Absolutely Simple Prymians of Trigonal Curves

As usual, Z,Q,C denote the ring of integers, the field of rational numbers and the field of complex numbers respectively. Let us fix a primitive cube root of unity ζ3 = −1+ √ −3 2 ∈ C. Let Q(ζ3) = Q( √ −3) be the third cyclotomic field and Z[ζ3] = Z +Z · ζ3 its ring of integers. We write λ for the (principal) maximal ideal (1− ζ3) · Z[ζ3] of Z[ζ3]. It is known [11, Th. 5 on p. 176] (see also [5]) that for all positive integers m different from 2 there exists a m-dimensional complex abelian variety, whose endomorphism ring is Z[ζ3]. Shimura’s proof is purely complex-analytic and not constructive; roughly speaking it deals with points of the corresponding moduli space that do not belong to a countable union of subvarieties of positive codimension. In this paper we discuss a geometric approach to an explicit construction of those abelian varieties via jacobians, prymians and Galois theory. In order to explain our approach, let us start with the following definitions. Let f(x) ∈ C[x] be a polynomial of degree n ≥ 4 without multiple roots. Let Cf,3 be a smooth projective model of the smooth affine curve y = f(x). It is well known ([2], pp. 401-402, [15], Prop. 1 on p. 3359, [7], p. 148) that the genus g(Cf,3) of Cf,3 is n− 1 if 3 does not divide n and n− 2 if it does. In both cases g(Cf,3) ≥ 3 is not congruent to 2 modulo 3. The map (x, y) 7→ (x, ζ3y) gives rise to a non-trivial birational automorphism δ3 : Cf,3 → Cf,3 of period 3. By functoriality, δ3 induces the linear operator in the space of differentials of the first kind