These lectures concern the arithmetic of modular curves, and in particular the geometry of integral models of modular curves in the neighborhood of their singular points. These singularities only appear modulo p. If X is a modular curve and x is a singular point, then the nature of the singularity is measured by the completed local ring ÔX,x. Let us first review the basics of integral models of...

Fix integers r, k, g with r > k > 0 and g ≥ 2. Let X be an integral projective curve with g := pa(X) and E a rank r torsion free sheaf on X which is a flat limit of a family of locally free sheaves on X. Here we prove the existence of a rank k subsheaf A of E such that r(deg(A)) ≥ k(deg(E)) − k(r − k)g. We show that for every g ≥ 9 there is an integral projective curve X,X not Gorenstein, and a...

An adaptive regularized level set method for image segmentation is proposed. A weighted p x Dirichlet integral is presented as a geometric regularization on zero level curve, which is used to diminish the influence of image noise on level set evolution while ensuring the active contours not to pass through weak object boundaries. The idea behind the new energy integral is that the amount of reg...

We consider the structure of rational points on elliptic curves in Weierstrass form. Let x(P ) = AP /B 2 P denote the xcoordinate of the rational point P then we consider when BP can be a prime power. Using Faltings’ Theorem we show that for a fixed power greater than 1, there are only finitely many rational points. Where descent via an isogeny is possible we show, with no restrictions on the p...

We consider the structure of rational points on elliptic curves in Weierstrass form. Let x(P ) = AP /B 2 P denote the xcoordinate of the rational point P then we consider when BP can be a prime power. Using Faltings’ Theorem we show that for a fixed power greater than 1, there are only finitely many rational points with this property. Where descent via an isogeny is possible we show, with no re...

OBJECTIVE To investigate differences in tongue images of subjects with and without hyperuricemia. MATERIALS AND METHODS This population-based case-control study was performed in 2012-2013. We collected data from 46 case subjects with hyperuricemia and 46 control subjects, including results of biochemical examinations and tongue images. Symmetrical Haar-like features based on integral images w...

Let C be an integral curve in Pk , with k an algebraically closed field. In the main result of this paper we prove that the Hilbert function of its general plane section C ∩H is of decreasing type, extending to any characteristic a result proved in the case char k = 0 by Maggioni and Ragusa. Moreover, the proof given in this paper does not depend on the uniform position property. We also prove ...

Let f be a weight 2 normalized newform on the congruence subgroup Γ0(N) with integral Fourier coefficients. There are two important numerical invariants attached to f : its congruence number and its modular degree. By definition, the congruence number of f is the largest integer Df such that there exists a weight 2 cusp form on Γ0(N), with integral coefficients, which is orthogonal to f with re...

• Review of vector fields and differential equations in Rn . Let X be a vector field on a nonempty open subset of Rn . An integral curve of X with initial condition p0 is a differentiable path c : (a,b) → Rn , defined on an open interval containing 0 such that c(0) = p0 and c ′(t ) = X (c(t )) for all t ∈ (a,b). Thus to the vector field X one associates a differential equation ṗ = X (p). A solu...