A singular K3 surface related to sums of consecutive cubes

We study the surface arising from the diophantine equation $m^3+(m+1)^3+...+(m+k-1)^3=l^2$. It turns out that this is a $K3$ surface with Picard number 20. We stduy its aritmetic properties in detail. We construct elliptic fibrations on it, and we find a parametric solution to the original equation. Also, we determine the Hasse-Weil zeta function of the surface over $Q$

Elliptic K3 surfaces associated with the product of two elliptic curves: Mordell-Weil lattices and their fields of definition

To a pair of elliptic curves, one can naturally attach two K3 surfaces: the Kummer surface of their product and a double cover of it, called the Inose surface. They have prominently featured in many interesting constructions in algebraic geometry and number theory. There are several more associated elliptic K3 surfaces, obtained through base change of the Inose surface; these have been previously studied by Kuwata. We give an explicit description of the geometric Mordell-Weil groups of each of these elliptic surfaces in the generic case (when the elliptic curves are non-isogenous). In the non-generic case, we describe a method to calculate explicitly a finite index subgroup of the Mordell-Weil group, which may be saturated to give the full group. Our methods rely on several interesting group actions, the use of rational elliptic surfaces, as well as connections to the geometry of low degree curves on cubic and quartic surfaces. We apply our techniques to compute the full Mordell-Weil group in several examples of arithmetic interest, arising from isogenous elliptic curves with complex multiplication, for which these K3 surfaces are singular.Comment: 44 pages. Final version. To appear in Nagoya Math.

Born-Infeld Lagrangian using Cayley-Dickson algebras

We rewrite the Born-Infeld Lagrangian, which is originally given by the determinant of a $4 \times 4$ matrix composed of the metric tensor $g$ and the field strength tensor $F$, using the determinant of a $(4 \cdot 2^n) \times (4 \cdot 2^n)$ matrix $H_{4 \cdot 2^{n}}$. If the elements of $H_{4 \cdot 2^{n}}$ are given by the linear combination of $g$ and $F$, it is found, based on the representation matrix for the multiplication operator of the Cayley-Dickson algebras, that $H_{4 \cdot 2^{n}}$ is distinguished by a single parameter, where distinguished matrices are not similar matrices. We also give a reasonable condition to fix the paramet

Analysis of magnetic characteristics of three-phase reactor made of grain-oriented silicon steel

Flux and iron loss distributions of three-phase reactor are analyzed using the finite element method considering 2-D B-H curves and iron losses in arbitrary directions which are measured up to high flux density. It is shown that the total iron loss of reactor yoke does not change so much by the yoke dimension, although the local iron loss is increased when the width of yoke is decreased. The experimental verification of flux and iron loss distributions are also carried out </p