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

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Denote the corresponding element of Pic 2 (C) by D. We will choose ∞ + + ∞ − asa representative for D. Since D is the canonical class, the Riemann Roch Theoremimplies that any other element of Pic 2 (C) contains precisely one effective divisor P + Qfixed as a pair by Gal( ¯Q/Q) (see [16, Chapter 1, Section 1]). We will identify any suchelement of Pic 2 (C) with its unique effective divisor. We can identify Pic 0 (C) with Pic 2 (C)via addition by D. Doing so identifies 0 ∈ Pic 0 (C) with D. We will in general express<strong>points</strong> on the Jacobian other than 0 in the form P + Q − D, where P and Q are notconjugate under the hyperelliptic involution.Let P 0 be a rational point on C. Sometimes we will want to express <strong>points</strong> in theJacobian in the form P ′ + Q ′ − 2P 0 . We will do it in the following way. Let P + Q − Dbe an element in J(C). Consider the divisor D ′ = P + Q − D + P 0 − ¯P 0 ∈ Pic 0 (C). Ingeneral, D ′ will not be linearly equivalent to 0. Let P ′ +Q ′ be the only degree 2 effectivedivisor such that P ′ +Q ′ −D is linearly equivalent to D ′ . Then the divisor P ′ +Q ′ −2P 0is linearly equivalent to D ′ + D − 2P 0 , which is linearly equivalent to P + Q − P 0 − ¯P 0 ,which is linearly equivalent to P + Q − D.The Jacobian of the curve J(C) can also be thought of as an algebraic varietywhose <strong>points</strong> correspond to the elements of Pic 0 (C). In [16, Chapter 2] the authorsdescribe explicitly this variety as the intersection of 72 quadrics in P 15 and they alsogive explicit maps taking divisors of the form P + Q − D to the algebraic variety.The model of the Jacobian variety in P 15 is a large object, difficult to manipulate.There is another variety associated to J(C) that retains much of the information of theJacobian which is much simpler, the Kummer surface K = K(C). This is a quarticsurface in P 3 . We take its definition from [16, Chapter 3]. We define K as the projectivelocus in P 3 of the map given byP + Q − D ∈ J(C) ↦→ (k 1 , k 2 , k 3 , k 4 ) = (1, x + u, xu, (F 0 (x, u) − 2yv)/(x − u) 2 ),where P = (x, y), Q = (u, v), P ≠ Q, andF 0 (x, u) = 2f 0 +f 1 (x+u)+2f 2 xu+f 3 xu(x+u)+2f 4 (xu) 2 +f 5 (xu) 2 (x+u)+2f 6 (xu) 3 .9
The Kummer surface is then given by a quartic equationR(k 1 , k 2 , k 3 )k 2 4 + S(k 1 , k 2 , k 3 )k 4 + T (k 1 , k 2 , k 3 ) = 0, (2.1.1)where R, S and T are homogeneous of degree 2, 3 and 4 respectively. These polynomialscan be found in Chapter 3 of [16]. We can extend the map to pairs of <strong>points</strong>P, Q that include the <strong>points</strong> at infinity, and to pairs of the form P, P . The values arethe following, according to Flynn and Smart [22]. For pairs P, P where P is not one ofthe <strong>points</strong> at infinity, k 1 , k 2 , k 3 are as above, and k 4 is uniquely determined by Equation(2.1.1). If the pair is of the form (x, y), ∞ ± , then k 1 = 0, k 2 = 1, k 3 = x 1 , k 4 =f 5 x 2 + 2f 6 x 3 1 − (±2y√ f 6 ). The pair ∞ + , ∞ − is mapped to (0, 0, 0, 1) and if the pairconsists of two equal <strong>points</strong> at infinity, the corresponding point on the Kummer surfaceis (0, 0, 1, k 4 ) where k 4 can be determined from equation (2.1.1). We remark that theJacobian variety is a double cover of the Kummer surface (see [16, Chapter 3, Section8]).Finally, we state the Mordell–Weil Theorem. The proof for the general theoremon Abelian varieties over number fields can be found in Serre’s book [39, Chapter 4].Theorem 2.1.1. The group of <strong>points</strong> on J(C) defined over Q is a finitely generatedabelian group.2.2 Bounds for the height differenceWe define a naive height on K as the restriction of the naive height in P 3 : let (x 0 , x 1 , x 2 , x 3 )be the homogeneous coordinates of a point in P 3 (Q). Multiplying by a nonzero constant,we can assume that the x i s are all integers having 1 as their greatest common divisor.We define the (exponential) naive height of (x 0 , x 1 , x 2 , x 3 ) as max(|x 0 |, |x 1 |, |x 2 |, |x 3 |).We will normally consider the logarithmic naive height, that is, the logarithm of the naiveheight.Definition. The naive height or logarithmic height of a point Q on the Jacobian variety10
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 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
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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 ],
Page 84 and 85:
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
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
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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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