Results 11 to 20 of about 27 (27)

How to generate all integral triangles containing a given angle

International Journal of Mathematics and Mathematical Sciences, Volume 24, Issue 8, Page 569-572, 2000., 2000
We present an elementary prescription based on the rational secant method for generating all the integral triangles containing a given angle of rational cosine. This is a direct generalization of the ancient problem of finding all the Pythagorean triples. As an example, we discuss a specific equation studied by Diophantus of Alexandria, which turns out
Nelson Petulante, Ifeoma Kaja
wiley   +1 more source

$E_{6}$ AND THE ARITHMETIC OF A FAMILY OF NON-HYPERELLIPTIC CURVES OF GENUS 3

Forum of Mathematics, Pi, 2015
We study the arithmetic of a family of non-hyperelliptic curves of genus 3 over the field $\mathbb{Q}$ of rational numbers. These curves are the nearby fibers of the semi-universal deformation of a simple singularity of type $E_{6}$. We show that average
JACK A. THORNE
doaj   +1 more source

NON-ARCHIMEDEAN YOMDIN–GROMOV PARAMETRIZATIONS AND POINTS OF BOUNDED HEIGHT

Forum of Mathematics, Pi, 2015
We prove an analog of the Yomdin–Gromov lemma for $p$-adic definable sets and more broadly in a non-Archimedean definable context. This analog keeps track of piecewise approximation by Taylor polynomials, a nontrivial aspect in the totally disconnected ...
RAF CLUCKERS, GEORGES COMTE, FRANÇOIS LOESER   +2 more
doaj   +1 more source

Complete verification of strong BSD for many modular abelian surfaces over ${\mathbf {Q}}$

Forum of Mathematics, Sigma
We develop the theory and algorithms necessary to be able to verify the strong Birch–Swinnerton-Dyer Conjecture for absolutely simple modular abelian varieties over ${\mathbf {Q}}$ . We apply our methods to all 28 Atkin–Lehner quotients of $X_0(
Timo Keller, Michael Stoll
doaj   +1 more source

Rational torsion points on abelian surfaces with quaternionic multiplication

Forum of Mathematics, Sigma
Let A be an abelian surface over ${\mathbb {Q}}$ whose geometric endomorphism ring is a maximal order in a non-split quaternion algebra. Inspired by Mazur’s theorem for elliptic curves, we show that the torsion subgroup of $A({\mathbb {Q}})$
Jef Laga, Ciaran Schembri, Ari Shnidman, John Voight   +3 more
doaj   +1 more source

Rank jumps and multisections of elliptic fibrations on K3 surfaces

Forum of Mathematics, Sigma
We consider the countably many families $\mathcal {L}_d$ , $d\in \mathbb {N}_{\geq 2}$ , of K3 surfaces admitting an elliptic fibration with positive Mordell–Weil rank. We prove that the elliptic fibrations on the very general member of these
Alice Garbagnati, Cecília Salgado
doaj   +1 more source

The Manin–Peyre conjecture for smooth spherical Fano varieties of semisimple rank one

Forum of Mathematics, Sigma
The Manin–Peyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type T as well as some higher-dimensional smooth spherical Fano varieties.
Valentin Blomer, Jörg Brüdern, Ulrich Derenthal, Giuliano Gagliardi   +3 more
doaj   +1 more source

Average Analytic Ranks of Elliptic Curves over Number Fields

Forum of Mathematics, Sigma
We give a conditional bound for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field K are modular and have L-functions which satisfy the ...
Tristan Phillips
doaj   +1 more source

Improvements on dimension growth results and effective Hilbert’s irreducibility theorem

Forum of Mathematics, Sigma
We sharpen and generalize the dimension growth bounds for the number of points of bounded height lying on an irreducible algebraic variety of degree d, over any global field.
Raf Cluckers, Pierre Dèbes, Yotam I. Hendel, Kien Huu Nguyen, Floris Vermeulen   +4 more
doaj   +1 more source

The groups of points on abelian varieties over finite fields

Open Mathematics, 2010
Rybakov Sergey
doaj   +1 more source