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

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DeclarationsI declare that, except where acknowledged, the material contained in this thesis is myown work and it has not been submitted elsewere for the purpose of obtaining an academicdegree.Chapter 2 consists on standard material which is needed in the rest of the work.The material in Chapter 3 has been accepted for publication in The InternationalJournal of Number Theory.Homero Renato Gallegos Ruiz.December 2010.viii
AbstractLet C : Y 2 = a n X n + · · · + a 0 be a hyperelliptic curve with the a i rational integers,n ≥ 5, and the polynomial on the right irreducible. Let J be its Jacobian. Let S be afinite set of rational primes. In this thesis we give explicit methods for finding all of the<strong>integral</strong> and S-<strong>integral</strong> <strong>points</strong> on C. The work consists of the following parts.1. We give a completely explicit upper bound for the size of the S-<strong>integral</strong> <strong>points</strong> onthe model C, provided we know at least one rational point on C and a Mordell–Weilbasis for J(Q). There is a refinement of the Mordell–Weil sieve that can then beused to determine all the S-<strong>integral</strong> <strong>points</strong> on the curve.2. In the case the curve has genus 2 and the polynomial defining the curve hasreal roots only, we reduce the upper bound for the size of the <strong>integral</strong> <strong>points</strong> tomanageable proportions using linear forms in hyperelliptic logarithms. We thenfind all of the <strong>integral</strong> <strong>points</strong> on C by a direct search.3. We give an algorithm for the computation of hyperelliptic logarithms of real <strong>points</strong>on genus 2 curves defined by a polynomial having real roots only. This is neededfor 2.We illustrate the practicality of the method by finding all the <strong>integral</strong> <strong>points</strong> onthe curve Y 2 = f(X) = X 5 − 5X 3 − X 2 + 3X + 1, and all the S-<strong>integral</strong> <strong>points</strong> on thecurve Y 2 − Y = X 5 − X for the set S of the first 22 primes.ix
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 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
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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
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curves with the point at infinity a
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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
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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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