A note on integral points on elliptic curves

Mark Watkins

Displaying similar documents to “A note on integral points on elliptic curves”

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...

Optimal curves differing by a 5-isogeny

Dongho Byeon, Taekyung Kim (2014)

Acta Arithmetica

Similarity:

For i = 0,1, let $E_{i}$ be the $X_{i} (N)$ -optimal curve of an isogeny class of elliptic curves defined over ℚ of conductor N. Stein and Watkins conjectured that E₀ and E₁ differ by a 5-isogeny if and only if E₀ = X₀(11) and E₁ = X₁(11). In this paper, we show that this conjecture is true if N is square-free and is not divisible by 5. On the other hand, Hadano conjectured that for an elliptic curve E defined over ℚ with a rational point P of order 5, the 5-isogenous curve E’ := E/⟨P⟩ has a rational...

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.

Vector bundles on plane cubic curves and the classical Yang–Baxter equation

Igor Burban, Thilo Henrich (2015)

Journal of the European Mathematical Society

Similarity:

In this article, we develop a geometric method to construct solutions of the classical Yang–Baxter equation, attaching a family of classical $r$ -matrices to the Weierstrass family of plane cubic curves and a pair of coprime positive integers. It turns out that all elliptic $r$ -matrices arise in this way from smooth cubic curves. For the cuspidal cubic curve, we prove that the obtained solutions are rational and compute them explicitly. We also describe them in terms of Stolin’s classication...

Infinite rank of elliptic curves over $ℚ^{a b}$

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 $E (ℚ^{a b})$ 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 $E (K ℚ^{a b})$ 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 $K ℚ^{a b}$ .

Rational Points on Certain Hyperelliptic Curves over Finite Fields

Maciej Ulas (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Let K be a field, a,b ∈ K and ab ≠ 0. Consider the polynomials g₁(x) = xⁿ+ax+b, g₂(x) = xⁿ+ax²+bx, where n is a fixed positive integer. We show that for each k≥ 2 the hypersurface given by the equation $S_{k}^{i} : u ² = \prod_{j = 1}^{k} g_{i} (x_{j})$ , i=1,2, contains a rational curve. Using the above and van de Woestijne’s recent results we show how to construct a rational point different from the point at infinity on the curves $C_{i} : y ² = g_{i} (x)$ , (i=1,2) defined over a finite field, in polynomial time.

An elliptic surface of Mordell-Weil rank $8$ over the rational numbers

Charles F. Schwartz (1994)

Journal de théorie des nombres de Bordeaux

Similarity:

Néron showed that an elliptic surface with rank $8$ , and with base $B = P_{1} ℚ$ , and geometric genus $= 0$ , may be obtained by blowing up $9$ points in the plane. In this paper, we obtain parameterizations of the coefficients of the Weierstrass equations of such elliptic surfaces, in terms of the $9$ points. Manin also describes bases of the Mordell-Weil groups of these elliptic surfaces, in terms of the $9$ points ; we observe that, relative to the Weierstrass form of the equation, $Y^{2} = X^{3} + A X^{2} + B X + C$ ...

Small exponent point groups on elliptic curves

Florian Luca, James McKee, Igor E. Shparlinski (2006)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let $E$ be an elliptic curve defined over $F_{q}$ , the finite field of $q$ elements. We show that for some constant $η > 0$ depending only on $q$ , there are infinitely many positive integers $n$ such that the exponent of $E (F_{q^{n}})$ , the group of $F_{q^{n}}$ -rational points on $E$ , is at most $q^{n} exp (- n^{η / log log n})$ . This is an analogue of a result of R. Schoof on the exponent of the group $E (F_{p})$ of $F_{p}$ -rational points, when a fixed elliptic curve $E$ is defined over $ℚ$ and the prime $p$ tends to infinity.

Frobenius nonclassicality with respect to linear systems of curves of arbitrary degree

Nazar Arakelian, Herivelto Borges (2015)

Acta Arithmetica

Similarity:

For each integer s ≥ 1, we present a family of curves that are $_{q}$ -Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case s=2, we give necessary and sufficient conditions for such curves to be $_{q}$ -Frobenius nonclassical with respect to the linear system of conics. In the $_{q}$ -Frobenius nonclassical cases, we determine the exact number of $_{q}$ -rational points. In the remaining cases, an upper bound for the number of $_{q}$ -rational points will follow from Stöhr-Voloch...

The Analytic Rank of a Family of Jacobians of Fermat Curves

Tomasz Jędrzejak (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

We study the family of curves $F_{m} (p) : x^{p} + y^{p} = m$ , where p is an odd prime and m is a pth power free integer. We prove some results about the distribution of root numbers of the L-functions of the hyperelliptic curves associated to the curves $F_{m} (p)$ . As a corollary we conclude that the jacobians of the curves $F_{m} (5)$ with even analytic rank and those with odd analytic rank are equally distributed.