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Abstract

The integral t-motivic cohomology and the class module of a (rigid analytically trivial) Anderson t-motive were introduced by the first author in [Gaz22b]. This paper is devoted to their determination in the particular case of tensor powers of the Carlitz t-motive, namely, the function field counterpart A––(n) of Tate twists Z(n). We find out that these modules are in relation with fundamental objects of function field arithmetic: integral t-motivic cohomology governs linear relations among Carlitz polylogarithms, its torsion is expressed in terms of the denominator of Bernoulli-Carlitz numbers and the Fitting ideal of class modules is a special zeta value. We also express the regulator of A––(n) for positive n in terms of generalized Carlitz polylogarithms; after establishing their algebraic relations using difference Galois theory together with the Anderson-Brownawell-Papanikolas criterion, we prove that the regulator is an isomorphism if, and only if, n is prime to the characteristic.