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

More documents

Recommendations

Info

Once we find r linearly independent elements in J(Q) we test if they, togetherwith the torsion <strong>points</strong>, generate J(Q)/2J(Q). We do it by testing if they are not thedouble of another point on J(Q). We can test whether a point of infinite order P is notthe double of a point Q ∈ J(Q) using heights. If P = 2Q, then ĥ(Q) = 1/4ĥ(P ), as wecan see from the definition of ĥ. We then can look for <strong>points</strong> Q on J(Q) of canonicalheight at most 1/4ĥ(P ) and see whether 2Q = P or not.2.4 The infinite descentThe procedure that computes a set of generators for the full Mordell–Weil group out ofa set of generators for J(Q)/2J(Q) is known as infinite descent. There are differentways of performing the infinite descent (see [22, 46]). We will use a simple method dueto Zagier [41, Proposition 7.2].Theorem 2.4.1 (Zagier). Let S c = {P ∈ J(Q) : ĥ(P ) ≤ c}. If S c contains a full set ofrepresentatives for J(Q)/mJ(Q) (m > 1), then J(Q) is generated by S c .Proof. The proof in [41] also applies in the hyperelliptic case. We repeat it here forcompleteness.Let J 1 be the subgroup of J(Q) generated by S c . Suppose that J 1 ≠ J(Q).Then choose a Q ∈ J(Q), with Q /∈ J 1 of minimal height.This is possible, sinceĥ(J(Q)) is a discrete subset of R (see last remark in Section 2.2). Since Q /∈ S c wehave ĥ(Q) > c. Choose P ∈ S c with P ≡ Q (mod mJ(Q)), so P = Q + mR for someR ∈ J(Q). We note that R /∈ J 1 because of our assumption on Q.Now note thatĥ(R) = 1 m 2 ĥ(P − Q) ≤ 2 (ĥ(P ) + ĥ(Q))m2 ≤ 2 m 2 (c + ĥ(Q)) < 4 m 2 ĥ(Q)≤ ĥ(Q),since m ≥ 2. Since R /∈ J 1 this is a contradiction to the minimality of Q and henceproves the proposition.15
Once we know a set of generators for J(Q)/2J(Q), we compute the heights ofrepresentatives of every class modulo 2J(Q). Let c be the maximum of those heights.Then, the set S c satisfies the conditions of the theorem and we know it generates thefull Mordell–Weil group. If the height of one of the generators is very large, or if the rankis very large, then Zagier’s theorem might be impractical, and one would need to usethe lattice enlargement procedures and the sieving techniques from [22, 46].Example 2.4.2. Let C be the genus 2 curve defined by the equationy 2 = 4x 5 − 4x + 1.This curve is equivalent to that in Theorem 1.1 of [15]. There the authors compute theMordell–Weil group of the curve, and we check here the computations to illustrate themethods presented in this chapter.Using MAGMA we find some rational <strong>points</strong> on the curve.∞, (−1, ±1), (0, ±1), (1, ±1, 1), (2, ±11),(3, ±31), (1/4, ±1/16), (−15/16, ±679/512), (30, ±9859).MAGMA’s command TorsionSubgroup reveals that the Mordell–Weil group is torsionfree. Using the command ReducedBasis we find a set of generators for the subgroupgenerated by the divisors P − Q, where P, Q are in the list we have just found. A list ofindependent generators for the subgroup is(0, 1) − ∞, (1, 1) − ∞, (−1, 1) − ∞.Hence J(Q) has rank at least 3. Now we use the command RankBounds to see that anupper bound obtained from the information of the 2-Selmer group of J(Q) is 3. ThenJ(Q) has rank 3. We now want to see that none of the generators is the double ofsome divisor.We first compute an upper bound for the height difference using theHeightConstant command. We find that µ 2 is about 2.17. We compute the canonicalheights of the divisors above with the Height command and see that they are respectivelyabout 0.16, 0.60 and 0.79. If the double of a divisor D is one of the divisors above,16
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 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
show all

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

Create successful ePaper yourself

Delete template?

Save as template?