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Rank of elliptic curves associated to Brahmagupta quadrilaterals

Farzali Izadi, Foad Khoshnam, Arman Shamsi Zargar (2016)

Colloquium Mathematicae

We construct a family of elliptic curves with six parameters, arising from a system of Diophantine equations, whose rank is at least five. To do so, we use the Brahmagupta formula for the area of cyclic quadrilaterals (p³,q³,r³,s³) not necessarily representing genuine geometric objects. It turns out that, as parameters of the curves, the integers p,q,r,s along with the extra integers u,v satisfy u⁶+v⁶+p⁶+q⁶ = 2(r⁶+s⁶), uv = pq, which, by previous work, has infinitely many integer solutions.

Ranks of quadratic twists of elliptic curves

Mark Watkins, Stephen Donnelly, Noam D. Elkies, Tom Fisher, Andrew Granville, Nicholas F. Rogers (2014)

Publications mathématiques de Besançon

We report on a large-scale project to investigate the ranks of elliptic curves in a quadratic twist family, focussing on the congruent number curve. Our methods to exclude candidate curves include 2-Selmer, 4-Selmer, and 8-Selmer tests, the use of the Guinand-Weil explicit formula, and even 3-descent in a couple of cases. We find that rank 6 quadratic twists are reasonably common (though still quite difficult to find), while rank 7 twists seem much more rare. We also describe our inability to find...

Rational fixed points for linear group actions

Pietro Corvaja (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We prove a version of the Hilbert Irreducibility Theorem for linear algebraic groups. Given a connected linear algebraic group G , an affine variety V and a finite map π : V → G , all defined over a finitely generated field κ of characteristic zero, Theorem 1.6 provides the natural necessary and sufficient condition under which the set π ( V ( κ ) ) contains a Zariski dense sub-semigroup Γ ⊂ G ( κ ) ; namely, there must exist an unramified covering p : G ˜ → G and a map θ : G ˜ → V such that π ∘ θ = p . In the case κ = ℚ , G = 𝔾 a is the additive group, we reobtain the...

Rational periodic points for quadratic maps

Jung Kyu Canci (2010)

Annales de l’institut Fourier

Let K be a number field. Let S be a finite set of places of K containing all the archimedean ones. Let R S be the ring of S -integers of K . In the present paper we consider endomorphisms of ℙ 1 of degree 2 , defined over K , with good reduction outside S . We prove that there exist only finitely many such endomorphisms, up to conjugation by PGL 2 ( R S ) , admitting a periodic point in ℙ 1 ( K ) of order > 3 . Also, all but finitely many classes with a periodic point in ℙ 1 ( K ) of order 3 are parametrized by an irreducible curve.

Rational points on a subanalytic surface

Jonathan Pila (2005)

Annales de l’institut Fourier

Let X ⊂ ℝ n be a compact subanalytic surface. This paper shows that, in a suitable sense, there are very few rational points of X that do not lie on some connected semialgebraic curve contained in X .

Currently displaying 921 – 940 of 1274