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Register a new userFixed points and homology of superelliptic Jacobians
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Let $eta: C_{f,N}to mathbb{P}^1$ be a cyclic cover of $mathbb{P}^1$ of degree $N$ which is totally and tamely ramified for all the ramification points. We determine the group of fixed points of the cyclic group $mathbf{mu}_Ncong mathbb{Z}/Nmathbb{Z}$ acting on the Jacobian $J_N:=Jac(C_{f,N})$. For each $ell$ distinct from the characteristic of the base field, the Tate module $T_ell J_N$ is shown to be a free module over the ring $mathbb{Z}_ell[T]/(sum_{i=0}^{N-1}T^i)$. We also calculate the degree of the induced polarization on the new part $J_N^{new}$ of the Jacobian.
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Let $K$ be a field of prime characteristic $p$, $n>4 $ an integer, $f(x)$ an irreducible polynomial over $K$ of degree $n$, whose Galois group is either the full symmetric group $S_n$ or the alternating group $A_n$. Let $l$ be an odd prime different from $p$, $Z[zeta_l]$ the ring of integers in the $l$th cyclotomic field, $C_{f,l}:y^l=f(x)$ the corresponding superelliptic curve and $J(C_{f,l})$ its jacobian. We prove that the ring of all endomorphisms of $J(C_{f,l})$ coincides with $Z[zeta_l]$ if $J(C_{f,l})$ is an ordinary abelian variety and $(l,n) e (5,5)$.
Let K be a field of characteristic zero, f(x) be a polynomial with coefficients in K and without multiple roots. We consider the superelliptic curve C_{f,q} defined by y^q=f(x), where q=p^r is a power of a prime p. We determine the Hodge group of the simple factors of the Jacobian of C_{f,q}.
In this paper we study the Coleman-Oort conjecture for superelliptic curves, i.e., curves defined by affine equations $y^n=F(x)$ with $F$ a separable polynomial. We prove that up to isomorphism there are at most finitely many superelliptic curves of fixed genus $ggeq 8$ with CM Jacobians. The proof relies on the geometric structures of Shimura subvarieties in Siegel modular varieties and the stability properties of Higgs bundles associated to fibred surfaces.
We show that Connes B-operator on a cyclic differential graded k-module M is a model for the canonical circle action on the geometric realization of M. This implies that the negative cyclic homology and the periodic cyclic homology of a differential graded category can be identified with the homotopy fixed points and the Tate fixed points of the circle action on its Hochschild complex.
We introduce and study higher order Jacobian ideals, higher order and mixed Hessians, higher order polar maps, and higher order Milnor algebras associated to a reduced projective hypersurface. We relate these higher order objects to some standard graded Artinian Gorenstein algebras, and we study the corresponding Hilbert functions and Lefschetz properties.
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