We give a precise description of the homology group of the Fermat curve as a cyclic module over a group ring. As an application, we prove the freeness of the profinite homology of the Fermat tower. This allows us to define measures, an equivalent of Anderson's adelic beta functions, in a manner similar to Ihara's definition of $\ell$-adic universal power series for Jacobi sums. We give a simple proof of the interpolation property using a motivic decomposition of the Fermat curve.

We generalize a result of Anderson by showing that torsion points of certain orders cannot lie on a theta divisor in the Jacobians of hyperelliptic images of Fermat curves. The proofs utilize the explicit geometry of hyperelliptic Jacobians and their formal groups at the origin.

This paper is concerned with the arithmetic of curves of the form vp=us(1−u), where p is a prime with $p$ ≥ 5 and s is an integer such that 1 ≤ s ≤ p−2. The Jacobians of these curves admit complexion by a primitive p-th root of unity ζ. We find explicit rational functions on these curves whose divisors are p-multiples of divisors representing (1-ζ)2 - and (1-ζ)3-division points on the corresponding Jacobians. This also gives an effective version of a theorem of Greenberg.