Generalized Jacobians and explicit descents

0.3 MB·

by  Brendan Creutz

GENERALIZED JACOBIANS AND EXPLICIT DESCENTS
BRENDAN CREUTZ
6
Abstract. We developa cohomologicaldescriptionofvariousexplicitdescents intermsof
1
0 generalized Jacobians, generalizing the known description for hyperelliptic curves. Specifi-
2 cally,givenanintegerndividingthedegreeofsomereducedeffectivedivisormonacurveC,
n weshowthatmultiplicationbynonthegeneralizedJacobianJm factorsthroughanisogeny
a ϕ : Am → Jm whose kernel is naturally the dual of the Galois module (Pic(Ck)/m)[n]. By
J
geometricclassfieldtheory,thiscorrespondstoanabeliancoveringofCk :=C×SpeckSpec(k)
4 ofexponentnunramifiedoutsidem. Then-coveringsofC parameterizedbyexplicitdescents
2
are the maximal unramified subcoverings of the k-forms of this ramified covering.

]
T
1. Introduction
N
. Suppose f(x,y) is a binary form of degree d over a field k of characteristic not equal to
h
t 2. Pencils of quadrics with discriminant form f(x,y) have been studied in [BSD63,Cas62,
a
m Cre01,BG,Wan,BGW,BGWb]. When d is even, the SL (k)/µ -orbits of pairs (A,B) with
d 2
[ discriminant form f(x,y) correspond to a collection of 2-coverings of the hyperelliptic curve

C : z2 = f(x,y). When k = Q these coverings are used in [Bha] and [BGWb] to compute
1
v the average size of the 2-Selmer set of C, and of the torsor J1 parameterizing divisor classes
5 of degree 1, respectively, from which they deduce the fantastic result that most hyperelliptic
4
curves over Q have no rational points.
4
6 The same collection of coverings can also be described in terms of the k-algebra L :=
0
k[x]/f(x,1). This description was used in [BS09] and [Cre13] to compute 2-Selmer sets
.
1 of C and J1, respectively, for individual hyperelliptic curves. A key step in both [Cre13]
0
6 and [BGWb] is to check that this collection of coverings is large enough to contain the
1 locally soluble 2-coverings (under suitable hypotheses on C). In [BGWb] this is achieved
:
v by identifying the collection of coverings as the unramified subcoverings of k-forms of the
i
X maximal abelian covering of exponent 2 unramified outside the pair of points at infinity on
r the affine model of z2 = f(x,y), a characterization that is quite natural in light of the use
a
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 interms ofgeneralized Jacobians, generalizing thedescription forhyperelliptic
curves given in [PS97]. Specifically, given an integer n dividing the degree of some reduced
effectivedivisormonacurveC, weshowthatmultiplicationbynonthegeneralizedJacobian
Jm factors through an isogeny ϕ : Am → Jm whose kernel is naturally the dual of the Galois
module (Pic(C )/m)[n]. By geometric class field theory, this corresponds to an abelian
k
covering of C := C × Spec(k) of exponent n unramified outside m. The n-coverings
k Speck
of C parameterized by the explicit descents mentioned above are the maximal unramified
subcoverings of the k-forms of this ramified covering.
1
This description unifies the methods of explicit descent on curves (and/or their J1) de-
scribed in [MSS96,Sta05,BS09,Cre13,Cre14,BPS], and provides new insights into the de-
scents on the corresponding Jacobians (For example, Lemma 2.9, Corollary 2.11 and Re-
mark 3.4). It also allows us to show that the corresponding collection of coverings of J1
contains the locally solvable coverings and, in particular, that the ‘descent map’ in [BPS]
could be used to compute 2-Selmer sets of J1 for non-hyperelliptic curves of genus ≥ 2.
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 [BGWb] for the hyperelliptic case). The
results in Propositions 3.10 and 3.12 would then have implications for the structure of the
space of orbits. Thorne has recently made progress understanding the situation for non-
hyperelliptic 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 Gal = Gal(k/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 ´etale k-scheme ∆ = Spec(L) we use
Res = Res to denote the restriction of scalars functor taking L-schemes to k-schemes.
∆ L/k
2. The modulus setup
Defintion 2.1. Let C be a nice curve over k. A modulus setup for C is a pair (n,m)
consistingof a positive integern notdivisible bythe 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 ℓ := deg(m)/n. We are primarily interested in
the following examples.
M.1 Suppose π : C → P1 is a double cover which is not ramified over ∞ ∈ P1. Let n = 2
and m = π∗∞.
M.2 Suppose C is a plane cubic curve. Let n = 3 and let m be any triple of distinct
colinear points.
M.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.
M.4 Suppose C is a plane quartic curve. Let n = 2 and let m be any quadruple of distinct
colinear points.
M.5 Generalizing the previous example, suppose C is any nice curve, n = 2 and m is a
canonical divisor. Then m = 2g −2 and ℓ = g −1.
We may view m as a finite ´etale subscheme m = SpecM ⊂ 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 Pic and Pic be the commutative group schemes over k
C Cm
representing the Picard functors of C and Cm. There is an exact sequence of commutative
group schemes over k,
(2.1) 0 → T → Pic → Pic → 0,
Cm C
2
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.
Lemma 2.2. T = ResmGm/Gm is the quotient by the diagonal embedding of Gm, and there
is an exact sequence of finite group schemes
Res1 µ
m n N
1 −→ −→ T[n] −→ µ −→ 1,
n
µ
n
where the map N is induced by the norm map ResmGm → Gm and Res1mµn is the kernel of
N : Resmµn → µn.
Proof. The first statement, that T = ResmGm/Gm, follows from well known results on the
structure of generalized Jacobians (see [Ser88, §V Prop. 7]). The inclusion map Res1 G →
m m
ResmGm induces a surjective map onto ResmGm/Gm with kernel µm. This gives the middle
rows of the following commuative and exact diagram.
µ // Res1 µ //// T[n]
n(cid:127)_ m n
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
1 // µ // Res1 G // ResmGm // 1
m m m Gm
n n n
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
1 // µ // Res1 G // ResmGm // 1
m m m Gm
m/n n
(cid:15)(cid:15)(cid:15)(cid:15) (cid:15)(cid:15)
µ // 1
n
(cid:3)
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 2.3. Given a modulus setup (n,m) there is a commutative group scheme A over k
and isogenies ψ : Pic → A and ϕ : A → Pic such that ker(ψ) = Res1mµn ⊂ T[n] and
Cm Cm µn
ϕ◦ψ = [n]. Moreover, we have a commutative and exact diagram
1 // T′ // A // Pic // 0
C
ϕ ϕ n
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
1 // T // Pic // Pic // 0.
Cm C
where T′ is a torus and T′[ϕ] ≃ µ .
n
Proof. By Lemma 2.2, Pic contains a finite group scheme isomorphic to Res1 µ . The
Cm m n
quotient of Pic by this subgroup scheme yields an isogeny ψ : Pic → A. The existence
Cm Cm
of ϕ follows from the fact that ker(ψ) is contained in the kernel of multiplication by n. Since
3
ker(ψ) ⊂ T, A is an extension of Pic . The rest of the assertions follow from the diagram in
C
(cid:3)
Lemma 2.2.
Remark 2.4. When n = m = deg(m) = 2, we have that T[n] ≃ µ . Hence ψ is the identity
n
map on A = Pic and ϕ is multiplication by 2.
Cm
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 = Specm. We use Divm(C) to dentote the
divisors of C that have support disjoint from m.
Lemma 2.5. Let A be as defined in Lemma 2.3. Then
×
Pic (k) = Div(C )/{div(f) : f ∈ k(C ) },
C k k
PicCm(k) = Divm(Ck)/{div(f) : f ∈ k(Ck)×, f(m) = 1},
A(k) = Divm(Ck)/{div(f) : f ∈ k(Ck)×, f(m) ∈ Res1mµn}.
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 = ResmGm/Gm ⊂ PicCm
are represented by divisors of functions which do not vanish on m,
{div(f) : f ∈ k(C ), f(m) ∈ M×}
T(k) = k .
{div(f) : f ∈ k(C )×, f(m) = 1}
k
The description of A(k) in the statement then follows from the fact that A is the quotient
of Pic by the image of Res1 µ in T. The final statement follows easily from the fact that
Cm m n
ϕ◦ψ is multiplication by n on Pic . (cid:3)
Cm
2.3. Component groups. The component groups of Pic ,Pic and A are all isomorphic
Cm C
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,
′
(2.3) 1 → T → Am → J → 0.
In particular, Am is geometrically connected.
We label the components
(2.4) Pic = Ji, Pic = Ji , A = Ai ,
C Cm m m
iG∈Z iG∈Z iG∈Z
so that the superscripts denote the image under the degree map. To ease notation we also
denote the degree 0 components by J = J0, Jm = Jm0 and Am = A0m. For any i ∈ Z, Ji and
Jmi are torsors under J and Jm, respectively.
Let m′ ∈ 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
Pic m−1 Pic m−1 A m−1
(2.5) J := ZmC′ = Ji, Jm := ZmC′m = Jmi , Am := Zm′ = Aim
iG=0 iG=0 iG=0
4
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 m′. However, the map PicC → J depends only on m.
Recall that ℓ := m/n. From the definitions in (2.5) and Lemma 2.3 one easily obtains the
following commutative and exact diagram.
(2.6) µ µ
n(cid:127)_ n(cid:127)_
(cid:15)(cid:15) (cid:15)(cid:15) 1
0 // Am[ϕ](cid:31)(cid:127) // Am[ϕ] ℓdeg // Z/n // 0
(cid:15)(cid:15)(cid:15)(cid:15) (cid:15)(cid:15)(cid:15)(cid:15) 1
0 // J[n](cid:31)(cid:127) // J[n] ℓdeg // Z/n // 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) = m′−m.
Given D1,D2 ∈ Jm[n], choose representative divisors D1,D2 ∈ Divm(Ck), and let di =
deg(D )/ℓ. There exist unique functions h′ ∈ k(C )× such that nD = div(h ) + d m′ and
i i k i i i
h′(m) = 1. Set h = fdih′, so that nD = div(h )+d m. Define:
i i i i i
hordPD1 ×
(2.7) e(D ,D ) = (−1)d1d2 (−1)n(ordPD1)(ordPD2) 2 (P) ∈ k .
1 2 hordPD2
P∈YC(k) 1
Lemma 2.6. This gives a Galois equivariant bilinear pairing e : Jm[n]×Jm[n] → µn.
Proof. Thismaybecheckedexactlyasin[PS97,Section7](oneneedonlyreplacethefunction
x there with the function f in the definition above). (cid:3)
Lemma 2.7. The pairing in Lemma 2.6 induces nondegenerate Galois equivariant pairings,
e : Am[ϕ]×Am[ϕ] → µn,
e : Am[ϕ]×J[n] → µn,
e : J[n]×J[n] → µ .
n
The induced pairing on J[n]×J[n] coincides with the Weil pairing.
Remark 2.8. The definition of e given above depends on the choice of m′ in (2.5) and the
function f with div(f) = m′−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 Res1 µ and T[n] with respect to
m n
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 D ∈ T[n]. Then D is represented by D = div(f) for some f ∈ k(C )× with
1 1 1 k
f(m) ∈ Resmµn. Since nD1 = div(fn) and fn(m) = 1 we must use h1 = fn in the definition
5
of the pairing. Suppose D ∈ J[n] and let D ,h ,d be as in the definition of the pairing.
2 2 2 2
Then we have
hordPf
e(D ,D ) = (−1)n(ordPf)(ordPD2) 2 (P)
1 2
fnordPD2
Y
P∈C(k)
hordPf
= (−1)(ordPf)(ordPh2+d2ordPm) 2 (P) (since nD = div(h )+d m.)
fordPh2+d2ordPm 2 2 2
Y
P∈C(k)
= (−1)d2(ordPf)(ordPm)f−d2ordPm(P) (by Weil reciprocity)
Y
P∈C(k)
= f−d2ordPm(P) (since f(m) is invertible)
Y
P∈C(k)
= N(f(m))−d2 ,
where N denotes the induced norm ResmGm → Gm. From this one easily sees that Res1mµn
lies in the kernel of the pairing and that T[n] pairs trivially with the degree 0 subgroup,
Jm[n] ⊂ Jm[n]. (cid:3)
Taking Galois cohomology of (2.6) yields a commutative and exact diagram
(2.8) k×/k×n k×/k×n
(cid:15)(cid:15) (cid:15)(cid:15)
Z/nZ δ′ // H1(Am[ϕ]) // H1(Am[ϕ])deg/ℓ // H1(Z/n)
(cid:15)(cid:15) (cid:15)(cid:15)
Z/nZ δ // H1(J[n]) // H1(J[n]) deg/ℓ // H1(Z/n)
Υ Υ′
(cid:15)(cid:15) (cid:15)(cid:15)
Br(k)[n] Br(k)[n]
Lemma 2.9. The images of δ(1) and δ′(1) in H1(J) and H1(Am) are the classes of Jℓ and
Aℓ , respectively. The maps Υ and Υ′ are given by
m
Υ(ξ) = ξ ∪ δ(1)
e
′ ′
Υ(ξ) = ξ ∪ δ (1),
e
where ∪ denotes the cup product induced by the pairing of Lemma 2.7.
e
Proof. The extensions
Jℓ : 0 → J[n] → J[n] → Z/nZ → 0, and
n
Jmℓ,n : 0 → Am[ϕ] → Am[ϕ] → Z/nZ → 0
represent classes
[Jℓ] ∈ Ext1 (Z/nZ,J[n]) ∼= H1(J[n])
n Z/nZ[Galk]
[Jmℓ,n] ∈ Ext1Z/nZ[Galk](Z/nZ,Am[ϕ]) ∼= H1(Am[ϕ]).
6
Let δ and δ′ denote the coboundary maps in the Galois cohomology of Jℓ and Jℓ , re-
n m,n
spectively. Then δ(1) = [Jℓ] and δ′(1) = [Jℓ ]. Let δ∨ and δ′∨ denote the coboundries
n m,n
of the extensions (Jℓ)∨ and (Jℓ )∨ obtained by dualizing and let ǫ : J[n] ∼= J[n]∨ and
n m,n
ǫ′ : J[n] ∼= Am[ϕ]∨ be the isomorphisms induced by the e-pairings of Lemma 2.7.
By [NSW08, Corollary 1.4.6] the following diagrams are commutative.
H1(J[n]) ǫ∗ // H1(J[n]∨) δ∨(•)∪1 // H2(µ )
m
•∪δ(1) •∪δ(1) (−1)2
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
H2(J[n]⊗J[n]) (ǫ⊗id)∗// H2(J[n]∨ ⊗J[n]) eval∗ // H2(µ )
m
H1(J[n]) ǫ′∗ // H1(Am[ϕ]∨) δ′∨(•)∪1 // H2(µm)
•∪δ′(1) •∪δ′(1) (−1)2
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
H2(J[n]⊗Am[ϕ]) (ǫ′⊗id)//∗H2(Am[ϕ]∨ ⊗Am[ϕ]) eval∗ // H2(µm)
The composition along the top row is the map Υ (resp. Υ′), 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.
(cid:3)
2.5. Brauer class of a k-rational divisor class. Given a nice curve X, there is a well
known exact sequence
(2.9) 0 → Pic(X) −→ Pic (k) −Θ→X Br(k)
X
(see [Lic69]). The map Θ gives the obstruction to a k-rational divisor class being repre-
X
sented by a k-rational divisor.
Lemma 2.10. 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) = ℓ·Θ (x).
C
Proof. The image of d is isotropic with respect to the Weil-pairing cup product ∪ . This
e
gives a commutative diagram of pairings
∪ : H1(J[n]) × H1(J[n]) → Br(k)
e
d→ ← =
h, i : J(k) × H1(J) → Br(k)
By a result of Lichtenbaum (see the proof of [Lic69, Corollary 1]) we have that hx,[J1]i =
Θ (x). By the previous lemma we have
C
Υ◦d(x) = d(x)∪δ(1) = hx,[Jℓ]i = ℓ·hx,[J1]i = ℓ·Θ (x).
n C
(cid:3)
Corollary 2.11. If
(1) the period of C divides ℓ, or
7
(2) k is a local or global field and gcd(m,g −1) divides ℓ,
then Υ◦d = 0.
Proof. The image of Θ : J(k) → Br(k) is isomorphic to the cokernel of Pic0(C) → J(k),
C
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 ℓ 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). (cid:3)
3. The descent setup
We recall the following definition from [BPS].
Defintion 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 ´etale k-scheme
∆ = SpecL, 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), β ∈ Div(C ) is a divisor such that nβ −m principal. So the class of β in J lies
δ k δ δ
in J[n]. This gives rise to a commutative and exact diagram,
(3.1) Res0 Z/nZ(cid:31)(cid:127) // Res Z/nZdeg //// Z/nZ
∆ ∆
(cid:15)(cid:15) (cid:15)(cid:15) 1
J[n](cid:31)(cid:127) // J[n] ℓdeg //// Z/nZ
Defintion 3.2. We say that (n,∆,β) is an n-descent setup if the vertical maps in (3.1) are
surjective.
The following examples show that all of the modulus setups in setups considered in Sec-
tion 2 are associated to an n-descent setup. Details for Example D.1 and Example D.2 may
be found in [BPS, Examples 6.9], while Example D.3 is considered in [Cre14].
D.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 M.1.
D.2 Suppose C is any curve of genus ≥ 2. We obtain a 2-descent setup for C by taking
∆ to be the Gal -set of odd theta characteristics. By [BPS, Proposition 5.8] there
k
is some β ∈ Div(C × ∆) such that [β ] = δ for δ ∈ ∆(k). We can take m to be a
δ
canonical divisor and thus recover the modulus setup in Example M.5.
D.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 M.3.
8
D.4 More generally, suppose C is a genus one curve of degree m in Pm−1 and n | m. There
is a Gal -invariant subset of flexes of size n2 whose differences represent the n-torsion
k
points. For any such flex x, ℓ.x is linearly equivalent to a divisor D := P +···+P
x 1 ℓ
with P ∈ C such that nD is a hyperplane section Furthermore, these D may be
i x x
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 3.3. In [BPS, Section 6] it is shown that a choice of m and fm ∈ k(C×∆)× yields
a homomorphism
L×
fm : Pic(C) −→
k×L×n
related to the connecting homomorphism d : J(k) → H1(J[n]) by a commutative diagram
Pic0(C)(cid:31)(cid:127) // Pic(C) fm // L×
k×L(cid:127)×_ n
(cid:15)(cid:15) (cid:15)(cid:15)
J(k) d // H1(J[n]) ǫ // H1 Res∆µn .
(cid:16) µn (cid:17)
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′[ϕ]) // H1(Am[ϕ]) // H1(J[n]) Υ // H2(T′[ϕ])
ǫ
(cid:15)(cid:15) (cid:15)(cid:15)
k×/k×n // L×/L×n // H1 Res∆µn // Br(k)[n]
(cid:16) µn (cid:17)
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 3.4. Let J(k) ⊂ J(k) be the subgroup which maps under ǫ◦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 2.10. For example, when C is a
•
non-hyperelliptic curve of genus 3 over a global field with descent setup as in Example D.2,
we see that J(k) = J(k).
•
3.1. ϕ- and n-coverings.
Defintion 3.5. Suppose φ : A → B is an isogeny of semiabelian varieties over k and T
is a B-torsor. We say π : T′ → T is a φ-covering of T if there exist isomorphisms a,b of
k-varieties fitting into a commutative diagram
T′ b // A
k k
π φ
(cid:15)(cid:15) (cid:15)(cid:15)
T a // B .
k k
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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 J1 and n-coverings of J1. The
m
pullback of a ϕ-covering T → J1 along the canonical map (C−m) → J1 sending a geometric
m m
point x to the class of the divisor x in J1(k) ⊂ Pic (k) yields an unramified covering of
m Cm
(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.
Defintion 3.6. 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 J1 along the canonical map (C −m) → J1. A morphism π : X → C is an
m m
n-covering of C if it is the pullback of an n-covering of J1 along the canonical map C → J1.
By Galois theory, the field extension of k(C ) corresponding to a ϕ-covering is the com-
k
positum 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(C ) represents a point of order
k
n in J[n], then there exists a function h ∈ k(C )× such that div(h ) = nD−dm for some
D k D
d ∈ Z. The corresponding extension of k(C ) is obtained by adjoining an n-th root of h .
k D
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(J1) → Covn(C)
and Covϕ(J1) → Covϕ(C) are bijections that respect this action. There is also a canonical
m
map Covϕ(C) → Covn(C), which associates to a ϕ-covering of C the maximal unramified
intermediate covering of C. Let Covn(C) denote the image of this map, and Covn(J1) the
0 0
corresponding subset of Covn(J1). Thus, Covn consists of isomorphism classes of n-coverings
0
that may be lifted to a ϕ-covering.
Remark 3.7. Suppose (2,m) is a modulus setup for C : z2 = f(x,y), a double cover of P1
as in Example D.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 J1.
Theorem 22 and the discussion of Section 5 in [BGWb] shows that the isomorphism
classes of 2-coverings of J1 that arise in this way are precisely those in Cov2(J1).
0
(2) Section 3 of [BS09] gives an explicit construction of a collection of 2-coverings of C
from the set H (notation as in [BS09]). Comparing Lemma 3.9 below with the proof
k
of [BS09, Theorem 3.4] shows that the collection of coverings they produce is precisely
Cov2(C). It follows from this that Cov2(J1) also coincides with the set Cov (J1/k)
0 0 good
defined in [Cre13, Section 6].
Remark 3.8. Suppose (n,m) is a modulus setup for a genus one curve as in Example D.3.
In Section 4 we show that the set Covn(C) defined in this paper coincides with that in
0
[Cre14, Definition 3.3].
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