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

Igor Burban; Thilo Henrich

Displaying similar documents to “Vector bundles on plane cubic curves and the classical Yang–Baxter equation”

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

Ricardo P. Conceição (2013)

Acta Arithmetica

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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

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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

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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

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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.

Non-special projectively normal line bundles on general k-gonal curves

E. Ballico (2004)

Annales Polonici Mathematici

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Let X be a sufficiently general smooth k-gonal curve of genus g and R ∈ Pic(X) the degree k spanned line bundle. We find an optimal integer z > 0 such that the line bundle $R^{\otimes z}$ is very ample and projectively normal.

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

Bo-Hae Im, Michael Larsen (2013)

Acta Arithmetica

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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 $K ℚ^{a b}$ .

$𝒟$ -bundles and integrable hierarchies

David Ben-Zvi, Thomas Nevins (2011)

Journal of the European Mathematical Society

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We study the geometry of $𝒟$ -bundles—locally projective $𝒟$ -modules—on algebraic curves, and apply them to the study of integrable hierarchies, specifically the multicomponent Kadomtsev–Petviashvili (KP) and spin Calogero–Moser (CM) hierarchies. We show that KP hierarchies have a geometric description as flows on moduli spaces of $𝒟$ -bundles; in particular, we prove that the local structure of $𝒟$ -bundles is captured by the full Sato Grassmannian. The rational, trigonometric, and elliptic solutions...

The automorphism group of ${\bar{M}}_{0, n}$

Andrea Bruno, Massimiliano Mella (2013)

Journal of the European Mathematical Society

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The paper studies fiber type morphisms between moduli spaces of pointed rational curves. Via Kapranov’s description we are able to prove that the only such morphisms are forgetful maps. This allows us to show that the automorphism group of ${\bar{M}}_{0, n}$ is the permutation group on $n$ elements as soon as $n \geq 5$ .

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

Nazar Arakelian, Herivelto Borges (2015)

Acta Arithmetica

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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...

A note on integral points on elliptic curves

Mark Watkins (2006)

Journal de Théorie des Nombres de Bordeaux

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We investigate a problem considered by Zagier and Elkies, of finding large integral points on elliptic curves. By writing down a generic polynomial solution and equating coefficients, we are led to suspect four extremal cases that still might have nondegenerate solutions. Each of these cases gives rise to a polynomial system of equations, the first being solved by Elkies in 1988 using the resultant methods of , with there being a unique rational nondegenerate solution. For the second...

Rational Points on Certain Hyperelliptic Curves over Finite Fields

Maciej Ulas (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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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.

The Analytic Rank of a Family of Jacobians of Fermat Curves

Tomasz Jędrzejak (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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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.