Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture

Most, if not all, unconditional results towards the abc-conjecture rely ultimately on classical Baker's method. In this article, we turn our attention to its elliptic analogue. Using the elliptic Baker's method, we have recently obtained a new upper bound for the height of the S-integral points on an elliptic curve. This bound depends on some parameters related to the Mordell-Weil group of the curve. We deduce here a bound relying on the conjecture of Birch and Swinnerton-Dyer, involving classical, more manageable quantities. We then study which abc-type inequality over number fields could be derived from this elliptic approach.Comment: 20 pages. Some changes, the most important being on Conjecture 3.2, three references added ([Mas75], [MB90] and [Yu94]) and one reference updated [BS12]. Accepted in Bull. Brazil. Mat. So

On the spatial distribution of solutions of decomposable form equations

Abstract.We study the distribution in space of the integral solutions to an integral decompos-able form equation, by considering the images of these solutions under central projection onto a unit ball. If we think of the solutions as stars in the night sky, we ask what constellations are visible from the earth (the unit ball). Answers are given for a large class of examples which are then illustrated using the software packages KANT and Maple. These pictures highlight the accuracy of our predictions and arouse interest in cases not covered by our results. Within the range of applicability of our results lie solutions to norm form equations and units in abelian group rings. Thus our theory has a lot to say about where these interesting objects can be found and what they look like

On The Spatial Distribution Of Solutions Of Decomposable Form Equations

. We study the distribution in space of the integral solutions to an integral decomposable form equation, by considering the images of these solutions under central projection onto a unit ball. If we think of the solutions as stars in the night sky, we ask what constellations are visible from the earth (the unit ball). Answers are given for a large class of examples which are then illustrated using the software packages KANT and Maple. These pictures highlight the accuracy of our predictions and arouse interest in cases not covered by our results. Within the range of applicability of our results lie solutions to norm form equations and units in abelian group rings. Thus our theory has a lot to say about where these interesting objects can be found and what they look like. Introduction Let F (x) = F (x 1 ; : : : ; xn ) 2 Z[x 1 ; : : : ; xn ] denote a decomposable form. This is a homogeneous polynomial with coefficients in Z which factorises over C as a product of linear forms. We assum..

Early B Cell Factor 1 Regulates B Cell Gene Networks by Activation, Repression and Transcription - Independent Poising of Chromatin

The transcription factor early B cell factor-1 (Ebf1) is a key determinant of B lineage specification and differentiation. To gain insight into the molecular basis of Ebf1 function in early-stage B cells, we combined a genome-wide ChIP sequencing analysis with gain- and loss-of-function transcriptome analyses. Among 565 genes that are occupied and transcriptionally regulated by Ebf1, we identified large sets involved in (pre)-B cell receptor and Akt signaling, cell adhesion, and migration. Interestingly, a third of previously described Pax5 targets was found to be occupied by Ebf1. In addition to Ebf1-activated and -repressed genes, we identified targets at which Ebf1 induces chromatin changes that poise the genes for expression at subsequent stages of differentiation. Poised chromatin states on specific targets could also be established by Ebf1 expression in T cells but not in NIH 3T3 cells, suggesting that Ebf1 acts as a “pioneer” factor in a hematopoietic chromatin context

Latency type-specific distribution of epigenetic marks at the alternative promoters Cp and Qp of Epstein-Barr virus

Transcripts for the Epstein-Barr virus (EBV)-encoded nuclear antigens are initiated at the alternative promoters Wp, Cp and Qp. Although the host cell-dependent activity of Cp is regulated by DNA methylation, Qp is unmethylated independently of its activity. Because histone modifications affect the chromatin structure, we compared the levels of diacetylated histone H3, tetraacetylated histone H4 and histone H3 dimethylated on lysine 4 (H3K4me2) at Cp and Qp, in well characterized cell lines representing the major EBV latency types. We found an activity-dependent histone code: acetylated histones marked active Cp, whereas active Qp was selectively enriched both in acetylated histones and H3K4me2. We concluded that active (but not silent) Cp and Qp are located to 'acetylation islands' in latent, episomal EBV genomes, similar to the active chromatin domains of the human genome

Transcription factor EBF1 is essential for the maintenance of B cell identity and prevention of alternative fates in committed cells

The transcription factors EBF1 and Pax5 have been linked to activation of the B cell lineage program and irreversible loss of alternative lineage potential (commitment), respectively. Here we conditionally deleted Ebf1 in committed pro-B cells after transfer into alymphoid mice. We found that those cells converted into innate lymphoid cells (ILCs) and T cells with variable-diversity-joining (VDJ) rearrangements of loci encoding both B cell and T cell antigen receptors. As intermediates in lineage conversion, Ebf1-deficient CD19+ cells expressing Pax5 and transcriptional regulators of the ILC and T cell fates were detectable. In particular, genes encoding the transcription factors Id2 and TCF-1 were bound and repressed by EBF1. Thus, both EBF1 and Pax5 are required for B lineage commitment by repressing distinct and common determinants of alternative cell fates

Sensory Map Transfer to the Neocrotex Relies on Pretarget Ordering of Thalamic Axons

Sensory maps, such as the representation of mouse facial whiskers, are conveyed throughout the nervous system by topographic axonal projections that preserve neighboring relationships between adjacent neurons. In particular, the map transfer to the neocortex is ensured by thalamocortical axons (TCAs), whose terminals are topographically organized in response to intrinsic cortical signals. However, TCAs already show a topographic order early in development, as they navigate toward their target. Here, we show that this preordering of TCAs is required for the transfer of the whisker map to the neocortex. Using Ebf1 conditional inactivation that specifically perturbs the development of an intermediate target, the basal ganglia, we scrambled TCA topography en route to the neocortex without affecting the thalamus or neocortex. Notably, embryonic somatosensory TCAs were shifted toward the visual cortex and showed a substantial intermixing along their trajectory. Somatosensory TCAs rewired postnatally to reach the somatosensory cortex but failed to form a topographic anatomical or functional map. Our study reveals that sensory map transfer relies not only on positional information in the projecting and target structures but also on preordering of axons along their trajectory, thereby opening novel perspectives on brain wiring