Superelliptic Jacobians Research Articles

On torsion of superelliptic Jacobians

We give a structure theorem for the $m$-torsion of the Jacobian of a general superelliptic curve $y^m=F(x)$. We study existence of torsion on curves of the form $y^q=x^p-x+a$ over finite fields of characteristic $p$. We apply those results to bound from below the Mordell-Weil ranks of Jacobians of certain superelliptic curves over $\mathbb Q$.

The rational points on certain Abelian varieties over function fields

In this paper, we consider Abelian varieties over function fields that arise as twists of Abelian varieties by cyclic covers of irreducible quasi-projective varieties. Then, in terms of Prym varieties of the cyclic covers, we prove a structure theorem on their Mordell–Weil group. Our results give an explicit method to construct elliptic curves, hyper- and super-elliptic Jacobians that have large ranks over function fields of certain varieties.

Low-dimensional factors of superelliptic Jacobians

Given a polynomial f ∈ C(x), we consider the family of superelliptic curves y d = f (x) and their Jacobians Jd for varying integers d. We show that for any integer g the number of abelian varieties up to isogeny of dimension ≤g which appear in any Jd is finite and their multiplicities are bounded.

Fixed points and homology of superelliptic Jacobians

Let $$\eta : C_{f,N}\rightarrow \mathbb {P}^1$$ 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 covering group $${{\mathrm{Aut}}}(\eta )\simeq \mathbb {Z}/ N \mathbb {Z}$$ acting on the Jacobian $$J_N:={{\mathrm{Jac}}}(C_{f,N})$$ . For each prime $$\ell $$ 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 study the subvarieties of $$J_N$$ and calculate the degree of the induced polarization on the new part $$J_N^\mathrm {new}$$ of the Jacobian.

Centers of Hodge groups of superelliptic Jacobians

We give lower bounds for dimensions of the centers of Hodge groups of superelliptic Jacobians. In the “generic case” this bound is precise. We also discuss dimensions of simple abelian subvarieties of superelliptic Jacobians.

Hodge groups of certain superelliptic jacobians

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}.

Ranks of Jacobians in towers of function fields

.2. Endomorphismalgebrasof superelliptic Jacobians2.1. Notation. In this section, we collect some results on the endomorphism al-gebras of certain superelliptic Jacobians. Throughout the paper, k will denote analgebraic closure of k.If X and Y are abelian varieties over k, then we write End(X) and Hom(X,Y)for the corresponding ring and group of homomorphisms over k.Let f be a non-constant polynomial with coefficients in k and without multipleroots. We write m for the degree of f, R

Endomorphisms of superelliptic jacobians

Let K be a field of characteristic zero, n ≥ 5 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group contains a doubly transitive simple non-abelian group. Let p be an odd prime, \({\mathbb{Z}[\zeta_p]}\) the ring of integers in the pth cyclotomic field, Cf, p : yp = f(x) the corresponding superelliptic curve and J(Cf, p) its jacobian. Assuming that either n = p + 1 or p does not divide n(n − 1), we prove that the ring of all endomorphisms of J(Cf, p) coincides with \({\mathbb{Z}[\zeta_p]}\) . The same is true if n = 4, the Galois group of f(x) is the full symmetric group S4 and K contains a primitive pth root of unity.

Endomorphisms of superelliptic jacobians

Let K be a field of characteristic zero, n ≥ 5 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group contains a doubly transitive simple non-abelian group. Let p be an odd prime, $${\mathbb{Z}[\zeta_p]}$$ the ring of integers in the pth cyclotomic field, C f, p : y p = f(x) the corresponding superelliptic curve and J(C f, p ) its jacobian. Assuming that either n = p + 1 or p does not divide n(n − 1), we prove that the ring of all endomorphisms of J(C f, p ) coincides with $${\mathbb{Z}[\zeta_p]}$$ . The same is true if n = 4, the Galois group of f(x) is the full symmetric group S 4 and K contains a primitive pth root of unity.