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Register a new userEndomorphisms of ordinary superelliptic jacobians
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Authors
Yuri G. Zarhin
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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 $(l,n) e (5,5)$.
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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}.
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.
This paper has been withdrawn by the author due to a crucial argument error at p.10.
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.
Let $Omega$ be the complement of a connected, essential hyperplane arrangement. We prove that every dominant endomorphism of $Omega$ extends to an endomorphism of the tropical compactification $X$ of $Omega$ associated to the Bergman fan structure on the tropicalization of $Omega$. This generalizes a previous result by Remy, Thuillier and the second author which states that every automorphism of Drinfelds half-space over a finite field $mathbb{F}_q$ extends to an automorphism of the successive blow-up of projective space at all $mathbb{F}_q$-rational linear subspaces. This successive blow-up is in fact the minimal wonderful compactification by de Concini and Procesi, which coincides with $X$ by results of Feichtner and Sturmfels. Whereas the previous proof is based on Berkovich analytic geometry over the trivially valued finite ground field, the generalization discussed in the present paper relies on matroids and tropical geometry.
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