Jacobians with with automorphisms of prime order
Yuri G. Zarhin
TL;DR
The paper addresses when a principally polarized abelian variety with a prime-order automorphism $δ$ of order $p>2$ is not a Jacobian. It introduces the multiplicity function $\mathbf{a}_{X,δ}$ describing how $δ$ acts on $Ω^1(X)$ and defines notions of well-rounded, admissible, and strongly admissible; using fixed-point data on a hypothetical curve, it shows explicit relations between $\mathbf{a}_{\mathcal{J},δ}$ and a fixed-point function $\mathbf{b}$, proving obstructions to Jacobian realizations. A constructive inverse result shows how to realize prescribed $\mathbf{a}_{\mathcal{J},δ}$ via curves of the form $y^p=f(x)$ and corresponding CM-type Jacobians, with detailed analyses for the $p=3$ case and for CM abelian varieties. The paper then develops criteria showing that self-powers of certain CM abelian varieties (e.g., with endomorphism ring containing $\mathbb{Z}[ζ_p]$) cannot be Jacobians, providing explicit bounds and applications to plane superelliptic curves, and offering explicit counts of strongly admissible data in CM and $p=3$ cases. Overall, it furnishes a framework to certify non-Jacobian behavior from the eigenstructure of $δ$ and fixed-point data, with concrete constructions and nonexistence results for Jacobians in families of self-products.
Abstract
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.