An explicit algebraic family of genus-one curves violating the Hasse principle

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

B

JORN

P

OONEN

An explicit algebraic family of genus-one curves

violating the Hasse principle

Journal de Théorie des Nombres de Bordeaux, tome

13, n

o

_{1 (2001),}

p. 263-274

<http://www.numdam.org/item?id=JTNB_2001__13_1_263_0>

© Université Bordeaux 1, 2001, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

An

_{explicit algebraic family}

of

_genus-one

curves

violating

the

Hasse

_principle

par BJORN POONEN

RÉSUMÉ. Nous montrons que pour tout t ~

_Q,

la courbe

3 (x + y + z)3 = 0

de P2 est une courbe de genre 1 _qui ne satisfait _pas au _principe de Hasse. On donne un modèle de Weierstrass

_explicite

_pour sa

jacobienne.

Le groupe de Shafarevich-Tate de chacune des ces

jacobiennes

contient un _sous-groupe

_isomorphe

à

_Z/3

x

Z/3.

ABSTRACT. We prove that for any t ~

_Q,

the curve

+

9y3 +

10z3

+ 12

(t2 + 82 t2 + 22)

3 (x

+ y +

z)3

= 0 in P2 is a _genus 1 curve

_violating

the Hasse

_principle.

An

_explicit

Weierstrass model for its Jacobian Et is

_given.

The

Shafarevich-Tate group of each Et contains a

subgroup isomorphic

to

_Z/3

x

Z/3.

1. Introduction

One _says that a

_variety

X over

_Q

violates the Hasse

_principle

if

X(Qv) =I 0

for all

_completions

_(av

of

_Q

_(i.e.,

R and

_Qp

for all

_primes

p)

but

_X(Q)

=

0.

Hasse

_proved

that

_degree

2

_{hypersurfaces}

_{in Pn}

_satisfy

the Hasse

_principle.

In

_particular,

if X is a _{genus 0 curve,} then X satisfies

the Hasse

_principle,

since the anticanonical

_embedding

of X is a conic in

p2.

Around

_1940,

Lind

_[Lin]

and

_{(independently,}

but

_shortly

_later)

Re-ichardt

_[Re]

discovered

_examples

of genus 1 curves over

_Q

that violate

the Hasse

_principle,

such as the

nonsingular

projective

model of the affine curve

Manuscrit recu le 27 octobre 1999.

This research was _{supported by} National Science Foundation grant DMS-9801104, an Alfred

Later,

Selmer

_[Se]

gave

examples

of

diagonal plane

cubic curves

(also

of

genus

1)

violating

the Hasse

principle, including

in p2.

O’Neil

_{[O’N, §6.5]}

constructs an

_interesting

_example

of an

_algebraic

fam-ily

of _{genus 1} curves each

having

Qp-points

for all

_p

oo. Some fibers in

her

_family

violate the Hasse

_{principle, by}

_failing

to have a

Q-point.

In

other

_words,

these fibers

_represent

nonzero elements of the

Shafarevich-Tate groups of their Jacobians.

In

_[CP],

Colliot-Thélène and the

_present

author _{prove, among} other

things,

the existence of nonisotrivial families of _{genus 1} curves over the

base

_{P 1,}

smooth over a dense _open

subset,

such that the fiber over each

rational

_point

of

Pl

is a smooth

plane

cubic

violating

the Hasse

principle.

In more concrete

_terms,

this

_implies

that there exists a

family

of

plane

cu-bics

_depending

on a

_{parameter t,}

such that the

j-invariant

is a nonconstant

function of

_t,

and such that

_substituting

_any rational number for t results

in a smooth

plane

cubic over

_{Q violating}

the Hasse

_principle.

The _purpose of this _paper is to

_produce

an

_{explicit example}

of such a

family.

Our

_{example, presented}

as a

_family

of cubic curves in

p2

with

homogeneous

coordinates _{x, ~, z,} is

Remark. Noam Elkies

_pointed

out to me that the existence of nontrivial

families with constant

_j-invariant

could be

_easily

deduced from

_previously

known results. In Case I of the

_proof

of Theorem X.6.5 in

_[Si],

one finds a

proof

(based

on ideas of Lind and

Mordell)

that

2y2

= 1-

Nx4

_represents

a

nontrivial element of

_{III [2]}

of its

_Jacobian,

when N - 1

_{(mod 8)}

is a

_prime

for which 2 is not a

quartic

residue. The same

_argument

works if N =

c4N’

where c E

_Q*

and N’ is a

_product

of

_primes

_{p - 1}

(mod

8),

provided

that

2 is not a

quartic

residue for at least one _p

appearing

with odd

_exponent

in N’. One can check that if N

= a4

+

16b4

for _{some a,} b E

_Q

n

_z*,

then

N has this form. One can now substitute rational functions for a and b

mapping

into

_Z2,

with

_a/b

not constant. For

_instance,

the choices

a = 1 +

2/(t2

+ t +

₁₎

and b = 1 lead _to the

family

of _{genus 1} curves of

j-invariant

1728

violating

the Hasse

principle.

2. The cubic surface construction

Let us review

briefly

the construction in

[CP].

Swinnerton-Dyer

[SD]

the Hasse

_principle;

choose one. If L is a line in

p3 meeting

V in

_exactly

3

_{geometric points,}

and W denotes the

_blowup

of V

_along

V n

_L,

then

projection

from L induces a fibration W -&#x3E;

Pl

whose fibers are

hyperplane

sections of V.

_Moreover,

if L is

_{sufficiently general,}

then W -&#x3E;

Pl

will be

a Lefschetz

pencil,

_meaning

that the

only singularities

of fibers are nodes.

In

_fact,

for most

_L,

all fibers will be either smooth

_plane

cubic _curves, or

cubic curves with a

single

node.

For some N &#x3E;

_1,

the above construction can be done with models over

Spec

so that for each

prime

ptN,

reduction mod _p

yields

a

family

of

plane

cubic curves each smooth or with a

_single

node. One then _proves that

if

_p$’N,

each fiber above an

Fp-point

has a smooth

_Fp-point,

so Hensel’s

Lemma constructs a

_Qp-point

on the fiber

_Wt

of W ~

Pl

above _any

t E

P’(Q)-There is no reason that such

Wt

should have

_Qp-points

for

piN,

but

the existence of

_Qp-points

on V

implies

that at least for t in a

_nonempty

p-adically

open subset

Up

of

Pl(Qp),

Wt(Qp)

will be

nonempty.

We obtain the desired

_{family by base-extending}

W --&#x3E;

Pl

_by

a rational function

f :

P1 ~ pl

such that for each

_piN.

More details of this construction can be found in

~CP~.

3. Lemmas

Lemma 1. Let V be a smooth cubic

surface

in

p3

over an

algebraically

closed

_field

k. Let L be a line in

p3 intersecting

V in

_exactly

3

points.

Let

W be the

_{blowup of}

V at these

_points.

Let W -

Pl

be the

_{fibration of}

W

by plane

cubics induced

_by

the

_projection

_{P3 ~ L ~}

Pl

_from

L. Assume that

some

fiber of

7r : W --&#x3E;

Pl

is smooth. Then at most 12

fibers

are

singular,

and

_if

there are

_exactly

_12,

each is a nodal

_plane

cubic.

We

_give

two

_approaches

towards this

_result,

one via

explicit

calculations

with the discriminant of a

_ternary

cubic

form,

and the other via Euler

char-acteristics. The first has the

_advantage

of

_requiring

much less

_machinery,

but we

_complete

this

_{proof only}

under the

_assumption

that L does not meet

any of the 27 lines on V.

(With

more

work,

one could

probably

_prove the

general

case

_too,

but we have not tried too

_seriously,

since the

_special

case

proved

is all we need for our

application,

and also since the second

proof

works

_generally.)

The second

_proof

can be

interpreted

as

_explaining

the

order of

_vanishing

of the discriminant of the

_family

in terms of the Euler characteristic of a bad fiber.

be the

_generic

_ternary

cubic

_form,

with indeterminates _{ao, ... , ag} as

coeffi-cients. Let

_H(x,

_y,

_z)

be the Hessian of

_F,

_{i. e.,}

the determinant of the 3 x 3

matrix of second

_partial

derivatives of F. Let A be

2-93-3

times the de-terminant of the 6 x 6 obtained

by

writing

each of

_{âF/ây, aF/az,}

âH/âx,

in terms of the basis

_x2,

_xy,

_y2,

_{xz, yz,}

z2.

_(This

is

a

special

case of a classical formula for the resultant of three

_quadratic

forms

in three

_variables,

which is

_reproduced

in

_[St],

for

_instance.)

One

_computes

that A is a

homogeneous polynomial

of

degree

12 in

Z [ao, ... , ag].

If we

_specialize

F to the

_{homogenization}

_{of y2 -}

_(x3

+ Ax

+

_B),

we find

that A becomes the usual discriminant

_-16(4A3

+

27B2)

of the

_elliptic

curve

[Si,

_p.

50].

Because A is an invariant for the action of

GL3(k),

it follows that F

gives

the usual discriminant for _any

_elliptic

curve in

general

Weierstrass form

at least in characteristic zero, and hence in any characteristic.

Every

smooth

_plane

cubic is

_projectively

_equivalent

to such an

elliptic

_curve, so

A is

_nonvanishing

whenever the curve F = 0 is smooth.

On the other

_hand,

if F = 0 is

singular

at

(0 :

0 :

1),

_so that _{a7 = a8 =}

ag =

0,

we

_compute

that A becomes 0.

Again

using

the invariance of

A,

we deduce that A = 0 if and

only

if F = 0 is

singular.1

Since a cubic surface is

isomorphic

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

W -~

P2.

Taking

the

product

of this

with the fibration _map W --~

Pl

_yields

a

morphism W :

W -~

p2

x

pl

birational onto its

_image.

The

_{morphism T}

_separates

_points,

since W ~

p2

_separates

_points

_except

for 9 lines which contract to

_points,

and these

project

isomorphically

to the second factor

_P’,

because

_by

_assumption

L does not meet the 6 lines contracted

_by

the

_morphism

V -~

P2.

To

verify

that T

_{separates tangent vectors,}

we need

only

observe that at a

point

P E W on one of the 9

_lines,

a

_tangent

vector transverse to the line

through P

maps to a nonzero

_tangent

vector at the

image point

in

p2,

while a

_tangent

vector at P

_along

the line _maps to the zero

_tangent

vector at the

_{image point}

in

p2

but to a nonzero

_tangent

vector at the

image

point

in

P 1.

Hence T is a closed

_immersion,

so we _{may view} the

_given

family

W -~

Pl

as a

family

of curves in

P2.

By

assumption,

there exists a

fiber of W -~

Pl

that is a smooth cubic curve. It follows that the divisor

W in

P 2

x

P 1

is of

_type

_{( 3,1 ) ,}

and hence W is

_given

_by

a

_{bihomogeneous}

equation

q(xo,

xl, x2, x3;

to,

tl)

= 0 of

degree

3 in _{xo, xl, x2, X3} and of

degree

1 in

_to,

_tl. In other

_words,

we _{may view} the fibration W -

Pl

as a

_family

of

cubic

_plane

curves where the coefhcients ao, ... , ag are linear

_polynomials

in

the

_homogeneous

coordinates

_to,

tl on the base

Pl -

Hence A for this

_family

is a

homogeneous polynomial

of

degree

12 in

to, tl,

and it is

nonvanishing

because of the

_assumption

that at least one fiber is smooth. Thus at most

12 fibers are

singular.

To finish the

_proof,

we need

_only

show that if a fiber is

_singular

and not

with just

a

single

node,

then

bl )

vanishes to order at least 2 at the

corresponding

point

on

P 1.

To prove

this,

we enumerate the combinatorial

1 Facts such as this are _{undoubtedly classical,} at least over _C, but it seems easier to reprove

possibilities

for a

plane

cubic,

corresponding

to the

degrees

of the factors

of the cubic

_polynomial:

see

_Figure

1.

In the "three lines" _case, after a linear

change

of variables with constant

coefficients we _may assume that the three intersection

_points

on the bad

fiber are

(1 :

0 :

0), (0 :

1 :

0),

and

(0 : 0 : 1),

so that the fiber is xyz = 0.

We

_compute

that if G is a

_general

_ternary

cubic and t is an

_{indeterminate,}

representing

the uniformizer at a

_point

on the base

P ,

then A for

is divisible

_by

t3.

In the "conic + line" _case, we assume that the two

intersection

_points

are the

_points

(1 :

0 :

0),

(0 :

1 :

0)

so that the conic

is a

rectangular hyperbola

and the line is the line z = 0 "at

infinity."

We

can then translate the center of the

_hyperbola

to

_{(0 :}

0 :

₁₎

and scale to

assume that the fiber is

(xy -

z2)z

= 0. This

time, A

for

(xy -

z2)z

₊ tG

is divisible

_by

t2.

In the

_"cuspidal

cubic" and "conic +

_tangent"

cases, we _may

change

coordinates so that the

_singularity

is at

_{(0 :}

0 :

₁₎

and the line x = 0 is

tangent

to the branches of the curve

there,

so

_{that a5 = a6 =}

_{a7 = a8 =}

ag = 0. In the

remaining

cases, "line + double

_line,"

"concurrent

_lines,"

and

_"triple

_line,"

we _may move a

point

of

multiplicity

3 to

(0 :

0 :

1),

and

again

we will have at least a5 = a6 = a7 = a8 =

a9 = 0. We check that

A vanishes to order 2 in all five of these cases,

by substituting aZ - tb2

for i =

5, fi, ?, 8, 9

in the

generic

formula for

A,

and

verifying

that the

specialized A

is divisible

_by

t2.

D

Second

_{proof of}

Lemma 1. Let _p be the characteristic of

_k,

and choose a p. Let

denote the Euler characteristic. Since V is

_isomorphic

to the

_blowup

of

p2

at 6

_points,

and W is the

_blowup

of V at 3

_points,

On the other

_{hand, combining}

the

_{Leray spectral}

_sequence

with the

_{Grothendieck-Ogg-Shafarevich}

formula

_([Ra,

Th6or6me

_1]

or

[Mi,

where

_W’T1

is the

_{generic fiber, Wt}

is the fiber

_{above t,}

and

is the

_alternating

sum of the Swan conductors of

Ft)

considered

as a

representation

of the inertia group at t of the base Since

W17

is a

smooth curve of _{genus g} =

1,

=

2 - 2g

= 0. If t _E is such that

Wt

is

_smooth,

then all terms within the brackets on the

right

side of

(1)

are

0,

so the sum is finite. The Swan conductor of

is trivial. Hence

₍₁₎

becomes

Since

_{sWt(Hlt(W17,Ft))}

is a

dimension,

it is

_nonnegative,

so the lemma will

follow from the

_following

claim: if

_Wt

is

_singular,

&#x3E; 1 with

_equality

if and

_only

if

Wt

is a nodal cubic. To _prove

this,

we

again

check the cases

listed in

_Figure

l.The Euler characteristic for

_each,

which is

_unchanged

if

we _pass to the associated reduced scheme

_C,

is

_computed

_using

the formula

where a : C - C is the normalization of

C, gCZ

i is the genus of the i-th

component

of

_C,

and

_Csing

is the set of

_singular

_points

of C. For

_example,

for the "conic +

_tangent,"

formula

₍₂₎

_gives

Lemma 2.

_If

_F(x,

_y,

_z)

E

_Fp[x,

y,

z]

is a nonzero

homogeneous

cubic

poly-nomial such that F does not

_{factor completely}

into linear

_factors

over

_FP,

then the subscheme X

_of

p2

_{defined by}

F = 0 has _a smooth

Fp-point.

Proof.

The

_polynomial

F must be

_squarefree,

since otherwise F would

fac-tor

_completely.

Hence X is reduced. If X is a smooth cubic _curve, then it

is of _genus

_1,

and

0

by

the Hasse bound.

Otherwise, enumerating

possibilities

as in

Figure

lshows that X is a

nodal or

_{cuspidal cubic,}

or a union of a line and a conic. The Galois action on

_components

is

_trivial,

because when there is more than _one, the

over

_Fp

to

Pl

with at most two

_{geometric points}

deleted. But

_{#P1(Fp) &#x3E;}

3,

so there remains a smooth

Fp-point

on X. D

4. The

_example

We will _carry out the _{program in} Section 2 with the cubic surface

in

P3.

Cassels and

_Guy

_[CG]

_proved

that V violates the Hasse

_principle.

Let L be the line

_{x + ~ ~- z = w = 0.}

The intersection V fl L as a subscheme

of L

Pl

with

_{homogeneous coordinates x, y}

is defined

_by

which has discriminant 242325 =

33.52.359 = ,A

0,

so the intersection consists

of three distinct

_{geometric points.}

This remains true in characteristic _p,

provided

that

_{p 0}

_~3,

_5,

_359}.

The

_projection

V --+

Pl

from L is

given

by

the rational function u :=

+ y +

z)

on V.

Also,

W is the surface in

p3

x

pl given

_by

the

((x,

y, z,

w);

(uo,

ul))-bihomogeneous

equations

The

_{morphism W - Pl}

is

_simply

the

_projection

to the second

_factor,

and the fiber

_Wu

above u E

_{Q =}

A (Q)

C

_{P (Q)}

can also be written as the

plane

cubic

- n n n n

The

_{dehomogenization}

defines an affine _open subset in

A2 of Wu.

Eliminating x

and _y from the

equations

shows that this affine

_variety

is

_singular

when u E

_Q

satisfies

The fiber above u = 0 is

smooth,

_so

by

Lemma

1,

the 12 values of u

satisfying

(5)

give

the

_only

_points

in above which the fiber

_Wu

is

_singular,

and moreover each of these

singular

fibers is a nodal cubic.

(Alternatively,

one could calculate that A of the first

proof

of Lemma 1

for

₍₄₎

_{equals - 2431354}

times the

_polynomial

_(5).

One can

easily verify

on

V.)

The

polynomial

(5)

is irreducible over

Q,

so

_W~

is smooth for all

u

The discriminant of

₍₅₎

is

2~.3~.5~.359~.

Fix a

_prime

p ~

{2,

_{3, 5,}

359},

and a

_place

Q

--+

_Fp.

The 12

singular

u-values in

P (Q)

reduce _to 12

distinct

_singular

u-values in

_P1(Fp)

for the

_family

W -&#x3E;

Pl

defined

_by

the

two

_equations

₍₃₎

over

Fp.

Moreover,

the fiber above u = 0 is smooth in

characteristic _p.

_By

Lemma

_1,

all the fibers of W -~

Pl

in characteristic _p

are smooth

plane

cubics or nodal

plane

cubics.

By

Lemma 2 and Hensel’s

Lemma, Wu

has a

_Qp-point

for all u E

P~(Qp).

Proposition

3.

_If

u E

_(a

_satisfies

u - 1

_(mod

_pZp)

_for

_p E

_{2,

3,

₅₁

and

~u E

_Z359,

then the

_fiber

Wu

has a

_{Qp -point}

for

all

_completions

_Qp,

_p

oo.

Proof.

Existence of real

_points

is

_automatic,

since

_W~,

is a

plane

curve of

odd

_degree.

Existence of

_Qp-points

for

_{p g}

_{2,3,5,359}

was

_proved

_just

above the statement of

_Proposition

3.

Consider _p = 359. A Grbbner basis calculation shows that there do not

exist al, a2,

bl, b2,

cl,

C2, U

E

_F359

such that

and

are identical. Hence Lemma 2

applies

to show that for

_{any U}

E

_F359,

the

plane

cubic defined

_by

₍₆₎

over

F359

has a smooth

_F35g-point,

and Hensel’s

Lemma

_implies

that

_Wu

has a

_Qssg-point

at least when u E

_Z359

·

When t6 = 1

_(mod

_5Z5),

the curve reduced modulo

_5,

consists of three lines

_through

P :=

(1 :

0 :

-1)

E

P2(F5),

so it does

not

_satisfy

the conditions of Lemma

_2,

but one of the

_lines,

_namely

_,y =

-2(x + y + z),

is defined over

F5,

and _every

F5-point

on this line

_except

P

is smooth on

W-.

Hence

Wu

has a

Q5-point.

The same

_argument

shows that

Wu

has a

Q2-point

whenever u - 1

(mod

2Z2),

since the curve reduced modulo 2 is

x3+y3

=

0,

which contains

x + y = 0.

Finally,

when u - 1

_{(mod 3Z3),}

the

_point

_{(1 : 2 : 1)}

satisfies the

equa-tion

₍₄₎

modulo

_32,

and Hensel’s Lemma

_gives

a

_point

_{(xo :}

2 :

₁₎

E

_{Wu (Q3)}

with

_{(mod 3Z3).}

This

_completes

the

_proof.

D

We now seek a nonconstant rational function

Pl

-~

Pl

that _maps

P ( Qp )

for p

E

_{3,5,359},

the function

has the desired

_property.

_Substituting

into

_(4),

we see that

has

_Qp-points

for all

_p

oo. On the other

_hand,

Xt(Q) =

0,

because

V(Q)

=

0.

Finally,

the existence of nodal fibers in the

family implies

as

in

_[CP]

that the

_j-invariant

of the

_family

has

_poles,

and hence is

noncon-stant.

5. The Jacobians

For t E

_Q,

let

_Et

denote the Jacobian of

_Xt.

The _papers

_[AP]

and

_[RVT]

each contain a

proof

that the classical formulas of Salmon for invariants of a

_plane

cubic

_yield

coefficients of a Weierstrass model of the Jacobian. We

used a GP-PARI

implementation

of these

by

Fernando

Rodriguez-Villegas,

available

_{electronically}

at

ftp://www.ma.utexas.edu/pub/villegas/gp/inv-cubic.gp

to show that our

Et

has a Weierstrass

model y2

= x3

+ Ax + B where

Because the nonexistence of rational

_points

on V is

explained

_by

a

Brauer-Manin

_obstruction,

Section 3.5 and in

_particular

_Proposition

3.5 of

_[CP]

show that there exists a second

_family

of _{genus 1} curves

Y

with the same

Jacobians such that the Cassels-Tate

_pairing

satisfies

_(Xt,

_Yt)

=

1/3

for all t E

_Q.

In

_particular,

for all t E

_Q,

the Shafarevich-Tate group

contains a

_{subgroup isomorphic}

to

_Z/3

x

_Z/3.

Although

we will not find

_explicit

_equations

for the second

_{family here,}

we can at least outline how this

_might

be

_done,

_following

_[CP],

_except

_using

Galois

_cohomology

over number fields and

Q(t)

wherever

possible

in

place

of 6tale

_cohomology

over _open subsets of

P1:

1. Let k =

Q( 3)

and

= V

xQ k.

On _pages 66-67 of

[CKS]

an

element

_Ak

of

_Br(Vk)

_giving

a Brauer-Manin obstruction over k is

of

_Ak

in

_H2(Gal(K/k),

_K(V)*)

for a certain finite abelian extension

K of k. We _may

_{map ~}

to an element of

H 2(k, Q (V) *).

2.

_Apply

the corestriction

coreSk/Q

to

Ak

to obtain an element A E

Br(V)

giving

a Brauer-Manin obstruction for V over

_Q.

(See

the

proof

of Lemme

_4(ii)

in

_~CKS~.)

In

_practice,

all elements of Brauer

groups are to be

_{represented by 2-cocycles analogous}

to

_~,

and all

operations

are

actually performed

on these

cocycles.

3. Pull back A under the

_morphisms

X ~

_V,

where

_X.~

over

Q (t)

is the

_generic

fiber of our final

_family

X ~

Pl

of _{genus 1 curves, to}

obtain an element of

_Br(X~).

4. Let

_X,.,

denote Find the

_image

of in Pic

by

writing

the divisor of the

_2-cocycle

_representing

the

_image

of

_Aq

in

Q (t) (Xq ) * )

as the

coboundary

of a

1-cochain,

which becomes a

1-cocycle

representing

an element of

H’ (Q (t), Pic X,7).

5. Observe that the

_newly

discovered

_1-cocycle

_actually

takes values in

Pico

=

E,7 (Q (t)),

where

_E,7

is the Jacobian of over

_Q(t).

6. Reconstruct the

_{principal homogeneous}

space

_Y,7

of

_Eq

over

Q(t)

from

this

_1-cocycle,

_by

_computing

the function field of

_Y,7

as in Section X.2

of [Si].

7. Find the minimal model of over

PQ,

if

desired,

to obtain a model

smooth over the same _open subset of

Pl

as X.

Acknowledgements

I thank Ahmed Abbes for

_explaining

the formula

₍₁₎

to me, Antoine Ducros for

_making

a comment that led to a

_{simplification}

of the rational

function used at the end of Section

_4,

and Noam Elkies for

_suggesting

the first

_proof

of Lemma 1 and the remark at the end of the introduction.

I thank Bernd Sturmfels for the

_expression

for the discriminant A of a

plane

cubic as a 6 x 6

_determinant,

and for

_providing

_Maple

code for it. I

thank also Bill McCallum and Fernando

_{Rodriguez-Villegas,}

for

_providing

Salmon’s formulas for invariants of

_plane

cubics in electronic form. The calculations for this _paper were

mostly

done

using

Maple,

Mathematica,

and GP-PARI on a Sun Ultra 2. The values of A and B in Section 5

and their

_{transcription}

into LaTeX were checked

_by

_pasting

the LaTeX

formulas into

_{Mathematica, plugging}

them into the formulas for the

j-invariant from

_GP-PARI,

and

_comparing

the result

_against

the

_j-invaxiant

of

Xt

as

_computed

_{directly by}

Mark van

Hoeij’s Maple package

"IntBasis"

at

http://www.math.fsu.edu/"’hoeij/compalg/IntBasis/W dex.html

References

[AP] S.Y. AN, S.Y. _KIM, D. _MARSHALL, S. _MARSHALL, W. _McCALLUM, A. _PERLIS, Jacobians

of genus one curves. _Preprint, 1999.

[CG] J.W.S. _CASSELS, M.J.T. _GUY, On the Hasse _{principle for} cubic _surfaces. Mathematika

13 _(1966), 111-120.

[CKS] J.-L. COLLIOT-THÉLÈNE, D. KANEVSKY, J.-J. SANSUC, Arithmétique des surfaces

cu-biques diagonales, pp. 1-108 in Diophantine approximation and transcendence theory

(Bonn, 1985), Lecture Notes in Math. 1290, Springer, Berlin, 1987.

[CP] J.-L. COLLIOT-THÉLÈNE, B. _{POONEN, Algebraic families of} nonzero elements of Shafarevich-Tate groups. J. Amer. Math. Soc. 13 _{(2000), 83-99.}

[Lin] C.-E. LIND, Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins. _{Thesis, University} of Uppsala, 1940.

[Ma] Yu.I. MANIN, Cubic forms. Translated from the Russian by M. Hazewinkel, Second

edi-tion, North-Holland, Amsterdam, 1974.

[Mi] J. _MILNE, Étale cohomology. Princeton Univ. _{Press, Princeton, N.J.,} 1980.

[O’N] C. O’NEIL, Jacobians _of curves _of genus one. _Thesis, Harvard _University, 1999.

[Ra] M. RAYNAUD, Caractéristique d’Euler-Poincaré d’un _faisceau et cohomologie des variétés abéliennes. Séminaire _{Bourbaki, Exposé} 286 (1965).

[Re] H. _{REICHARDT, Einige} im Kleinen überall lösbare, im Grossen unlösbare diophantische Gleichungen. J. Reine Angew. Math. 184 _(1942), 12-18.

[RVT] F. RODRIGUEZ-VILLEGAS, J. _TATE, On the Jacobian of plane cubics. in _preparation, 1999.

[Se] E. SELMER, The Diophantine equation ax3 + by3 + cz3 = 0. Acta Math. 85 _(1951),

203-362; 92 _(1954), 191-197.

[Si] J. SILVERMAN, The arithmetic of elliptic curves. Graduate Texts in Mathematics _106,

Springer-Verlag, New York-Berlin, 1986.

[St] B. STURMFELS, Introduction to resultants. _Applications of computational algebraic

geom-etry (San Diego, CA, 1997), 25-39, Proc. _{Sympos. Appl.} Math. _53, Amer. Math. Soc.,

Providence, RI, 1998.

[SD] H.P.F. _{SWINNERTON-DYER,} Two _special cubic _surfaces. Mathematika 9 _(1962), 54-56.

Bjorn POONEN

Department of Mathematics

University of California

Berkeley, CA 94720-3840 USA