Jacobians and intermediate Jacobians with additional symmetries

We study principally polarized complex abelian varieties $(X,\lambda)$ of positive dimension $g$ that admit an automorphism $\delta$ of prime order $p>2$, whose set of fixed points $X^{\delta}$ is finite.  Such triples $(X,\lambda,\delta)$ exist if and only if $(p-1)$ divides $2g$.  

By functoriality, $\delta$ acts on the $g$-dimensional complex vector space $\Omega^1(X)$ of  differentials of the first kind on $X$ as a diagonalizable linear operator, whose spectrum (the set of eigenvalues)  consists of primitive $p$th roots of unity with certain multiplicities. Let $\mathbf{a}_X$ be the corresponding multiplicity function on the set $\mu_p^{*}$ of all primitive $p$th roots of unity. (In particular, $\mathbf{a}_X(\zeta)=0$ if and only if $\zeta$ is {\sl not} an eigenvalue of $\delta: \Omega^1(X) \to \Omega^1(X)$.)  It is known that $\mathbf{a}_X$ is {\sl well rounded}, i.e., 

$$\mathbf{a}_X(\zeta)+\mathbf{a}_X(1/\zeta)=2g/(p-1) \ \forall \zeta \in \mu_p^{*}.$$

We describe explicitly all functions on  $\mu_p^{*}$ that can be realized as the multiplicity functions in the case when $(X,\lambda)$ are canonically polarized jacobians of smooth projective curves of genus $g$. It turns out that not all well rounded nonnegative integer-valued  functions  occur as multiplicity functions that arise from jacobians. As an application, we sketch another proof of the (already known) fact that intermediate jacobians of certain cubic threefolds are not isomorphic (as principally polarized abelian varieties) to jacobians of curves. As another application, we  prove that certain Prym varieties  are not  isomorphic to jacobians of curves.

References:

• Yu. G. Zarhin, Cubic surfaces and cubic threefolds, jacobians and intermediate jacobians. In: Algebra, Arithmetic and Geometry (Manin Festschrift). Progress in Math. 270, Birkhäuser, 2009, pp. 687--691.
• Yu. G. Zarhin, Absolutely simple  Prymians of trigonal curves.  Proceedings of the Steklov Institute of Math. 264 (2009), 204--215.
• Yu. G. Zarhin, Jacobians with automorphisms of prime order. arXiv:2109.06794 [math.AG]