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

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As mentioned earlier, Step (I’) has to be completely reworked from Step (I) for the S-<strong>integral</strong> setting.i. We replace some of the bounds of Bugeaud and Győry [13] for the size of systemsof fundamental units and the size of solutions to S-unit equations with more recentand sharper bounds due to Győry and Yu [27].ii. We combine Matveev’s bounds for linear forms in logarithms [31] with Yu’s boundsfor linear forms in p-adic logarithms [53] and an idea due to Voutier [49] to get anupper bound for the size of solutions of a particular type of S-unit equations.iii. In [15] the authors use an upper bound for the regulator of a number field basedon a theorem of Landau. The theorem of Landau actually bounds the product ofthe regulator and the class number, which we need to bound in our case.iv. Our bounds for the size of the S-<strong>integral</strong> solutions on (1.0.1) depend on the classnumber of some number field. We use an upper bound for the class number dueto Lenstra [28].The improvements on the first step are the following. First, our constants aresmaller than those in [15], and do not depend on parameters related to Lehmer’s problem.This is due to the improvement on systems of fundamental S-units we use. Moreover,though Voutier’s idea had been already used to give effective upper bounds forthe size of S-<strong>integral</strong> solutions in [12], the bounds were inexplicit.Our bounds arecompletely explicit and they are based on more recent estimates for linear forms inlogarithms than those used in [12].Step (II’)-(a) in its present form can only be used to find <strong>integral</strong> <strong>points</strong> on Cwhen the genus of the curve is 2, and when the polynomial defining the curve has realroots only. The generalisation of the method to polynomials with complex roots defininggenus 2 curves is work in progress.Step (II’)-(b) is practically unchanged with respect to step (II) above, and wemerely summarise what we need for it from [15].3
We need the following assumptions for all steps:a. The knowledge of at least one rational point P 0 on C.b. The knowledge of a Mordell–Weil basis for J(Q) where J is the Jacobian of C.For the second step we also need to assume that the canonical height ĥ :J(Q) → R is explicitly computable and that we have explicit upper and lower boundsfor the differenceµ 1 ≤ h(D) − ĥ(D) ≤ µ 2. (1.0.2)For step (II’)-(a) we further require that generators of J(Q) are divisors supported by<strong>points</strong> on C(R) only.The assumption that we know a point on the curve brings simplifications to ourmethod. As remarked in [15], if we do not know any rational point on the curve, it ispossible that there are no rational <strong>points</strong> at all. This can be proved using the techniquesof Bruin and Stoll [9, 10, 11].We have made most of the necessary computations with the computational algebrasystem MAGMA [5]. Some computations have been performed with the Mathematicasystem [52].The thesis is arranged as follows. Chapter 2 outlines the existing methods forthe computation of the bounds for the height difference, and the computation of theMordell–Weil group. It consists on standard material, but it is needed to set up notationwhich will be used throughout the thesis. Chapter 3 is concerned with the computationof the upper bounds for the size of the S-<strong>integral</strong> <strong>points</strong> on the affine model 1.0.1.It also explains how one can completely determine all the S-<strong>integral</strong> <strong>points</strong> using thevariant of the Mordell–Weil sieve found in [15]. Chapter 5 deals with the reduction of theupper bounds obtained in Chapter 3 for the particular case of <strong>integral</strong> <strong>points</strong> on genus 2hyperelliptic curves. The method requires the computation of the periods of the curve,and hyperelliptic logarithms of degree 0 divisors on C to high precision. In Chapter 4 we4
Page 1 and 2: S-integral <strong
Page 3: 3.3 Systems of fundamental units .
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 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 ′
Page 72 and 73:
then F 1 is precisely of degree 2 a
Page 74 and 75:
Lemma 4.2.10 (III-3-a). We have z 1
Page 76 and 77:
and∫ x0wS(z 1 (x))z ′ 1 (x)dxy
Page 78 and 79:
intervals whose endpoints</
Page 80 and 81:
C to the integrals
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for [U n , V n ] and [W n , U n ],
Page 84 and 85:
Since I n converges, T n converges
Page 86 and 87:
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 √⎧
Page 92 and 93:
and⎧⎪⎨ −1, (a n,i = b and x
Page 94 and 95:
algorithm used for the multiplicati
Page 96 and 97:
interested in. The roots of f are m
Page 98 and 99:
assuming that the polynomial f defi
Page 100 and 101:
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
Page 112 and 113:
for the divisors D 1 = (−1, 1)
Page 114 and 115:
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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