Related papers: Superelliptic jacobians

We give explicit uniform bounds for several quantities relevant to the study of Galois representations attached to elliptic curves $E/\mathbb Q$. We consider in particular the subgroup of scalars in the image of Galois, the first Galois…

If K/F is a finite abelian Galois extension of global fields whose Galois group has exponent t, we prove that there exists a short exact sequence that has as a consequence that if t is square free, then Dec(K/F)=Br_{t}(K/F) which we use to…

In this long survey article we show that the theory of elliptic and hyperelliptic curves can be extended naturally to all superelliptic curves. We focus on automorphism groups, stratification of the moduli space $\mathcal{M}_g$, binary…

Let $\mathcal{X}$ be an irreducible algebraic curve defined over a finite field $\mathbb{F}_q$ of characteristic $p>2$. Assume that the $\mathbb{F}_q$-automorphism group of $\mathcal{X}$ admits as an automorphism group the direct product of…

In this article, we show that for any non-isotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as possible form a density-$1$ subset. Our…

In this paper we obtain new quantitative forms of Hilbert's Irreducibility Theorem. In particular, we show that if $f(X, T_1, \ldots, T_s)$ is an irreducible polynomial with integer coefficients, having Galois group $G$ over the function…

Consider a normal projective variety $X$, a linear algebraic subgroup $G$ of Aut($X$), and the field $K$ of $G$-invariant rational functions on $X$. We show that the subgroup of Aut($X$) that fixes $K$ pointwise is linear algebraic. If $K$…

We prove that for any number field $K$ and any fixed genus $g \geq 2$, there are infinitely many non-isomorphic hyperelliptic curves of genus $g$ over $K$ whose Jacobians have rank over $K$ equal to each of 0, 1, or 2. As an example of our…

We prove that if $f:X \rightarrow A$ is a morphism from a smooth projective variety $X$ to an abelian variety $A$ over a number field $K$, and $G$ is a subgroup of automorphisms of $X$ satisfying certain properties, and if a prime $p$…

The two-dimensional Jacobian Conjecture says that a $\mathbb{C}$-algebra endomorphism $F:\mathbb{C}[x,y] \to \mathbb{C}[x,y]$ that has an invertible Jacobian is an automorphism. We show that if a $\mathbb{C}$-algebra endomorphism…

We study the arithmetic of abelian varieties over $K=k(t)$ where $k$ is an arbitrary field. The main result relates Mordell-Weil groups of certain Jacobians over $K$ to homomorphisms of other Jacobians over $k$. Our methods also yield…

We show that there is essentially a unique elliptic curve $E$ defined over a cubic Galois extension $K$ of $\mathbb Q$ with a $K$-rational point of order 13 and such that $E$ is not defined over $\mathbb Q$.

In this paper we prove the assertion that the number of monic cubic polynomials $F(x) = x^3 + a_2 x^2 + a_1 x + a_0$ with integer coefficients and irreducible, Galois over $\mathbb{Q}$ satisfying $\max\{|a_2|, |a_1|, |a_0|\} \leq X$ is…

In this paper, we describe the construction of superelliptic curves with a rational point of prescribed order on their jacobians. The construction is based on Hensel's Lemma and produces for a given integer $N$ a superelliptic curve of…

For each odd prime power q, we construct an infinite sequence of rational functions f(X) in F_q(X), each of which is exceptional, which means that for infinitely many n the map c-->f(c) induces a bijection of P^1(F_{q^n}). Moreover, each of…

Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation, \rho_E : Gal(\bar{k}/k) \to GL_2(\hat{Z}). For a fixed number field k, we describe the image of \rho_E for a…

Let $K$ be a finitely generated extension of $\mathbb{Q}$. We consider the family of $\ell$-adic representations ($\ell$ varies through the set of all prime numbers) of the absolute Galois group of $K$, attached to $\ell$-adic cohomology of…

We study the universal family of odd hyperelliptic curves of genus $g \geq 1$ over $\mathbb{Q}$. We relate the heights of $\mathbb{Q}$-points of Jacobians of curves in this family to the reduction theory of the representation of…

A cubic Galois polynomial is a cubic polynomial with rational coefficients that defines a cubic Galois field. Its discriminant is a full square and its roots $x_1,x_2,x_3$ (enumerated in some order) are real. There exists (and only one)…

Let K be an algebraically closed field of characteristic zero and let f(x,y) be a nonzero polynomial of K[x,y]. We prove that if the generic element of the family $(f-\lambda)\_{\lambda}$ is a rational polynomial, and if the Jacobian J(f,g)…