Related papers: From Picard groups of hyperelliptic curves to clas…

For an elliptic curve $E/\mathbb{Q}$ we show that there are infinitely many cyclic sextic extensions $K/\mathbb{Q}$ such that the Mordell-Weil group $E(K)$ has rank greater than the subgroup of $E(K)$ generated by all the $E(F)$ for the…

We prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in one variable over a finite field is finiteley generated.

Consider the elliptic curve $E$ given by the Weierstrass equation $y^2 = x^3 - 11x - 14$, which has complex multiplication by the order of conductor $2$ inside $\mathbb{Z}[i]$. It was recently observed in a paper of Daniels and…

By reformulating and extending results of Elkies, we prove some results on $\mathbb Q$-curves over number fields of odd degree. We show that, over such fields, the only prime isogeny degrees~$\ell$ which an elliptic curve without CM may…

Let $E$ be an ordinary elliptic curve over a finite field and $g$ be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class…

It has been conjectured that every algebraic curve may be defined either over its field of moduli or over an extension of degree two of it. In this paper we provide a negative answer to it by giving examples of hyperelliptic curves which…

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…

Given a hyperelliptic curve $C$ of genus $g$ over a number field $K$ and a Weierstrass model $\mathscr{C}$ of $C$ over the ring of integers ${\mathcal O}_K$ (i.e. the hyperelliptic involution of $C$ extends to $\mathscr{C}$ and the quotient…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $K$ be a number field of degree four that is Galois over $\mathbb{Q}$. The goal of this article is to classify the different isomorphism types of $E(K)_{\text{tors}}$.

A curve over a field k is pointless if it has no k-rational points. We show that there exist pointless genus-3 hyperelliptic curves over a finite field F_q if and only if q < 26, that there exist pointless smooth plane quartics over F_q if…

An abelian variety $A/K$ is heavenly at $\ell$ if the extension $K(A[\ell^\infty])/K(\mu_{\ell^{\infty}}\!)$ is both pro-$\ell$ and unramified away from $\ell$. It is known that for a fixed quadratic field $K$, the number of $K$-isomorphism…

Let $C$ be an algebraic curve of genus $g$ and $L$ a line bundle over $C$. Let $\mathcal{MS}_C(n,L)$ and $\mathcal{MO}_C(n,L)$ be the moduli spaces of $L$-valued symplectic and orthogonal bundles respectively, over $C$ of rank $n$. We…

The dimension of spaces of global sections for line bundles on semistable curves parametrized by the compactified Picard scheme is studied. The theorem of Riemann is shown to hold. The theorem of Clifford is shown to hold in the following…

We determine all groups which occur as torsion subgroups of $\mathbb Q$-curves defined over number fields of degrees $3$, $5$ and $7$. In particular, we prove that every torsion subgroup of a $\mathbb Q$-curve defined over a number field of…

We study a particular plane curve over a finite field whose normalization is of genus 0. The number of rational points of this curve achieves the Aubry-Perret bound for rational curves. The configuration of its rational points and a…

We observe that there are elliptic curves over number fields all of whose quadratic twists must have positive rank, assuming the Birch-Swinnerton-Dyer conjecture. We give a classification of such curves in terms of their local behaviour,…

Let $E/\bbq$ be an elliptic curve defined over $\bbq$ with conductor $N$ and $\gq$ the absolute Galois group of an algebraic closure $\bar{\bbq}$ of $\bbq$. We prove that for every $\sigma\in \gq$, the Mordell-Weil group $E(\oqs)$ of $E$…

In this paper, we initiate a systematic study of entanglements of division fields from a group theoretic perspective. For a positive integer $n$ and a subgroup $G\subseteq \text{GL}_2(\mathbb{Z}/{n}\mathbb{Z})$ with surjective determinant,…

Fix a hyperelliptic curve $C/\mathbb{Q}$ of genus $g$, and consider the number fields $K/\mathbb{Q}$ generated by the algebraic points of $C$. In this paper, we study the number of such extensions with fixed degree $n$ and discriminant…

There is a modular curve X'(6) of level 6 defined over Q whose Q-rational points correspond to j-invariants of elliptic curves E over Q for which Q(E[2]) is a subfield of Q(E[3]). In this note we characterize the j-invariants of elliptic…