Displaying similar documents to “An elliptic surface of Mordell-Weil rank 8 over the rational numbers”

Infinite rank of elliptic curves over a b

Bo-Hae Im, Michael Larsen (2013)

Acta Arithmetica

Similarity:

If E is an elliptic curve defined over a quadratic field K, and the j-invariant of E is not 0 or 1728, then E ( a b ) has infinite rank. If E is an elliptic curve in Legendre form, y² = x(x-1)(x-λ), where ℚ(λ) is a cubic field, then E ( K a b ) has infinite rank. If λ ∈ K has a minimal polynomial P(x) of degree 4 and v² = P(u) is an elliptic curve of positive rank over ℚ, we prove that y² = x(x-1)(x-λ) has infinite rank over K a b .

On the average value of the canonical height in higher dimensional families of elliptic curves

Wei Pin Wong (2014)

Acta Arithmetica

Similarity:

Given an elliptic curve E over a function field K = ℚ(T₁,...,Tₙ), we study the behavior of the canonical height h ̂ E ω of the specialized elliptic curve E ω with respect to the height of ω ∈ ℚⁿ. We prove that there exists a uniform nonzero lower bound for the average of the quotient ( h ̂ E ω ( P ω ) ) / h ( ω ) over all nontorsion P ∈ E(K).

Some examples of 5 and 7 descent for elliptic curves over Q

Tom Fisher (2001)

Journal of the European Mathematical Society

Similarity:

We perform descent calculations for the families of elliptic curves over Q with a rational point of order n = 5 or 7. These calculations give an estimate for the Mordell-Weil rank which we relate to the parity conjecture. We exhibit explicit elements of the Tate-Shafarevich group of order 5 and 7, and show that the 5-torsion of the Tate-Shafarevich group of an elliptic curve over Q may become arbitrarily large.

Elliptic curves over function fields with a large set of integral points

Ricardo P. Conceição (2013)

Acta Arithmetica

Similarity:

We construct isotrivial and non-isotrivial elliptic curves over q ( t ) with an arbitrarily large set of separable integral points. As an application of this construction, we prove that there are isotrivial log-general type varieties over q ( t ) with a Zariski dense set of separable integral points. This provides a counterexample to a natural translation of the Lang-Vojta conjecture to the function field setting. We also show that our main result provides examples of elliptic curves with an explicit...

On the uniqueness of elliptic K3 surfaces with maximal singular fibre

Matthias Schütt, Andreas Schweizer (2013)

Annales de l’institut Fourier

Similarity:

We explicitly determine the elliptic K 3 surfaces with section and maximal singular fibre. If the characteristic of the ground field is different from 2 , for each of the two possible maximal fibre types, I 19 and I 14 * , the surface is unique. In characteristic 2 the maximal fibre types are I 18 and I 13 * , and there exist two (resp. one) one-parameter families of such surfaces.

Convex integration and the L p theory of elliptic equations

Kari Astala, Daniel Faraco, László Székelyhidi Jr. (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

This paper deals with the L p theory of linear elliptic partial differential equations with bounded measurable coefficients. We construct in two dimensions examples of weak and so-called very weak solutions, with critical integrability properties, both to isotropic equations and to equations in non-divergence form. These examples show that the general L p theory, developed in [1, 24] and [2], cannot be extended under any restriction on the essential range of the coefficients. Our constructions...

Existence of positive radial solutions for the elliptic equations on an exterior domain

Yongxiang Li, Huanhuan Zhang (2016)

Annales Polonici Mathematici

Similarity:

We discuss the existence of positive radial solutions of the semilinear elliptic equation ⎧-Δu = K(|x|)f(u), x ∈ Ω ⎨αu + β ∂u/∂n = 0, x ∈ ∂Ω, ⎩ l i m | x | u ( x ) = 0 , where Ω = x N : | x | > r , N ≥ 3, K: [r₀,∞) → ℝ⁺ is continuous and 0 < r r K ( r ) d r < , f ∈ C(ℝ⁺,ℝ⁺), f(0) = 0. Under the conditions related to the asymptotic behaviour of f(u)/u at 0 and infinity, the existence of positive radial solutions is obtained. Our conditions are more precise and weaker than the superlinear or sublinear growth conditions. Our discussion is based on the...

Existence of a renormalized solution of nonlinear degenerate elliptic problems

Youssef Akdim, Chakir Allalou (2014)

Applicationes Mathematicae

Similarity:

We study a general class of nonlinear elliptic problems associated with the differential inclusion β ( u ) - d i v ( a ( x , D u ) + F ( u ) ) f in Ω where f L ( Ω ) . The vector field a(·,·) is a Carathéodory function. Using truncation techniques and the generalized monotonicity method in function spaces we prove existence of renormalized solutions for general L -data.

Fonctions biharmoniques adjointes

Emmanuel P. Smyrnelis (2010)

Annales Polonici Mathematici

Similarity:

The study of the equation (L₂L₁)*h = 0 or of the equivalent system L*₂h₂ = -h₁, L*₁h₁ = 0, where L j ( j = 1 , 2 ) is a second order elliptic differential operator, leads us to the following general framework: Starting from a biharmonic space, for example the space of solutions (u₁,u₂) of the system L₁u₁ = -u₂, L₂u₂ = 0, L j ( j = 1 , 2 ) being elliptic or parabolic, and by means of its Green pairs, we construct the associated adjoint biharmonic space which is in duality with the initial one.

An alternative way to classify some Generalized Elliptic Curves and their isotopic loops

Lucien Bénéteau, M. Abou Hashish (2004)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

The Generalized Elliptic Curves ( GECs ) are pairs ( Q , T ) , where T is a family of triples ( x , y , z ) of “points” from the set Q characterized by equalities of the form x . y = z , where the law x . y makes Q into a totally symmetric quasigroup. Isotopic loops arise by setting x * y = u . ( x . y ) . When ( x . y ) . ( a . b ) = ( x . a ) . ( y . b ) , identically ( Q , T ) is an entropic GEC and ( Q , * ) is an abelian group. Similarly, a terentropic GEC may be characterized by x 2 . ( a . b ) = ( x . a ) ( x . b ) and ( Q , * ) is then a Commutative Moufang Loop ( CML ) . If in addition x 2 = x , we have Hall GECs and ( Q , * ) is an exponent 3 CML . Any...