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

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To ease notation we write A 1 + A 2 log(max{h(ε 1 ), h(ε 2 ), 1}) for the expression thatattains the maximum in the last inequality, where we take A 2 to be 0 if the maximum isattained in one of the first two cases. Note thath(ε i ) ≤ h(ν i ε i ) + h(ν 1 ) ≤ h(ν i ε i ) + H ∗ ,since ε i = ν i ε i /ν i . We derive as in [15] thath(τ i ± τ j ) ≤ 2 log 2 + 3 h(κ) + h(α) + h(x).Hence,( )τ1 − τ 2h≤ A 1 + A 2 log(A 3 + h(x)), (3.7.2)τ 2 − τ 3where A 3 = H ∗ + 2 log 2 + 3 h(κ) + h(α). We also apply Proposition 3.6.1 to the unitequation−(τ 1 + τ 2 ) + (τ 3 + τ 1 ) + (τ 2 − τ 3 ) = 0,to obtain precisely the same bound for h(τ1 +τ 2τ 2 −τ 3). Using the identitywe obtain that( ) ( )τ1 − τ 2 τ1 + τ 2·= κ 1(α 2 − α 1 )τ 2 − τ 3 τ 2 − τ 3 (τ 2 − τ 3 ) 2 ,h(τ 2 − τ 3 ) ≤log 2 + h(κ)2+ h(α) + A 1 + A 2 log(A 3 + h(x)).We derive as in [15] thath(x) ≤ 7 log 2 + 5 h(α) + 3 h(κ) + 4 h(τ 2 − τ 3 ).Thush(x) ≤ 9 log 2 + 9 h(α) + 5 h(κ) + 4A 1 + 4A 2 log(A 3 + h(x)).If A 2 = 0 we obtain (3.7.1) directly. Otherwise, using Lemma 9.1 of [15] we completethe proof of the theorem.41
We use Theorem 3.7.1 in conjunction with Lemmas 3.1.1 and 3.1.2 to obtainan upper bound for the size of all S-<strong>integral</strong> <strong>points</strong> on a hyperelliptic curve withmodel (3.1.1).Remark. We use here the notation of Section 3.1. In the even degree case, the setK is the union of the sets ɛB ɛ , where ɛ ∈ E. The set B ɛ consists of square free rationalintegers supported by primes dividing a∆ Norm K/Q (ɛ) ∏ p∈Sp. Hence, the fieldsK 1 , K 2 , K 3 and L defined in the statement of the previous theorem are the same forall κ ∈ ɛB ɛ .In order to get the desired upper bound for the size of all S-<strong>integral</strong><strong>points</strong> we only need to compute the bound for the κ ∈ ɛB ɛ that maximises h(κ) andNorm Q(αi ,α j )/Q(κ i (α i − α j )), namely κ = bɛ where b is the largest square-free rationalinteger dividing a∆ Norm K/Q (ɛ) ∏ p∈Sp. Thus, though in the even degree case the setK is much bigger than in the odd degree case, the computation of the upper bound forthe size of the S-<strong>integral</strong> <strong>points</strong> is essentially not more complicated than that in the odddegree case.Example 3.7.2. This is a continuation of Example 2.4.2. Let C be the curve defined byY 2 = 4X 5 − 4X + 1,and let S be the set of the first 22 primes. We transform the equation intoC : 2y 2 = x 5 − 16x + 8, (3.7.3)via the change of variables y = 2Y and x = 2X which preserves S-<strong>integral</strong>ity. Thecurve C given by the model (3.7.3) is the same curve as the one in the proof of Theorem1.1 of [15]. The authors compute an upper bound for the size of the <strong>integral</strong> <strong>points</strong> onthe curve. We will here compute a bound for the S-<strong>integral</strong> <strong>points</strong>. Denote by J theJacobian of C. We had seen in Example 2.4.2 that J(Q) is free of rank 3. A set ofgenerators for J(Q) is given byD 1 = (0, 2) − ∞, D 2 = (2, 2) − ∞, D 3 = (−2, 2) − ∞.42
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 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 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
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 ) ∈
Page 122 and 123:
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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