Universiteit Leiden Mathematisch Instituut

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Universiteit Leiden

Mathematisch Instituut

Master Thesis

Density of Rational Points on a Family of Diagonal Quartic Surfaces

Thesis Advisor Candidate

Prof. Ronald M. van Luijk Dino Festi

Academic Year 2011–2012

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Introduction

Finding integer solutions of a system of equations with integer coefficients only is one of the most ancient mathematical problems: the first attempt to solve such a problem can be found in India, around 800 b.C. In the third century we can find the first large study about this problem, by Diophantus of Alexandria: this is why now we call this kind of problems diophantine problems.

In his studies, Diophantus treated only some particular equations, and did not de- velop a general theory about these equations.

Even though mathematicians reached many new results about diophantine problems throughout history, a formulation of a general theory about the problem is still an unfulfilled aim.

In the last centuries a geometric approach to the problems turned out to be advan- tagious: in this approach, the so called K3 surfaces play an important role.

The K3 surfaces are the 2-dimensional analogues of elliptic curves in the sense that their canonical sheaf is trivial (see [19, 20] for more details).The K3 surfaces are just in between surfaces that are geometrically relatively easy in some technical sense and surfaces that are geometrically complicated. Smooth quartic surfaces in P³ are examples of K3 surfaces. Little is known about the arithmetic of these surfaces. It is for instance not known whether there exists a K3 surface over the rational numbers (or any number field) on which the set of rational points is neither empty nor dense.

In 2010, Logan, McKinnon and van Luijk gave in [9] an interesting result about density of rational points on a large family of projective diagonal quartic surfaces, namely:

Theorem (Theorem 3.4). Let a, b, c, d ∈ Q^× be nonzero rational numbers with abcd square. Let P= (x0 : y₀: z₀ : w₀) be a rational point on the surface

V: ax⁴+ by⁴+ cz⁴+ dw⁴ = 0,

and suppose that all the coordinates of P are nonzero and that P does not lie on

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any of the 48 lines of the surface. Then the set of rational points of the surface is dense in both the Zariski and the real analitic topology.

To prove this result, the authors consider two elliptic fibrations of the surface, and they show that any smooth fiber with at least one rational point, viewed as elliptic curve has four rational 2-torsion points. Using the hypotheses concerning Pthey show that then P does not lie on an intersection of two singular fibers and it has infinite order on at least one of the smooth fibers passing through it. Then they use this point and the group structure of the fiber to get infinitely many other rational points on the fiber. For each of these points they apply the same argument, but with respect to the other fibration, deducing the Zariski density of the set of rational points on V.

In this thesis we will use a similar argument, but for a more specific case, not covered by the previous theorem. The case is less general, since the family we consider is given by two parameters instead of the four (in fact three) of the family considered in [9], but we get a similar result assuming weaker hypotheses.

Recall that we will always assume the existence of at least one rational point on the surface. We can state our result as follows:

Theorem. Let c1, c2be two nonzero rationals and W be the surface defined as W: x⁴− 4c²₁y⁴− c₂z⁴− 4c₂w⁴= 0.

Let P= (x0 : y0 : z0: w0) be a rational point on W with x0and y0both nonzero.

If |2c₁| is a square in Q^×, then also assume that z₀, w₀are not both zero. Then the set of rational points on the surface is Zariski dense.

To prove this result we consider two elliptic fibration of W defined over the rationals and we show that any smooth fiber with at least one rational point, viewed as an elliptic curve, has at most one nontrivial rational 2-torsion point and no other rational torsion points. Using the hypotheses concerning P we show that then P does not lie on an intersection of two singular fibers and it has infinite order on at least one of the smooth fibers passing through it. Then, using the same argument as in [9] we deduce the Zariski density of the set of rational points on W.

Another similar result can be found in [7]: in this paper Elkies shows that on the surface x⁴+y⁴+z⁴−w⁴ = 0 the set of rational points is dense in both the Zariski and real topology. This surface is neither in our family nor in the family considered in [18].

In [18], van Luijk focuses on the set of rational points on a surface which do not ensure the Zariski density of the set of rational points. He obtained the following result:

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Theorem (Theorem 2.2). Let k be a number field and let k be an algebraic closure of k. Let V be a projective smooth surface over k. For each integer d there exists an explicitly computable closed subset Z ⊆ V such that for each field extension K of k of degree at most d over Q and for each twist W of V, with corresponding isomorphism φ: W_k 7→ V_k, the set W(K) is Zariski dense in W as soon as it contains any point outsideφ⁻¹(Z).

In our work, in 4.2, we give an explicit (although very simple) example of such a subset for our case.

We start our work introducing the family of diagonal quartic surfaces together with two rational fibrations. We also study the lines on the surface and give some results about the image of the lines on the surface under the fibrations.

For a good comprehension of the fibration it is crucial to study their fibers. This is what we do in the second chapter. First we consider only a special case, for c1 = ¹₂ and c2 = 1; then we study the general case, showing that our fibrations are actually elliptic fibrations. We compute the j-invariant and discriminant of the smooth fibers, and give an explicit list of the singular fibers.

Proposition 2.2.4 is crucial in providing the explicit example for Theorem 2.2 in [18].

In the next chapter we study the torsion subgroup of the fibers with a rational point, viewed as elliptic curves. At the end of the chapter we can finally state theorem 3.5.2, which gives an explicit description of the torsion subgroup. This is the most important part of our work.

Thanks to the result given in chapter 3, in chapter 4 we state and prove The- orem 4.1.1, which represents our main result, together with a straightforward Corollary. In the proof we apply the same argument used in the proof of The- orem 3.4 in [9] even if our result does not represent a special case of that theorem.

In the fifth and last chapter we introduce another family of projective diagonal quartic surfaces in order to apply our results to this family as well. This chapter is uncomplete and it will represent a stimulus for our further works.

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Chapter 1 A family of projective diagonal quartic surfaces

Consider the following family of diagonal quartic surfaces in P³(Q), named A148 in [4, A1, pag. 135]:

Wc₁,c2: x⁴− 4c²₁y⁴− c₂z⁴− 4c₂w⁴= 0 (1.1) where c₁, c₂ ∈ Q^×. When c₁ and c₂ are clear from the context, we will denote Wc₁,c2 by simply W.

1.1 Fibrations defined over Q

The equation defining the surface gives raise to a natural fibration defined over Q:

indeed we have that

x⁴− 4c²₁y⁴ = c2(z⁴+ 4w⁴) and hence

(x²− 2c₁y²)(x²+ 2c1y²)= c2(z²+ 2zw + 2w²)(z²− 2zw+ 2w²)

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so that we can consider the following fibrations from W to P¹:

ψ1: (x : y : z : w) 7→ (x²− 2c₁y²: z²+ 2zw + 2w²)= (c2(z²− 2zw+ 2w²) : x²+ 2c1y²), (1.2) ψ₂: (x : y : z : w) 7→ (x²− 2c₁y²: z²− 2zw+ 2w²)= (c2(z²+ 2zw + 2w²) : x²+ 2c1y²)

(1.3) Proposition 1.1.1. The fibrations defined in (1.2) and (1.3) are well defined on W.

Proof. Start considering ψ₁. We can use the same arguments to prove the proposition in the case of ψ2.

We have to show that the quantities x²− 2c₁y², z²+ 2zw + 2w², z²− 2zw+ 2w²and x²+ 2c1y²are not all zero for any (x : y : z : w) on W. Consider P= (x0 : y₀: z₀ : w0) on W and assume that

x²₀− 2c₁y²₀ = 0 = z²₀+ 2z0w₀+ 2w²₀.

Recall that since P is an element of the projective space, its coordinates cannot be all zero. Now notice that x²− 2c₁y² and x² + 2c1y² have no common factors in Q[x, y]; the same holds for z²+ 2zw + 2w²and z²− 2zw+ 2w²in Q[z, w]. From

z²₀+ 2z0w₀+ 2w²₀= 0

it follows then that either z₀= 0 = w0or z²₀− 2z₀w0+ 2w²₀ , 0.

If z²₀− 2z₀w₀+ 2w²₀ , 0 then we are done.

So assume z₀ = w0: then

z²₀− 2z₀w0+ 2w²0= 0

and at least one of x₀ and y₀ is nonzero. But from x²₀ − 2c₁y²₀ = 0 we have that then both are nonzero. It follows that x²₀+ 2c1y²₀ is nonzero, since x²− 2c₁y²and

x²+ 2c1y²have no common factors in Q[x, y].

Now we will prove a property of these two fibrations that will allow us to translate the results obtained for ψ1into results for ψ2 (and viceversa).

Proposition 1.1.2. Letψ₁andψ₂ the fibration from W to P¹defined as in in(1.2) and(1.3). Consider the automorphism of P³defined by:

χ: (x : y : z : w) 7→ (x : y : z : −w).

The automorphismχ induces an automorphism of W; with an abuse of notation we callχ the automorphism of W. The automorphism χ makes the following diagram

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commute.

W ^χ ^//

ψ1

W

ψ2

P¹ P¹

Proof. Trivial.

1.2 Lines on W

Now we will find the equations of the 48 lines lying on W.

Let W = Wc₁,−c2 ⊂ P³ be the surface of the family A148 given by the equation (1.1):

x⁴− 4c²₁y⁴+ c2z⁴+ 4c2w⁴ = 0.

The surface W contains 48 lines, Lk, given by the following equations:

Ll+4 j=











x = −√

2γ1ζ₈^2ly z = −√

2ζ₈^{2 j}⁺¹w, L16+l+4 j=











x = −γ2ζ₈^2l⁺¹z y = −^γ_γ²₁ζ₈^{2 j}w , L32+l+4 j=











x = −√

2γ2ζ₈^2l⁺¹w y = −^√^γ_2γ²

1ζ₈^{2 j}w , where ζ₈ is a primitive 8-th root of unity, √

2 = ζ8 − ζ₈³ and l, j ∈ {0, 1, 2, 3};

furthermore γ₁ and γ₂ are elements of Q such that γ²₁ = c1 and γ⁴₂ = c2, and let i= ζ₈².

If we define

α = x²− 2c₁y², β = z²+ 2zw + 2w², α = x²+ 2c1y²,

β = z²− 2zw+ 2w²;

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then it follows that on W we have αα= −c2ββ,

and let ψj, with j = 1, 2, be the fibrations from W to P¹ defined as in (1.2) and (1.3), given by:

ψ₁: (x : y : z : w) 7→ (α : β)= (−c2β : α), (1.4) ψ₂: (x : y : z : w) 7→ (α : β)= (−c2β : α). (1.5) We want to see where the 48 lines are mapped by those fibrations.

1.3 Lines mapped by ψ

₁

In this section we study the image of the 48 lines on the surface W via the fibrations ψ1and ψ2.

It is very easy to see that the four lines given by the conditions α = 0, β , 0 or, equivalently, α= 0, β = 0, namely L4, L₆, L₈, L₁₀are mapped to (0 : 1), and in fact the fiber above (0 : 1) is given by the union of these 4 lines. To show this, it is enough to recall that

α = x²− 2c₁y² = (x − √

2γ1y)(x+ √ 2γ1y), β = z²− 2zw+ 2w² = (z − ζ8

√

2w)(z+ ζ₈³√ 2w),

and from αα = −c2ββ it follows that α = 0, β , 0 is equivalent to α = 0, β = 0, from which the equations of our lines follow.

With an analogous argument we show that the fiber above the point (1 : 0) is given by the union of the lines L1, L3, L13, L15. In this case the conditions are α , 0, β = 0, i.e. α = 0, β = 0.

We can summarize these results in the following Proposition.

Proposition 1.3.1. The fiber ofψ1above(0 : 1) is the union of the lines L4, L6, L8, L10. The fiber ofψ₁above(1 : 0) is the union of the lines L₁, L₃, L₁₃, L₁₅.

Both fibers have type I₄, i.e. the four lines of each fiber form a tetragon: each line of the fiber intersects the next line cyclically (see also [16, p. 365]).

In order to study the case of the other lines it is useful to recall that the surface W defined in (1.1) is a K3 surface (see [4, II.2.2]), and to prove the following Lemma.

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Lemma 1.3.2. Let s and t be two rationals, not both zero. Then the fiber of ψ₁ above(s : t) on W is linearly equivalent to the fiber above (0 : 1).

Proof. The fiber F of ψ₁above (s : t) is given by the equations:











t(x²− 2c₁y²) = s(z²+ 2zw + 2w²) c₂t(z²− 2zw+ 2w²) = s(x²+ 2c1y²) . Let F0be the fiber of ψ1 above (0 : 1) and consider then the function

f = t(x²− 2c₁y²) − s(z²+ 2zw + 2w²)

α .

The divisor of f is F − F0and the statement follows.

This reult holds more in general for any map from V to P¹.

Let F₀ denote the fiber of ψ₁above (0 : 1); from Lemma 1.3.2 it follows that, for any fiber F of ψ₁and any k ∈ 0, . . . , 47, the following identity of intersection numbers holds: F · Lk = F0· Lk.

On the other hand, it is very easy to compute the 48 × 48 matrix A = (aj,k)_{0≤ j,k≤47} where aj,k = Lj· Lk, since we have to compute the intersection number of lines, and so it can only be either 0 if the two lines does not intersect, or 1 if they intersect;

for any j we have that the self-intersection number Lj · Lj equals −2, using [8, Prop.V.1.5, pag. 361] and recalling that by definition the canonical divisor of a K3 surface is K = 0. It also turns out that rank(A) = 20.

Proposition 1.3.3. Let j ∈ {0, . . . , 47} be such that Lj is not in the fiber of ψ₁ above(0 : 1) nor (1 : 0) (see 1.3.1).

If j ≤15 then Ljis surjectively mapped to P¹viaψ1with a2-to-1 correspondence.

If j ≥16 then Ljis surjectively mapped to P¹viaψ₁with a1-to-1 correspondence.

Proof. Let s be a non zero rational and let F denote the fiber of ψ₁ above (s : 1).

To show that any line is surjectively mapped to P¹ it is enough to show that the intersection number F · Ljis greater than zero. But by 1.3.2 we have that F · Lj = F₀· Lj, and by 1.3.1 we know that F₀ = L4+ L6+ L8+ L10. So by bilinearity of the intersection pairing we have that

F · Lj = L4· Lj+ L6· Lj+ L8· Lj+ L10· Lj = a4, j+ a6, j+ a8, j+ a10, j

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for any j as in the hypotheses and s ∈ Q^×, where the ai, j0

s are the entries of the intersection number matrix A computed before.

If j ≤ 15 the sum above turns out to be 2: hence any fiber F has two intersections with Lj, i.e. Ljis surjectively mapped to P¹via ψ₁ with a 2-to-1 correspondence.

If j ≥ 16 then the sum above turns out to be 1: hence any fiber F intersects Ljin a point, i.e. Ljis surjectively mapped to P¹via ψ₁with a 1-to-1 correspondence.

1.4 Lines mapped by ψ

₂

Thanks to 1.1.2 it is easy to restate the results of the prevoius section for ψ2. In any proposition we just have to substitute the line Lk with the line χ(Lk), where χ is defined as in 1.1.2.

In this way we get the following Propositions.

Proposition 1.4.1. The fiber ofψ₂above(0 : 1) is the union of the lines L₀, L₂, L₁₂, L₁₄. The fiber ofψ₂above(1 : 0) is the union of the lines L₅, L₇, L₉, L₁₁.

Proposition 1.4.2. Let Ljwith j ∈ {0, . . . , 47} be a line on W not in the fiber of ψ₂ above(0 : 1) nor (1 : 0) (see 1.4.1).

If j ≤15 then Ljis surjectively mapped to P¹viaψ₂with a2-to-1 correspondence.

If j ≥16 then Ljis surjectively mapped to P¹viaψ2with a1-to-1 correspondence.

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Chapter 2 Elliptic fibers

Let W be a surface defined as in (1.1),

W = Wc₁,c2: x⁴− 4c²₁y⁴− c₂z⁴− 4c₂w⁴ = 0 with its elliptic fibrations ψj, ji = 1, 2 defined as in (1.2) and (1.3),

ψ₁: (x : y : z : w) 7→ (x²− 2c₁y²: z²+ 2zw + 2w²)= (c2(z²− 2zw+ 2w²) : x²+ 2c1y²), ψ₂: (x : y : z : w) 7→ (x²− 2c₁y²: z²− 2zw+ 2w²)= (c2(z²+ 2zw + 2w²) : x²+ 2c1y²).

Let F be a smooth fiber of ψ₁. In this chapter we will see that F has genus 1 and we will find the Weierstrass equation and the j-invariant of the Jacobian of F. If we assume that F is smooth and there is a rational point on it, then the Jacobian of Fis isomorphic to F; taking the given rational point as nutral element, F inherits the structure of an elliptic curve.

In order to do this we will follow the procedure presented in [1].

First we will consider the case for W = W¹₂,1 and then the general case for W = Wc₁,c2 where c₁, c2run in Q^×.

2.1 W = W

¹

2,1

Let W = W¹₂,1 be the surface defined as in (1.1). Namely:

W: x⁴− y⁴ = z⁴+ 4w⁴,

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and consider the two fibrations from W to P¹defined as in (1.2) and (1.3):

ψ₁: (x : y : z : w) 7→ (x²− y² : z²+ 2zw + 2w²)= (z²− 2zw+ 2w², x²+ y²), ψ₂: (x : y : z : w) 7→ (x²− y² : z²− 2zw+ 2w²)= (z²+ 2zw + 2w², x²+ y²).

We start considering the fibers of ψ₁. Let F denote the fiber of ψ₁above the point (s : 1); then F is given by:











x²− y² = s(z²+ 2zw + 2w²) z²− 2zw+ 2w² = s(x²+ y²) ⇐⇒











x²− y²− sz²− 2szw − 2sw² = 0 sx²+ sy²− z²+ 2zw − 2w² = 0. Notice that by the equations above we can deduce that F has genus 1, since they are intersection of two quadrics in P³. Let U and V denote the quadric forms given by

U(x, y, z, w) = x²− y²− sz²− 2szw − 2sw², (2.1) V(x, y, z, w) = sx²+ sy²− z²+ 2zw − 2w². (2.2) Let A and B be two 4 × 4 square symmetric matrices such that

U(x, y, z, w)= (x, y, z, w) · A ·^t(x, y, z, w), V(x, y, z, w)= (x, y, z, w) · B ·^t(x, y, z, w).

Then we have that

A=







1 0 0 0

0 −1 0 0

0 0 −s −s

0 0 −s −2s







, (2.3)

B=







s 0 0 0

0 s 0 0

0 0 −1 1

0 0 1 −2







. (2.4)

Let∆, Θ, Φ, Θ⁰, ∆⁰be defined by the following identity:

det(λA+ B) = ∆λ⁴+ Θλ³+ Φλ²+ Θ⁰λ + ∆⁰. (2.5) (Doing the easy computations) we find that they are

∆ = −s² Θ = −6s Φ = s⁴− 1 Θ⁰= 6s³

∆⁰= s².

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Then we define the following quantities:

a0 = ∆ a₁ = −Θ

4 a2 = Φ

6 a3 = −Θ⁰

4 a₄ = ∆⁰

h= a0a4− 4a1a3+ 3a²2

k= a0a2a4+ 2a1a2a3− a₀a²₃− a₄a²₁− a³₂.

By the result in [1, III.3], we have that the Jacobian of F is isomorphic to the elliptic curve E defined by

y²= x³− 4hx − 16k, namely:

E: y²= x³− s⁸+ 94s⁴+ 1

3 x+ 2s¹²− 582s⁸+ 582s⁴− 2

27 . (2.6)

If we assume that there is a rational point P on F , then F is isomorphic to its Jacobian, hence to E. In this way we can look at (F, P) as an elliptic curve, isomorphic to E. The j-invariant of E is given by:

j= 2(s⁸+ 94s⁴+ 1)³

s⁴(s²− 2s − 1)²(s²+ 2s − 1)²(s⁴+ 6s²+ 1)². (2.7) and whose discriminant d is

d= 2048s⁴(s²− 2s − 1)²(s²+ 2s − 1)²(s⁴+ 6s²+ 1)²= 0 (2.8) So in our case, finding the values of s such that the fiber above (s : 1) is singular means finding the roots of

s⁴(s²− 2s − 1)²(s²+ 2s − 1)²(s⁴+ 6s²+ 1)² = 0

Since we already computed the fibers above (0 : 1) and (1 : 0), we can reduce to consider the following equation:

(s²− 2s − 1)(s²+ 2s − 1)(s⁴+ 6s²+ 1) = 0 (2.9)

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whose roots are {±1 ± √

2, i(±1 ± √ 2)}.

So we can conclude that in the case of the fibration ψ1we have exactly 10 singular fibers, namely the fibers above (1 : 0), (0 : 1), (s : 1) where s ∈ {±1 ± √

2, i(±1 ±

√

2)}. The fibers above the latter eight points turn out to have type I₂, as we will see in the next section.

Beacuse of the involution χ in 1.1.2, the same result holds for the fibers of ψ₂

2.2 W = W

^c1,c₂

Now we will to do the same computations in the general setting: take W defined by:

W = W^c1,c2: x⁴− 4c²₁y⁴− c₂z⁴− 4c₂w⁴ = 0 with its elliptic fibrations ψj, j = 1, 2 defined by:

And consider the point (s : 1) ∈ P¹ with s , 0. Then the fiber of ψ1 above (s : 1), say F, is given by:











x²− 2c₁y² = s(z²+ 2zw + 2w²) c₂(z²− 2zw+ 2w²) = s(x²+ 2c1y²) ⇐⇒











x²− 2c₁y²− sz²− 2szw − 2sw² = 0 sx²+ 2sc1y²− c₂z²+ 2c2zw −2c₂w² = 0. Notice that the fiber has genus 1, since it is give by the intersection of two quadrics

in P³.

If we denote by U and V the quadric forms given by

U(x, y, z, w)= x²− 2c₁y²− sz²− 2szw − 2sw², (2.10) V(x, y, z, w)= sx²+ 2sc1y²− c₂z²+ 2c2zw −2c₂w². (2.11) then the matrices A, B defined as in the previous section, are given by

A=







1 0 0 0

0 −2c₁ 0 0

0 0 −s −s

0 0 −s −2s







, (2.12)

B=







s 0 0 0

0 2sc₁ 0 0

0 0 −c₂ c2

0 0 c₂ −2c₂







. (2.13)

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It follows that

∆ = −2s²c1

Θ = −12sc1c₂ Φ = 2s⁴c1− 2c1c²₂ Θ⁰ = 12c1c2s³

∆⁰ = 2s²c1c²₂; and finally

h= s⁸c²₁+ 94s⁴c²₁c²₂+ c²₁c⁴₂ 3

k= s¹²c³₁+ 291s⁸c³₁c₂²− 291s⁴c³₁c⁴₂+ c³₁c⁶₂

27 .

and so, by the result in [1, III.3], the Jacobian of F is isomorphic to the curve E given by:

E = E(s, c1, c2) : y² = x³− 4hx − 16k

= x³− 4

3(s⁸c²₁+ 94s⁴c²₁c²₂+ c²₁c⁴₂)x+ 16

27(s¹²c³₁+ 291s⁸c³₁c²₂− 291s⁴c³₁c⁴₂+ c³₁c⁶₂)

= x³− 4c²₁

3 (s⁸+ 94s⁴c²₂+ c⁴₂)x+ 16c³₁

27 (s¹²+ 291s⁸c²₂− 291s⁴c⁴₂+ c⁶₂) Moving the rational 2-torsion point (^4c¹^(s

4−c²₂)

3 , 0) to (0, 0) we can write:

E: y² = x³+ 4c1(c²₂− s⁴)x²+ 4c²₁(c⁴₂− 34s⁴c²₂+ s⁸)x. (2.14) If we assume that on F there is a rational point, then F is isomorphic to its Jaco- bian, hence to E.

In other words, we have shown the following theorem:

Theorem 2.2.1. Let W be the surface defined as in (1.1) and ψ₁its fibration defined in(1.2). Let s be a non zero rational and F the fiber of ψ1 above(s : 1) on W.

Then the Jacobian of F is isomorphic over the rationals to the elliptic curve given by:

y² = x³+ 4c1(c²₂− s⁴)x²+ 4c²₁(c⁴₂− 34s⁴c²₂+ s⁸)x.

Corollary 2.2.2. Let W, ψ₁, s and F defined as in 2.2.1, and assume there is a rational point on F.

Then F is isomorphic over the rationals to the elliptic curve given by:

y² = x³+ 4c1(c²₂− s⁴)x²+ 4c²₁(c⁴₂− 34s⁴c²₂+ s⁸)x.

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From now until the end of the chapter assume that F admits a rational point.

Then we may ask when F is a singular curve, and we already know that the fibers above (0 : 1) and (1 : 0) are singular curves (namely, union of four lines of type I₄, as we will see). In order to answer this question it is useful to compute the discriminant of E in terms of s.

The j-invariant of E is:

j= j(s, c2)= 2(s⁸+ 94s⁴c²₂+ c⁴₂)³

c²₂s⁴(s⁴− 6s²c2+ c²₂)²(s⁴+ 6s²c2+ c²₂)², (2.15) and the discriminant d is:

d = d(s, c2)= 2¹⁷s⁴c⁶₁c⁴₂(s⁴− 6s²c₂+ c²₂)²(s⁴+ 6s²c₂+ c²₂)². (2.16) Notice that the roots of both the j-invariant and the discrminant of E are indipen- dent of c₁.

Finding the values of s = s(c2) for which F is singular means finding the roots of the equation (in the variable s over Q(c2))

c⁶₂s⁴(s⁴− 6s²c₂+ c²₂)²(s⁴+ 6s²c₂+ c²₂)² = 0.

Recalling that we assumed s, c₂ , 0 this is equivalent to solving the equation (s⁴− 6s²c2+ c²2)(s⁴+ 6s²c2+ c²2)= 0.

The set of roots of the above equation is: {(±1 ± √

2)γ²₂, i(±1 ± √ 2)γ²₂}.

So we can conclude that in the case of the fibration ψ₁we have exactly 10 singular fibers, namely the fibers above (1 : 0), (0 : 1), (s : 1) with

s ∈ {(±1 ±

√

2)γ²₂, i(±1 ± √ 2)γ²₂}, where γ₂is such that γ⁴₂ = c2(see 1.2).

But not all of these fibers admit rational points.

In fact let P = (x0 : y0 : z0 : w0) be a rational point of F, then ψ1(P) will have rational coordinates: hence ψ₁(P) , (s : 1) with s ∈ {(±1 ± √

2)γ₂², i(±1 ± √ 2)γ²₂}.

Indeed for any s in that set, s is not rational: for example let s = (1 + √

2)γ₂²and assume it is rational. Then

s²= (1 + √

2)²γ⁴₂ = (1 + √ 2)²c₂, from which it follows that

3+ 2√ 2= s²

c2

.

The righthand side of the above identity is a rational, while the lefthand side is not, getting a contradiction. With the same argument we show that also the other roots cannot be rationals. We claim that the following Proposition holds.

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Proposition 2.2.3. Let W and ψ₁ defined as before. Assume that |2c₁| is not a square in Q and let F be a fiber of ψ1on W. Assume F has a rational point. Then F is not singular.

If2c₁is a square in Q then there are exactly two rational points on the fiber above (0 : 1) and this is the only singular fiber with rational points.

If −2c₁ is a square in Q then there are exactly two rational points on the fiber above(1 : 0) and this is the only singular fiber with rational points.

Proof. We have already seen that the only fibers that may be singular are the fibers above (1 : 0), (0 : 1), (s : 1) with s ∈ {(±1 ±

√

2)γ²₂, i(±1 ± √

2)γ₂²} and that the fibers above (s : 1) admit no rational points.

Now assume that |2c₁| is not a square in Q. We need to check that also the fibers above (1 : 0) and (0 : 1) have no rational points.

Indeed first assume that P = (x0 : y₀ : z₀ : w₀) is a rational point of W sent to (0 : 1) via ψ₁. Then we have that 0= x²₀− 2c₁y²₀, but since 2c₁is not a square the equality can hold only if x0 = y0 = 0. Recalling that P lies on W it follows that

c₂(z⁴₀+ 4w⁴₀)= 0 which implies that z0 = w0 = 0, getting a contradiction.

Now assume that P is sent to (1 : 0) via ψ₁: then z²₀ + 2z0w₀+ 2w²₀ = 0, which implies that z₀ = w0 = 0. But P is on W, then it follows that

0= x⁴₀− 4c²₁y⁴₀= (x²₀+ 2c1y²₀)(x²₀− 2c₁y²₀),

from which we can conclude, since none of ±2c₁ is a rational square, that x₀ = y0 = 0. Then we get another contradiction. In this way we have proved that if

|2c₁| is not a square in Q then the fibers above (1 : 0) and (0 : 1) have no rational points.

Assume now that 2c₁ is a rational square: then the points (±√

2c₁ : 1 : 0 : 0) are rational points on W sent to (0 : 1). We can use the same argument as before to show that the fiber above (1 : 0) has no rational points.

Finally, assume that −2c₁is a rational square: then the points (±√

−2c₁: 1 : 0 : 0) are rational points on W sent to (1 : 0). As before we can show that there are no

rational points on the fiber above (0 : 1).

As in the previous section, taking ψ₂ instead of ψ₁ one gets exactly the same results, thanks to the involution χ defined in the proof of 1.1.2.

It may be interesting investigate the intersection points of the fibers of ψ₁and ψ2 above (0 : 1) and (1 : 0).

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Let F₀and F∞be the fibers of ψ₁above (0 : 1) and (1 : 0) respectively; let G₀and G∞ be the analogue for ψ2. Then the following result holds:

Lemma 2.2.4. The intersection of the fibers above (0 : 1) and (1 : 0) are the following:

F₀∩ G₀ = {(±√

2γ₁ : 1 : 0 : 0)}, F₀∩ G∞ = {(0 : 0 : √

2ζ₈: 1), (0 : 0 : −

√

2ζ₈³ : 1)}, F∞∩ G₀ = {(0 : 0 : −√

2ζ₈ : 1), (0 : 0 :

√

2ζ₈³ : 1)}, F∞∩ G∞ = {(±√

2γ₁i: 1 : 0 : 0)}, where i andγ₁are defined as in section 1.2.

Proof. • F₀ ∩ G₀: recall that F₀ is given by the conditions α = β = 0 while G0is given by the conditions α= β = 0. Then their intersection is given by the conditions α = β = β = 0. The conditions β = β = 0 imply z = w = 0.

The condition α= 0 gives the desired result.

• F₀∩ G∞: recall that G∞is given by the conditions α= β = 0; then the intersection with F₀ is given by the conditions α= α = β = 0. From α = α = 0 it follows that x= y = 0; from β = 0 the desired conclusion does.

• F∞∩ G₀: as in the case of F₀∩ G∞, but with β in the place of β.

• F∞∩ G∞: as in the case of F₀∩ G₀, but with α in the place of α.

Notice that, using Lemma 2.2.4, we can see that the points in F0 ∩ G∞ and F∞∩ G₀are not rational for any choice of c₁; the points in F₀∩ G₀are rational if and only if 2c₁is a rational square; the points in F∞∩ G∞are rationals if and only if −2c1is a rational square.

Although not in this work, it is often very useful to know which type the singular fibers have.

Proposition 2.2.5. Let F0 and F∞ be the singular fibers of ψ1 above(0 : 1) and (1 : 0) respectively. Then they both have type I₄.

Let s be an element of {(±1 ±√

2)γ²₂, i(±1 ± √

2)γ₂²}, and let Fsdenote the singular fiber ofψ1above(s : 1). Than Fshas type I2.

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Proof. We explicitely computed the fibers F₀and F∞, and so it is easy to see that they have type I4.

To show that the fiber Fshas type I₂it is enough to recall the table in [16, p. 365]

and to notice that the valuation of the j-invariant and the discriminant of Fs at s is −2 and 2 respectively, as one can deduce from (2.15) and (2.16) respectively.

Looking at the table, we can conclude that Fshas type I₂.

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Chapter 3 Torsion Points

Let W be a surface defined as in (1.1),

W = Wc₁,c2: x⁴− 4c²₁y⁴− c₂z⁴− 4c₂w⁴ = 0 with its elliptic fibrations ψi, i = 1, 2 defined as in (1.2) and (1.3),

In this chapter we will study the subgroup of the group of rational points formed by the rational torsion points on each but finitely many fibers with at least one rational point on it. We will treat only the fibers of ψ₁, but thanks to Proposi- tion 1.1.2 all the results hold for the fibers of ψ2as well.

Our claim is that on each smooth fiber with at least one rational point (used as 0 to make the fiber an elliptic curve), the rational torsion subgroup of the fiber is isomorphic to Z/2Z. We will see that in order to prove this, using Mazur’s theorem, it is enough to show that on these fibers there is only one non trivial rational 2-torsion point and no nontrivial rational 3,4 and 5-torsion points.

3.1 2-torsion points

We will start our study considering the fibers of ψ₁, but because of the involution χ the same results hold for ψ2as well.

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Assume there is a rational point on W, call it P = (x0 : y₀ : z₀ : w₀) ∈ W(Q), such that ψ1(P)= (s : 1) for some nonzero rational s (recall that we already know what are the fibers above (0 : 1) and (1 : 0) ).

Let F denote the fiber of ψ₁ passing through P; then we have an isomorphism between F and the elliptic curve E given in 2.2.2 sending P to the zero element of E. So we can think about F, P as an elliptic curve having P as zero element. With an abuse of notation, we will write E when we want to consider F, P as an elliptic curve, and we will write simply F when we consider F just as a curve.

Thanks to the result of the previous chapter, we could easily compute the rational 2-torsion points of E using its Weierstrass equation computed in 2.2.2. We prefer to use another argument, to better understand the arithmetic of these surfaces.

Consider the following isomorphisms of W defined by:

σ: (x : y : z : w) 7→ (−x : y : z : w), (3.1) τ: (x : y : z : w) 7→ (x : −y : z : w). (3.2) They both respect the fibration, i.e. the following diagrams commute for j= 1, 2.

W ^σ ^//

ψj

W

ψj

W ^τ ^//

ψj

W

ψj

P¹ P¹ P¹ P¹

By the definition of σ and τ it is easy to see that σ(P), τ(P), σ ◦ τ(P) ∈ F(Q).

Notice that σ is an automorphism of F, but not of E, since it does not fix the point P. Recall the following results:

Lemma 3.1.1. Let F be defined as before, P a rational point on F and let E denote (F, P) viewed as elliptic curve over Q.

Consider the following short sequence

0 ^//E(Q) ^T ^//Aut_Q(F) ^Υ ^//Aut_Q(E) ^//0.

where T is the map sending a point Q of E(Q) to the translation (using the group structure on E) by Q; the map Υ sends an automorphism of F, say ω, to the automorphism of E given by T_−ω(P)◦ω.

The sequence is exact. In fact it splits.

Proof. The injectivity of the map T is trivial, as well as the surjectivity of the map Υ. So we just need to show that ImT = kerΥ.

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Let ω ∈ ImT , then ω is a translation, say ω= TQ. Then Υ(ω) = T−ω(P)◦ω

= T−Q◦ TQ

= idE

Now assume that ω ∈ kerΥ. It implies that T−ω(P)◦ω = id^E and hence ω = (T−ω(P))⁻¹ = Tω(P).

Hence ω is the translation by the point ω(P).

To show that the sequence splits just notice that Aut_Q(E) is contained in Aut_Q(F).

With this the proof is completed.

Proposition 3.1.2. Let E be an elliptic curve defined over Q, then

Aut_Q(E) '











{±1} if j(E) , 0, 1728 {±1, ±i} if j(E)= 1728 hζ₆i if j(E)= 0

where j(E) denotes the j-invariant of E and ζ₆is a primitive 6-th root of unity.

Proof. See [14], pag. 104, Corollary III.10.2.

We are interested in the image of σ and τ in Aut_Q(E).

Lemma 3.1.3. Let E = (F, P) be smoth a fiber of W with at least a rational point viewed as elliptic curve over Q, and let σ and τ be the elements of AutQ(F) defined as in(3.1) and (3.2).

Then there are rational points R_σand R_τon F such that, for any point Q on F we have:

σ(Q) = Rσ− Q, τ(Q) = Rτ− Q.

In particular R_σ = σ(P) and Rτ= τ(P).

Proof. We start considering σ.

First notice that since σ is an involution, it has order 2, and so, by 3.1.2, its image underΥ can only be either 1 or −1. So we have only these two cases:

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• σ is mapped to 1: then σ is a translation by a point, and since σ is an involution, it is a translation by a 2-torsion point; in this case σ does not admit fixed points. Recall that σ cannot be the identity, since it does not fix P.

• σ is mapped to −1: then by Lemma 3.1.1 we have that there is an Rσ∈ E(Q) such that for any Q ∈ E, σ(Q) = Rσ− Q.

Let R⁰ be a point on E(Q) such that 2R⁰ = Rσ, then we get that σ(R⁰) = R⁰, hence the fixed points for σ are given by R⁰ + E[2], i.e. the points of the form R⁰+ T where T is a 2-torsion point of E.

But now notice that σ admits fixed points if and only if F ∩ {x= 0} , ∅, indeed σ(x : y : z : w) = (x : y : z : w) ⇐⇒ (−x : y : z : w) = (x : y : z : w) ⇐⇒ x = 0 But actually F ∩ {x = 0} , ∅ (for example consider the point (0 : ^√^ξ_2γ

1 : z0 : w0), where ξ ∈ Q is a square root of −s ), and so we have that σ is mapped to −1.

Using the same argument but considering F ∩ {y = 0} we can show that also τ is mapped to −1.

Hence we can conclude that there are R_σ, R_τ, ∈ E_Q such that for any Q ∈ E we have that σ(Q) = Rσ − Q and τ(Q) = Rτ − Q. Notice that Rσ = σ(P) and

Rτ = τ(P).

Now consider στ= σ ◦ τ:

στ(Q) = σ(Rτ− Q)= (Rσ− Rτ)+ Q;

hence στ is a translation, but it is also an involution: indeed στ(x : y : z : w) = (−x : −y : z : w).

Then it is a translation by a 2-torsion point, so we can conclude that

T0 := Rσ− R_τ= στ(P) = (−x0 : −y0 : z : w) ∈ E(Q)[2]. (3.3)

In order to find the other two 2-torsion points take the following two maps from W to W, which also respect the fibration ψ1:

ρ₁: (x : y : z : w) 7→ (

√ 2x :

√

2y : 2w : z) ρ₂: (x : y : z : w) 7→ (−

√ 2x : −

√

2y : 2w : z).

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They are both involutions but they also have fixed points, namely the points such that z= √

2w for ρ1and the points such that z= −√

2w for ρ2. So, as before, there are R₁, R₂ ∈ E such that:

ρ₁(Q)= R1− Q, where R₁ = ρ1(Q), (3.4) ρ₂(Q)= R2− Q, where R₂ = ρ1(Q). (3.5) Notice that R1and R2are nonrational since ρ1and ρ2map P to a non rational point:

indeed ρ₁(P) = (√

2x₀: √

2y₀ : 2w₀ : z₀) is rational if and only if x₀ = y0 = 0, but in this case P would be sent either to (0 : 1) or (1 : 0), while we assumed that F were a fiber above (s : 1) with s nonzero rational; the same argument works for ρ₂ as well.

Moreover, looking at their definitions, we see that ρ₁ρ₂ = στ, from which it follows that R1− R2= T0. But consider ρ1σ:

ρ₁σ(Q) = ρ1(R_σ− Q)= (R1− R_σ)+ Q; (3.6) so ρ₁σ is an involution (since composition of two commuting involutions) and a translation, then it is a translation by a 2-torsion point. We can hence conclude that:

T₁ := (R1− Rσ)= ρ1σ(P) = (−√ 2x₀:

√

2y₀ : 2w₀ : z₀) ∈ E[2]. (3.7) The last torsion point is then T₂ := T1+ T0.

To compute it explicitely we can use the same argument used to compute T₁, but considering ρ₂σ(= ρ1τ). Then we have that:

T₂ = ρ2σ(P) = (√ 2x₀: −

√

2y₀ : 2w₀ : z₀) (= ρ1τ(P)). (3.8) To check that in fact T0+ T1+ T2 = P, recalling the definition of these points, it is enough to check that

(ρ₂σ)(ρ₁σ)(στ) = idF,

but this is straightforward using the definitions of σ, τ, ρ₁, ρ₂. So we have found that

E[2]= {P, T0, T1, T2}.

Recalling that P is rational, (3.3) shows that T₀ ∈ E(Q)[2]; instead (3.7) and (3.8) show that T₁ and T₂ are not rational. Hence we have proved the following Theorem.

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Theorem 3.1.4. Let E = (F, P) be a smooth fiber of ψ1on W with a rational point P, viewed as elliptic curve. Then

E(Q)[2] = {P, T0} ' Z/2Z.

whereστ(P) = T0 = Rσ− R_τ, with Rσ, Rτdefined as in 3.1.3.

Proof. We already proved that E(Q)[2] = {P, T0}. Then to show that {P, T₀} ' Z/2Z it is enough to show that P , T0. Let P ∈ W be the point with coordinates (x0 : y0 : z0 : w0), then we have seen that T0 = (−x0 : −y0 : z0 : w0). It follows that P= T0if and only if either x₀ = y0 = 0 or z0 = w0 = 0. But for each of these two cases it follows that then P lies on a singular fiber, while we assumed that P

was on a smooth fiber.

3.2 4-torsion points

Let s ∈ Q^×, and let F denote the fiber on W of ψ1above (s : 1). Assume that on F there is a rational point. Then by Corollary 2.2.2 we have that F is isomorphic to the elliptic curve E given by

E: y² = x³+ 4c1(c²₂− s⁴)x²+ 4c²₁(c⁴₂− 34s⁴c²₂+ s⁸)x=: g(x).

So looking for the 4-torsion points on F is equivalent to looking for the 4-torsion points on E. Notice that g(x)= xg1(x), where

g₁(x)= x²+ 4c1(c²₂− s⁴)x+ 4c²1(c⁴₂− 34s⁴c²₂+ s⁸).

Let f₄(x) be the 4-division polynomial of E:

f4(x)= 8xh1(x)h2(x)h3(x), (3.9) where

h1(x)= g1(x),

h₂(x)= x²− 4c²₁(s⁸− 34s⁴c²₂+ c⁴₂),

h₃(x)= x⁴+ 8c1(c²₂− s⁴)x³+ 24c²₁(s⁸− 34s⁴c²₂+ c⁴₂)x²+ + 32c³₁(−s¹²+ 35s⁸c²₂− 35s⁴c⁴₂+ 32c⁶₂)x+

+ 16c⁴₁(s¹⁶− 68s¹²c²₂+ 1158s⁸c⁴₂− 68s⁴c⁶₂− c⁸₂).

Universiteit Leiden Mathematisch Instituut