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We explain how recent work on 3-descent and 4-descent for elliptic curves over Q can be combined to search for generators of the Mordell-Weil group of large height. As an application we show that every elliptic curve of prime conductor in…

This is the fourth of a series of papers studying real algebraic threefolds, but the methods are mostly independent from the previous ones. Let $f:X\to C$ be a map of a smooth projective real algebraic 3-fold to a curve $C$ whose general…

Let $V_1$ be the Fano threefold given as a hypersurface of degree 6 in $P(1,1,1,2,3)$ (over a number field $K$). Then there exists a finite extension $K'/K$ such that the set of $K'$-rational points of $X$ is Zariski dense.

Let $M$ be the moduli space of rank $2$ stable bundles with fixed determinant of degree $1$ on a smooth projective curve $C$ of genus $g\ge 2$. When $C$ is generic, we show that any elliptic curve on $M$ has degree (respect to…

We establish some upper and lower bounds of the rational topological complexity for certain classes of elliptic spaces. Our techniques permit us in particular to show that the rational topological complexity coincides with the dimension of…

We study deformation theory of elliptic fibre bundles over curves in positive characteristics. As applications, we give examples of non-liftable elliptic surfaces in charactertic two and three, which answers a question of Katsura and Ueno.…

Let $E$ be an elliptic curve over a number field $K$. Descent calculations on $E$ can be used to find upper bounds for the rank of the Mordell-Weil group, and to compute covering curves that assist in the search for generators of this…

We study rationality problems for smooth complete intersections of two quadrics. We focus on the three-dimensional case, with a view toward understanding the invariants governing the rationality of a geometrically rational threefold over a…

This is a paper in smooth $4$-manifold topology, inspired by the N\'{e}ron-Lang Theorem in number theory. More precisely, we prove that a smooth version $\MW(\pi)$ of Mordell-Weil group of an elliptic fibration $\pi:M\to\Pb^1$ is finitely…

In a previous paper, the author examined the possible torsions of an elliptic curve over the quadratic fields $\mathbb Q(i)$ and $\mathbb Q(\sqrt{-3})$. Although all the possible torsions were found if the elliptic curve has rational…

We survey our contributions on the classification of elliptic fibrations on K3 surfaces with a non-symplectic involution. We place them in the more general framework of K3 surfaces with an involution without any hypothesis on its fixed…

We consider the problem of counting the number of rational points on the family of Kummer surfaces associated with two non-isogenous elliptic curves. For this two-parameter family we prove Manin's unity, using the presentation of the Kummer…

We study tilings of the plane composed of two repeating tiles of different assigned areas relative to an arbitrary periodic lattice. We classify isoperimetric configurations (i.e., configurations with minimal length of the interfaces) both…

We prove that elliptic K3 surfaces over a number field which admit a second elliptic fibration satisfy the potential Hilbert property. Equivalently, the set of their rational points is not thin after a finite extension of the base field.…

A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. Recently it was shown that the Diophantine equations describing such a cuboid…

We study the surface area of an ellipsoid in n-dimensional Euclidean space as the function of the lengths of their major semi-axes. We write down an explicit formula as an integral over the unit sphere, use the formula to derive convexity…

The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the set of closed geodesics is dense in the space of geodesics.

A K3 surface over a number field has infinitely many rational points over a finite field extension. For K3 surfaces of degree 2, arising as double covers of $\mathbb{P}^2$ branched along a smooth sextic curve, we give a bound for the degree…

A new, simple method to approach enumerative questions about rational curves on rational surfaces is described. Applications include a short proof of Kontsevich's formula for plane curves and a the solution of the analogous problem for the…

Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the…