Related papers: Minimal number of points with bad reduction for el…

We consider the Minimum Convex Partition problem: Given a set P of n points in the plane, draw a plane graph G on P, with positive minimum degree, such that G partitions the convex hull of P into a minimum number of convex faces. We show…

Let $K$ be a local field of residue characteristic $p$. Let $C$ be a curve over $K$ whose minimal proper regular model has totally degenerate semi-stable reduction. Under certain hypotheses, we compute the prime-to-$p$ rational torsion…

Consider the projections of a finite set $A\subset R^n$ onto the coordinate hyperplanes. How small can the sum of the sizes of these projections be, given the size of $A$? In a different form, this problem has been studied earlier in the…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. For a quadratic number field $K$ and an odd prime number $p$, let $L$ be a $\mathbb{Z}_p$-extension of $K$. We prove that $E(L)_{\text{tors}}=E(K)_{\text{tors}}$ when $p>5$. It enables…

Mordell curves over a number field $K$ are elliptic curves of the form $ y^2 = x^3 + c$, where $c \in K \setminus \{ 0 \}$. Let $p \geq 5$ be a prime number, $K$ a number field such that $[K:\mathbb{Q}] \in \{ 2p, 3p \}$ and let $E$ be a…

We prove regularity estimates for weak solutions to the Dirichlet problem for a divergence form elliptic operator. We give $L^p$ estimates for the second derivative for $p<2$. Our work generalizes results due to Miranda [28].

This paper deals with a class of singularly perturbed nonlinear elliptic problems $(P_\e)$ with subcritical nonlinearity. The coefficient of the linear part is assumed to concentrate in a point of the domain, as $\e\to 0$, and the domain is…

Let $E$ be an elliptic curve and $\pi:E\to\mathbb{P}^{1}$ a standard double cover identifying $\pm P\in E$. It is known that for some torsion points $P_{i}\in E$, $1\leq i\leq4$, the cross ratio of $\{\pi(P_{i})\}_{i=1}^{4}$ is independent…

We study the rigidity of the local conditions in two well-known local-global principles for elliptic curves over number fields. In particular, we consider a local-global principle for torsion due to Serre and Katz, and one for isogenies due…

Let $P$ be a set of points in general position in the plane. Join all pairs of points in $P$ with straight line segments. The number of segment-crossings in such a drawing, denoted by $\crg(P)$, is the \emph{rectilinear crossing number} of…

We overview a series of recent works addressing numerical simulations of partial differential equations in the presence of some elements of randomness. The specific equations manipulated are linear elliptic, and arise in the context of…

This article deals with quantitative error analysis resulting from ellipsometric data obtained from measurement on curved surfaces including the influence of non-collimated beams. Numerical model based on the combination of geometrical and…

In a recent preprint, F. Calegari has shown that for $\ell = 2, 3, 5$ and 7 there exist 2-dimensional surjective representations $\rho$ of $\Gal(\bar{\Q}/\Q)$ with values in $\F_\ell$ coming from the $\ell$-torsion points of an elliptic…

The main purpose of the note is to exclude the existence of certain submaximal curves in fake projective planes. This will lead to lower bounds on multipoint Seshadri constants of `fake' $\mathcal{O}(1)$ on fake projective planes.

We examine the notion of strongly non-zero points and use it as a tool in the study of several types of elliptic pseudoprimes. Moreover, we give give some probabilistic results about the existence of strong elliptic pseudoprimes for a…

In this paper we determine the quadratic points on the modular curves X_0(N), where the curve is non-hyperelliptic, the genus is 3, 4 or 5, and the Mordell--Weil group of J_0(N) is finite. The values of N are 34, 38, 42, 44, 45, 51, 52, 54,…

Let $E_1, \ldots, E_s $ be $s$, not necessary distinct, elliptic curves over $\mathbb{Q}$. We give upper bounds on the frequency of $s$-tuples of points in $E_1(\mathbb{Q})\times \ldots \times E_s(\mathbb{Q})$ whose denominators or…

We study the problem of computing a minimum equivalent digraph (also known as the problem of computing a strong transitive reduction) and its maximum objective function variant, with two types of extensions. First, we allow to declare a set…

We consider nonconforming methods for symmetric elliptic problems and characterize their quasi-optimality in terms of suitable notions of stability and consistency. The quasi-optimality constant is determined and the possible impact of…

We study the cone of non-negative polynomials on generalized elliptic curves. We show that the zero set of every extreme ray has dense real points. If a generalized elliptic curve is embedded via a complete linear system, then we show that…