprising connection between the proprety of a monic irreducible polynomial $\mathfrak p$ to be Wieferich and the $\mathfrak p$-adic valuation of special $L$-values of Drinfeld modules. This generalizes a theorem of Thakur for the Carlitz module. We also study statistical distributions of Wieferich primes, proving in particular that a place of degree $d$ is Wieferich with the expected probability $q^{-d}$ when we average over large enough sets of Drinfeld modules.
	X. Caruso, A. Leudière Algorithms for computing norms and characteristic polynomials on general Drinfeld modules à paraître à Math. Comp., 44 pages 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 new formula expressing the characteristic polynomial of the Frobenius as a reduced norm in a central simple algebra.
	X. Caruso, Q. Gazda Computation of classical and $v$-adic $L$-series of $t$-motives à paraître à Research in Number Theory, 21 pages We design an algorithm for computing the $L$-series associated to an Anderson $t$-motives, exhibiting quasilinear complexity with respect to the target precision. Based on experiments, we conjecture that the order of vanishing at $T=1$ of the $v$-adic $L$-series of a given Anderson $t$-motive with good reduction does not depend on the finite place $v$.
	D. Ayotte, X. Caruso, A. Leudière, J. Musleh Drinfeld modules in SageMath ACM Communications in Computer Algebra 57 (2023), 65–71 We present the first implementation of Drinfeld modules fully integrated in the SageMath ecosystem. First features will be released with SageMath 10.0.