GENERALIZED JACOBIANS AND EXPLICIT DESCENTS
BRENDAN CREUTZ
1. Introduction
Suppose f (x, y) is a binary form of degree d over a field k of characteristic not equal to
2. Pencils of quadrics with discriminant form f (x, y) have been studied in [BSD63, Cas62,
Cre01, BG, Wan, BGW13a, BGW13b]. When d is even, the SLd (k)/µ2 -orbits of pairs (A, B)
with discriminant form f (x, y) correspond to a collection of 2-coverings of the hyperelliptic
curve C : z 2 = f (x, y). When k = Q these coverings are used in [Bha] and [BGW13b]
to compute the average size of the 2-Selmer set of C, and of the torsor J 1 parameterizing
divisor classes of degree 1, respectively, from which they deduce the fantastic result that
most hyperelliptic curves over Q have no rational points.
Previously, the same collection of coverings had been described in terms of the k-algebra
L := k[x]/f (x, 1) and used in [BS09] and [Cre13] to compute 2-Selmer sets of C and J 1 ,
respectively, for individual hyperelliptic curves. A key step in both [Cre13] and [BGW13b]
is to check that the collection of coverings used is large enough to contain the locally soluble
2-coverings (under suitable hypotheses). In [BGW13b] this is achieved by identifying the
collection of coverings as the unramified subcoverings of k-forms of the maximal abelian
covering of exponent 2 unramified outside the pair of points at infinity on the affine model
of z 2 = f (x, y), a characterization that is quite natural in light of the use of generalized
Jacobians in [PS97].
Meanwhile the theory of explicit descents has expanded to incorporate descriptions of
Selmer sets for all curves (the case of non-hyperelliptic curves of genus at least 2 in [BPS]
and curves of genus 1 in [Cre14]). In this paper we develop a cohomological description of
these descents in terms of generalized Jacobians, generalizing the description for hyperelliptic
curves given in [PS97]. Specifically, given an integer n dividing the degree of some reduced
effective divisor m on a curve C, we show that multiplication by n on the generalized Jacobian
Jm factors through an isogeny ϕ : Am → Jm whose kernel is naturally the dual of the Galois
module (Pic(Ck )/m)[n]. By geometric class field theory, this corresponds to an abelian
covering of Ck := C ×Spec k Spec(k) of exponent n unramified outside m. The n-coverings
of C parameterized by the explicit descents mentioned above are the maximal unramified
subcoverings of the k-forms of this ramified covering.
This description unifies the methods of explicit descent on curves (and/or their J 1 ) described in [MSS96, Sta05, BS09, Cre13, Cre14, BPS], and provides new insights into the descents on the corresponding Jacobians (For example, Lemma 2.10, Corollary 2.12 and Remark 3.5). It also allows us to show that the corresponding collection of coverings of J 1
contains the locally solvable coverings and, in particular, that the ‘descent map’ in [BPS]
could be used to compute 2-Selmer sets of J 1 for non-hyperelliptic curves of genus ≥ 2.
Date: December 2015.
1
We expect this will be of relevance to future efforts to compute these Selmer sets on
average. Namely, it should be possible to identify this collection of coverings with the orbits
in some coregular representation (as is done in [BGW13b] for the hyperelliptic case). The
results in Propositions 3.11 and 3.13 would then have implications for the structure of the
space of orbits. Thorne has recently made progress understanding the situation for nonhyperelliptic genus 3 curves with a marked rational point [Tho15, Tho]. It is our hope that
the results of this paper may shed light on the corresponding situation when there are no
rational points.
1.1. Notation. Throughout this paper n is an integer and k is a field of characteristic
not divisible by n, with separable closure k and absolute Galois group Galk = Gal(k/k).
We will use C to denote a nice curve over k, i.e. a smooth, projective and geometrically
integral k-variety of dimension 1. For a nonempty finite étale k-scheme ∆ = Spec(L) we use
Res∆ = ResL/k to denote the restriction of scalars functor taking L-schemes to k-schemes.
2. The modulus setup
Definition 2.1. Let C be a nice curve over k. A modulus setup for C is a pair (n, m)
consisting of a positive integer n not divisible by the characteristic of k, and a reduced effective
divisor m ∈ Div(C) of degree m, with n dividing m.
Given a modulus setup (n, m) we define ` := deg(m)/n.
Example 2.2. We are primarily interested in the following examples.
(1) Suppose π : C → P1 is a double cover which is not ramified over ∞ ∈ P1 . Let n = 2
and m = π ∗ ∞.
(2) Suppose C is a plane cubic curve. Let n = 3 and let m be any triple of distinct
colinear points.
(3) Generalizing the previous example, suppose C is a genus one curve of degree m in
Pn−1 . Take m to be any reduced hyperplane section and take n to be a divisor of m.
(4) Suppose C is a plane quartic curve. Let n = 2 and let m be any quadruple of distinct
colinear points.
(5) Generalizing the previous example, suppose C is any nice curve, n = 2 and m is a
canonical divisor. Then m = 2g − 2 and ` = g − 1.
We may view m as a finite étale subscheme m = Spec M ⊂ C, or as a modulus in the sense
of geometric class field theory (see [Ser88]). Let Cm denote the singular curve associated
to m as in [Ser88, IV.4]. Let PicC and PicCm be the commutative group schemes over k
representing the Picard functors of C and Cm . There is an exact sequence of commutative
group schemes over k,
(2.1)
0 → T → PicCm → PicC → 0 ,
where T is a an algebraic torus. The restriction of (2.1) to the identity components is an
exact sequence of semiabelian varieties,
(2.2)
1 → T → Jm → J → 0 ,
where Jm is the generalized Jacobian of C associated to the modulus m and J is the usual
Jacobian of C.
2
Lemma 2.3. T = Resm Gm /Gm is the quotient by the diagonal embedding of Gm , and there
is an exact sequence of finite group schemes
Res1m µn
N
−→ T [n] −→ µn −→ 1 ,
µn
1 −→
where the map N is induced by the norm map Resm Gm → Gm and Res1m µn is the kernel of
N : Resm µn → µn .
Proof. The first statement, that T = Resm Gm /Gm , follows from well known results on the
structure of generalized Jacobians (see [Ser88, §V Prop. 7]). The inclusion map Res1m Gm →
Resm Gm induces a surjective map onto Resm Gm /Gm with kernel µm . This gives the middle
rows of the following commuative and exact diagram.
/
µn _
1
/
µm
Res1m
n
/
1
/
µm
T [n]
/ Resm Gm
/
/
Res1m µn
Gm
n
/
1
/
1
Gm
n
/ Resm Gm
Res1m Gm
Gm
n
m/n
/
µn
1
The exact sequence in the statement of the lemma follows by applying the snake lemma. 2.1. The isogeny associated to a modulus setup.
Lemma 2.4. Given a modulus setup (n, m) there is a commutative group scheme A over k
1
and isogenies ψ : PicCm → A and ϕ : A → PicCm such that ker(ψ) = Resµmn µn ⊂ T [n] and
ϕ ◦ ψ = [n]. Moreover, we have a commutative and exact diagram
1
/
/
T0
ϕ
1
/
T
/
/
ϕ
PicCm
/
PicC
A
/
0
n
PicC
/
0.
where T 0 is a torus and T 0 [ϕ] ' µn .
Proof. By Lemma 2.3, PicCm contains a finite group scheme isomorphic to Res1m µn . The
quotient of PicCm by this subgroup scheme yields an isogeny ψ : PicCm → A. The existence
of ϕ follows from the fact that ker(ψ) is contained in the kernel of multiplication by n. Since
ker(ψ) ⊂ T , A is an extension of PicC . The rest of the assertions follow from the diagram in
Lemma 2.3.
Remark 2.5. When n = m = deg(m) = 2, we have that T [n] ' µn . Hence ψ is the identity
map on A = PicCm and ϕ is multiplication by 2.
3
2.2. Description using divisor classes. A function f ∈ k(C)× that is regular away from
m gives, by evaluation, an element f (m) ∈ M = Spec m. We use Divm (C) to dentote the
divisors of C that have support disjoint from m.
Lemma 2.6. Let A be as defined in Lemma 2.4. Then
PicC (k) = Div(Ck )/{div(f ) : f ∈ k(Ck )× } ,
PicCm (k) = Divm (Ck )/{div(f ) : f ∈ k(Ck )× , f (m) = 1} ,
A(k) = Divm (Ck )/{div(f ) : f ∈ k(Ck )× , f (m) ∈ Res1m µn } .
Moreover, ϕ : A → PicCm is induced by multplication by n on Divm (Ck ).
Proof. The first two statements are well known (see [Ser88]; note that f (m) = 1 if and only if
f ≡ 1 mod m, since m is reduced). The k-points of the subgroup T = Resm Gm /Gm ⊂ PicCm
are represented by divisors of functions which do not vanish on m,
×
{div(f ) : f ∈ k(Ck ), f (m) ∈ M }
T (k) =
.
{div(f ) : f ∈ k(Ck )× , f (m) = 1}
The description of A(k) in the statement then follows from the fact that A is the quotient
of PicCm by the image of Res1m µn in T . The final statement follows easily from the fact that
ϕ ◦ ψ is multiplication by n on PicCm .
2.3. Component groups. The component groups of PicCm , PicC and A are all isomorphic
to Z, the isomorphism being given by the degree map on divisor classes. The degree 0
component of A is a semiabelian variety Am fitting into an exact sequence,
1 → T 0 → Am → J → 0 .
(2.3)
In particular, Am is geometrically connected.
We label the components
G
G
Jmi ,
Ji ,
PicCm =
(2.4)
PicC =
i∈Z
i∈Z
A=
G
Aim ,
i∈Z
so that the superscripts denote the image under the degree map. To ease notation we also
denote the degree 0 components by J = J 0 , Jm = Jm0 and Am = A0m . For any i ∈ Z, J i and
Jmi are torsors under J and Jm , respectively.
Let m0 ∈ Divm (C) be an effective reduced k-rational divisor linearly equivalent to and
with disjoint support from m (which exists by Bertini’s theorem, provided k has sufficiently
many elements). Each of the quotient groups
(2.5)
J :=
m−1
G
PicC
=
Ji ,
Zm0
i=0
Jm :=
m−1
G
PicCm
=
Jmi ,
Zm0
i=0
Am :=
m−1
G
A
=
Aim
Zm0
i=0
has m components. It is not generally true that all effective divisors linearly equivalent to
and disjoint from m give the same class in PicCm , so the quotient maps PicCm → Jm and
A → Am may depend on m0 . However, the map PicC → J depends only on m.
4
Recall that ` := m/n. From the definitions in (2.5) and Lemma 2.4 one easily obtains the
following commutative and exact diagram.
(2.6)
µn _
0
0
/
µn _
Am [ϕ] /
J[n]

/

/
Am [ϕ]
J [n]
1
`
deg
1
`
deg
/
Z/n
/
Z/n
/
/
0
0
2.4. Extended Weil pairings. We now define a Galois equivariant and nondegenerate
bilinear pairing e : Am [ϕ]×Am [ϕ] → µn that induces nondegenerate pairings on Am [ϕ]×J [n]
and J[n] × J[n] via the maps in (2.6).
To begin, define a pairing on Jm [n] as follows. Fix f ∈ k(C)× such that div(f ) = m0 − m.
Given D1 , D2 ∈ Jm [n], choose representative divisors D1 , D2 ∈ Divm (Ck ), and let di =
deg(Di )/`. There exist unique functions h0i ∈ k(Ck )× such that nDi = div(hi ) + di m0 and
h0 (m) = 1. Set hi = f di h0i , so that nDi = div(hi ) + di m. Define:
(2.7)
e(D1 , D2 ) = (−1)d1 d2
Y
(−1)n(ordP D1 )(ordP D2 )
P ∈C(k)
P D1
hord
×
2
(P ) ∈ k .
ordP D2
h1
Lemma 2.7. This gives a Galois equivariant bilinear pairing e : Jm [n] × Jm [n] → µn .
Proof. This may be checked exactly as in [PS97, Section 7] (one need only replace the function
x there with the function f in the definition above).
Lemma 2.8. The pairing in Lemma 2.7 induces nondegenerate Galois equivariant pairings,
e : Am [ϕ] × Am [ϕ] → µn ,
e : Am [ϕ] × J [n] → µn ,
e : J[n] × J[n] → µn .
The induced pairing on J[n] × J[n] coincides with the Weil pairing.
Remark 2.9. The definition of e given above depends on the choice of m0 in (2.5) and the
function f with div(f ) = m0 − m. However, as shown in the proof below, the induced pairings
on Am [ϕ] × J [n] and J[n] × J[n] do not depend on these choices.
Proof. We will show that the orthogonal complements of Res1m µn and T [n] with respect to
e are Jm [n] and Jm [n], respectively. This is enough to ensure that e induces the pairings
stated. The pairing induced on J[n] is evidently the Weil pairing (see A.4 in the appendix),
which is known to be nondegenerate. Nondegeneracy of the other pairings follows.
Let D1 ∈ T [n]. Then D1 is represented by D1 = div(f ) for some f ∈ k(Ck )× with
f (m) ∈ Resm µn . Since nD1 = div(f n ) and f n (m) = 1 we must use h1 = f n in the definition
of the pairing. Suppose D2 ∈ J [n] and let D2 , h2 , d2 be as in the definition of the pairing.
5
Then we have
ordP f
Y
n(ordP f )(ordP D2 ) h2
e(D1 , D2 ) =
(−1)
(P )
f n ordP D2
P ∈C(k)
=
Y
(−1)(ordP f )(ordP h2 +d2 ordP m)
P ∈C(k)
=
Y
P f
hord
2
f ordP h2 +d2 ordP m
(−1)d2 (ordP f )(ordP m) f −d2 ordP m (P )
(P )
(since nD2 = div(h2 ) + d2 m.)
(by Weil reciprocity)
P ∈C(k)
=
Y
f −d2 ordP m (P )
(since f (m) is invertible)
P ∈C(k)
= N (f (m))−d2 ,
where N denotes the induced norm Resm Gm → Gm . From this one easily sees that Res1m µn
lies in the kernel of the pairing and that T [n] pairs trivially with the degree 0 subgroup,
Jm [n] ⊂ Jm [n].
Taking Galois cohomology of (2.6) yields commutative and exact diagram
k × /k ×n
(2.8)
δ0
Z/nZ
δ
Z/nZ
/
k × /k ×n
/
H1 (Am [ϕ])
/
H1 (J[n])
/
H1 (Am [ϕ])
Υ
H1 (J [n])
Br(k)[n]
deg /`
/
H1 (Z/n)
deg /`
/
H1 (Z/n)
Υ0
Br(k)[n]
Lemma 2.10. The images of δ(1) and δ 0 (1) in H1 (J) and H1 (Am ) are the classes of J ` and
A`m , respectively. The maps Υ and Υ0 are given by
Υ(ξ) = ξ ∪e δ(1)
Υ0 (ξ) = ξ ∪e δ 0 (1) ,
where ∪e denotes the cup product induced by the pairing of Lemma 2.8.
Proof. The extensions
Jn` : 0 → J[n] → J [n] → Z/nZ → 0 , and
`
Jm,n
: 0 → Am [ϕ] → Am [ϕ] → Z/nZ → 0
represent classes
[Jn` ] ∈ Ext1Z/nZ[Galk ] (Z/nZ, J[n]) ∼
= H1 (J[n])
[J ` ] ∈ Ext1
(Z/nZ, Am [ϕ]) ∼
= H1 (Am [ϕ]) .
m,n
Z/nZ[Galk ]
6
`
, reLet δ and δ 0 denote the coboundary maps in the Galois cohomology of Jn` and Jm,n
`
0
`
∨
0∨
spectively. Then δ(1) = [Jn ] and δ (1) = [Jm,n ]. Let δ and δ denote the coboundries
`
of the extensions (Jn` )∨ and (Jm,n
)∨ obtained by dualizing and let : J[n] ∼
= J[n]∨ and
0
∨
∼
: J [n] = Am [ϕ] be the isomorphisms induced by the e-pairings of Lemma 2.8.
By [NSW08, Corollary 1.4.6] the following diagrams are commutative.
∗
H1 (J[n])
H1 (J[n]∨ )
•∪δ(1)
H2 (J[n] ⊗ J[n])
(⊗id)∗
/
0∗
H1 (J [n])
/
/
(0 ⊗id)∗
/
/
H2 (µm )
•∪δ(1)
H2 (J[n]∨ ⊗ J[n])
H1 (Am [ϕ]∨ )
•∪δ 0 (1)
H2 (J [n] ⊗ Am [ϕ])
δ ∨ (•)∪1
eval∗
/
H2 (µm )
δ 0∨ (•)∪1
/
H2 (µm )
•∪δ 0 (1)
H2 (Am [ϕ]∨ ⊗ Am [ϕ])
(−1)2
eval∗
/
(−1)2
H2 (µm )
The composition along the top row is the map Υ (resp. Υ0 ), while the path from the top-left
to the bottom-right along the bottom row agrees with the description given in the statement
of the lemma.
2.5. Brauer class of a k-rational divisor class. Given a nice curve X, there is a well
known exact sequence
(2.9)
Θ
X
0 → Pic(X) −→ PicX (k) −→
Br(k)
(see [Lic69]). The map ΘX gives the obstruction to a k-rational divisor class being represented by a k-rational divisor.
Lemma 2.11. Let d : J(k) → H1 (k, J[n]) denote the connecting homomorphism in the
Kummer sequence. For any x ∈ J(k) we have Υ ◦ d(x) = ` · ΘC (x).
Proof. The image of d is isotropic with respect to the Weil-pairing cup product ∪e . This
gives a commutative diagram of pairings
H1 (J[n]) × H1 (J[n]) → Br(k)
d
h, i :
J(k)
× H1 (J) → Br(k)
=
←
→
∪e :
By a result of Lichtenbaum (see the proof of [Lic69, Corollary 1]) we have that hx, [J 1 ]i =
ΘC (x). By the previous lemma we have
Υ ◦ d(x) = d(x) ∪ δ(1) = hx, [Jn` ]i = ` · hx, [J 1 ]i = ` · ΘC (x) .
Corollary 2.12. If
(1) the period of C divides `, or
7
(2) k is a local or global field and gcd(m, g − 1) divides `,
then Υ ◦ d = 0.
Proof. The image of ΘC : J(k) → Br(k) is isomorphic to the cokernel of Pic0 (C) → J(k),
which is annihilated by the period of C ([PS97, Prop. 3.2]). Over a local field, the period of
C divides g − 1 ([PS97, Prop. 3.4]). Since the period also divides m = deg(m), (2) implies
that ` is divisible by the period locally. Hence Υ ◦ d = 0 locally. This must also be true
globally by the local-global principle for Br(k).
3. The descent setup
We recall the following definition from [BPS].
Definition 3.1. A descent setup for C for a nice curve C is a triple (n, ∆, β) consisting of
a positive integer n not divisible by the characteristic of k, a nonempty finite étale k-scheme
∆ = Spec L, and a divisor β ∈ Div(C × ∆) such that nβ = m × ∆ + div(fm ) for some
m ∈ Div(C) and fm ∈ k(C × ∆)× .
Suppose (n, ∆, β) is a descent setup for C. If the divisor m appearing in the definition is
reduced, then (n, m) is a modulus setup, which we say is associated to (n, ∆, β). For each
δ ∈ ∆(k), βδ ∈ Div(Ck ) is a divisor such that nβδ − m principal. So the class of βδ in J lies
in J [n]. This gives rise to a commutative and exact diagram,
(3.1)
Res0∆ Z/nZ J[n]


/
Res∆ Z/nZ
/
J [n]
deg
1
`
deg
//
Z/nZ
//
Z/nZ
Definition 3.2. We say that (n, ∆, β) is an n-descent setup if the vertical maps in (3.1) are
surjective.
Example 3.3. The following show that all of the modulus setups in Example 2.2 are associated to an n-descent setup. Details for (1) and (2) may be found in [BPS, Examples 6.9],
while (3) is considered in [Cre14].
(1) Suppose C is a double cover of P1 which is not ramified over ∞. Let ∆(k) be the set
of ramification points and take β to be the diagonal embedding of ∆ in C × ∆. Then
(2, ∆, β) is a 2-descent setup. Taking m be the pullback of ∞ ∈ Div(P1 ) we recover
the modulus setup in Example 2.2 (1).
(2) Suppose C is any curve of genus ≥ 2. We obtain a 2-descent setup for C by taking
∆ to be the Galk -set of odd theta characteristics. By [BPS, Proposition 5.8] there
is some β ∈ Div(C × ∆) such that [βδ ] = δ for δ ∈ ∆(k). We can take m to be a
canonical divisor and thus recover the modulus setup in Example 2.2 (5).
(3) Suppose C is a genus one curve of degree n in Pn−1 (or equivalently, a genus one
curve together with the linear equivalence class of k-rational divisor of degree n). We
obtain an n-descent setup by taking ∆ to be the set of n2 flex points (i.e. points x ∈ C
such that n.x is a hyperplance section) and β to be the diagonal embedding of ∆ in
C × ∆. Taking m to be a generic hyperplane section recovers the modulus setup in
Example 2.2 (3).
8
(4) More generally, suppose C is a genus one curve of degree m in Pm−1 and n | m. There
is a Galk -invariant subset of flexes of size n2 whose differences represent the n-torsion
points. For any such flex x, `.x is linearly equivalent to a divisor Dx := P1 + · · · + P`
with Pi ∈ C such that nDx is a hyperplane section Furthermore, these Dx may be
chosen so as to give a divisor β ∈ Div(C × ∆). This gives an n-descent setup with
corresponding modulus setup (n, m), where m is a generic hyperplane section. In the
case n = 2, m = 4 this is done explicitly in [Sta05].
Remark 3.4. [BPS, Section 6] shows how a choice of m and fm ∈ k(C × ∆)× yields a
homomorphism
L×
fm : Pic(C) −→ × ×n
k L
related to the connecting homomorphism d : J(k) → H1 (J[n]) by a commutative diagram
Pic0 (C) 
/
/
d
J(k)
fm
Pic(C)
1
H (J[n])
/
/
H
1
L×
k× L ×n
_
Res∆ µn
µn
.
The injective map on the right comes from dualizing (3.1) and taking Galois cohomology to
obtain a commutative and exact diagram,
(3.2)
H1 (T 0 [ϕ])
k × /k ×n
/
/
H1 (Am [ϕ])
/
/
L× /L×n
Υ
H1 (J[n])
H1
/
H2 (T 0 [ϕ])
Res∆ µn
µn
/
Br(k)[n]
From this we see that the isogeny ϕ : Am → Jm naturally arises in the context of explicit
n-desecent as its kernel is the Cartier dual of J [n].
Remark 3.5. Let J(k)◦ ⊂ J(k) be the subgroup which maps under ◦ d into the subgroup
L× /k × L×n . This is the largest subgroup of J(k) to which one can extend the fm map to a map
which is compatible with the connecting homomorphism and takes values in L× /k × L×n (cf.
[PS97, Section 10]). We can determine J(k)◦ using Lemma 2.11. For example, when C is a
non-hyperelliptic curve of genus 3 over a global field with descent setup as in example 3.3(2),
we see that J(k)◦ = J(k).
3.1. ϕ- and n-coverings.
Definition 3.6. Suppose φ : A → B is an isogeny of semiabelian varieties over k and T
is a B-torsor. We say π : T 0 → T is a φ-covering of T if there exist isomorphisms a, b of
k-varieties fitting into a commutative diagram
Tk0
/
b
Ak
π
Tk
φ
a
9
/
Bk .
Two φ-coverings of T are isomorphic if they are isomorphic in the category of T -schemes.
Suppose (n, m) is a modulus setup for a nice curve C over k. The isogenies ϕ : Am → Jm
and n : J → J give rise to the notions of ϕ-coverings of Jm1 and n-coverings of J 1 . The
pullback of a ϕ-covering T → Jm1 along the canonical map (C − m) → Jm1 sending a geometric
point x to the class of the divisor x in Jm1 (k) ⊂ PicCm (k) yields an unramified covering of
(C − m). Corresponding to this is a unique (up to isomorphism) morphism π : Y → C of
smooth projective curves over k which is unramifed outside m.
Definition 3.7. Suppose (n, m) is a modulus setup for a nice curve C over k. A morphism
π : Y → C of nice curves is a ϕ-covering of C if it is the unique extension of the pullback of
a ϕ-covering of Jm1 along the canonical map (C − m) → Jm1 . A morphism π : X → C is an
n-covering of C if it is the pullback of an n-covering of J 1 along the canonical map C → J 1 .
By Galois theory, the field extension of k(Ck ) corresponding to a ϕ-covering is the compositum of the extensions corresponding to the index n subgroups of Am [ϕ], or equivalently,
to the points of order n in the Cartier dual J [n]. If D ∈ Div(Ck ) represents a point of order
n in J [n], then there exists a function hD ∈ k(Ck )× such that div(hD ) = nD − dm for some
d ∈ Z. The corresponding extension of k(Ck ) is obtained by adjoining an n-th root of hD .
In particular, n-coverings of C are the k-forms of the maximal unramfied abelian covering
of C of exponent n, while ϕ-coverings of C are (examples of) abelian coverings of exponent
n and conductor m.
For φ an isogeny, let Covφ (V ) denote the set of isomorphism classes of φ-coverings of
V . When nonempty, Covφ (V ) is a principal homogeneous space for the group H1 (k, ker(φ))
acting by twisting. By geometric class field theory the canonical maps Covn (J 1 ) → Covn (C)
and Covϕ (Jm1 ) → Covϕ (C) are bijections that respect this action. There is also a canonical
map Covϕ (C) → Covn (C), which associates to a ϕ-covering of C the maximal unramified
intermediate covering of C. Let Covn0 (C) denote the image of this map, and Covn0 (J 1 ) the
corresponding subset of Covn (J 1 ). Thus, Covn0 consists of isomorphism classes of n-coverings
that may be lifted to a ϕ-covering.
Remark 3.8. Suppose (2, m) is a modulus setup for C : z 2 = f (x, y), a double cover of P1
as in Example 3.3(1).
(1) Given a pair of symmetric bilinear forms (A, B) such that disc(Ax − By) = f (x, y)
the Fano variety of maximal linear subspaces contained in the base locus of the pencil
of quadrics generated by (A, B) may be given the structure of a 2-covering of J 1 .
Theorem 22 and the discussion of Section 5 in [BGW13b] shows that the isomorphism
classes of 2-coverings of J 1 that arise in this way are precisely those in Cov20 (J 1 ).
(2) Section 3 of [BS09] gives an explicit construction of a collection of 2-coverings of C
from the set Hk (notation as in [BS09]). Comparing Lemma 3.10 below with the proof
of [BS09, Theorem 3.4] shows that the collection of coverings they produce is precisely
Cov20 (C). It follows from this that Cov20 (J 1 ) also coincides with the set Covgood (J 1 /k)
defined in [Cre13, Section 6].
Remark 3.9. Suppose (n, m) is a modulus setup for a genus one curve as in Example 3.3(2).
In Section 4 we show that the set Covn0 (C) defined in this paper coincides with that in
[Cre14, Definition 3.3].
10
Lemma 3.10. Suppose (n, m) is a modulus setup associated to an n-descent setup (n, ∆, β)
and that π : X → C is an n-covering. The class of (X, π) in Covn (C) lies in Covn0 (C) if
and only if π ∗ βδ is linearly equivalent to a k-rational divisor, for some δ ∈ ∆(k).
Proof. Suppose π : X → C lifts to a ϕ-covering Y → C. The subfield k(X) ⊂ k(Y )
corresponds to the subgroup µn = T 0 [ϕ] ⊂ Am [ϕ]. The extension k(X) ⊂ k(Y ) is therefore
obtained by adjoining to k(X) an n-th root of a function f such that div(f ) = nD − π ∗ dm,
for some d ∈ Z and f ∈ k(X)× . Furthermore, we can arrange that d = 1. Indeed, we must
have gcd(n, d) = 1, otherwise there would be a proper unramified intermediate extension of
k(X) ⊂ k(Y ). Hence π ∗ m = nD + div(f ) for some D ∈ Div(X) and f ∈ k(X)× . Recall
that nβ − m × ∆ = div(fm ). So, for any δ ∈ ∆(k), the function h := f /π ∗ (fm,δ ) ∈ k(Xk )×
has divisor n(D − π ∗ βδ ). Since adjoining an nth root of h to k(Xk ) gives an unramified
intermediate field of k(Xk ) ⊂ k(Yk ), we must have h ∈ k(Xk )×n . This shows that D − π ∗ βδ
is principal.
For the other direction, suppose D ∈ Div(X) is a k-rational divisor linearly equivalent to
∗
π βδ . Then div(π ∗ fm,δ ) = nπ ∗ βδ − π ∗ m = nD − π ∗ m + div(f ), for some f ∈ k(Xk )× . Thus,
the divisor nD − π ∗ m ∈ Div(X) is principal and k-rational. By Hilbert’s Theorem 90 it is
the divisor of some k-rational function g ∈ k(X)× . Let Y → X be the covering obtained by
adjoining an
pn-th root of g to k(X). Over k we see that k(Y ) is the compositum of k(Xk )
and k(Ck )( n fm,δ ), so Y → C is a ϕ-covering of C.
Proposition 3.11. Suppose (n, m) is a modulus setup for C associated to an n-descent setup
(n, ∆, β). The following are equivalent.
(1) The class of Jm1 in H1 (k, Jm ) is divisible by ϕ.
(2) There exists a ϕ-covering of Jm1 .
(3) There exists a ϕ-covering of C.
(4) Covϕ (C) 6= ∅.
(5) Covn0 (C) 6= ∅.
(6) There exists an n-covering π : X → C with the property that π ∗ βδ is linearly equivalent to a k-rational divisor, for some δ ∈ ∆(k).
(7) The maximal unramified abelian covering of Ck of exponent n descends to k and the
image of the k-rational divisor class π ∗ βδ in Br(k) under the map ΘX of (2.9) lies
in the image of the map Υ of (2.8), for every maximal unramified abelian covering
π : X → C of exponent n and every δ ∈ ∆(k).
Proof. There exists a ϕ-covering of (Jm1 )k . The Galois descent obstruction to defining this
over k is the image in H2 (k, Am [ϕ]) of the class of this covering under the map
H0 k, H1 (Jm1 )k , Am [ϕ] → H2 (k, Am [ϕ])
from the Hochschild-Serre spectral sequence (cf. [Sko01, Section 2.2]). This class coincides
with the image of [Jm1 ] under the coboundary arising from the exact sequence
0 → Am [ϕ] → Am → Jm → 0
(see [Sko01, Lemma 2.4.5]). This proves the equivalence of (1) and (2), while the equivalence
of (2) and (3) follows from geometric class field theory. The equivalences (3) ⇔ (4) ⇔ (5)
follow immediately from the definitions, and (3) ⇔ (6) is given by Lemma 3.10.
11
It remains to prove (6) ⇔ (7). An n-covering π : X → C is a k-form of the maximal
unramified abelian covering of exponent n, which we may assume exists. Then, for any
δ, δ 0 ∈ ∆(k) the divisors π ∗ βδ and π ∗ βδ0 are linearly equivalent. Indeed βδ − βδ0 represents a
class in J[n]. It follows that the class of π ∗ βδ in Pic(Xk ) is fixed by Galk . The image of this
class in Br(k) is trivial if and only if the class can be represented by a k-rational divisor.
Since the set of all isomorphism classes of n-coverings of C is a principal homogeneous space
for H1 (k, J[n]) under the action of twisting, the equivalence of (6) and (7) follows from the
next lemma.
Lemma 3.12. Suppose π : X → C is an n-covering and πξ : Xξ → C is the twist by the
cocycle ξ ∈ Z 1 (J[n]). Then for any δ ∈ ∆(k),
Υ([ξ]) = ΘXξ (πξ∗ βδ ) − ΘX (π ∗ βδ ).
Proof. There is an isomorphism of coverings ρ : X ξ → X with the property that σ ρ ◦ ρ−1 =
Tξσ ∈ Aut(X/Ck ) is translation by ξσ ∈ J[n], for every σ ∈ Galk . Let W = πξ∗ βδ and
W 0 := ρ∗ (W ) = π ∗ βδ . These represent Galois invariant divisor classes, hence, for any
σ ∈ Galk there are functions fσ ∈ k(Xξ )× and gσ ∈ k(X)× with div(fσ ) = σ W − W and
div(gσ ) = σ W 0 − W 0 . The classes in Br(k) of W and W 0 are given by the 2-cocycles
σ
σ
fτ · fσ
gτ · gσ
a(σ,τ ) =
and a0(σ,τ ) =
,
fστ
gστ
×
×
both of which take values in k . Since fσ /ρ∗ gσ ∈ k , the computation
a(σ,τ )
a(σ,τ )
fτ
fσ ρ∗ gστ σ (ρ∗ gτ )
σ
·
=
=
·
·
a0(σ,τ )
ρ∗ (a0(σ,τ ) )
ρ ∗ gτ
ρ∗ gσ fστ ρ∗ (σ gτ )
{z
}
|
coboundary
×
shows that ΘXξ (W ) − ΘX (W 0 ) is represented by the 2-cocycle η ∈ Z 2 (Galk , k ) defined by
σ
(ρ∗ gτ )
gτ ◦ σ ρ
.
=
σg ◦ ρ
ρ∗ (σ gτ )
τ
σ
η(σ,τ ) =
σg
◦T
Using that (ρ−1 )∗ is the identity on k ⊂ k(Y ) and that σ ρ◦ρ−1 = Tξσ we have η(σ,τ ) = τσ gτ ξσ .
We recognize this as the Weil pairing η(σ,τ ) = en (σ Pτ , ξσ ), where Pτ ∈ J[n] is the class
represented by the divisor τ βδ − βδ (see (A.3)). The cocycle Pτ ∈ Z 1 (Galk , J[n]) represents
[Jn` ]. So η(σ,τ ) represents the e-pairing cup product [Jn` ] ∪e [ξ] = [ξ] ∪e [Jn` ] which is equal to
Υ(ξ) by Lemma 2.10.
3.2. Soluble coverings. For an isogeny φ, let Covφsol (V ) denote the set of isomorphism
classes of φ-coverings U → V with U (k) 6= ∅. When k is a global field, let Selφ (V ) denote
the set of isomorphism classes of φ-coverings of V that are soluble everywhere locally.
Proposition 3.13. The group H1 (k, J[n]) acts on the set Covn (J 1 /k) by twisting. This gives
rise to simply transitive actions of:
(1) H1 (k, J[n]) on Covn (J 1 ), when [J 1 ] is divisible by n;
(2) H1 (k, J[n]) on Covn (C), when [J 1 ] is divisible by n;
(3) ker(Υ) on Covn0 (J 1 ), when [Jm1 ] is divisible by ϕ;
(4) ker(Υ) on Covn0 (C), when [Jm1 ] is divisible by ϕ;
12
(5) J(k)/nJ(k) on Covnsol (J 1 ), when J 1 (k) 6= ∅;
(6) J(k)/nJ(k) on Covnsol (J 1 ) ∩ Covn0 (J 1 ), when C(k) 6= ∅;
and, assuming k is a global field, of
(7) Seln (J) on Seln (J 1 ), when [J 1 ] ∈ nX(J);
(8) Seln (J) on Seln (J 1 ) ∩ Covn0 (J 1 ), when [J 1 ] ∈ nX(J) and C is locally soluble.
Corollary 3.14. Suppose C(k) 6= ∅, then [Jm1 ] is divisible by ϕ.
Proof. This follows from (6) since under our assumption Covnsol (J 1 ) 6= ∅.
Corollary 3.15. Suppose that k is a global field and C is everywhere locally solvable. Then
Seln (J 1 ) ⊂ Covn0 (J 1 ).
Proof. If Seln (J 1 ) = ∅ there is nothing to prove. Otherwise, [J 1 ] ∈ nX(J) in which the
result follows from (7) and (8).
Proof of Proposition 3.13.
(1) First note that n-coverings are J[n]-torsors. As in [Sko01, Section 2.2], the low degree
terms of Hochschild-Serre spectral sequence give an exact sequence
∂
1
1
0 → H1 (k, J[n]) → Hét
(J 1 , J[n]) → H0 (k, Hét
(Jk1 , J[n])) → H2 (k, J[n]) .
(2)
(3)
(4)
(5)
(6)
(7)
(8)
There exists an n-covering of Jk1 and the image of its class under ∂ is the obstruction to
the existence of an n-covering of J 1 . This obstruction coincides with the coboundary
of [J 1 ] arising from the exact sequence 0 → J[n] → J → J → 0 (see [Sko01, Lemma
2.4.5]). In particular, if [J 1 ] is divisible by n, then Covn (J 1 ) 6= ∅. In this case
H1 (k, J[n]) acts simply transitively on Covn (J 1 ) by exactness of the sequence above.
It follows from geometric class field theory that the map Covn (J 1 ) → Covn (C) given
by pullback is a bijection which respects the action of H1 (k, J[n]), so (1) ⇒ (2).
This follows from Proposition 3.11(5) and Lemma 3.12.
This follows from (3) by pullback.
If J 1 (k) 6= ∅, then Covnsol (J 1 ) 6= ∅ (since in this case [J 1 ] = 0 in H1 (k, J) is divisible
by n). The difference of any two soluble n-coverings has trivial image in H1 (k, J),
hence must lie in the image of the Kummer map J(k)/nJ(k) → H1 (k, J[n]).
This follows from (5) once we show that Covn0 (J 1 ) contains Covnsol (J 1 ). By assumption
there is some x ∈ C(k), and hence a lift of x to a point x0 ∈ X(k) on some n-covering
π : X → C. This is the pullback of some n-covering T → J 1 , which is necessarily
soluble. Since X(k) 6= ∅ we have Pic(X) = PicX (k). Proposition 3.11(4) shows that
[T → J 1 ] ∈ Covn0 (J 1 ) 6= ∅ and that [Jm1 ] is divisible by ϕ. By (3), Covn0 (J 1 ) is the
orbit of [T → J 1 ] under ker(Υ), and by (5) Covnsol (J 1 ) is the orbit of [T → J 1 ] under
J(k)/nJ(k). It thus suffices to prove that the image of J(k) under the Kummer map
is contained in the kernel of Υ. This follows from Lemma 2.11 since our assumption
that C(k) 6= ∅ implies that ΘC is the zero map.
Since [J 1 ] ∈ nX(J), we have that Seln (J 1 ) 6= ∅. One then argues as in (5) (everywhere locally) to see that the difference of two locally soluble n-coverings gives an
element of Seln (J).
This follows from (7) once we show that Seln (J 1 ) ⊂ Covn0 (J 1 ). Suppose T → J 1
is a locally soluble n-covering and π : X → C is the pullback. Applying (6) over
the completions kv of k we see that ΘX (π ∗ βδ ) has trivial image in Br(kv ). But then
13
ΘX (π ∗ βδ ) must be trivial in Br(k) by the local-global principle for the Brauer group.
Hence (X, π) ∈ Covn0 (C). Then T → J 1 lies in Covn0 (J 1 ) as desired.
3.3. A descent map. Suppose (n, m) is a modulus setup associated to a descent setup
(n, ∆, β) for C. Recall from Remark 3.4 that a choice for fm ∈ k(C × ∆)× induces a map
fm : Pic(C) → L× /k × L×n . The condition in Lemma 3.10 allows us to define a ‘descent map’
f˜m : Covn0 (C) → L× /k × L×n with the property that for any extension K/k and Q ∈ C(K),
(3.3)
f˜m (C 0 , π) = fm (π(Q)) in
(L ⊗ K)×
.
K × (L ⊗ K)×n
This map is compatible with the action of ker(Υ) on Covn0 (C) and the image of ker(Υ) in
L× /k × L×n under the map in (3.2). In particular, the action of Pic0 (C) on Covn0 (C) via
d
Pic0 (C) ⊂ J(k) → ker(Υ) is compatible with the action of fm Pic0 (C) on the image of f˜m
inside L× /k × L×n . In the situations of Example 3.3 (1) and (3), details of this construction
can be found in [Cre13, Prop. 5.4] and [Cre14, Theorem 5.2], respectively. The general case
can be carried out in the same way.
Composing f˜m with the pullback map Covn0 (J 1 ) → Covn0 (C) one obtains a map
g̃m : Covn0 (J 1 ) → L× /k × L×n .
If Q ∈ C(k) 6= ∅, then Proposition 3.13(6) shows that Covnsol (J 1 ) ⊂ Covn0 (J 1 ), and compatibility with the action of ker(Υ) shows that the image of Covnsol (J 1 ) under g̃m is the orbit
under fm (J(k)/nJ(k)) of fm (Q). Similarly if k is a global field and C is everywhere locally
soluble, then Proposition 3.13(8) shows Seln (J 1 ) ⊂ Covn0 (J 1 ); its image under g̃m is contained
in a set which can be computed from the local images of Pic1 (Ckv ). In [Cre13] one finds
examples (in the case of hyperelliptic curves) where this is used to prove that Seln (J 1 ) = ∅
thus concluding that [J 1 ] ∈
/ nX(J). This illustrates the value of Proposition 3.13 from the
perspective of explicit descents; namely it shows that the set of coverings Covn0 (J 1 ) is large
enough to be of interest for such arithmetic applications.
In general, the map Covn0 (J 1 ) → L× /k × L×n will not be injective, so the subset cut out
by the local conditions may be larger than the image of Seln (J 1 ) (One says that we are
computing a ‘fake Selmer set’ for J 1 ). There are ways to deal with this (at least in principle;
see Remark 3.21).
3.4. Norm conditions. When C : z 2 = f (x) is a hyperelliptic curve with m the divisor
above ∞ ∈ Div(P1 ), the 2-divisibility of [Jm1 ] is equivalent to the vanishing of the class of the
leading coefficient of f (x) in k × /k ×2 NL/k (L× ). In this section we consider generalizations of
this condition.
If the diagonal embedding Z/nZ ,→ Res∆ Z/nZ is contained in the kernel of the map
×
Res∆ Z/nZ → J [n],
P then #∆ = nr for some r and there exists a function g ∈ k(C)
such that div(g) = δ∈∆ δ − rm. From the descent setup (n, ∆, β), we
P have a function fmn∈
×
k(C ×∆) such that div(fm ) = nβ −m×∆. Then div(NL/k (fm )) = n δ∈∆ −nrm = div(g ).
Hence NL/k (fm ) = cg n for some c ∈ k × . Different choices for fm and g would modify c by
an element of k ×n NL/k (L× ). Hence the class of c in k × /k ×n NL/k (L× ) is determined by the
descent setup (n, ∆, β).
14
Proposition 3.16. Suppose that the diagonal embedding Z/nZ ,→ Res∆ Z/nZ is contained
in the kernel of the map Res∆ Z/nZ → J [n] and let c ∈ k × be as defined above. If [Jm1 ] is
divisible by ϕ, then the class of c in k × /k ×n NL/k (L× ) is trivial.
Proof. Suppose [Jm1 ] is divisible by ϕ. By Proposition 3.11 there exists a ϕ-covering π : C 00 →
C. There exists α ∈ L× and f ∈ k(C 00 × ∆)× such that αf n = π ∗ fm . Taking norms we find
NL/k (α)NL/k (f n ) = π ∗ NL/k (fδ ) = cπ ∗ g n , where g ∈ k(C)× is the function involved in the
definition of c. From this we see that c ∈ k ×n NL/k (L× ).
When the diagonal embedding of Z/nZ is equal to the kernel of the map Res∆ Z/nZ →
J [n], the norm condition in the proposition is also sufficient for ϕ-divisibility of [Jm1 ]. Indeed,
in this case, dualizing (3.1) one finds an exact sequence 0 → Am [ϕ] → Res∆ µn → µn → 1
which leads to an injective map k × /NL/k (L× )k ×n ,→ H2 (Am [ϕ]) and one can show that the
class of c maps to ∂[Jm1 ] [BGW13b, Theorem 24].
In general, the kernel of Res∆ Z/nZ → J [n] may be larger than (or fail to contain) the
diagonal embedding of Z/nZ. As described in [BPS, Appendix] one can always find a finite
étale k-scheme ∆0 = Spec(L0 ) and a correspondence τ : ∆ 99K ∆0 which induces a surjection
of Res∆0 Z/nZ onto the kernel. We recall that a correspondence τ : E 99K E 0 between finite
k-schemes is a homomorphism ResE Z → ResE 0 Z.
Lemma 3.17. There exist finite étale k-schemes ∆0 = Spec(L0 ), ∆00 = Spec(L00 ) and correspondences τ : ∆ 99K ∆0 , τ 0 : ∆0 99K ∆00 and τ 00 : ∆ → ∆00 with τ 0 ◦ τ = nτ 00 that induce an
τ0
τ
∗
∗
exact sequence 0 → Am [ϕ] → Res∆ µn −→
Res∆0 µn −→
Res∆00 µn .
Proof. The map Am [ϕ] → Res∆ µn comes from dualizing Res∆ Z/nZ → J [n]. For the
existence of τ, τ 0 , τ 00 we refer the reader to [BPS, §A.1, §A.2].
Example 3.18. If C is a hyperelliptic curve with ∆ its set of Weierstrass points, we can take
∆0 = Spec(k), ∆00 = ∅ and τ to be the correspondence inducing the norm NL/k : Res∆ µn →
µn .
Example 3.19. Suppose C is a cubic curve with ∆ the set of flex points. Take ∆00 = ∆
and take ∆0 to be the Galk -set consisting of the twelve lines passing through three distinct
flexes. The incidence relations between flexes and lines define correspondences τ : ∆ 99K ∆0
and σ : ∆0 99K ∆. The compositions of the norm maps from Res∆ µn and Res∆0 µn to µn
with the diagonal embedding µn ,→ Res∆ µn are induced by correspondences ρ : ∆ 99K ∆ and
ρ0 : ∆0 99K ∆. Set τ 0 = σ − ρ0 and τ 00 = 1 − ρ. The proof of [Cre10, Lemma 5.3] shows that
these correspondences satisfy the conditions of Lemma 3.17.
Let τ, τ 0 , τ 00 be as in Lemma 3.17 and define maps
υ(`) = (`n , τ∗ (`)) ,
υ : Res∆ Gm → Res∆ Gm × Res∆0 Gm ;
0
υ : Res∆ Gm × Res∆0 Gm → Res∆0 Gm × Res∆00 Gm ;
0
0
υ (`, ` ) =
`0n τ∗0 (`0 )
,
τ∗ (`) τ∗00 (`)
.
By Lemma 3.17, there is an exact sequence
(3.4)
υ0
υ
0 → Am [ϕ] → Res∆ Gm −→ Res∆ Gm × Res∆0 Gm −→ Res∆0 Gm × Res∆00 Gm .
Proposition 3.20. With notation as above, let U = image(υ) and U 0 = image(υ 0 ).
(1) H1 (U ) ' U 0 (k)/υ 0 (L× × L0× ).
15
(2) There exists g ∈ k(C × ∆0 )× such that the functions c0 = τ∗ (fm )/g n and c00 =
τ∗00 (fm )/τ∗0 (g) are constant and c = (c0 , c00 ) ∈ U 0 (k) ⊂ L0× × L00× .
(3) The class of c in H1 (U ) depends only on (n, ∆, β).
(4) The class of c in H1 (U ) is trivial if [Jm1 ] is divisible by ϕ.
Proof.
(1) This follows immediately from the definitions.
(2) One can check that τ∗ (fm ) is n times a principal divisor (see [BPS, Lemma A.9]).
Hence there exist c0 ∈ L0× and g ∈ k(C × ∆0 )× such that c0 = g n /τ∗ (fm ). Using that
τ 0 ◦τ = nτ we see that τ∗00 (fm ) and τ∗0 (g) have the same divisor, whence the existence of
c00 = τ∗0 (g)/τ∗00 (fm ) ∈ L00× . To see that c ∈ U 0 (k) simply note that c = υ 0 (fm (x), g(x)),
for any x ∈ C(k).
(3) Modifying fm or g by a scalar leaves the class of c modulo υ 0 (L× × L0× ) unchanged.
(4) This proved in the same way as in Proposition 3.16.
Remark 3.21.
(1) The exact sequence (3.4) yields an injective map H1 (U ) ,→ H2 (Am [ϕ]). We expect
that this sends the class of c to ∂[Jm1 ], where ∂ denotes the coboundary coming from
the exact sequence 0 → Am [ϕ] → Am → Jm → 0. In particular we expect the converse
of Proposition 3.20(4) holds.
(2) As described in the appendix to [BPS], one can extend the map fm : Pic(C) →
L× /k × L×n to a map Pic(C) → U (k)/υ(L× ). Using this one can extend the descent map described in Section 3.3 to an injective map Covn0 (J 1 ) → U (k)/υ(L× ),
thus also allowing one to compute the (full) n-Selmer set of J 1 . In the case of genus
one curves this is described in detail in [Cre14].
4. Index and divisibility of torsors under elliptic curves
Let T be a torsor under an elliptic curve E. We define the index of T to be the least
positive degree of a k-rational divisor on T . The index I of T and the order P of T in
H1 (k, E) are known to satisfy P | I | P 2 , and over number fields all pairs of integers (P, I)
satisfying these relations are known to occur [CS10].
Proposition 4.1. Let [C] be a torsor under an elliptic curve E with underlying curve C.
The following are equivalent.
(1) There exists a torsor [C 0 ] ∈ H1 (k, E) of index dividing n2 such that n[C 0 ] = [C].
(2) The curve C admits a modulus setup (n, m) with n = deg(m) such that [Jm1 ] is divisible
by ϕ in H1 (k, Jm ).
Remark 4.2. In [Cre] it is shown that condition (2) is satisfied when C is a locally solvable
curve over a global field k and the action of Galk on J[n] is sufficiently generic. In particular,
when k = Q, it holds when n = pr is any prime power with p > 7.
Our proof of Proposition 4.1 will make use of the following interpretation of the elements
of H1 (k, E[n]) taken from [CFO+ 08].
Definition 4.3. A torsor divisor class pair (T, Z) consists of a J-torsor T and a k-rational
divisor class Z ∈ PicT (k). Two torsor divisor class pairs (T, Z) and (T 0 , Z 0 ) are isomorphic
if there is an isomorphism of torsors s : T → T 0 such that s∗ Z 0 = Z.
16
The automorphism group of the pair (E, n.0E ) can be identified with E[n], and every pair
(T, Z) with deg(Z) = n can be viewed as a twist of (E, n.0E ) ([CFO+ 08, Lemmas 1.7 and
1.8]). It follows that the torsor divisor class pairs of degree n, viewed as twists of (E, n.0E ),
are parameterized by the group H1 (k, E[n]).
Lemma 4.4. Suppose (T 0 , Z 0 ) is a torsor divisor class pair representing a lift of the class
of (T, Z) under the map n∗ : H1 (k, E[n2 ]) → H1 (k, E[n]). The Brauer classes associated
to the k-rational divisor classes Z 0 and Z satisfy n[Z 0 ] = [Z] in Br(k). In particular, Z is
represented by a k-rational divisor if Z 0 is.
Proof. Suppose the class of (T 0 , Z 0 ) is represented by a 1-cocycle ξσ ∈ Z 1 (E[n2 ]). Let fσ , gσ ∈
∗
k(E)× be functions such that div(fσ ) = τξ∗σ [n]∗ 0E − [n]∗ 0E and div(gσ ) = τnξ
n.0E − n.0E .
σ
Comparing divisors we see that we may scale by a constant to arrange that fσn = gσ ◦ [n].
Moreover, using that ξσ is a cocycle, we see that the coboundaries of the 1-cochains (σ 7→ fσ )
×
and (σ 7→ gσ ) give 2-cocyclces F, G ∈ Z 2 (k ) satisfying F n = G.
To prove the lemma one shows that F and G represent the Brauer classes correpsonding to
0
Z and Z, respectively. By [CFO+ 08, Prop. 1.32], the pair (gσ , nξσ ) denotes a lift of nξσ to the
theta group corresponding to the torsor divisor class pair (E, n.0E ). Then [CFO+ 08, Prop.
2.2] shows that [G] = [Z]. In the same way we see that (fσ , ξσ ) gives a lift of ξσ to the theta
group corresponding to (E, [n]∗ 0E ) ' (E, n2 .0E ) and so [F ] = [Z 0 ].
Proof of Proposition 4.1. (2) ⇒ (1). Suppose (2) holds and let (n, ∆, β) be the n-descent
setup corresponding to m. By Proposition 3.11 there is an n-covering π : C 0 → C such that
π ∗ βδ is linearly equivalent to a k-rational divisor for some δ ∈ ∆(k). The genus one curve
C 0 can be endowed with a torsor structure so that n[C 0 ] = [C] in H1 (k, E). Moreover, the
index of [C 0 ] divides deg(π ∗ βδ ) = n2 .
2
(1) ⇒ (2). Suppose (1) holds and let Z 0 ∈ Picn (C 0 ). Consider the torsor divisor class pair
([C 0 ], Z 0 ). The image of this class under n∗ : H1 (k, E[n2 ]) → H1 (k, E[n]) is represented by a
pair ([C], Z). By Lemma 4.4, Z ∈ Picn (C). By Riemann-Roch the divisor class Z contains
a reduced and effective divisor m of degree n. Then (n, m) is a modulus setup for C with
n = deg(m). The divisor m determines a map C → Pn−1 (which is an embedding for n > 2).
Let ∆ := {x ∈ C(k) : n.x ∼ m} and take β to be the diagonal embedding of ∆ in C × ∆.
Then (n, m) is associated to the n-descent setup (n, ∆, β).
The pair (C 0 , Z 0 ) corresponds to an n2 -covering of E, which we may assume factors through
the n-covering of E determined by (C, Z). In particular, there is a commutative diagram
C0
π0
s0
/
E
n
/
C
π
/
E
n
/
E
s
E
where s and s0 are isomorphisms defined over k which determine the E-torsor structures on
C and C 0 . Now [m] = Z = [s∗ n.0E ], so we must have s∗ 0E = βδ for some δ ∈ ∆(k). On
the other hand, Z 0 is the class of s0∗ n2 .0E = s0∗ [n]∗ 0E = π 0∗ s∗ 0E = π 0∗ βδ . As this class is
represented by a k-rational divisor, Proposition 3.11 shows that [Jm1 ] is divisible by ϕ.
17
A. Weil Pairings
In this appendix we recall the three equivalent definitions of the Weil pairing on the
n-torsion of a Jacobian variety that have been used in the paper.
A.1. The Weil pairing via the principal polarization. Suppose φ : A → B is an
isogeny of abelian varieties with dual isogeny φ̂ : B̂ → Â. There is a canonical isomorphism
β : B̂[φ̂] ' A[φ]∨ := Hom(A[φ], Gm ), and thus a nondegenerate pairing ẽφ : A[φ]×B̂[φ̂] → Gm
defined by eφ (x, y) = β(y)(x) (see [Mum70, §15 Theorem 1]). Applying this in the case
φ = [n] : A → A yields a pairing
ẽn : A[n] × Â[n] → Gm .
(A.1)
Now suppose C is a nice curve and let λ : J → Jˆ be the canonical principal polarization
of its Jacobian J. Using this one defines,
(A.2)
en : J[n] × J[n] → Gm ;
en (x, y) = ẽn (x, λ(y)).
A.2. The Weil pairing via Kummer theory. Let M be the maximal unramified abelian
extension of K = k(Ck ) of exponent n. Kummer theory gives a perfect pairing
κ : Gal(M/K) × (K × ∩ M ×n )/K ×n → µn .
There is an isomorphism r : J[n] ' Gal(M/K) of Galk -modules. Specifically, M is the
function field of an étale covering π : Y → Ck fitting into a cartesian diagram
/
Y
π
/
Ck
Jk
n
Jk .
Translation by a point of J[n] on Jk pulls back to give an automorphism of Y over Ck , and
hence an element of Gal(M/K). There is also an isomorphism s : J[n] ' (K × ∩ M ×n )/K ×n .
Explicitly, if P ∈ J[n] is represented by a divisor D ∈ Div(Ck ) such that nD = div(f ), then
there exist g ∈ M × such that g n = f and r(P ) = gK ×n . From [Mil86b, Section 16] and
[Mil86a, Remark 6.10] one deduces that the pairing in (A.2) can be written as,
(A.3)
en (x, y) = κ(r(x), s(y)) .
A.3. Weil’s definition of the pairing. Let D1 , D2 ∈ Div(Ck ) be divisors representing
P1 , P2 ∈ J[n] and let h1 , h2 ∈ k(Ck )× be functions such that div(hi ) = nDi . Then
(A.4)
en (P1 , P2 ) =
Y
(−1)m ordx (D) ordx (E)
x∈Ck (k)
In [How96] it is proved that this agrees with (A.2).
18
g ordx (D)
(x) .
f ordx (E)
References
[Bha] Manjul Bhargava, Most hyperelliptic curves over Q have no rational points, available at arXiv:
1308.0395. ↑1
[BG] Manjul Bhargava and Benedict Gross, The average size of the 2-Selmer group of Jacobians of
hyperelliptic curves having a rational Weierstrass point, available at arXiv:1208.1007. ↑1
[BGW13a] Manjul Bhargava, Benedict Gross, and Xiaoheng Wang, Arithmetic invariant theory II (2013).
↑1
, Pencils of quadrics and the arithmetic of hyperelliptic curves (2013). ↑1, 2, 10, 15
[BGW13b]
[BSD63] B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. I, J. Reine Angew. Math.
212 (1963), 7–25. ↑1
[BPS] Nils Bruin, Bjorn Poonen, and Michael Stoll, Generalized explicit descent and its application to
curves of genus 3. ↑1, 8, 9, 15, 16
[BS09] Nils Bruin and Michael Stoll, Two-cover descent on hyperelliptic curves, Math. Comp. 78 (2009),
no. 268, 2347–2370. ↑1, 10
[Cas62] J. W. S. Cassels, Arithmetic on curves of genus 1. IV. Proof of the Hauptvermutung, J. Reine
Angew. Math. 211 (1962), 95–112. ↑1
[CS10] Pete L. Clark and Shahed Sharif, Period, index and potential. III, Algebra Number Theory 4
(2010), no. 2, 151–174. ↑16
[Cre01] J. E. Cremona, Classical invariants and 2-descent on elliptic curves, J. Symbolic Comput. 31
(2001), no. 1-2, 71–87. Computational algebra and number theory (Milwaukee, WI, 1996). ↑1
[CFO+ 08] J. E. Cremona, T. A. Fisher, C. O’Neil, D. Simon, and M. Stoll, Explicit n-descent on elliptic
curves. I. Algebra, J. Reine Angew. Math. 615 (2008), 121–155. ↑16, 17
[Cre10] Brendan Creutz, Explicit second p-descent on elliptic curves (2010). Ph.D. thesis, Jacobs University. ↑15
, Second p-descents on elliptic curves, Math. Comp. 83 (2014), no. 285, 365–409, DOI
[Cre14]
10.1090/S0025-5718-2013-02713-5. MR3120595 ↑1, 8, 10, 14, 16
[Cre13]
, Explicit descent in the Picard group of a cyclic cover of the projective line, Algorithmic
number theory: Proceedings of the 10th Biennial International Symposium (ANTS-X) held in
San Diego, July 9–13, 2012 (Everett W. Howe and Kiran S. Kedlaya, eds.), Open Book Series,
vol. 1, Mathematical Science Publishers, 2013, pp. 295–315. ↑1, 10, 14
[Cre]
, Most binary forms come from a pencil of quadrics. preprint 2015. ↑16
[How96] Everett W. Howe, The Weil pairing and the Hilbert symbol, Math. Ann. 305 (1996), no. 2,
387–392. ↑18
[Lic69] Stephen Lichtenbaum, Duality theorems for curves over p-adic fields, Invent. Math. 7 (1969),
120–136. ↑7
[MSS96] J. R. Merriman, S. Siksek, and N. P. Smart, Explicit 4-descents on an elliptic curve, Acta Arith.
77 (1996), no. 4, 385–404. ↑1
[Mil86a] J. S. Milne, Jacobian varieties, Arithmetic geometry (Storrs, Conn., 1984), Springer, New York,
1986, pp. 167–212. ↑18
[Mil86b]
, Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986,
pp. 103–150. ↑18
[Mum70] David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay; Oxford University
Press, London, 1970. ↑18
[NSW08] Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, 2nd
ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical
Sciences], vol. 323, Springer-Verlag, Berlin, 2008. ↑7
[PS97] Bjorn Poonen and Edward F. Schaefer, Explicit descent for Jacobians of cyclic covers of the
projective line, J. Reine Angew. Math. 488 (1997), 141–188. ↑1, 5, 8, 9
[Ser88] Jean-Pierre Serre, Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117,
Springer-Verlag, New York, 1988. Translated from the French. ↑2, 3, 4
19
[Sko01] Alexei Skorobogatov, Torsors and rational points, Cambridge Tracts in Mathematics, vol. 144,
Cambridge University Press, Cambridge, 2001. ↑11, 13
[Sta05] Sebastian Stamminger, Explicit 8-descent on elliptic curves (2005). PhD Thesis, International
University Bremen. ↑1, 9
[Tho15] Jack A. Thorne, E6 and the arithmetic of a family of non-hyperelliptic curves of genus 3, Forum
Math. Pi 3 (2015), e1, 41. ↑2
, On the 2-Selmer groups of plane quartic curves with a marked point. preprint 2014. ↑2
[Tho]
[Wan] Xiaoheng Wang, Maximal linear spaces contained in the base loci of pencils of quadrics, available
at arXiv:1302.2385. ↑1
Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800,
Christchurch 8140, New Zealand
E-mail address: [email protected]
URL: http://www.math.canterbury.ac.nz/~bcreutz
20