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

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ContentsList of TablesivList of FiguresvAcknowledgmentsviDeclarationsviiiAbstractixChapter 1 Introduction 1Chapter 2 The Mordell–Weil group 72.1 Basic definitions and theorems . . . . . . . . . . . . . . . . . . . . . . . 82.2 Bounds for the height difference . . . . . . . . . . . . . . . . . . . . . . 102.3 Computing the rank of J(Q) – 2-descent. . . . . . . . . . . . . . . . . . 132.4 The infinite descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Chapter 3Upper bounds for the size of S-<strong>integral</strong> <strong>points</strong> on hyperellipticcurves 183.1 Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1.1 The Odd Degree Case . . . . . . . . . . . . . . . . . . . . . . . 203.1.2 The Even Degree Case . . . . . . . . . . . . . . . . . . . . . . . 223.2 S-integers and heights . . . . . . . . . . . . . . . . . . . . . . . . . . . 23ii
3.3 Systems of fundamental units . . . . . . . . . . . . . . . . . . . . . . . 273.4 Upper bounds for the regulator and the class number . . . . . . . . . . . 293.5 Linear forms in logarithms . . . . . . . . . . . . . . . . . . . . . . . . . 313.6 Upper bounds for the size of the solutions to S-unit equations . . . . . . 323.7 Upper bounds for the size of S-<strong>integral</strong> <strong>points</strong> on hyperelliptic curves . . 373.8 The Mordell–Weil sieve . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Chapter 4 Periods of genus 2 curves defined over the reals 504.1 Reduction of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2 Bost and Mestre’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . 554.3 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.3.1 A note on precision . . . . . . . . . . . . . . . . . . . . . . . . . 84Chapter 5 Reduction of the upper bound for the size of the <strong>integral</strong> <strong>points</strong> 885.1 Analytic Jacobians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.2 Computation of periods . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.3 The linear form in hyperelliptic logarithms . . . . . . . . . . . . . . . . . 985.3.1 The odd degree case . . . . . . . . . . . . . . . . . . . . . . . . 995.3.2 The even degree case . . . . . . . . . . . . . . . . . . . . . . . . 1055.4 Reduction of the upper bound, the LLL-algorithm . . . . . . . . . . . . . 110Bibliography 116iii
Page 1: S-integral <strong
Page 6 and 7: AcknowledgmentsTo the glory of my G
Page 8 and 9: DeclarationsI declare that, except
Page 10 and 11: Chapter 1IntroductionConsider the h
Page 12 and 13: As mentioned earlier, Step (I’) h
Page 14 and 15: present an algorithm of Bost and Me
Page 16 and 17: Chapter 2The Mordell-Weil groupThe
Page 18 and 19: Denote the corresponding element of
Page 20 and 21: of C is the logarithmic naive heigh
Page 22 and 23: For the computation of the canonica
Page 24 and 25: Once we find r linearly independent
Page 26 and 27: then it should have strictly lower
Page 28 and 29: integral p
Page 30 and 31: and such that all y(P i ) are non-z
Page 32 and 33: mapped to x − α (mod Q ∗ K ∗
Page 34 and 35: the ring of integers of K is the us
Page 36 and 37: Proof. Recall that ‖ε‖ υ = 1
Page 38 and 39: Proof. Note that for any place υ i
Page 40 and 41: We will also need a bound for the c
Page 42 and 43: a large field, we get one in terms
Page 44 and 45: We will compute a lower bound for t
Page 46 and 47: Choose nowδ = 2c 9 (n, d)H h(µ 1
Page 48 and 49: c ∗ 14 = 2H∗ d s 1+s 2 −2 P d
Page 50 and 51: To ease notation we write A 1 + A 2
Page 52 and 53:
Let f(x) = x 5 − 16x + 8. Let α
Page 54 and 55:
3.8 The Mordell-Weil sieveWe have g
Page 56 and 57:
of the first 22 primes. We transfor
Page 58 and 59:
The reader can find the MAGMA progr
Page 60 and 61:
The AGM of two positive real number
Page 62 and 63:
Note that sgn(t) does not change al
Page 64 and 65:
4.2 Bost and Mestre’s algorithmIn
Page 66 and 67:
Lemma 4.2.4 (I-3). Let P, Q ∈ R[X
Page 68 and 69:
and[U, V ] = −2R∆(P, Q, R),[V,
Page 70 and 71:
andU(x) = [Q, R](x) = (b + b ′
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then F 1 is precisely of degree 2 a
Page 74 and 75:
Lemma 4.2.10 (III-3-a). We have z 1
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and∫ x0wS(z 1 (x))z ′ 1 (x)dxy
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intervals whose endpoints</
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C to the integrals
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for [U n , V n ] and [W n , U n ],
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Since I n converges, T n converges
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and a sum of integral</stro
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2 = a 1a ′ 1 − b 1b ′ 1 + √
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• Else do- PutandIH i = θ √⎧
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and⎧⎪⎨ −1, (a n,i = b and x
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algorithm used for the multiplicati
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interested in. The roots of f are m
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assuming that the polynomial f defi
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the g × 2g matrix⎛∫∫∫ ⎞A
Page 102 and 103:
curves with the point at infinity a
Page 104 and 105:
We are now interested in the numeri
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conformed ourselves with estimates
Page 108 and 109:
Let P be an integral</stron
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We will now find an upper bound for
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for the divisors D 1 = (−1, 1)
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Proof. Since c 16 > 0, then X(P ) i
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now want to estimate the height of
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and for all x ≤ −c 16max{ ∫ x
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andφ(2D i )2= (d i,1 , d i,2 ) ∈
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Again, by our assumption, the right
Page 124 and 125:
The reader can find the MAGMA progr
Page 126 and 127:
[8] Nils Bruin, Chabauty methods us
Page 128 and 129:
[30] Serge Lang, Algebraic number t
Page 130:
[51] Michel Waldschmidt, Diophantin
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S-integral points on hyperelliptic curves Homero Renato Gallegos ...

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