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

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then it should have strictly lower height. Using the upper bound for the height differencewe look for all <strong>points</strong> of naive logarithmic height less than ĥ((−1, 1))+µ 2. We use againthe command ReducedBasis to obtain independent generators for the group generatedby the <strong>points</strong> found in our search, and we obtain the same generators. Hence, none ofthe original divisors was of the form 2D, and they generate J(Q)/2J(Q). Finally, wecompute the heights of the representatives of each coset, and apply Zagier’s theoremto find a complete set of generators of J(Q). We see that the Mordell–Weil group isgenerated by the divisors above.17
Chapter 3Upper bounds for the size ofS-<strong>integral</strong> <strong>points</strong> on hyperellipticcurvesThe material presented here is a slightly expanded version of that in my paper [25],which is to appear in The International Journal of Number Theory.Consider the hyperelliptic curve with affine modelC : Y 2 = a n X n + a n−1 X n−1 + · · · + a 0 , (3.0.1)where a 0 , . . . , a n are rational integers, a n ≠ 0, n ≥ 5, where the polynomial on theright-hand side is irreducible. Let S be a finite set of rational primes. A rational numberx = p/q, p, q ∈ Z, (p, q) = 1, is an S-integer if q is either 1 or it is divisible by primes inS only. The size (or height) of x, x ≠ 0 is defined by max{log|a|, log|b|}. In this chapterwe give a method for explicitly computing an upper bound for the size of the S-<strong>integral</strong><strong>points</strong> on hyperelliptic curves (3.0.1). Our strategy is based in part on the ideas in [15](see Chapter 1 for a detailed explanation of the similarities and the differences betweenthe two approaches).We use a variant of Baker’s method to transform the problem of finding the S-18
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 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 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
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algorithm used for the multiplicati
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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
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
Page 116 and 117:
now want to estimate the height of
Page 118 and 119:
and for all x ≤ −c 16max{ ∫ x
Page 120 and 121:
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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