An explicit algebraic family of genus-one curves ... - MIT Mathematics
AN EXPLICIT ALGEBRAIC FAMILY OF GENUS-ONE CURVES VIOLATING THE HASSE PRINCIPLE
BJORN POONEN Abstra t.
We prove that for any t 2 Q, the urve 5x3 + 9y3 + 10z 3 + 12
t2 + 82 3 t2 + 22
(x + y + z )3 = 0
in P2 is a genus 1 urve violating the Hasse prin iple. An expli it Weierstrass model for its Ja obian Et is given. The Shafarevi h-Tate group of ea h Et ontains a subgroup isomorphi to Z=3 Z=3.
1.
Introdu tion
One says that a variety X over Q violates the Hasse prin iple if X (Qv ) 6= ; for all
ompletions Qv of Q (i.e., Rn and Qp for all primes p) but X (Q) = ;. Hasse proved that degree 2 hypersurfa es in P satisfy the Hasse prin iple. In parti ular, if X is a genus 0
urve,2 then X satis es the Hasse prin iple, sin e the anti anoni al embedding of X is a oni in P . Around 1940, Lind [Lin℄ and (independently, but shortly later) Rei hardt [Re℄ dis overed examples of genus 1 urves over Q that violate the Hasse prin iple, su h as the nonsingular proje tive model of the aÆne urve 2y2 = 1 17x4 : Later, Selmer [Se℄ gave examples of diagonal plane ubi urves (also of genus 1) violating the Hasse prin iple, in luding 3x3 + 4y3 + 5z3 = 0 in P2. O'Neil [O'N, x6.5℄ onstru ts an interesting example of an algebrai family of genus 1 urves ea h having Qp-points for all p 1. Some bers in her family violate the Hasse prin iple, by failing to have a Q-point. In other words, these bers represent nonzero elements of the Shafarevi h-Tate groups of their Ja obians. In [CP℄, Colliot-Thelene and the present author prove,1 among other things, the existen e of non-isotrivial families of genus 1 urves over the base P , smooth over a dense open subset, 1 su h that the ber over ea h rational point of P is a smooth plane ubi violating the Hasse prin iple. In more on rete terms, this implies that there exists a family of plane ubi s depending on a parameter t, su h that the j -invariant is a non- onstant fun tion of t, and su h that substituting any rational number for t results in a smooth plane ubi over Q violating the Hasse prin iple. Date : Mar h 18, 2000.
This resear h was supported by National S ien e Foundation grant DMS-9801104, an Alfred P. Sloan Fellowship, and a David and Lu ile Pa kard Fellowship. 1
2
BJORN POONEN
The purpose of this paper is to produ e an expli it example of su h a family. Our example, presented as a family of ubi urves in P2 with homogeneous oordinates x; y; z, is 2 t + 82 3 3 3 3 5x + 9y + 10z + 12 t2 + 22 (x + y + z)3 = 0: Remark . Noam Elkies pointed out to me that the existen e of non-trivial families with
onstant j -invariant ould be easily dedu ed from previously known results. In Case I of the proof of Theorem X.6.5 in [Si℄, one nds a proof (based on ideas of Lind and Mordell) that 2 4 2y = 1 Nx represents a nontrivial element of X[2℄ of its Ja obian, when N 14 (mod 8) is a prime for whi h 2 is not a quarti residue. The same argument works if N = N 0 where
2 Q and N 0 is a produ t of primes p 1 (mod 8), provided that 2 is not a quarti residue for at least one p appearing with odd exponent in N 0 . One an he k that if N = a4 + 16b4 for some a; b 2 Q \ Z2, 1then N has this form. One an now substitute rational fun tions for a and b 2mapping P (Q2) into Z2, with a=b not onstant. For instan e, the hoi es a = 1 + 2=(t + t + 1) and b = 1 lead to the family 2y2 = 1 (t2 + t + 3)4 + 16(t2 + t + 1)4 x4 of genus 1 urves of j -invariant 1728 violating the Hasse prin iple. 2. Let us review brie y the onstru tion in [CP℄. Swinnerton-Dyer [SD℄ proved that there 3 exist smooth
ubi surfa es V in P over Q violating the Hasse prin iple; hoose one. If L is a line in P3 meeting V in exa tly 3 geometri points, and W denotes the blowup of V along 1 V \ L, then proje tion from L indu es a bration W ! P whose bers are hyperplane se tions of V . Moreover, if L is suÆ iently general, then W ! P1 will be a Lefs hetz pen il, meaning that the only singularities of bers are nodes. In fa t, for most L, all bers will be either smooth plane ubi urves, or ubi urves with a single node. For some N 1, the above onstru tion an be done with models over Spe Z[1=N ℄ so that for ea h prime p6 jN , redu tion mod p yields a family of plane ubi urves ea h smooth or with a single node. One then proves that if p6 jN , ea h ber above an Fp-point has a1 smooth Fp-point,1 so Hensel's Lemma onstru ts a Qp-point on the ber Wt of W ! P above any t 2 P (Q). There is no reason that su h Wt should have Qp-points for pjN , but the existen e of1 Qppoints on V implies that at least for t in a nonempty p-adi ally open subset Up of P (Qp), Wt (Qp ) will be nonempty. We obtain the desired family by base-extending W ! P1 by a rational fun tion f : P1 ! P1 su h that f (P1(Qp)) Up for ea h pjN . More details of this onstru tion an be found in [CP℄. 3. The ubi surfa e onstru tion
Lemmas
Let V be a smooth ubi surfa e in P3 over an algebrai ally losed eld k . Let L be a line in P3 interse ting V in exa tly 3 points. Let W be the blowup of V at these points. Let W ! P1 be the bration of W by plane ubi s indu ed by the proje tion P3 n L ! P1 from L. Assume that some ber of : W ! P1 is smooth. Then at most 12 bers are singular, and if there are exa tly 12, ea h is a nodal plane ubi . Lemma 1.
We give two approa hes towards this result, one via expli it al ulations with the dis riminant of a ternary ubi form, and the other via Euler hara teristi s. The rst has
FAMILY OF GENUS-ONE CURVES
3
smooth cubic e=0
nodal cubic e=1
cuspidal cubic e=2
conic + line e=2
conic + tangent e=3
line + double line e=3
three lines e=3
concurrent lines e=4
triple line e=2
Plane ubi urves with their Euler hara teristi s. the advantage of requiring mu h less ma hinery, but we omplete this proof only under the assumption that L does not meet any of the 27 lines on V . (With more work, one ould probably prove the general ase too, but we have not tried too seriously, sin e the spe ial ase proved is all we need for our appli ation, and also sin e the se ond proof works generally.) The se ond proof an be interpreted as explaining the order of vanishing of the dis riminant of the family in terms of the Euler hara teristi of a bad ber. First proof of Lemma 1, assuming that L does not meet the 27 lines. Let F (x; y; z ) = a0x3 + a1x2y + a2xy 2 + a3 y 3 + a4x2z + a5xyz + a6y2z + a7xz2 + a8yz2 + a9z3 be the generi ternary ubi form, with indeterminates a0; : : : ; a9 as oeÆ ients. Let H (x; y; z) be the Hessian of F , i.e., the determinant of the 3 3 matrix of se ond partial derivatives of F . Let be 2 9 3 3 times the determinant of the 6 6 obtained by writing ea h of F=x, F=y , F=z , H=x, H=y , H=z in terms of the basis x2, xy , y 2, xz , yz , z 2 . (This Figure 1.
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BJORN POONEN
is a spe ial ase of a lassi al formula for the resultant of three quadrati forms in three variables, whi h is reprodu ed in [St℄, for instan e.) One omputes that is a homogeneous polynomial of degree 12 in Z[a0; : : : ; a9℄. If we spe ialize F to the homogenization of y 2 (x3 + Ax + B ), we nd that be omes the usual dis riminant 16(4A3 + 27B 2) of the ellipti urve [Si, p. 50℄. Be ause is an invariant for the a tion of GL3(k), it follows that F gives the usual dis riminant for any ellipti urve in general Weierstrass form y 2 + a1xy + a3 y = x3 + a2 x2 + a4 x + a6 at least in hara teristi zero, and hen e in any hara teristi . Every smooth plane ubi is proje tively equivalent to su h an ellipti urve, so is nonvanishing whenever the urve F = 0 is smooth. On the other hand, if F = 0 is singular at (0 : 0 : 1), so that a7 = a8 = a9 = 0, we ompute that be omes 0.1 Again using the invarian e of , we dedu e that = 0 if and only if F = 0 is singular. Sin e a ubi surfa e is isomorphi to P2 blown up at 6 points ([Ma, Theorem IV.24.4℄, for example), and W is the blowup of V at 3 points, we obtain a birational morphism 2 1 W ! P . Taking the produ t of this with the bration map W ! P yields a morphism :W! P2 P1 birational onto its image. The morphism separates points, sin e 2 W ! P separates points ex ept for 9 lines whi h ontra t to points, and these proje t isomorphi ally to the se ond fa tor P2 1, be ause by assumption L does not meet the 6 lines
ontra ted by the morphism V ! P . To verify that separates tangent ve tors, we need only observe that at a point P 2 W on one of the 9 lines, a tangent ve tor2 transverse to the line through P maps to a nonzero tangent ve tor at the image point in P , while 2a tangent ve tor at P along the line maps to the zero tangent ve tor at the image point in P but to a nonzero tangent ve tor at the image point in P1 . Hen e is a losed immersion, so we may view the given family W ! P1 as a family of urves in P2 . By assumption, there exists a 1 ber of W ! P that is a smooth ubi urve. It follows that the divisor W in P2 P1 is of type (3; 1), and hen e W is given by a bihomogeneous equation q(x0; x1; x2; x3; t0; t1) = 0 of degree 3 in x0;1x1; x2; x3 and of degree 1 in t0; t1. In other words, we may view the bration W ! P as a family of ubi plane urves where the oeÆ ients a0; : : : ; a9 are linear polynomials in the homogeneous oordinates t0; t1 on the base P1. Hen e for this family is a homogeneous polynomial of degree 12 in t0; t1, and it is nonvanishing be ause of the assumption that at least one ber is smooth. Thus at most 12 bers are singular. To nish the proof, we need only show that if a ber is singular and not with just a single node, then (b0; b1) vanishes to order at least 2 at the orresponding point on P1. To prove this, we enumerate the ombinatorial possibilities for a plane ubi , orresponding to the degrees of the fa tors of the ubi polynomial: see Figure 1. In the \three lines" ase, after a linear hange of variables with onstant oeÆ ients we may assume that the three interse tion points on the bad ber are (1 : 0 : 0), (0 : 1 : 0), and (0 : 0 : 1), so that the ber is xyz = 0. We ompute that if G is a general ternary1
ubi and t is an indeterminate, representing the uniformizer at a point on the base P , 3 then for xyz + tG is divisible by t . In the \ oni + line" ase, we assume that the two interse tion points are the points (1 : 0 : 0), (0 : 1 : 0) so that the oni is a re tangular 1Fa ts su h as this are undoubtedly lassi al, at least over C, but it seems easier to reprove them than to nd a suitable referen e.
FAMILY OF GENUS-ONE CURVES
5
hyperbola and the line is the line z = 0 \at in nity." We an then translate the enter of the hyperbola2 to (0 : 0 : 1) and s ale to2 assume that the ber is (xy z2)z = 0. This time, for (xy z )z + tG is divisible by t . In the \ uspidal ubi " and \ oni + tangent" ases, we may hange oordinates so that the singularity is at (0 : 0 : 1) and the line x = 0 is tangent to the bran hes of the urve there, so that a5 = a6 = a7 = a8 = a9 = 0. In the remaining ases, \line + double line," \ on urrent lines," and \triple line," we may move a point of multipli ity 3 to (0 : 0 : 1), and again we will have at least a5 = a6 = a7 = a8 = a9 = 0. We he k that vanishes to order 2 in all ve of these ases, by substituting ai = tbi for i = 52; 6; 7; 8; 9 in the generi formula for , and verifying that the spe ialized is divisible by t . Se ond proof of Lemma 1. Let p be the hara teristi of k , and hoose a prime ` 6= p. Let (V ) =
2X dim V
( 1)i dimF Hei t(V; F`) `
i=0
denote the Euler hara teristi . Sin e V is isomorphi to the blowup of P2 at 6 points, and W is the blowup of V at 3 points, (W ) = (P2 ) + 6 + 3 = 3 + 6 + 3 = 12: On the other hand, ombining the Leray spe tral sequen e Hp(P1; Rq F`) =) Hp+q (W; F`) with the Grothendie k-Ogg-Shafarevi h formula ([Ra, Theoreme 1℄ or [Mi, Theorem 2.12℄) yields X (1) (W ) = (W ) (P1) + [ (Wt) (W ) swt(Het(W ; F`))℄ ; t2P1 (k)
where W is the generi ber, Wt is the ber above t, and swt(Het(W ; F` )) :=
2 X i=0
( 1)i swt(Hiet(W ; F` ))
i is the alternating sum of the Swan ondu tors of H et (W ; F` ) onsidered as a representation of the inertia group at t of the1 base P1. Sin e W is a smooth urve of genus g = 1, (W ) = 2 2g = 0. If t 2 P (k ) is su h that Wt is smooth, then all terms within the bra kets on the right side of (1) are 0, so the sum is nite. The Swan ondu tor of H0et(W ; F` ) = H2et(W ; F` ) = F` is trivial. Hen e (1) be omes X (Wt) + swt (H1et (W ; F` )) ; 12 = t: Wt is singular
Sin e swt(He1t(W ; F`)) is a dimension, it is nonnegative, so the lemma will follow from the following laim: if Wt is singular, (Wt) 1 with equality if and only if Wt is a nodal ubi . To prove this, we again he k the ases listed in Figure 1. The Euler hara teristi for ea h,
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BJORN POONEN
whi h is un hanged if we pass to the asso iated redu ed s heme C , is omputed using the formula X (2) (C ) = (2 2gC~ ) + #Csing # 1 (Csing); i ~ where : C ! C is the normalization of C , gC~ is the genus of the i-th omponent of C~ , and Csing is the set of singular points of C . For example, for the \ oni + tangent," formula (2) gives 2 X 3 = (2 2 0) + 1 2: i=1 Lemma 2. If F (x; y; z ) 2 Fp [x; y; z ℄ is a nonzero homogeneous ubi polynomial su h that F does not fa tor ompletely into linear fa tors over Fp , then the subs heme X of P2 de ned by F = 0 has a smooth Fp -point. Proof. The polynomial F must be squarefree, sin e otherwise F would fa tor ompletely. Hen e X is redu ed. If X is a smooth ubi urve, then it is of genus 1, and X (Fp) 6= ; by the Hasse bound. Otherwise, enumerating possibilities as in Figure 1 shows that X is a nodal or uspidal
ubi , or a union of a line and a oni . The Galois a tion on omponents is trivial, be ause when there is more than one, 1the omponents have di erent degrees. There is an open subset of X isomorphi over Fp to P with at most two geometri points deleted. But #P1(Fp) 3, so there remains a smooth Fp-point on X . 4. We will arry out the program in Se tion 2 with the ubi surfa e V : 5x3 + 9y 3 + 10z 3 + 12w3 = 0 in P3. Cassels and Guy [CG℄ proved that V violates the Hasse prin iple. Let L be the line x + y + z = w = 0. The interse tion V \ L as a subs heme of L = P1 with homogeneous
oordinates x; y is de ned by 5x3 + 9y3 10(x + y)3; whi h has dis riminant 242325 = 33 52 359 6= 0, so the interse tion onsists of three distin t geometri points. This remains true in hara teristi p, provided that p 62 f3; 5; 359g. The proje tion V 99 K P1 from 3L is given by the rational fun tion u := w=(x + y + z) 1 on V . Also, W is the surfa e in P P given by the ((x; y; z; w);(u0; u1))-bihomogeneous equations (3) W : 5x3 + 9y 3 + 10z 3 + 12w3 = 0 u0w = u1(x + y + z ): The morphism W1 ! P1 is1 simply the proje tion to the se ond fa tor, and the ber Wu above u 2 Q = A (Q) P (Q) an also be written as the plane ubi (4) Wu : 5x3 + 9y 3 + 10z 3 + 12u3 (x + y + z )3 = 0: The dehomogenization h(x; y ) = 5x3 + 9y 3 + 10 + 12u3 (x + y + 1)3 ; i
i
The example
FAMILY OF GENUS-ONE CURVES
de nes an aÆne open subset in A2 of Wu . Eliminating x and y from the equations h h = =0 h= x y
7
shows that this aÆne variety is singular when u 2 Q satis es (5) 2062096u12 + 6065760u9 + 4282200u6 + 999000u3 + 50625 = 0: The ber above u = 0 is smooth, so by Lemma 1, the 12 values of u satisfying (5) give the only points in P1(Q) above whi h the ber Wu is singular, and moreover ea h of these singular bers is a nodal ubi . (Alternatively, one ould al ulate that of the rst proof of Lemma 1 for (4) equals 2431354 times the polynomial (5). One an easily verify that in any
hara teristi p 62 f2; 3; 5g, L does not meet any of 1the 27 lines on V .) The polynomial (5) for50 all u42 P (Q). is irredu ible over Q, so Wu is146smooth 92 The dis riminant of (5) is 2 3 5 359 . Fix a prime p 62 f2; 3; 5; 359g, and a pla e Q 99 K Fp . The 12 singular u-values in P1 (Q) redu e to 12 distin t singular u-values in P1 (Fp ) for the family W ! P1 de ned by the two equations (3) over Fp . Moreover, the ber above u = 0 is smooth in hara teristi p. By Lemma 1, all the bers of W ! P1 in
hara teristi p are smooth plane ubi s or nodal plane ubi s. By Lemma 2 and Hensel's Lemma, Wu has a Qp-point for all u 2 P1(Qp). Proposition 3. If u 2 Q satis es u 1 (mod pZp ) for p 2 f2; 3; 5g and u 2 Z359, then the ber Wu has a Qp -point for all ompletions Qp , p 1. Proof. Existen e of real points is automati , sin e Wu is a plane urve of odd degree. Existen e of Qp-points for p 62 f2; 3; 5; 359g was proved just above the statement of Proposition 3. Consider p = 359. A Grobner basis al ulation shows that there do not exist a1, a2, b1, b2, 1 , 2, u 2 F359 su h that (6) 5x3 + 9y3 + 10z3 + 12u3(x + y + z)3 and (5x + a1y + a2z)(x + b1y + b2z)(x + 1y + 2z) are identi al. Hen e Lemma 2 applies to show that for any u 2 F359, the plane ubi de ned by (6) over F359 has a smooth F359-point, and Hensel's Lemma implies that Wu has a Q359-point at least when u 2 Z359. When u 1 (mod 5Z5), the urve redu ed modulo 5, W u : 4y 3 + 2(x + y + z )3 = 0;
onsists of three lines through P := (1 : 0 : 1) 2 P2(F5), so it does not satisfy the onditions of Lemma 2, but one of the lines, namely y = 2(x + y + z), is de ned over F5, and every F5 -point on this line ex ept P is smooth on W u . Hen e Wu has a Q5 -point. The same argument shows that Wu has a Q2-point whenever u 1 (mod 2Z2), sin e the
urve redu ed modulo 2 is x3 + y3 = 0, whi h ontains x + y = 0. Finally, when u 1 (mod 3Z3), the point (1 : 2 : 1) satis es the equation (4) modulo 32, and Hensel's Lemma gives a point (x0 : 2 : 1) 2 Wu (Q3) with x0 1 (mod 3Z3 ). This
ompletes the proof.
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BJORN POONEN
1 1 1 We now seek a non- onstant rational fun tion P that maps P (Qp) into 1+ pZp for P ! 22 = 1 for p 2 f3; 5; 359g, the fun tion p 2 f2; 3; 5g and into Z359 for p = 359. Sin e p 60 = t2 + 82 u=1+ 2 t + 22 t2 + 22 has the desired property. Substituting into (4), we see that 2 t + 82 3 3 3 3 (7) Xt : 5x + 9y + 10z + 12 2 (x + y + z)3 = 0 t + 22 has Qp-points for all p 1. On the other hand, Xt (Q) = ;, be ause V (Q) = ;. Finally, the existen e of nodal bers in the family implies as in [CP℄ that the j -invariant of the family has poles, and hen e is non- onstant. 5. For t 2 Q, let Et denote the Ja obian of Xt . The papers [AKMMMP℄ and [RVT℄ ea h
ontain a proof that the lassi al formulas of Salmon for invariants of a plane ubi yield
oeÆ ients of a Weierstrass model of the Ja obian. We used a GP-PARI implementation of these by Fernando Rodriguez-Villegas, available ele troni ally at The Ja obians
ftp://www.ma.utexas.edu/pub/villegas/gp/inv- ubi .gp
to show that our Et has a Weierstrass model y2 = x3 + Ax + B where A = 145800(t2 + 82)3 (t2 + 22) and B = 6129675t12 96155100t10 + 359349979500t8 + 65556113292000t6 + 4990338518958000t4 + 180317231391182400t2 + 2572729234128532800: Be ause the non-existen e of rational points on V is explained by a Brauer-Manin obstru tion, Se tion 3.5 and in parti ular Proposition 3.5 of [CP℄ show that there exists a se ond family of genus 1 urves Yt with the same Ja obians su h that the Cassels-Tate pairing satis es hXt ; Yti = 1=3 for all t 2 Q. In parti ular, for all t 2 Q, the Shafarevi h-Tate group X(Et) ontains a subgroup isomorphi to Z=3 Z=3. Although we will not nd expli it equations for the se ond family here, we an at least outline how this might be done, following [CP℄, ex ept using Galois ohomology over number 1 elds and Q(t) wherever p possible in pla e of etale ohomology over open subsets of P : (1) Let k = Q( 3) and Vk = V Q k. On pages 66{67 of [CTKS℄ an element Ak of Br(Vk ) giving a Brauer-Manin obstru tion over2 k is des ribed byparameters ; for a 2- o y le representing the image of Ak in H (Gal(K=k); K (V2 ) ) for a ertain nite abelian extension K of k. We may map to an element of H (k; Q(V )). (2) Apply the orestri tion oresk=Q to Ak to obtain an element A 2 Br(V ) giving a Brauer-Manin obstru tion for V over Q. (See the proof of Lemme 4(ii) in [CTKS℄.) In pra ti e, all elements of Brauer groups are to be represented by 2- o y les analogous to , and all operations are a tually performed on these o y les. (3) Pull ba k A under the morphisms X ! X ! V , where X over Q(t) is the generi ber of our nal family X ! P1 of genus 1 urves, to obtain an element A of Br(X ).
FAMILY OF GENUS-ONE CURVES
9
(4) Let X denote X Q(t) Q(t). Find the image of A in H1(Q(t); Pi X ), by writing the divisor of the 2- o y le representing the image of A in H2(Q(t); Q(t)(X )) as the oboundary of a 1- o hain, whi h be omes a 1- o y le representing an element of H1(Q(t); Pi X ). (5) Observe that the newly dis overed 1- o y le a tually takes values in Pi 0 X = E (Q(t)), where E is the Ja obian of X over Q(t). (6) Re onstru t the prin ipal homogeneous spa e Y of E over Q(t) from this 1- o y le, by omputing the fun tion eld of Y as in Se tion X.2 of [Si℄. (7) Find the minimal model of Y over P1Q, if desired, to obtain a model smooth over the same open subset of P1 as X . A knowledgements
I thank Ahmed Abbes for explaining the formula (1) to me, Antoine Du ros for making a omment that led to a simpli ation of the rational fun tion used at the end of Se tion 4, and Noam Elkies for suggesting the rst proof of Lemma 1 and the remark at the end of the introdu tion. I thank Bernd Sturmfels for the expression for the dis riminant of a plane ubi as a 6 6 determinant, and for providing Maple ode for it. I thank also Bill M Callum and Fernando Rodriguez-Villegas, for providing Salmon's formulas for invariants of plane ubi s in ele troni form. The al ulations for this paper were mostly done using Maple, Mathemati a, and GP-PARI on a Sun Ultra 2. The values of A and B in Se tion 5 and their trans ription into LaTeX were he ked by pasting the LaTeX formulas into Mathemati a, plugging them into the formulas for the j -invariant from GP-PARI, and
omparing the result against the j -invariant of Xt as omputed dire tly by Mark van Hoeij's Maple pa kage \IntBasis" at http://www.math.fsu.edu/~hoeij/ ompalg/IntBasis/index.html
for a few values of t 2 Q.
Referen es
[AKMMMP℄ S. Y. An, S. Y. Kim, D. Marshall, S. Marshall, W. M Callum, A. Perlis, Ja obians of genus one urves, preprint, 1999. [CG℄ J. W. S. Cassels and M. J. T. Guy, On the Hasse prin iple for ubi surfa es, Mathematika 13 (1966), 111{120. [CTKS℄ J.-L. Colliot-Thelene, D. Kanevsky, et J.-J. Sansu , Arithmetique des surfa es ubiques diagonales, pp. 1{108 in Diophantine approximation and trans enden e theory (Bonn, 1985), Le ture Notes in Math., 1290, Springer, Berlin, 1987. [CP℄ J.-L. Colliot-Thelene and B. Poonen, Algebrai families of nonzero elements of Shafarevi h-Tate groups, J. Amer. Math. So . 13 (2000), 83{99. [Lin℄ C.-E. Lind, Untersu hungen uber die rationalen Punkte der ebenen kubis hen Kurven vom Ges hle ht Eins, Thesis, University of Uppsala, 1940. [Ma℄ Yu. I. Manin, Cubi forms, Translated from the Russian by M. Hazewinkel, Se ond edition, NorthHolland, Amsterdam, 1974. [Mi℄ J. Milne, Etale
ohomology, Prin eton Univ. Press, Prin eton, N.J., 1980. [O'N℄ C. O'Neil, Ja obians of urves of genus one, Thesis, Harvard University, 1999. [Ra℄ M. Raynaud, Cara teristique d'Euler-Poin are d'un fais eau et ohomologie des varietes abeliennes, Seminaire Bourbaki, Expose 286 (1965). [Re℄ H. Rei hardt, Einige im Kleinen uberall losbare, im Grossen unlosbare diophantis he Glei hungen, J. Reine Angew. Math. 184 (1942), 12{18. [RVT℄ F. Rodriguez-Villegas and J. Tate, On the Ja obian of plane ubi s, in preparation, 1999.
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[Se℄ E. Selmer, The diophantine equation ax3 + by3 + z 3 = 0, A ta Math. 85 (1951), 203{362 and 92 (1954), 191{197. [Si℄ J. Silverman, The arithmeti of ellipti urves, Graduate Texts in Mathemati s 106, Springer-Verlag, New York-Berlin, 1986. [St℄ B. Sturmfels, Introdu tion to resultants, Appli ations of omputational algebrai geometry (San Diego, CA, 1997), 25{39, Pro . Sympos. Appl. Math. 53, Amer. Math. So ., Providen e, RI, 1998. [SD℄ H. P. F. Swinnerton-Dyer, Two spe ial ubi surfa es, Mathematika 9 (1962), 54{56. Department of Mathemati s, University of California, Berkeley, CA 94720-3840, USA
E-mail address : poonenmath.berkeley.edu
BJORN POONEN Abstra t.
We prove that for any t 2 Q, the urve 5x3 + 9y3 + 10z 3 + 12
t2 + 82 3 t2 + 22
(x + y + z )3 = 0
in P2 is a genus 1 urve violating the Hasse prin iple. An expli it Weierstrass model for its Ja obian Et is given. The Shafarevi h-Tate group of ea h Et ontains a subgroup isomorphi to Z=3 Z=3.
1.
Introdu tion
One says that a variety X over Q violates the Hasse prin iple if X (Qv ) 6= ; for all
ompletions Qv of Q (i.e., Rn and Qp for all primes p) but X (Q) = ;. Hasse proved that degree 2 hypersurfa es in P satisfy the Hasse prin iple. In parti ular, if X is a genus 0
urve,2 then X satis es the Hasse prin iple, sin e the anti anoni al embedding of X is a oni in P . Around 1940, Lind [Lin℄ and (independently, but shortly later) Rei hardt [Re℄ dis overed examples of genus 1 urves over Q that violate the Hasse prin iple, su h as the nonsingular proje tive model of the aÆne urve 2y2 = 1 17x4 : Later, Selmer [Se℄ gave examples of diagonal plane ubi urves (also of genus 1) violating the Hasse prin iple, in luding 3x3 + 4y3 + 5z3 = 0 in P2. O'Neil [O'N, x6.5℄ onstru ts an interesting example of an algebrai family of genus 1 urves ea h having Qp-points for all p 1. Some bers in her family violate the Hasse prin iple, by failing to have a Q-point. In other words, these bers represent nonzero elements of the Shafarevi h-Tate groups of their Ja obians. In [CP℄, Colliot-Thelene and the present author prove,1 among other things, the existen e of non-isotrivial families of genus 1 urves over the base P , smooth over a dense open subset, 1 su h that the ber over ea h rational point of P is a smooth plane ubi violating the Hasse prin iple. In more on rete terms, this implies that there exists a family of plane ubi s depending on a parameter t, su h that the j -invariant is a non- onstant fun tion of t, and su h that substituting any rational number for t results in a smooth plane ubi over Q violating the Hasse prin iple. Date : Mar h 18, 2000.
This resear h was supported by National S ien e Foundation grant DMS-9801104, an Alfred P. Sloan Fellowship, and a David and Lu ile Pa kard Fellowship. 1
2
BJORN POONEN
The purpose of this paper is to produ e an expli it example of su h a family. Our example, presented as a family of ubi urves in P2 with homogeneous oordinates x; y; z, is 2 t + 82 3 3 3 3 5x + 9y + 10z + 12 t2 + 22 (x + y + z)3 = 0: Remark . Noam Elkies pointed out to me that the existen e of non-trivial families with
onstant j -invariant ould be easily dedu ed from previously known results. In Case I of the proof of Theorem X.6.5 in [Si℄, one nds a proof (based on ideas of Lind and Mordell) that 2 4 2y = 1 Nx represents a nontrivial element of X[2℄ of its Ja obian, when N 14 (mod 8) is a prime for whi h 2 is not a quarti residue. The same argument works if N = N 0 where
2 Q and N 0 is a produ t of primes p 1 (mod 8), provided that 2 is not a quarti residue for at least one p appearing with odd exponent in N 0 . One an he k that if N = a4 + 16b4 for some a; b 2 Q \ Z2, 1then N has this form. One an now substitute rational fun tions for a and b 2mapping P (Q2) into Z2, with a=b not onstant. For instan e, the hoi es a = 1 + 2=(t + t + 1) and b = 1 lead to the family 2y2 = 1 (t2 + t + 3)4 + 16(t2 + t + 1)4 x4 of genus 1 urves of j -invariant 1728 violating the Hasse prin iple. 2. Let us review brie y the onstru tion in [CP℄. Swinnerton-Dyer [SD℄ proved that there 3 exist smooth
ubi surfa es V in P over Q violating the Hasse prin iple; hoose one. If L is a line in P3 meeting V in exa tly 3 geometri points, and W denotes the blowup of V along 1 V \ L, then proje tion from L indu es a bration W ! P whose bers are hyperplane se tions of V . Moreover, if L is suÆ iently general, then W ! P1 will be a Lefs hetz pen il, meaning that the only singularities of bers are nodes. In fa t, for most L, all bers will be either smooth plane ubi urves, or ubi urves with a single node. For some N 1, the above onstru tion an be done with models over Spe Z[1=N ℄ so that for ea h prime p6 jN , redu tion mod p yields a family of plane ubi urves ea h smooth or with a single node. One then proves that if p6 jN , ea h ber above an Fp-point has a1 smooth Fp-point,1 so Hensel's Lemma onstru ts a Qp-point on the ber Wt of W ! P above any t 2 P (Q). There is no reason that su h Wt should have Qp-points for pjN , but the existen e of1 Qppoints on V implies that at least for t in a nonempty p-adi ally open subset Up of P (Qp), Wt (Qp ) will be nonempty. We obtain the desired family by base-extending W ! P1 by a rational fun tion f : P1 ! P1 su h that f (P1(Qp)) Up for ea h pjN . More details of this onstru tion an be found in [CP℄. 3. The ubi surfa e onstru tion
Lemmas
Let V be a smooth ubi surfa e in P3 over an algebrai ally losed eld k . Let L be a line in P3 interse ting V in exa tly 3 points. Let W be the blowup of V at these points. Let W ! P1 be the bration of W by plane ubi s indu ed by the proje tion P3 n L ! P1 from L. Assume that some ber of : W ! P1 is smooth. Then at most 12 bers are singular, and if there are exa tly 12, ea h is a nodal plane ubi . Lemma 1.
We give two approa hes towards this result, one via expli it al ulations with the dis riminant of a ternary ubi form, and the other via Euler hara teristi s. The rst has
FAMILY OF GENUS-ONE CURVES
3
smooth cubic e=0
nodal cubic e=1
cuspidal cubic e=2
conic + line e=2
conic + tangent e=3
line + double line e=3
three lines e=3
concurrent lines e=4
triple line e=2
Plane ubi urves with their Euler hara teristi s. the advantage of requiring mu h less ma hinery, but we omplete this proof only under the assumption that L does not meet any of the 27 lines on V . (With more work, one ould probably prove the general ase too, but we have not tried too seriously, sin e the spe ial ase proved is all we need for our appli ation, and also sin e the se ond proof works generally.) The se ond proof an be interpreted as explaining the order of vanishing of the dis riminant of the family in terms of the Euler hara teristi of a bad ber. First proof of Lemma 1, assuming that L does not meet the 27 lines. Let F (x; y; z ) = a0x3 + a1x2y + a2xy 2 + a3 y 3 + a4x2z + a5xyz + a6y2z + a7xz2 + a8yz2 + a9z3 be the generi ternary ubi form, with indeterminates a0; : : : ; a9 as oeÆ ients. Let H (x; y; z) be the Hessian of F , i.e., the determinant of the 3 3 matrix of se ond partial derivatives of F . Let be 2 9 3 3 times the determinant of the 6 6 obtained by writing ea h of F=x, F=y , F=z , H=x, H=y , H=z in terms of the basis x2, xy , y 2, xz , yz , z 2 . (This Figure 1.
4
BJORN POONEN
is a spe ial ase of a lassi al formula for the resultant of three quadrati forms in three variables, whi h is reprodu ed in [St℄, for instan e.) One omputes that is a homogeneous polynomial of degree 12 in Z[a0; : : : ; a9℄. If we spe ialize F to the homogenization of y 2 (x3 + Ax + B ), we nd that be omes the usual dis riminant 16(4A3 + 27B 2) of the ellipti urve [Si, p. 50℄. Be ause is an invariant for the a tion of GL3(k), it follows that F gives the usual dis riminant for any ellipti urve in general Weierstrass form y 2 + a1xy + a3 y = x3 + a2 x2 + a4 x + a6 at least in hara teristi zero, and hen e in any hara teristi . Every smooth plane ubi is proje tively equivalent to su h an ellipti urve, so is nonvanishing whenever the urve F = 0 is smooth. On the other hand, if F = 0 is singular at (0 : 0 : 1), so that a7 = a8 = a9 = 0, we ompute that be omes 0.1 Again using the invarian e of , we dedu e that = 0 if and only if F = 0 is singular. Sin e a ubi surfa e is isomorphi to P2 blown up at 6 points ([Ma, Theorem IV.24.4℄, for example), and W is the blowup of V at 3 points, we obtain a birational morphism 2 1 W ! P . Taking the produ t of this with the bration map W ! P yields a morphism :W! P2 P1 birational onto its image. The morphism separates points, sin e 2 W ! P separates points ex ept for 9 lines whi h ontra t to points, and these proje t isomorphi ally to the se ond fa tor P2 1, be ause by assumption L does not meet the 6 lines
ontra ted by the morphism V ! P . To verify that separates tangent ve tors, we need only observe that at a point P 2 W on one of the 9 lines, a tangent ve tor2 transverse to the line through P maps to a nonzero tangent ve tor at the image point in P , while 2a tangent ve tor at P along the line maps to the zero tangent ve tor at the image point in P but to a nonzero tangent ve tor at the image point in P1 . Hen e is a losed immersion, so we may view the given family W ! P1 as a family of urves in P2 . By assumption, there exists a 1 ber of W ! P that is a smooth ubi urve. It follows that the divisor W in P2 P1 is of type (3; 1), and hen e W is given by a bihomogeneous equation q(x0; x1; x2; x3; t0; t1) = 0 of degree 3 in x0;1x1; x2; x3 and of degree 1 in t0; t1. In other words, we may view the bration W ! P as a family of ubi plane urves where the oeÆ ients a0; : : : ; a9 are linear polynomials in the homogeneous oordinates t0; t1 on the base P1. Hen e for this family is a homogeneous polynomial of degree 12 in t0; t1, and it is nonvanishing be ause of the assumption that at least one ber is smooth. Thus at most 12 bers are singular. To nish the proof, we need only show that if a ber is singular and not with just a single node, then (b0; b1) vanishes to order at least 2 at the orresponding point on P1. To prove this, we enumerate the ombinatorial possibilities for a plane ubi , orresponding to the degrees of the fa tors of the ubi polynomial: see Figure 1. In the \three lines" ase, after a linear hange of variables with onstant oeÆ ients we may assume that the three interse tion points on the bad ber are (1 : 0 : 0), (0 : 1 : 0), and (0 : 0 : 1), so that the ber is xyz = 0. We ompute that if G is a general ternary1
ubi and t is an indeterminate, representing the uniformizer at a point on the base P , 3 then for xyz + tG is divisible by t . In the \ oni + line" ase, we assume that the two interse tion points are the points (1 : 0 : 0), (0 : 1 : 0) so that the oni is a re tangular 1Fa ts su h as this are undoubtedly lassi al, at least over C, but it seems easier to reprove them than to nd a suitable referen e.
FAMILY OF GENUS-ONE CURVES
5
hyperbola and the line is the line z = 0 \at in nity." We an then translate the enter of the hyperbola2 to (0 : 0 : 1) and s ale to2 assume that the ber is (xy z2)z = 0. This time, for (xy z )z + tG is divisible by t . In the \ uspidal ubi " and \ oni + tangent" ases, we may hange oordinates so that the singularity is at (0 : 0 : 1) and the line x = 0 is tangent to the bran hes of the urve there, so that a5 = a6 = a7 = a8 = a9 = 0. In the remaining ases, \line + double line," \ on urrent lines," and \triple line," we may move a point of multipli ity 3 to (0 : 0 : 1), and again we will have at least a5 = a6 = a7 = a8 = a9 = 0. We he k that vanishes to order 2 in all ve of these ases, by substituting ai = tbi for i = 52; 6; 7; 8; 9 in the generi formula for , and verifying that the spe ialized is divisible by t . Se ond proof of Lemma 1. Let p be the hara teristi of k , and hoose a prime ` 6= p. Let (V ) =
2X dim V
( 1)i dimF Hei t(V; F`) `
i=0
denote the Euler hara teristi . Sin e V is isomorphi to the blowup of P2 at 6 points, and W is the blowup of V at 3 points, (W ) = (P2 ) + 6 + 3 = 3 + 6 + 3 = 12: On the other hand, ombining the Leray spe tral sequen e Hp(P1; Rq F`) =) Hp+q (W; F`) with the Grothendie k-Ogg-Shafarevi h formula ([Ra, Theoreme 1℄ or [Mi, Theorem 2.12℄) yields X (1) (W ) = (W ) (P1) + [ (Wt) (W ) swt(Het(W ; F`))℄ ; t2P1 (k)
where W is the generi ber, Wt is the ber above t, and swt(Het(W ; F` )) :=
2 X i=0
( 1)i swt(Hiet(W ; F` ))
i is the alternating sum of the Swan ondu tors of H et (W ; F` ) onsidered as a representation of the inertia group at t of the1 base P1. Sin e W is a smooth urve of genus g = 1, (W ) = 2 2g = 0. If t 2 P (k ) is su h that Wt is smooth, then all terms within the bra kets on the right side of (1) are 0, so the sum is nite. The Swan ondu tor of H0et(W ; F` ) = H2et(W ; F` ) = F` is trivial. Hen e (1) be omes X (Wt) + swt (H1et (W ; F` )) ; 12 = t: Wt is singular
Sin e swt(He1t(W ; F`)) is a dimension, it is nonnegative, so the lemma will follow from the following laim: if Wt is singular, (Wt) 1 with equality if and only if Wt is a nodal ubi . To prove this, we again he k the ases listed in Figure 1. The Euler hara teristi for ea h,
6
BJORN POONEN
whi h is un hanged if we pass to the asso iated redu ed s heme C , is omputed using the formula X (2) (C ) = (2 2gC~ ) + #Csing # 1 (Csing); i ~ where : C ! C is the normalization of C , gC~ is the genus of the i-th omponent of C~ , and Csing is the set of singular points of C . For example, for the \ oni + tangent," formula (2) gives 2 X 3 = (2 2 0) + 1 2: i=1 Lemma 2. If F (x; y; z ) 2 Fp [x; y; z ℄ is a nonzero homogeneous ubi polynomial su h that F does not fa tor ompletely into linear fa tors over Fp , then the subs heme X of P2 de ned by F = 0 has a smooth Fp -point. Proof. The polynomial F must be squarefree, sin e otherwise F would fa tor ompletely. Hen e X is redu ed. If X is a smooth ubi urve, then it is of genus 1, and X (Fp) 6= ; by the Hasse bound. Otherwise, enumerating possibilities as in Figure 1 shows that X is a nodal or uspidal
ubi , or a union of a line and a oni . The Galois a tion on omponents is trivial, be ause when there is more than one, 1the omponents have di erent degrees. There is an open subset of X isomorphi over Fp to P with at most two geometri points deleted. But #P1(Fp) 3, so there remains a smooth Fp-point on X . 4. We will arry out the program in Se tion 2 with the ubi surfa e V : 5x3 + 9y 3 + 10z 3 + 12w3 = 0 in P3. Cassels and Guy [CG℄ proved that V violates the Hasse prin iple. Let L be the line x + y + z = w = 0. The interse tion V \ L as a subs heme of L = P1 with homogeneous
oordinates x; y is de ned by 5x3 + 9y3 10(x + y)3; whi h has dis riminant 242325 = 33 52 359 6= 0, so the interse tion onsists of three distin t geometri points. This remains true in hara teristi p, provided that p 62 f3; 5; 359g. The proje tion V 99 K P1 from 3L is given by the rational fun tion u := w=(x + y + z) 1 on V . Also, W is the surfa e in P P given by the ((x; y; z; w);(u0; u1))-bihomogeneous equations (3) W : 5x3 + 9y 3 + 10z 3 + 12w3 = 0 u0w = u1(x + y + z ): The morphism W1 ! P1 is1 simply the proje tion to the se ond fa tor, and the ber Wu above u 2 Q = A (Q) P (Q) an also be written as the plane ubi (4) Wu : 5x3 + 9y 3 + 10z 3 + 12u3 (x + y + z )3 = 0: The dehomogenization h(x; y ) = 5x3 + 9y 3 + 10 + 12u3 (x + y + 1)3 ; i
i
The example
FAMILY OF GENUS-ONE CURVES
de nes an aÆne open subset in A2 of Wu . Eliminating x and y from the equations h h = =0 h= x y
7
shows that this aÆne variety is singular when u 2 Q satis es (5) 2062096u12 + 6065760u9 + 4282200u6 + 999000u3 + 50625 = 0: The ber above u = 0 is smooth, so by Lemma 1, the 12 values of u satisfying (5) give the only points in P1(Q) above whi h the ber Wu is singular, and moreover ea h of these singular bers is a nodal ubi . (Alternatively, one ould al ulate that of the rst proof of Lemma 1 for (4) equals 2431354 times the polynomial (5). One an easily verify that in any
hara teristi p 62 f2; 3; 5g, L does not meet any of 1the 27 lines on V .) The polynomial (5) for50 all u42 P (Q). is irredu ible over Q, so Wu is146smooth 92 The dis riminant of (5) is 2 3 5 359 . Fix a prime p 62 f2; 3; 5; 359g, and a pla e Q 99 K Fp . The 12 singular u-values in P1 (Q) redu e to 12 distin t singular u-values in P1 (Fp ) for the family W ! P1 de ned by the two equations (3) over Fp . Moreover, the ber above u = 0 is smooth in hara teristi p. By Lemma 1, all the bers of W ! P1 in
hara teristi p are smooth plane ubi s or nodal plane ubi s. By Lemma 2 and Hensel's Lemma, Wu has a Qp-point for all u 2 P1(Qp). Proposition 3. If u 2 Q satis es u 1 (mod pZp ) for p 2 f2; 3; 5g and u 2 Z359, then the ber Wu has a Qp -point for all ompletions Qp , p 1. Proof. Existen e of real points is automati , sin e Wu is a plane urve of odd degree. Existen e of Qp-points for p 62 f2; 3; 5; 359g was proved just above the statement of Proposition 3. Consider p = 359. A Grobner basis al ulation shows that there do not exist a1, a2, b1, b2, 1 , 2, u 2 F359 su h that (6) 5x3 + 9y3 + 10z3 + 12u3(x + y + z)3 and (5x + a1y + a2z)(x + b1y + b2z)(x + 1y + 2z) are identi al. Hen e Lemma 2 applies to show that for any u 2 F359, the plane ubi de ned by (6) over F359 has a smooth F359-point, and Hensel's Lemma implies that Wu has a Q359-point at least when u 2 Z359. When u 1 (mod 5Z5), the urve redu ed modulo 5, W u : 4y 3 + 2(x + y + z )3 = 0;
onsists of three lines through P := (1 : 0 : 1) 2 P2(F5), so it does not satisfy the onditions of Lemma 2, but one of the lines, namely y = 2(x + y + z), is de ned over F5, and every F5 -point on this line ex ept P is smooth on W u . Hen e Wu has a Q5 -point. The same argument shows that Wu has a Q2-point whenever u 1 (mod 2Z2), sin e the
urve redu ed modulo 2 is x3 + y3 = 0, whi h ontains x + y = 0. Finally, when u 1 (mod 3Z3), the point (1 : 2 : 1) satis es the equation (4) modulo 32, and Hensel's Lemma gives a point (x0 : 2 : 1) 2 Wu (Q3) with x0 1 (mod 3Z3 ). This
ompletes the proof.
8
BJORN POONEN
1 1 1 We now seek a non- onstant rational fun tion P that maps P (Qp) into 1+ pZp for P ! 22 = 1 for p 2 f3; 5; 359g, the fun tion p 2 f2; 3; 5g and into Z359 for p = 359. Sin e p 60 = t2 + 82 u=1+ 2 t + 22 t2 + 22 has the desired property. Substituting into (4), we see that 2 t + 82 3 3 3 3 (7) Xt : 5x + 9y + 10z + 12 2 (x + y + z)3 = 0 t + 22 has Qp-points for all p 1. On the other hand, Xt (Q) = ;, be ause V (Q) = ;. Finally, the existen e of nodal bers in the family implies as in [CP℄ that the j -invariant of the family has poles, and hen e is non- onstant. 5. For t 2 Q, let Et denote the Ja obian of Xt . The papers [AKMMMP℄ and [RVT℄ ea h
ontain a proof that the lassi al formulas of Salmon for invariants of a plane ubi yield
oeÆ ients of a Weierstrass model of the Ja obian. We used a GP-PARI implementation of these by Fernando Rodriguez-Villegas, available ele troni ally at The Ja obians
ftp://www.ma.utexas.edu/pub/villegas/gp/inv- ubi .gp
to show that our Et has a Weierstrass model y2 = x3 + Ax + B where A = 145800(t2 + 82)3 (t2 + 22) and B = 6129675t12 96155100t10 + 359349979500t8 + 65556113292000t6 + 4990338518958000t4 + 180317231391182400t2 + 2572729234128532800: Be ause the non-existen e of rational points on V is explained by a Brauer-Manin obstru tion, Se tion 3.5 and in parti ular Proposition 3.5 of [CP℄ show that there exists a se ond family of genus 1 urves Yt with the same Ja obians su h that the Cassels-Tate pairing satis es hXt ; Yti = 1=3 for all t 2 Q. In parti ular, for all t 2 Q, the Shafarevi h-Tate group X(Et) ontains a subgroup isomorphi to Z=3 Z=3. Although we will not nd expli it equations for the se ond family here, we an at least outline how this might be done, following [CP℄, ex ept using Galois ohomology over number 1 elds and Q(t) wherever p possible in pla e of etale ohomology over open subsets of P : (1) Let k = Q( 3) and Vk = V Q k. On pages 66{67 of [CTKS℄ an element Ak of Br(Vk ) giving a Brauer-Manin obstru tion over2 k is des ribed byparameters ; for a 2- o y le representing the image of Ak in H (Gal(K=k); K (V2 ) ) for a ertain nite abelian extension K of k. We may map to an element of H (k; Q(V )). (2) Apply the orestri tion oresk=Q to Ak to obtain an element A 2 Br(V ) giving a Brauer-Manin obstru tion for V over Q. (See the proof of Lemme 4(ii) in [CTKS℄.) In pra ti e, all elements of Brauer groups are to be represented by 2- o y les analogous to , and all operations are a tually performed on these o y les. (3) Pull ba k A under the morphisms X ! X ! V , where X over Q(t) is the generi ber of our nal family X ! P1 of genus 1 urves, to obtain an element A of Br(X ).
FAMILY OF GENUS-ONE CURVES
9
(4) Let X denote X Q(t) Q(t). Find the image of A in H1(Q(t); Pi X ), by writing the divisor of the 2- o y le representing the image of A in H2(Q(t); Q(t)(X )) as the oboundary of a 1- o hain, whi h be omes a 1- o y le representing an element of H1(Q(t); Pi X ). (5) Observe that the newly dis overed 1- o y le a tually takes values in Pi 0 X = E (Q(t)), where E is the Ja obian of X over Q(t). (6) Re onstru t the prin ipal homogeneous spa e Y of E over Q(t) from this 1- o y le, by omputing the fun tion eld of Y as in Se tion X.2 of [Si℄. (7) Find the minimal model of Y over P1Q, if desired, to obtain a model smooth over the same open subset of P1 as X . A knowledgements
I thank Ahmed Abbes for explaining the formula (1) to me, Antoine Du ros for making a omment that led to a simpli ation of the rational fun tion used at the end of Se tion 4, and Noam Elkies for suggesting the rst proof of Lemma 1 and the remark at the end of the introdu tion. I thank Bernd Sturmfels for the expression for the dis riminant of a plane ubi as a 6 6 determinant, and for providing Maple ode for it. I thank also Bill M Callum and Fernando Rodriguez-Villegas, for providing Salmon's formulas for invariants of plane ubi s in ele troni form. The al ulations for this paper were mostly done using Maple, Mathemati a, and GP-PARI on a Sun Ultra 2. The values of A and B in Se tion 5 and their trans ription into LaTeX were he ked by pasting the LaTeX formulas into Mathemati a, plugging them into the formulas for the j -invariant from GP-PARI, and
omparing the result against the j -invariant of Xt as omputed dire tly by Mark van Hoeij's Maple pa kage \IntBasis" at http://www.math.fsu.edu/~hoeij/ ompalg/IntBasis/index.html
for a few values of t 2 Q.
Referen es
[AKMMMP℄ S. Y. An, S. Y. Kim, D. Marshall, S. Marshall, W. M Callum, A. Perlis, Ja obians of genus one urves, preprint, 1999. [CG℄ J. W. S. Cassels and M. J. T. Guy, On the Hasse prin iple for ubi surfa es, Mathematika 13 (1966), 111{120. [CTKS℄ J.-L. Colliot-Thelene, D. Kanevsky, et J.-J. Sansu , Arithmetique des surfa es ubiques diagonales, pp. 1{108 in Diophantine approximation and trans enden e theory (Bonn, 1985), Le ture Notes in Math., 1290, Springer, Berlin, 1987. [CP℄ J.-L. Colliot-Thelene and B. Poonen, Algebrai families of nonzero elements of Shafarevi h-Tate groups, J. Amer. Math. So . 13 (2000), 83{99. [Lin℄ C.-E. Lind, Untersu hungen uber die rationalen Punkte der ebenen kubis hen Kurven vom Ges hle ht Eins, Thesis, University of Uppsala, 1940. [Ma℄ Yu. I. Manin, Cubi forms, Translated from the Russian by M. Hazewinkel, Se ond edition, NorthHolland, Amsterdam, 1974. [Mi℄ J. Milne, Etale
ohomology, Prin eton Univ. Press, Prin eton, N.J., 1980. [O'N℄ C. O'Neil, Ja obians of urves of genus one, Thesis, Harvard University, 1999. [Ra℄ M. Raynaud, Cara teristique d'Euler-Poin are d'un fais eau et ohomologie des varietes abeliennes, Seminaire Bourbaki, Expose 286 (1965). [Re℄ H. Rei hardt, Einige im Kleinen uberall losbare, im Grossen unlosbare diophantis he Glei hungen, J. Reine Angew. Math. 184 (1942), 12{18. [RVT℄ F. Rodriguez-Villegas and J. Tate, On the Ja obian of plane ubi s, in preparation, 1999.
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BJORN POONEN
[Se℄ E. Selmer, The diophantine equation ax3 + by3 + z 3 = 0, A ta Math. 85 (1951), 203{362 and 92 (1954), 191{197. [Si℄ J. Silverman, The arithmeti of ellipti urves, Graduate Texts in Mathemati s 106, Springer-Verlag, New York-Berlin, 1986. [St℄ B. Sturmfels, Introdu tion to resultants, Appli ations of omputational algebrai geometry (San Diego, CA, 1997), 25{39, Pro . Sympos. Appl. Math. 53, Amer. Math. So ., Providen e, RI, 1998. [SD℄ H. P. F. Swinnerton-Dyer, Two spe ial ubi surfa es, Mathematika 9 (1962), 54{56. Department of Mathemati s, University of California, Berkeley, CA 94720-3840, USA
E-mail address : poonenmath.berkeley.edu
