Related papers: Hyperelliptic jacobians without complex multiplica…

We consider the problem of classifying quadruples $(K,E,m_1,m_2)$ where $K$ is a number field, $E$ is an elliptic curve defined over $K$ and $(m_1,m_2)$ is a pair of relatively prime positive integers for which the intersection $K(E[m_1])…

We prove an explicit surjectivity result for products of non-isotrivial, non-isogenous elliptic curves over a function field of arbitrary characteristic. This is by way of an isogeny degree bound in this setting, generated from bounds for…

Motivated by our arithmetic applications, we required some tools that might be of independent interest. Let $\mathcal E$ be an absolutely irreducible group scheme of rank $p^4$ over $\mathbb Z_p$. We provide a complete description of the…

Consider the Jacobian of a hyperelliptic genus two curve defined over a prime field of characteristic p and with complex multiplication. In this paper we show that the p-Sylow subgroup of the Jacobian is either trivial or of order p.

We introduce and study a class of field extensions that we call pre-Galois; viz. extensions that become Galois after some linearly disjoint Galois base change. Among them are geometrically Galois extensions of k(T), with k a field:…

For a prime \(p\ge 2\) and a number field K with p-class group of type (p,p) it is shown that the class, coclass, and further invariants of the metabelian Galois group \(G=Gal(F_p^2(K) | K)\) of the second Hilbert p-class field \(F_p^2(K)\)…

We characterise the integral affine plane curves over a finite field $k$ with the property that all but finitely many of their $\overline{k}$-points have coordinates that are $j$-invariants of elliptic curves with isomorphic endomorphism…

Inspired by experimental data, this paper investigates which isogeny classes of abelian varieties defined over a finite field of odd characteristic contain the Jacobian of a hyperelliptic curve. We provide a necessary condition by…

Over a global field (number field or function field of a curve over a finite field), theorems for the Galois cohomology of algebraic groups have long been known. For $F$ the function field of a curve over the formal series field…

We show that for an elliptic curve E defined over a number field K, the group E(A) of points of E over the adele ring A of K is a topological group that can be analyzed in terms of the Galois representation associated to the torsion points…

Let E/F be a quadratic number (resp. p-adic) field extension, and F' an odd degree cyclic field extension of F. We establish a base-change functorial lifting of automorphic (resp. admissible) representations from the unitary group U(3,E/F)…

For a non-CM elliptic curve $E$ over the rationals, the Galois action on its torsion points can be expressed in terms of a Galois representation $\rho_E : G \to GL_2(\hat{\mathbb{Z}})$, where $G$ is the absolute Galois group of the…

We show that if F is the rational numbers or a multiquadratic number field, p is 2,3, or 5, and K/F is a Galois extension of degree a power of p, then for elliptic curves E/Q ordered by height, the average dimension of the p-Selmer groups…

Let $k$ be an algebraically closed field of characteristic $p >0$. Suppose $g \geq 3$ and $0 \leq f \leq g$. We prove there is a smooth projective $k$-curve of genus $g$ and $p$-rank $f$ with no non-trivial automorphisms. In addition, we…

Let $G$ denote a linear algebraic group over $\mathbf{Q}$ and $K$ and $L$ two number fields. Assume that there is a group isomorphism of points on $G$ over the finite adeles of $K$ and $L$, respectively. We establish conditions on the group…

We show that every linear algebraic group over an algebraically closed field of characteristic zero is the differential Galois group of a regular singular linear differential equation with rational function coefficients.

The main purpose of this paper is to extend results on isomorphism types of the abelianized absolute Galois group $\mathcal G_K^{ab}$, where $K$ denotes imaginary quadratic field. In particular, we will show that if the class number $h_K$…

Let $k$ be an algebraically closed field of characteristic zero, $F$ be an algebraically closed extension of $k$ of transcendence degree one, and $G$ be the group of automorphisms over $k$ of the field $F$. The purpose of this note is to…

Let $K$ be a number field and $E_1, \ldots, E_n$ be elliptic curves over $K$, pairwise non-isogenous over $\overline{K}$ and without complex multiplication over $\overline{K}$. We study the image of the adelic representation of the absolute…

We show how for every integer n one can explicitly construct n distinct plane quartics and one hyperelliptic curve over the complex numbers all of whose Jacobians are isomorphic to one another as abelian varieties without polarization. When…