On the maximum number of rational points on singular curves over finite fields

We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve defined over Fq of geometric genus g and arithmetic genus $\pi$

Optimal and maximal singular curves

International audienceUsing an Euclidean approach, we prove a new upper bound for the number of closed points of degree 2 on a smooth absolutely irreducible projective algebraic curve defined over the finite field $\mathbb F_q$.This bound enables us to provide explicit conditions on $q, g$ and $\pi$ for the non-existence of absolutely irreducible projective algebraic curves defined over $\mathbb F_q$ of geometric genus $g$, arithmetic genus $\pi$ and with $N_q(g)+\pi-g$ rational points.Moreover, for $q$ a square, we study the set of pairs $(g,\pi)$ for which there exists a maximal absolutely irreducible projective algebraic curve defined over $\mathbb F_q$ of geometric genus $g$ and arithmetic genus $\pi$, i.e. with $q+1+2g\sqrt{q}+\pi-g$ rational points

New sextics of genus 6 and 10 attaining the Serre bound

We provide new examples of curves of genus 6 or 10 attaining the Serre bound. They all belong to the family of sextics introduced in [19] as a a generalization of the Wiman sextics [36] and Edge sextics [9]. Our approach is based on a theorem by Kani and Rosen which allows, under certain assumptions, to fully decompose the Jacobian of the curve. With our investigation we are able to update several entries in \url{http://www.manypoints.org} ([35])

Che-1 arrests human colon carcinoma cell proliferation by displacing HDAC1 from the p21WAF1/CIP1 promoter.

Che-1 is a recently identified human RNA polymerase II binding protein involved in the regulation of gene transcription and cell proliferation. We previously demonstrated that Che-1 inhibits the Rb growth-suppressing function by interfering with Rb-mediated HDAC1 recruitment on E2F target gene promoters. By hybridization of cancer profile arrays, we found that Che-1 expression is strongly down-regulated in several tumors, including colon and kidney carcinomas, compared with the relative normal tissues. Consistent with these data, Che-1 overexpression inhibits proliferation of HCT116 and LoVo human colon carcinoma cell lines by activation of the cyclin-dependent kinase inhibitor p21WAF1/Cip1 in a p53-independent manner and by promoting growth arrest at the G1 phase of the cell cycle. Che-1 activates p21WAF1/Cip1 by displacing histone deacetylase (HDAC)1 from the Sp1 binding sites of the p21WAF1/Cip1 gene promoter and accumulating acetylated histone H3 on these sites. Accordingly, Che-1-specific RNA interference negatively affects p21WAF1/Cip1 transactivation and increases cell proliferation in HCT116 cells. Taken together, our results indicate that Che-1 can be considered a general HDAC1 competitor and its down-regulation is involved in colon carcinoma cell proliferation

BACE1 influences clinical manifestations and central inflammation in relapsing remitting multiple sclerosis

Neurodegenerative and inflammatory processes influence the clinical course of multiple sclerosis (MS). The beta-site amyloid precursor protein cleaving enzyme 1 (BACE1) has been associated with cognitive dysfunction, amyloid deposition and neuroinflammation in Alzheimer's disease.We explored in a group of 50 patients with relapsing-remitting MS the association between the cerebrospinal fluid (CSF) levels of BACE1, clinical characteristics at the time of diagnosis and prospective disability after three-years follow-up. In addition, we assessed the correlations between the CSF levels of BACE 1, amyloid beta (A beta) 1-40 and 1-42, phosphorylated tau (pTau), lactate, and a set of inflammatory and anti-inflammatory molecules.BACE1 CSF levels were correlated positively with depression as measured with Beck Depression Inventory-Second Edition scale, and negatively with visuospatial memory performance evaluated by the Brief Visuospatial Memory Test-Revised. In addition, BACE CSF levels were positively correlated with Bayesian Risk Estimate for MS at onset, and with Expanded Disability Status Scale score assessed three years after diagnosis. Furthermore, a positive correlation was found between BACE1, amyloid beta 42/40 ratio (Spearman's r = 0.334, p = 0.018, n = 50), pTau (Spearman's r = 0.304, p = 0.032, n = 50) and lactate concentrations (Spearman's r = 0.361, p = 0.01, n = 50). Finally, an association emerged between BACE1 CSF levels and a group of pro and anti-inflammatory molecules, including interleukin (IL)-4, IL-17, IL-13, IL-9 and interferon-gamma.BACE1 may have a role in different key mechanisms such as neurodegeneration, oxidative stress and inflammation, influencing mood, cognitive disorders and disability progression in MS

Transient Receptor Potential Vanilloid 1 Modulates Central Inflammation in Multiple Sclerosis

Introduction: Disease course of multiple sclerosis (MS) is negatively influenced by proinflammatory molecules released by activated T and B lymphocytes and local immune cells. The endovanilloid system plays different physiological functions, and preclinical data suggest that transient receptor potential vanilloid type 1 (TRPV1) could modulate neuroinflammation in this disorder.Methods: The effect of TRPV1 activation on the release of two main proinflammatory cytokines, tumor necrosis factor (TNF) and interleukin (IL)-6, was explored in activated microglial cells. Furthermore, in a group of 132 MS patients, the association between the cerebrospinal fluid (CSF) levels of TNF and IL-6 and a single nucleotide polymorphisms (SNP) influencing TRPV1 protein expression and function (rs222747) was assessed.Results: In in vitro experiments, TRPV1 stimulation by capsaicin significantly reduced TNF and IL-6 release by activated microglial cells. Moreover, the anti-inflammatory effect of TRPV1 activation was confirmed by another TRPV1 agonist, the resiniferatoxin (RTX), whose effects were significantly inhibited by the TRPV1 antagonist, 5-iodoresiniferatoxin (5-IRTX). Vice versa, BV2 pre-treatment with 5-IRTX increased the inflammatory response induced by LPS. Moreover, in MS patients, a significant association emerged between TRPV1 SNP rs222747 and CSF TNF levels. In particular, the presence of a G allele, known to result in increased TRPV1 protein expression and function, was associated to lower CSF levels of TNF.Conclusions: Our results indicate that TRPV1 influences central inflammation in MS by regulating cytokine release by activated microglial cells. The modulation of the endovanilloid system may represent a useful approach to contrast neuroinflammation in MS

Number of rational points on singular curves over finite fields

On s'intéresse, dans cette thèse, à des questions concernant le nombre maximum de points rationnels d'une courbe singulière définie sur un corps fini, sujet qui, depuis Weil, a été amplement abordé dans le cas lisse. Cette étude se déroule en deux temps. Tout d'abord on présente une construction de courbes singulières de genres et corps de base donnés, possédant un grand nombre de points rationnels : cette construction, qui repose sur des notions et outils de géométrie algébrique et d'algèbre commutative, permet de construire, en partant d'une courbe lisse X, une courbe à singularités X', de telle sorte que X soit la normalisée de X', et que les singularités ajoutées soient rationnelles sur le corps de base et de degré de singularité prescrit. Ensuite, en utilisant une approche euclidienne, on prouve une nouvelle borne sur le nombre de points fermés de degré deux d'une courbe lisse définie sur un corps fini.La combinaison de ces résultats, à priori indépendants, permet notamment d'étudier le problème de savoir quand la borne d'Aubry-Perret, analogue de la borne de Weil dans le cas singulier, est atteinte. Cela nous amène de façon naturelle à l'étude des propriétés des courbes maximales et, lorsque la cardinalité du corps de base est un carré, à l'analyse du spectre des genres de ces dernières.In this PhD thesis, we focus on some issues about the maximum number of rational points on a singular curve defined over a finite field. This topic has been extensively discussed in the smooth case since Weil's works. We have split our study into two stages. First, we provide a construction of singular curves of prescribed genera and base field and with many rational points: such a construction, based on some notions and tools from algebraic geometry and commutative algebra, yields a method for constructing, given a smooth curve X, another curve X' with singularities, such that X is the normalization of X', and the added singularities are rational on the base field and with the prescribed singularity degree. Then, using a Euclidian approach, we prove a new bound for the number of closed points of degree two on a smooth curve defined over a finite field.Combining these two a priori independent results, we can study the following question: when is the Aubry-Perret bound (the analogue of the Weil bound in the singular case) reached? This leads naturally to the study of the properties of maximal curves and, when the cardinality of the base field is a square, to the analysis of the spectrum of their genera