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AN EXPLICIT ALGEBRAIC FAMILY OF GENUS-ONE CURVES
VIOLATING THE HASSE PRINCIPLE
BJORN POONEN
Abstrat.
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 priniple. An expliit Weierstrass model for its
Jaobian Et is given. The Shafarevih-Tate group of eah Et ontains a subgroup isomorphi
to Z=3 Z=3.
1.
Introdution
One says that a variety X over Q violates the Hasse priniple 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 hypersurfaes in P satisfy the Hasse priniple. In partiular, if X is a genus 0
urve,2 then X satises the Hasse priniple, sine the antianonial embedding of X is a oni
in P .
Around 1940, Lind [Lin℄ and (independently, but shortly later) Reihardt [Re℄ disovered
examples of genus 1 urves over Q that violate the Hasse priniple, suh as the nonsingular
projetive 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 priniple, inluding
3x3 + 4y3 + 5z3 = 0
in P2.
O'Neil [O'N, x6.5℄ onstruts an interesting example of an algebrai family of genus 1 urves
eah having Qp-points for all p 1. Some bers in her family violate the Hasse priniple,
by failing to have a Q-point. In other words, these bers represent nonzero elements of the
Shafarevih-Tate groups of their Jaobians.
In [CP℄, Colliot-Thelene and the present author prove,1 among other things, the existene
of non-isotrivial families of genus 1 urves over the
base P , smooth over a dense open subset,
1
suh that the ber over eah rational point of P is a smooth plane ubi violating the Hasse
priniple. In more onrete terms, this implies that there exists a family of plane ubis
depending on a parameter t, suh that the j -invariant is a non-onstant funtion of t, and
suh that substituting any rational number for t results in a smooth plane ubi over Q
violating the Hasse priniple.
Date : Marh 18, 2000.
This researh was supported by National Siene Foundation grant DMS-9801104, an Alfred P. Sloan
Fellowship, and a David and Luile Pakard Fellowship.
1
2
BJORN POONEN
The purpose of this paper is to produe an expliit example of suh 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 existene of non-trivial families with
onstant j -invariant ould be easily dedued 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 Jaobian, when N 14 (mod
8)
is a prime
for
whih
2
is
not
a
quarti
residue.
The
same
argument
works
if
N = N 0 where
2 Q and N 0 is a produt 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 hek that if N = a4 + 16b4
for some a; b 2 Q \ Z2, 1then N has this
form. One an now substitute rational funtions
for a and b 2mapping P (Q2) into Z2, with a=b not onstant. For instane, the hoies
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 priniple.
2.
Let us review briey the onstrution
in [CP℄. Swinnerton-Dyer [SD℄ proved that there
3
exist smooth
ubi surfaes V in P over Q violating the Hasse priniple; hoose one. If L is
a line in P3 meeting V in exatly 3 geometri points, and W denotes
the blowup of V along
1
V \ L, then projetion from L indues a bration W ! P whose bers are hyperplane
setions of V . Moreover, if L is suÆiently general, then W ! P1 will be a Lefshetz penil,
meaning that the only singularities of bers are nodes. In fat, 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 onstrution an be done with models over Spe Z[1=N ℄ so
that for eah prime p6 jN , redution mod p yields a family of plane ubi urves eah smooth
or with a single node. One then proves that if p6 jN , eah ber above an Fp-point has a1
smooth Fp-point,1 so Hensel's Lemma onstruts a Qp-point on the ber Wt of W ! P
above any t 2 P (Q).
There is no reason that suh Wt should have Qp-points for pjN , but the existene of1 Qppoints on V implies that at least for t in a nonempty p-adially open subset Up of P (Qp),
Wt (Qp ) will be nonempty. We obtain the desired family by base-extending W ! P1 by a
rational funtion f : P1 ! P1 suh that f (P1(Qp)) Up for eah pjN .
More details of this onstrution an be found in [CP℄.
3.
The ubi surfae onstrution
Lemmas
Let V be a smooth ubi surfae in P3 over an algebraially losed eld k . Let L
be a line in P3 interseting V in exatly 3 points. Let W be the blowup of V at these points.
Let W ! P1 be the bration of W by plane ubis indued by the projetion 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 exatly 12, eah is a nodal plane ubi.
Lemma 1.
We give two approahes towards this result, one via expliit alulations with the disriminant of a ternary ubi form, and the other via Euler harateristis. 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 harateristis.
the advantage of requiring muh less mahinery, 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, sine the speial ase
proved is all we need for our appliation, and also sine the seond proof works generally.)
The seond proof an be interpreted as explaining the order of vanishing of the disriminant
of the family in terms of the Euler harateristi 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 oeÆients. Let H (x; y; z)
be the Hessian of F , i.e., the determinant of the 3 3 matrix of seond partial derivatives of
F . Let be 2 9 3 3 times the determinant of the 6 6 obtained by writing eah 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 speial ase of a lassial formula for the resultant of three quadrati forms in three
variables, whih is reprodued in [St℄, for instane.) One omputes that is a homogeneous
polynomial of degree 12 in Z[a0; : : : ; a9℄.
If we speialize F to the homogenization
of
y 2 (x3 + Ax + B ), we nd that beomes
the usual disriminant 16(4A3 + 27B 2) of the ellipti urve [Si, p. 50℄. Beause is an
invariant for the ation of GL3(k), it follows that F gives the usual disriminant for any
ellipti urve in general Weierstrass form
y 2 + a1xy + a3 y = x3 + a2 x2 + a4 x + a6
at least in harateristi zero, and hene in any harateristi. Every smooth plane ubi is
projetively equivalent to suh 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 beomes 0.1 Again using the invariane of , we dedue that = 0 if and only if
F = 0 is singular.
Sine a ubi surfae 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 produt of this with the bration map W ! P yields a morphism
:W!
P2 P1 birational onto its image. The morphism separates points, sine
2
W ! P separates points exept for 9 lines whih ontrat to points, and these projet
isomorphially to the seond fator P2 1, beause by assumption L does not meet the 6 lines
ontrated by the morphism V ! P . To verify that separates tangent vetors, we need
only observe that at a point P 2 W on one of the 9 lines, a tangent vetor2 transverse to the
line through P maps to a nonzero tangent vetor at the image point in P , while 2a tangent
vetor at P along the line maps to the zero tangent
vetor at the image point in P but to a
nonzero tangent vetor at the image
point
in
P1 . Hene 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 hene 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 oeÆients
a0; : : : ; a9 are
linear polynomials in the homogeneous oordinates t0; t1 on the base P1. Hene for this
family is a homogeneous polynomial of degree 12 in t0; t1, and it is nonvanishing beause 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 fators of the ubi polynomial: see Figure 1.
In the \three lines" ase, after a linear hange of variables with onstant oeÆients we
may assume that the three intersetion 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
intersetion points are the points (1 : 0 : 0), (0 : 1 : 0) so that the oni is a retangular
1Fats suh as this are undoubtedly lassial, at least over C, but it seems easier to reprove them than to
nd a suitable referene.
FAMILY OF GENUS-ONE CURVES
5
hyperbola and the line is the line z = 0 \at innity." We an then translate the enter of
the hyperbola2 to (0 : 0 : 1) and sale 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 branhes of the urve
there, so that a5 = a6 = a7 = a8 = a9 = 0. In the remaining ases, \line + double line,"
\onurrent lines," and \triple line," we may move a point of multipliity 3 to (0 : 0 : 1),
and again we will have at least a5 = a6 = a7 = a8 = a9 = 0. We hek 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 speialized is divisible by t .
Seond proof of Lemma 1. Let p be the harateristi 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 harateristi. Sine 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 spetral sequene
Hp(P1; Rq F`) =) Hp+q (W; F`)
with the Grothendiek-Ogg-Shafarevih 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 ondutors
of
H
et (W ; F` ) onsidered as a representation
of the inertia group at t of the1 base P1. Sine W is a smooth urve of genus g = 1,
(W ) = 2 2g = 0. If t 2 P (k ) is suh that Wt is smooth, then all terms within the
brakets on the right side of (1) are 0, so the sum is nite. The Swan ondutor of
H0et(W ; F` ) = H2et(W ; F` ) = F`
is trivial. Hene (1) beomes
X
(Wt) + swt (H1et (W ; F` )) ;
12 =
t: Wt is singular
Sine 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 hek the ases listed in Figure 1. The Euler harateristi for eah,
6
BJORN POONEN
whih is unhanged if we pass to the assoiated redued sheme 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 suh that
F does not fator ompletely into linear fators over Fp , then the subsheme X of P2 dened
by F = 0 has a smooth Fp -point.
Proof. The polynomial F must be squarefree, sine otherwise F would fator ompletely.
Hene X is redued. 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 ation on omponents is trivial, beause
when there is more than one, 1the omponents have dierent 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 Setion 2 with the ubi surfae
V : 5x3 + 9y 3 + 10z 3 + 12w3 = 0
in P3. Cassels and Guy [CG℄ proved that V violates the Hasse priniple.
Let L be the line
x + y + z = w = 0. The intersetion V \ L as a subsheme of L = P1 with homogeneous
oordinates x; y is dened by
5x3 + 9y3 10(x + y)3;
whih has disriminant 242325 = 33 52 359 6= 0, so the intersetion onsists of three distint
geometri points. This remains
true in harateristi p, provided that p 62 f3; 5; 359g.
The projetion V 99 K P1 from 3L is given
by the rational funtion u := w=(x + y + z)
1
on V . Also, W is the surfae 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 projetion to the seond fator, 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
denes 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 satises
(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 whih the ber Wu is singular, and moreover eah of these
singular bers is a nodal ubi.
(Alternatively, one ould alulate that of the rst proof of
Lemma 1 for (4) equals 2431354 times the polynomial (5). One an easily verify that in any
harateristi 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 irreduible over Q, so Wu is146smooth
92
The disriminant of (5) is 2 3 5 359 . Fix a prime p 62 f2; 3; 5; 359g, and a plae
Q 99 K Fp . The 12 singular u-values in P1 (Q) redue to 12 distint singular u-values in
P1 (Fp ) for the family W ! P1 dened by the two equations (3) over Fp . Moreover, the
ber above u = 0 is smooth in harateristi p. By Lemma 1, all the bers of W ! P1 in
harateristi p are smooth plane ubis or nodal plane ubis. By Lemma 2 and Hensel's
Lemma, Wu has a Qp-point for all u 2 P1(Qp).
Proposition 3. If u 2 Q satises 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. Existene of real points is automati, sine Wu is a plane urve of odd degree. Existene 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 alulation shows that there do not exist a1, a2, b1,
b2, 1 , 2, u 2 F359 suh that
(6)
5x3 + 9y3 + 10z3 + 12u3(x + y + z)3
and
(5x + a1y + a2z)(x + b1y + b2z)(x + 1y + 2z)
are idential. Hene Lemma 2 applies to show that for any u 2 F359, the plane ubi
dened 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 redued 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 dened over F5, and every
F5 -point on this line exept P is smooth on W u . Hene Wu has a Q5 -point.
The same argument shows that Wu has a Q2-point whenever u 1 (mod 2Z2), sine the
urve redued modulo 2 is x3 + y3 = 0, whih ontains x + y = 0.
Finally, when u 1 (mod 3Z3), the point (1 : 2 : 1) satises 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 funtion
P that maps P (Qp) into 1+ pZp for
P !
22 = 1 for p 2 f3; 5; 359g, the funtion
p 2 f2; 3; 5g and into Z359 for p = 359. Sine
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) = ;, beause V (Q) = ;. Finally, the
existene of nodal bers in the family implies as in [CP℄ that the j -invariant of the family
has poles, and hene is non-onstant.
5.
For t 2 Q, let Et denote the Jaobian of Xt . The papers [AKMMMP℄ and [RVT℄ eah
ontain a proof that the lassial formulas of Salmon for invariants of a plane ubi yield
oeÆients of a Weierstrass model of the Jaobian. We used a GP-PARI implementation of
these by Fernando Rodriguez-Villegas, available eletronially at
The Jaobians
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:
Beause the non-existene of rational points on V is explained by a Brauer-Manin obstrution, Setion 3.5 and in partiular Proposition 3.5 of [CP℄ show that there exists a seond
family of genus 1 urves Yt with the same Jaobians suh that the Cassels-Tate pairing satises hXt ; Yti = 1=3 for all t 2 Q. In partiular, for all t 2 Q, the Shafarevih-Tate group
X(Et) ontains a subgroup isomorphi to Z=3 Z=3.
Although we will not nd expliit equations for the seond family here, we an at least
outline how this might be done, following [CP℄, exept using Galois ohomology over number
1
elds and Q(t) wherever
p possible in plae 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 obstrution over2 k is desribed byparameters ; for a
2-oyle 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 orestrition oresk=Q to Ak to obtain an element A 2 Br(V ) giving a
Brauer-Manin obstrution for V over Q. (See the proof of Lemme 4(ii) in [CTKS℄.) In
pratie, all elements of Brauer groups are to be represented by 2-oyles analogous
to , and all operations are atually performed on these oyles.
(3) Pull bak 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-oyle representing the image of A in H2(Q(t); Q(t)(X )) as
the oboundary
of a 1-ohain, whih beomes a 1-oyle representing an element
of H1(Q(t); Pi X ).
(5) Observe that the newly disovered 1-oyle atually takes values in Pi0 X =
E (Q(t)), where E is the Jaobian of X over Q(t).
(6) Reonstrut the prinipal homogeneous spae Y of E over Q(t) from this 1-oyle,
by omputing the funtion eld of Y as in Setion 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 .
Aknowledgements
I thank Ahmed Abbes for explaining the formula (1) to me, Antoine Duros for making
a omment that led to a simpliation of the rational funtion used at the end of Setion 4,
and Noam Elkies for suggesting the rst proof of Lemma 1 and the remark at the end
of the introdution. I thank Bernd Sturmfels for the expression for the disriminant of a plane ubi as a 6 6 determinant, and for providing Maple ode for it. I thank
also Bill MCallum and Fernando Rodriguez-Villegas, for providing Salmon's formulas for
invariants of plane ubis in eletroni form. The alulations for this paper were mostly
done using Maple, Mathematia, and GP-PARI on a Sun Ultra 2. The values of A and B in
Setion 5 and their transription into LaTeX were heked by pasting the LaTeX formulas
into Mathematia, plugging them into the formulas for the j -invariant from GP-PARI, and
omparing the result against the j -invariant of Xt as omputed diretly by Mark van Hoeij's
Maple pakage \IntBasis" at
http://www.math.fsu.edu/~hoeij/ompalg/IntBasis/index.html
for a few values of t 2 Q.
Referenes
[AKMMMP℄ S. Y. An, S. Y. Kim, D. Marshall, S. Marshall, W. MCallum, A. Perlis, Jaobians of genus
one urves, preprint, 1999.
[CG℄ J. W. S. Cassels and M. J. T. Guy, On the Hasse priniple for ubi surfaes, Mathematika 13 (1966),
111{120.
[CTKS℄ J.-L. Colliot-Thelene, D. Kanevsky, et J.-J. Sansu, Arithmetique des surfaes ubiques diagonales,
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BJORN POONEN
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Department of Mathematis, University of California, Berkeley, CA 94720-3840, USA
E-mail address : poonenmath.berkeley.edu
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