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

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For the computation of the canonical height and for finding generators of theMordell–Weil group J(Q) one only needs the upper bound µ 2 for the height difference.MAGMA’s package dealing with hyperelliptic curves does not implement the lower bound.Nevertheless, we will use the lower bound µ 1 obtained from Lemma 2.2.1 in Chapter 5to relate upper bounds for the naive height of the <strong>integral</strong> <strong>points</strong> on C to upper boundsfor linear forms in hyperelliptic logarithms. We will also need it in Section 3.8.Remark. The existence of the bounds µ 1 , µ 2 for the height difference implies that theset of <strong>points</strong>{P ∈ J(Q) : ĥ(P ) ≤ B}is finite for a given positive real number B. Then, the intersection of the image under ĥof J(Q) with the interval [0, B] ⊂ R is a finite set. It follows that ĥ(J(Q)) is a discretesubset of R2.3 Computing the rank of J(Q) – 2-descent.There is no algorithm known so far that provably determines the rank of the Mordell–Weil group J(Q). But many algorithms have been developed to give upper and lowerbounds for it ([37, 38, 45]). In many cases the bounds coincide and then one knows therank. In this section we outline the main ideas behind such bounds.The proof of the Mordell–Weil theorem looks at the quotient J(Q)/mJ(Q). Thefiniteness of the quotient implies that J(Q) is finitely generated. Normally, one looksat the case m = 2, as the multiplication by 2 map is available as an isogeny definedover Q and has low degree. Up to the computation of the torsion, one can deduce therank of J(Q) from the structure of J(Q)/2J(Q) once the torsion subgroup has beencomputed. The computation of generators for J(Q)/2J(Q) is known as 2-descent.Stoll gives an algorithm for the computation of the torsion subgroup J(Q) tors in[44]. The idea of the algorithm is first computing the group J(F p ) for some small primesp of good reduction. The order of the rational torsion subgroup must divide the greatest13
common divisor g of the orders of the groups J(F p ). Since the rational torsion subgroupis a finite abelian group, it suffices to compute the q-parts for all primes dividing theorder of the group. Only primes dividing g can occur. Then, for every prime q dividingg one takes a prime of good reduction p ≠ q. Then one tries to lift <strong>points</strong> on J(F p )[q n ]to J(Q). One does so lifting them first to J(Q p ) and finally a computation of heightsdecides whether the point can indeed be lifted or not.An upper bound for the rank is obtained from the 2-Selmer group, as explainedin [45]. Once we get an upper bound r ′ for the rank, we try to search for <strong>points</strong> on theJacobian until we get r ′ independent elements. If that is the case we then know that r ′equals the rank of the curve. We can search for <strong>points</strong> on the Jacobian up to certainheight c by looking for <strong>points</strong> on the Kummer surface with naive logarithmic height atmost c + µ 2 , as long as c + µ 2 is not too large (< 10).In order to test whether a finite set W ⊂ J(Q)/J(Q) tors is a linearly independentset in J(Q) we need the following definition.Definition. The height pairing on J(Q) is the bilinear form〈·, ·〉 : J(Q) × J(Q) → Rgiven by〈P, Q〉 = 1 (ĥ(P + Q) − ĥ(P ) − ĥ(Q)).2Let P 1 , . . . , P n be a sequence of <strong>points</strong> on J(Q). The height pairing matrix of the given<strong>points</strong> is the matrix with entries〈P i , P j 〉.The bilinearity of the height pairing follows from the standard fact that the canonicalheight is a positive definite quadratic form on J(Q). The determinant of the heightpairing matrix of a sequence of <strong>points</strong> on J(Q) is 0 if and only if the <strong>points</strong> are not anindependent set in J(Q)/J(Q) tors . Once we are able to compute explicitly the canonicalheight, we have a very useful tool to test for linear independence in J(Q) without theknowledge of an explicit basis.14
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 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
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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