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

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Chapter 1IntroductionConsider the hyperelliptic curve with affine modelC : Y 2 = a n X n + a n−1 X n−1 + · · · + a 0 , (1.0.1)where a 0 , . . . , a n are rational integers, a n ≠ 0, n ≥ 5. Let S be a finite set of rationalprimes. Siegel’s theorem [40] states that (1.0.1) has only finitely many S-<strong>integral</strong> solutions.Siegel’s theorem is superseded by Faltings’ theorem [20] that states that thereare only finitely many rational <strong>points</strong> on C. Both Siegel’s and Faltings’ results are ineffective.That is, their proofs do not lead to an algorithm to determine all the rational<strong>points</strong>.Using his theory of linear forms in logarithms, Baker [1] was the first to providean effective result concerning the size of the <strong>integral</strong> <strong>points</strong> on C. He showed that any<strong>integral</strong> point (X, Y ) on this affine model satisfiesmax(|X|, |Y |) ≤ exp exp exp{(n 10n H) n2 }, H = maxi=1,...,n {|a i|}.Baker’s result has been improved and extended to solutions in algebraic integers andS-integers by many authors (see [43, 48, 36, 2, 3, 12]). Despite the improvements, thebounds remain astronomical and often involve inexplicit constants.Recently, Bugeaud et al. [15] gave the first general practical method for explicitly1
computing <strong>integral</strong> <strong>points</strong> on affine models (1.0.1) of hyperelliptic curves. Their methodfalls into two steps.I. They give an upper bound for the size of <strong>integral</strong> <strong>points</strong>.II. They describe a variant of the Mordell–Weil sieve capable of searching up to thebounds found in the first step.In this thesis we give a method for explicitly computing S-<strong>integral</strong> <strong>points</strong> on hyperellipticcurves (1.0.1). As in [15], our strategy falls in two steps.I’. We compute a completely explicit upper bound for the size of S-<strong>integral</strong> <strong>points</strong>.This step is based in part on the ideas for step (I) above in [15]. The methodhas to be completely reworked for the S-<strong>integral</strong> setting. We not only extend step(I) to the S-<strong>integral</strong> case, but we also improve on the bounds given in [15] forthe size of the <strong>integral</strong> <strong>points</strong>. Nevertheless, the bounds are very large, and it isimpractical to do a search for S-<strong>integral</strong> <strong>points</strong> up to the bound. This material alsoforms a 22-page paper [25] accepted for publication in The International Journalof Number Theory.II’. We use one of the following methods.(a) We try to reduce the bound found in Step (I’) to manageable proportions,using lattice reduction. We show how the bound can be turned into an upperbound for the size of the coefficients of the linear combinations of the <strong>points</strong>on the Jacobian corresponding to <strong>integral</strong> <strong>points</strong> on the curve.We turnthe linear combination in the Mordell–Weil group J(Q) into linear forms inhyperelliptic logarithms. We give an upper bound for the linear forms and weuse it to reduce the upper bound for the coefficients of the linear combinationto manageable proportions.(b) We use step (II) above to sieve up to the bounds found in Step (I’).2
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 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
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Lemma 4.2.10 (III-3-a). We have z 1
Page 76 and 77:
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
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
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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
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[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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