Density of Rational Points on a Family of Diagonal Quartic Surfaces Thesis advisor Candidate Prof. Ronald M. van Luijk Dino Festi Universiteit Leiden - Universit` a di Padova June 21, 2012 Dino Festi Leiden - June 21, 2012
Table of Contents ◮ A family of diagonal quartic surfaces. ◮ Elliptic fibers. ◮ Torsion points on the fibers: ◮ 2-torsion points, ◮ 4-torsion points, ◮ 5-torsion points, ◮ 3-torsion points, ◮ Full torsion subgroup. ◮ Proof of the main theorem. Dino Festi Leiden - June 21, 2012
A family of diagonal quartic surfaces For any c 1 , c 2 ∈ Q × , let W c 1 ,c 2 = W be the surface given by ( x 2 − 2 c 1 y 2 )( x 2 + 2 c 1 y 2 ) = c 2 ( z 2 + 2 zw + 2 w 2 )( z 2 − 2 zw + 2 w 2 ) . Dino Festi Leiden - June 21, 2012
A family of diagonal quartic surfaces For any c 1 , c 2 ∈ Q × , let W c 1 ,c 2 = W be the surface given by ( x 2 − 2 c 1 y 2 )( x 2 + 2 c 1 y 2 ) = c 2 ( z 2 + 2 zw + 2 w 2 )( z 2 − 2 zw + 2 w 2 ) . Theorem Let c 1 , c 2 and W be as above. Let P = ( x 0 : y 0 : z 0 : w 0 ) be a rational point on W with x 0 and y 0 both nonzero. If | 2 c 1 | is a square in Q × , then also assume that z 0 , w 0 are not both zero. Then the set of rational points on the surface is Zariski dense. Dino Festi Leiden - June 21, 2012
A family of diagonal quartic surfaces For any c 1 , c 2 ∈ Q × , let W c 1 ,c 2 = W be the surface given by ( x 2 − 2 c 1 y 2 )( x 2 + 2 c 1 y 2 ) = c 2 ( z 2 + 2 zw + 2 w 2 )( z 2 − 2 zw + 2 w 2 ) . Theorem Let c 1 , c 2 and W be as above. Let P = ( x 0 : y 0 : z 0 : w 0 ) be a rational point on W with x 0 and y 0 both nonzero. If | 2 c 1 | is a square in Q × , then also assume that z 0 , w 0 are not both zero. Then the set of rational points on the surface is Zariski dense. Fibrations from W to P 1 : ψ 1 : ( x : y : z : w ) �→ ( x 2 − 2 c 1 y 2 : z 2 + 2 zw + 2 w 2 ) = ( c 2 ( z 2 − 2 zw + 2 w 2 ) : x 2 + 2 c 1 y 2 ) , ψ 2 : ( x : y : z : w ) �→ ( x 2 − 2 c 1 y 2 : z 2 − 2 zw + 2 w 2 ) = ( c 2 ( z 2 + 2 zw + 2 w 2 ) : x 2 + 2 c 1 y 2 ) . Dino Festi Leiden - June 21, 2012
Elliptic fibers ψ 1 : ( x : y : z : w ) �→ ( x 2 − 2 c 1 y 2 : z 2 +2 zw +2 w 2 ) = ( c 2 ( z 2 − 2 zw +2 w 2 ) : x 2 +2 c 1 y 2 ) . The fiber F of ψ 1 above ( s : 1), with s ∈ Q , is given by: � x 2 − 2 c 1 y 2 = s ( z 2 + 2 zw + 2 w 2 ) c 2 ( z 2 − 2 zw + 2 w 2 ) = s ( x 2 + 2 c 1 y 2 ) Dino Festi Leiden - June 21, 2012
Elliptic fibers ψ 1 : ( x : y : z : w ) �→ ( x 2 − 2 c 1 y 2 : z 2 +2 zw +2 w 2 ) = ( c 2 ( z 2 − 2 zw +2 w 2 ) : x 2 +2 c 1 y 2 ) . The fiber F of ψ 1 above ( s : 1), with s ∈ Q , is given by: � x 2 − 2 c 1 y 2 = s ( z 2 + 2 zw + 2 w 2 ) c 2 ( z 2 − 2 zw + 2 w 2 ) = s ( x 2 + 2 c 1 y 2 ) Most fibers have genus 1: intersection of two smooth quadrics in P 3 . The Jacobian of the fiber is isomorphic over Q to the elliptic curve given by: y 2 = x 3 + 4 c 1 ( c 2 2 − s 4 ) x 2 + 4 c 2 1 ( c 4 2 − 34 s 4 c 2 2 + s 8 ) x. Dino Festi Leiden - June 21, 2012
Elliptic fibers ψ 1 : ( x : y : z : w ) �→ ( x 2 − 2 c 1 y 2 : z 2 +2 zw +2 w 2 ) = ( c 2 ( z 2 − 2 zw +2 w 2 ) : x 2 +2 c 1 y 2 ) . The fiber F of ψ 1 above ( s : 1), with s ∈ Q , is given by: � x 2 − 2 c 1 y 2 = s ( z 2 + 2 zw + 2 w 2 ) c 2 ( z 2 − 2 zw + 2 w 2 ) = s ( x 2 + 2 c 1 y 2 ) Most fibers have genus 1: intersection of two smooth quadrics in P 3 . The Jacobian of the fiber is isomorphic over Q to the elliptic curve given by: y 2 = x 3 + 4 c 1 ( c 2 2 − s 4 ) x 2 + 4 c 2 1 ( c 4 2 − 34 s 4 c 2 2 + s 8 ) x. Its j -invariant and discriminant are 2( s 8 + 94 s 4 c 2 2 + c 4 2 ) 3 j = 2 ) 2 , 2 s 4 ( s 4 − 6 s 2 c 2 + c 2 2 ) 2 ( s 4 + 6 s 2 c 2 + c 2 c 2 2 ( s 4 − 6 s 2 c 2 + c 2 2 ) 2 ( s 4 + 6 s 2 c 2 + c 2 d = 2 17 s 4 c 6 1 c 4 2 ) 2 . Dino Festi Leiden - June 21, 2012
Elliptic fibers ψ 1 : ( x : y : z : w ) �→ ( x 2 − 2 c 1 y 2 : z 2 +2 zw +2 w 2 ) = ( c 2 ( z 2 − 2 zw +2 w 2 ) : x 2 +2 c 1 y 2 ) . The fiber F of ψ 1 above ( s : 1), with s ∈ Q , is given by: � x 2 − 2 c 1 y 2 = s ( z 2 + 2 zw + 2 w 2 ) c 2 ( z 2 − 2 zw + 2 w 2 ) = s ( x 2 + 2 c 1 y 2 ) Most fibers have genus 1: intersection of two smooth quadrics in P 3 . The Jacobian of the fiber is isomorphic over Q to the elliptic curve given by: y 2 = x 3 + 4 c 1 ( c 2 2 − s 4 ) x 2 + 4 c 2 1 ( c 4 2 − 34 s 4 c 2 2 + s 8 ) x. Its j -invariant and discriminant are 2( s 8 + 94 s 4 c 2 2 + c 4 2 ) 3 j = 2 ) 2 , 2 s 4 ( s 4 − 6 s 2 c 2 + c 2 2 ) 2 ( s 4 + 6 s 2 c 2 + c 2 c 2 2 ( s 4 − 6 s 2 c 2 + c 2 2 ) 2 ( s 4 + 6 s 2 c 2 + c 2 d = 2 17 s 4 c 6 1 c 4 2 ) 2 . Remark: If we assume that on the fiber there is at least one rational point then the fiber is isomorphic to its Jacobian. Dino Festi Leiden - June 21, 2012
Elliptic fibers Question 1: Are there singular fibers? Dino Festi Leiden - June 21, 2012
Elliptic fibers Question 1: Are there singular fibers? We have exactly ten singular fibers, for both ψ 1 and ψ 2 . Namely the fibers above (1 : 0) , (0 : 1) , ( s : 1) with √ √ 2) γ 2 2) γ 2 s ∈ S := { ( ± 1 ± 2 , i ( ± 1 ± 2 } , where γ 2 is such that γ 4 2 = c 2 . Dino Festi Leiden - June 21, 2012
Elliptic fibers Question 1: Are there singular fibers? We have exactly ten singular fibers, for both ψ 1 and ψ 2 . Namely the fibers above (1 : 0) , (0 : 1) , ( s : 1) with √ √ 2) γ 2 2) γ 2 s ∈ S := { ( ± 1 ± 2 , i ( ± 1 ± 2 } , where γ 2 is such that γ 4 2 = c 2 . Question 2: Are there rational points on the singular fibers? Dino Festi Leiden - June 21, 2012
Elliptic fibers Question 1: Are there singular fibers? We have exactly ten singular fibers, for both ψ 1 and ψ 2 . Namely the fibers above (1 : 0) , (0 : 1) , ( s : 1) with √ √ 2) γ 2 2) γ 2 s ∈ S := { ( ± 1 ± 2 , i ( ± 1 ± 2 } , where γ 2 is such that γ 4 2 = c 2 . Question 2: Are there rational points on the singular fibers? Not above the points ( s : 1), with s in S . There may be rational points on the fibers of ψ 1 and ψ 2 above the points (0 : 1) and (1 : 0), whose union is given by: � x 4 − 4 c 2 1 y 4 = 0 z 4 + 4 w 4 = 0 Dino Festi Leiden - June 21, 2012
Elliptic fibers Proposition 2.2.3 Let W and ψ 1 defined as before. If | 2 c 1 | is not a square in Q then there are no rational points on any singular fiber. If 2 c 1 is 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 − 2 c 1 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. The same holds for ψ 2 . Dino Festi Leiden - June 21, 2012
Elliptic fibers Proposition 2.2.3 Let W and ψ 1 defined as before. If | 2 c 1 | is not a square in Q then there are no rational points on any singular fiber. If 2 c 1 is 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 − 2 c 1 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. The same holds for ψ 2 . Lemma 2.2.4 The intersection of the fibers above (0 : 1) and (1 : 0) are the following: √ F 0 ∩ G 0 = { ( ± 2 c 1 : 1 : 0 : 0) } , √ √ 2 ζ 3 F 0 ∩ G ∞ = { (0 : 0 : 2 ζ 8 : 1) , (0 : 0 : − 8 : 1) } , √ √ 2 ζ 3 F ∞ ∩ G 0 = { (0 : 0 : − 2 ζ 8 : 1) , (0 : 0 : 8 : 1) } , √ F ∞ ∩ G ∞ = { ( ± − 2 c 1 : 1 : 0 : 0) } . Dino Festi Leiden - June 21, 2012
2-torsion points Any smooth fiber with at least one rational point, say P = ( x 0 : y 0 : z 0 : w 0 ), is isomorphic over the rationals to the elliptic curve y 2 = x 3 + 4 c 1 ( c 2 2 − s 4 ) x 2 + 4 c 2 1 ( c 4 2 − 34 s 4 c 2 2 + s 8 ) x. Dino Festi Leiden - June 21, 2012
2-torsion points Any smooth fiber with at least one rational point, say P = ( x 0 : y 0 : z 0 : w 0 ), is isomorphic over the rationals to the elliptic curve y 2 = x 3 + 4 c 1 ( c 2 2 − s 4 ) x 2 + 4 c 2 1 ( c 4 2 − 34 s 4 c 2 2 + s 8 ) x. This elliptic curve has only two rational 2-torsion points: O and (0 , 0). Dino Festi Leiden - June 21, 2012
2-torsion points Any smooth fiber with at least one rational point, say P = ( x 0 : y 0 : z 0 : w 0 ), is isomorphic over the rationals to the elliptic curve y 2 = x 3 + 4 c 1 ( c 2 2 − s 4 ) x 2 + 4 c 2 1 ( c 4 2 − 34 s 4 c 2 2 + s 8 ) x. This elliptic curve has only two rational 2-torsion points: O and (0 , 0). Rational 2-torsion Non rational 2-torsion points on the fiber points on the fiber √ √ P = ( x 0 : y 0 : z 0 : w 0 ) T 1 = ( − 2 x 0 : 2 y 0 : 2 w 0 : z 0 ) √ √ T 0 = ( − x 0 : − y 0 : z 0 : w 0 ) T 2 = ( 2 x 0 : − 2 y 0 : 2 w 0 : z 0 ) Dino Festi Leiden - June 21, 2012
4-torsion points Claim (Theorem 3.1.4): Let F be a smooth fiber of ψ 1 with a rational point P on it. Then (F,P), viewed as an elliptic curve, has no rational nontrivial 4-torsion points. Dino Festi Leiden - June 21, 2012
4-torsion points Claim (Theorem 3.1.4): Let F be a smooth fiber of ψ 1 with a rational point P on it. Then (F,P), viewed as an elliptic curve, has no rational nontrivial 4-torsion points. Recall that ( F, P ) is isomorphic over the rationals to the elliptic curve y 2 = x 3 + 4 c 1 ( c 2 2 − s 4 ) x 2 + 4 c 2 1 ( c 4 2 − 34 s 4 c 2 2 + s 8 ) x. Dino Festi Leiden - June 21, 2012
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