Bounds of the rank of the Mordell-Weil group of jacobians of hyperelliptic curves

In this article we extend work of Shanks and Washington on cyclic extensions, and elliptic curves associated to the simplest cubic fields. In particular, we give families of examples of hyperelliptic curves $C: y^2=f(x)$ defined over $\mathbb{Q}$, with $f(x)$ of degree $p$, where $p$ is a Sophie Germain prime, such that the rank of the Mordell--Weil group of the jacobian $J/\mathbb{Q}$ of $C$ is bounded by the genus of $C$ and the $2$-rank of the class group of the (cyclic) field defined by $f(x)$, and exhibit examples where this bound is sharp.Comment: 22 pages, To appear in J. Th\'eor. Nombres Bordeau

Towards a classification of entanglements of Galois representations attached to elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve, let $\overline{\mathbb{Q}}$ be a fixed algebraic closure of $\mathbb{Q}$, and let $G_{\mathbb{Q}}=\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group of $\mathbb{Q}$. The action of $G_{\mathbb{Q}}$ on the adelic Tate module of $E$ induces the adelic Galois representation $\rho_E\colon G_{\mathbb{Q}} \to \text{GL}(2,\widehat{\mathbb{Z}}).$ The goal of this paper is to explain how the image of $\rho_E$ can be smaller than expected. To this end, we offer a group theoretic categorization of different ways in which an entanglement between division fields can be explained and prove several results on elliptic curves (and more generally, principally polarized abelian varieties) over $\mathbb{Q}$ where the entanglement occurs over an abelian extension.Comment: 33 pages; final versio

Risk factors for cardiovascular disease in people with idiopathic pulmonary fibrosis: a population-based study

OBJECTIVE: People with idiopathic pulmonary fibrosis (IPF) have been shown to be at an increased risk for cardiovascular (CV) disease, but reasons for this are unknown. The aim of this study was to compare the prevalence of common CV risk factors in people with IPF and the general population and establish the incidence of ischemic heart disease (IHD) and stroke after the diagnosis of IPF, controlling for these risk factors. METHODS: We used data from a large, UK primary care database to identify incident cases of IPF and matched general-population control subjects. We compared the prevalence of risk factors for CV disease and prescription of CV medications in people with IPF (before diagnosis) with control subjects from the general population and assessed the incidence of IHD and stroke in people with IPF (after diagnosis) compared with control subjects. RESULTS: We identified 3,211 cases of IPF and 12,307 control subjects. Patients with IPF were more likely to have a record of hypertension (OR, 1.31; 95% CI, 1.19-1.44), and diabetes (OR, 1.20; 95% CI, 1.07-1.34) compared with control subjects; they were also more likely to have been prescribed several CV drugs. The rate of first-time IHD events was more than twice as high in patients than control subjects (rate ratio, 2.32; 95% CI, 1.85-2.93; P < .001), but the incidence of stroke was only marginally higher (P = .09). Rate ratios for IHD and stroke were not altered substantially after adjusting for CV risk factors. CONCLUSIONS: Several CV risk factors were more prevalent in people with IPF; however, this did not account for the increased rate of IHD in this group of patients