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Generalised Hermite constants, Voronoi theory and heights on flag varieties

Bertrand Meyer (2009)

Bulletin de la Société Mathématique de France

This paper explores the study of the general Hermite constant associated with the general linear group and its irreducible representations, as defined by T. Watanabe. To that end, a height, which naturally applies to flag varieties, is built and notions of perfection and eutaxy characterising extremality are introduced. Finally we acquaint some relations (e.g., with Korkine–Zolotareff reduction), upper bounds and computation relative to these constants.

Generators and integer points on the elliptic curve y² = x³ - nx

Yasutsugu Fujita, Nobuhiro Terai (2013)

Acta Arithmetica

Let E be an elliptic curve over the rationals ℚ given by y² = x³ - nx with a positive integer n. We consider first the case where n = N² for a square-free integer N. Then we show that if the Mordell-Weil group E(ℚ ) has rank one, there exist at most 17 integer points on E. Moreover, we show that for some parameterized N a certain point P can be in a system of generators for E(ℚ ), and we determine the integer points in the group generated by the point P and the torsion points. Secondly, we consider...

Generators and integral points on twists of the Fermat cubic

Yasutsugu Fujita, Tadahisa Nara (2015)

Acta Arithmetica

We study integral points and generators on cubic twists of the Fermat cubic curve. The main results assert that integral points can be in a system of generators in the case where the Mordell-Weil rank is at most two. As a corollary, we explicitly describe the integral points on the curve.

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