Jacobians with with automorphisms of prime order (original) (raw)

In this paper we study principally polarized abelian varieties that admit an automorphism of prime order p > 2. It turns out that certain natural conditions on the multiplicities of its action on the differentials of the first kind do guarantee that those polarized varieties are not jacobians of curves. 1. Principally polarized abelian varieties with automorphisms We write Z+ for the set of nonnegative integers, Q for the field of rational numbers and C for the field of complex numbers. We have Z+ ⊂ Z ⊂ Q ⊂ R ⊂ C where Z is the ring of integers and R is the field of real numbers. If B is a finite (may be, empty) set then we write #(B) for its cardinality. Let p be an odd prime and ζp ∈ C a primitive (complex) pth root of unity. It generates the multiplicative order p cyclic group μp of pth roots of unity. We write Z[ζp] and Q(ζp) for the pth cyclotomic ring and the pth cyclotomic field respectively. We have ζp ∈ μp ⊂ Z[ζp] ⊂ Q(ζp) ⊂ C. Let g ≥ 1 be an integer and (X,λ) a principa...