S-integral points on hyperelliptic curves Homero Renato Gallegos ...

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List of Figures5.1 An octag<strong>on</strong> defining a genus 2 surface. . . . . . . . . . . . . . . . . . . 905.2 The Riemann surface of a <strong>hyperelliptic</strong> curve. . . . . . . . . . . . . . . . 935.3 Paths below the cycles in the homology basis. . . . . . . . . . . . . . . . 935.4 Cycles above [a i , a i+1 ] compared to the chosen homology basis. . . . . 95v
Page 1 and 2: S-integral <strong
Page 3: 3.3 Systems of fundamental units .
Page 7 and 8: Agradecimientos 1Para la gloria y l
Page 9 and 10: AbstractLet C : Y 2 = a n X n + ·
Page 11 and 12: computing integral
Page 13 and 14: We need the following assumptions f
Page 15 and 16: 3. Baker’s method and the Mordell
Page 17 and 18: 2.1 Basic definitions and theoremsT
Page 19 and 20: The Kummer surface is then given by
Page 21 and 22: It can be shown (see [22]) that for
Page 23 and 24: common divisor g of the orders of t
Page 25 and 26: Once we know a set of generators fo
Page 27 and 28: Chapter 3Upper bounds for the size
Page 29 and 30: Section 3.8 explains how the varian
Page 31 and 32: 3.1.2 The Even Degree CaseWe are as
Page 33 and 34: Hence for α ∈ K ∗ , the produc
Page 35 and 36: and for any place υ ∈ M Klog‖
Page 37 and 38: Lemma 3.3.1. Define the constantsc
Page 39 and 40: Lemma 3.4.1. Let K be a number fiel
Page 41 and 42: Thenlog|Λ| > −c 7 (n, d)A 1 ·
Page 43 and 44: Then( )ν1 ε 1h ≤ max{c 10 , c 1
Page 45 and 46: Next assume that B > 2e max((s 1 +
Page 47 and 48: Theorem 3.7.1. Let S be a finite se
Page 49 and 50: Write ν 1 = ν ′ ε and ε 1 =
Page 51 and 52: We use Theorem 3.7.1 in conjunction
Page 53 and 54: If the point P is an integr
Page 55 and 56:
Set L 0 = BZ r . The second step re
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whereB 1 = 518674189692551307183008
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Chapter 4Periods of genus 2 curves
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Under the correspondences, every po
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where, in the first case x < a, and
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Remark. If P has degree ≤ 1, then
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Proof. Consider the matrix⎛⎞P (
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Let P, Q, R ∈ R[X] be three degre
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since the expression on the left eq
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for all x ∈ [c, c ′ ]. Define f
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Q(x)V (z 2 (x)) = 0, but since V (u
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and for x 0 ∈ [u, b ′ ]∫ x0bS
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phically with p. It can be given by
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and that similar relations hold for
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Lemma 4.2.13 (IV -2 and 3). The seq
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We start then with one point x ∈
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The following is Bost and Mestre’
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Procedure ALet x n,i ∈ (a n+1 , a
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where a n,i is the largest of a n+1
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INPUT:Six real numbers a < a ′ <
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eal roots given bya 1 = −1.960026
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Chapter 5Reduction of the upper bou
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presented in Chapter 4. Section 5.3
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Theorem 5.1.2 (Abel’s Theorem). L
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A 1 the lift of the path around a 1
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⎛⎜2 ∫ ⎞⎛a 3 √dta 2 f(t)
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where a is a root of f, x ≥ a, an
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surface. In order to explicitly com
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Similarly, if D 1 , . . . , D n are
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with |n i | ≤ ˜trM/2 + ˜t (reca
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integers if X, Y ∈ Z. Then, the p
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Let t be the order of the torsion s
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The inequality from Lemma 5.3.8 imp
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Let m 1 , . . . , m r , n 1 , n 2
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In the example we give at the end o
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Bibliography[1] A. Baker, Bounds fo
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[19] Régis Dupont, Moyenne arith
Page 129 and 130:
[40] Carl L. Siegel, Über einige A
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S-integral points on hyperelliptic curves Homero Renato Gallegos ...

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