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

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present an algorithm of Bost and Mestre [6] for the computation of the periods, and wework on an extension of it so that we can also compute the hyperelliptic logarithms.We now briefly describe the available tools for the solution of Diophantine hyperellipticequations. After Siegel’s proof of the finiteness of the number of S-<strong>integral</strong><strong>points</strong> on curves of higher genus, effective methods for finding all the S-<strong>integral</strong> <strong>points</strong>on hyperelliptic curves have been developed. Some of them are not general, and thosethat are general are often impractical or inexplicit.1. Chabauty’s method. Chabauty [17] proved that a curve of genus g ≥ 2 has afinite number of rational <strong>points</strong> whenever the rank of the Mordell–Weil group ofits Jacobian is strictly less than g. His proof often leads to a sharp bound on thenumber of <strong>points</strong> on C. If the bound is attained, then one has not only the <strong>integral</strong><strong>points</strong>, but all the rational <strong>points</strong>. This requires knowledge of generators for theMordell–Weil group or a subgroup of finite index. For notes on the applicationof this method see [16, Chapter 13] or [32]. The examples we present in thisthesis are genus 2 curves with Mordell–Weil rank ≥ 2, so Chabauty’s methoddoes not apply. There are also some variants of Chabauty’s method (as EllipticCurve Chabauty [7], [8], [23], [24]) that can determine all the rational <strong>points</strong> on ahyperelliptic curve, but they are also limited to curves of small rank.2. Bilu and Hanrot’s variant of Baker’s method Bilu and Hanrot [4] have produceda method to solve superelliptic Diophantine equations (i.e. of the formY m = f(X) with m ≥ 3) which computes an upper bound for the size of the<strong>integral</strong> <strong>points</strong>. Their method does not involve Thue or unit equations since theycompute directly a linear form in logarithms from a careful analysis of some numberfields. The upper bound they get is huge, but it is then reduced by an applicationof the LLL-algorithm. The authors claim that one can adjust the methodto the hyperelliptic case, but they do not do it explicitly. The method needs somecomputations of class groups of number fields of large degree (≥ 20) that in manycases in practice are beyond the present abilities of computer algebra.5
3. Baker’s method and the Mordell–Weil sieve. A third method is the one we havementioned before: Baker’s method combined with the variant of the Mordell–Weilsieve [15] to find the <strong>integral</strong> <strong>points</strong>. This is the more general method, as it doesnot depend on the genus, and it does not involve computations of unit groups,but only bounds on the regulator of large degree number fields which are easyto compute. Nevertheless, the computations for the Mordell–Weil sieve can beexpensive. Thus, one would like to reduce the upper bound obtained from Baker’smethod as much as possible, to sieve up to a not so large bound, or not to haveto sieve at all.6
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 12 and 13: As mentioned earlier, Step (I’) h
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
Page 82 and 83:
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
Page 88 and 89:
2 = a 1a ′ 1 − b 1b ′ 1 + √
Page 90 and 91:
• 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
Page 106 and 107:
conformed ourselves with estimates
Page 108 and 109:
Let P be an integral</stron
Page 110 and 111:
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
Page 118 and 119:
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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