Genus one curves defined by separated variable polynomials and a polynomial Pell equation

Avanzi, Roberto; Zannier, Umberto

doi:10.4064/aa99-3-2

Cited by 31 publications

(64 citation statements)

References 13 publications

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“…We 336 Francesco Pappalardi and Alfred J. van der Poorten [2] suppose a to be nonzero, say of degree m at least g + 1. We will see that necessarily degb = m -g -1, that d e g / = g, and that / has leading coefficient m. In our example, m = 6 and g = 1.…”

Section: ) F ^^ = Log(a(x) + B(x)/d(x~))mentioning

confidence: 99%

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Pseudo-elliptic integrals, units, and torsion

Pappalardi¹,

Poorten²

2005

J. Aust. Math. Soc.

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We remark on pseudo-elliptic integrals and on exceptional function fields, namely function fields defined over an infinite base field but nonetheless containing non-trivial units. Our emphasis is on some elementary criteria that must be satisfied by a squarefree polynomial D{x) whose square root generates a quadratic function field with non-trivial unit. We detail the genus 1 case.2000 Mathematics subject classification: primary 11J70, 11A65, 11J68.

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Section: ) F ^^ = Log(a(x) + B(x)/d(x~))mentioning

confidence: 99%

“…The case g = 1 is completely known over Q, see [18] and its references, or for example [2]. In particular, one knows by Mazur's Theorem [13] that the only possibilities for m are m = 2, 3 , .…”

Section: The Quartic Casementioning

confidence: 99%

Pseudo-elliptic integrals, units, and torsion

Pappalardi¹,

Poorten²

2005

J. Aust. Math. Soc.

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“…Moreover, since all the irreducible components of C h dominate the curve C, it follows that the set C h (K ) can be infinite only in the case in which the curve C has geometric genus at most one. In the case in which C has genus zero, results equivalent to special cases of this question have already been studied ( [1,2,17,19,22]). We shall analyze completely the case in which the genus of C is one and the curve C h has infinitely many rational points.…”

Section: Introductionmentioning

confidence: 99%

Finite Weil restriction of curves

Flynn

Testa

2014

Monatsh Math

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Given number fields L ⊃ K , smooth projective curves C defined over L and B defined over K , and a non-constant L-morphism h : C → B L , we denote by C h the curve defined over K whose K -rational points parametrize the L-rational points on C whose images under h are defined over K . We compute the geometric genus of the curve C h and give a criterion for the applicability of the Chabauty method to find the points of the curve C h . We provide a framework which includes as a special case that used in Elliptic Curve Chabauty techniques and their higher genus versions. The set C h (K ) can be infinite only when C has genus at most 1; we analyze completely the case when C has genus 1.

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“…Finally, notice that in the other paper by Avanzi and Zannier [30] was obtained the classification of curves h P,Q (x, y) of genus 1 under condition that GCD(deg P, deg Q) = 1. Observe that together with the Ritt theorem this gives a complete classification of polynomials such that GCD(deg P, deg Q) = 1 and the equation P • f = Q • g has non-constant meromorphic solutions.…”

Section: Proofmentioning

confidence: 99%

Algebraic curvesP(x) −Q(y) = 0 and functional equations

Pakovich¹

2011

Complex Variables and Elliptic Equations

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In this paper we give several conditions implying the irreducibility of the algebraic curve P (x)−Q(y) = 0, where P, Q are rational functions. We also apply the results obtained to the functional equations P (f ) = Q(g) and P (f ) = cP (g), where c ∈ C. For example, we show that for a generic pair of rational functions P, Q the first equation has no non-constant solutions f, g meromorphic on C whenever (deg P − 1)(deg Q − 1) ≥ 2.

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Genus one curves defined by separated variable polynomials and a polynomial Pell equation

Cited by 31 publications

References 13 publications

Pseudo-elliptic integrals, units, and torsion

Pseudo-elliptic integrals, units, and torsion

Finite Weil restriction of curves

Algebraic curvesP(x) −Q(y) = 0 and functional equations

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