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Let Q denote a fixed algebraic closure of the field Q of rational numbers. Let p be a prime with p ≥ 11 and ζ a primitive p-th root of 1 in Q. Let K = Q(ζ). For a ∈ {1, 2, ..., p − 2}, consider a smooth projective model Fp,a of the affine curve v = u(1 − u). The curves Fp,a are cyclic quotients of the Fermat curve of degree p and are called Fermat quotient curves. The curve Fp,a has genus (p−1)/2 and its Jacobian Jp,a admits complex multiplication by Z[ζ] which is induced by the automorphism ζ : (u, v) → (u, ζv) of Fp,a. By Section IV in [4], we know that the curve Fp,a is hyperelliptic if and only if a {1, (p − 1)/2, p − 2}. We will denote by S the set of values of a such that 1 ≤ a ≤ p − 2 and Fp,a is not hyperelliptic. The cusps on Fp,a are the elements of the set Cp,a = {(0, 0), (1, 0),∞}. Since the divisors of the rational functions u and 1 − u are p((0, 0) −∞) and p((1, 0) −∞), respectively, it follows that the difference of two points in Cp,a is a torsion point on Jp,a (we refer the reader to the work of Gross and Rohrlich in [9], where some deep related results are proven). For the rest of this paper, we fix a cusp c and the Albanese embedding of Fp,a in Jp,a given by P → [P − c]. The cuspidal torsion packet on Fp,a is the set Tp,a = Fp,a ∩ (Jp,a)tors, whose elements are called cuspidal torsion points. By the Manin-Mumford conjecture (first proved by Raynaud in [14]), Tp,a is a finite set. The purpose of this paper is to establish the following theorem:

Almost rational torsion points and the cuspidal torsion packet on Fermat quotient curves