1 Introduction
An algebraic variety Y over
$\mathbb {Q}$
violates the Hasse principle if
$Y(\mathbb {Q}) = \emptyset $
despite the fact that
$Y(\mathbb {Q}_p) \neq \emptyset $
for all completions
$\mathbb {Q}_p$
of
$\mathbb {Q}$
, including the archimedean completion
$\mathbb {Q}_{\infty } = \mathbb {R}$
. The Hasse-Minkowski theorem shows that quadrics in
$\mathbb {P}^n$
never violate the Hasse principle, but violations do exist in higher degree. Some early examples include the hyperelliptic curve
$2y^2 = x^4 - 17$
studied by Lind and Reichardt [Reference Lind24, Reference Reichardt33] and Selmer’s plane cubic
$3x^3 + 4y^3 + 5z^3 = 0$
[Reference Selmer35]. Each of these is a genus one curve C, and is therefore a torsor for its Jacobian, the elliptic curve
$E = \mathrm {Pic}^0(C)$
. The fact that C violates the Hasse principle means that it represents a nontrivial element
$[C]$
in the Tate-Shafarevich group 
parameterizing locally trivial E-torsors. The order of
$[C]$
in 
is, in these cases, equal to the minimum positive degree of an effective
$0$
-cycle – hence
$2$
in the first example and
$3$
in the second.
There are by now many other examples of nontrivial elements of Tate-Shafarevich groups of elliptic curves. However, it is an open question whether for every prime p there exists an elliptic curve
$E/\mathbb {Q}$
with a class of order p in 
. Geometrically, such E-torsors are realized as genus one curves
$C \subset \mathbb {P}^{p-1}_{\mathbb {Q}}$
contained in no hyperplane, which violate the Hasse principle. The lack of a systematic construction of order p elements is somewhat surprising, since heuristics of Delaunay predict that for a given prime p, the probability that a random elliptic curve E satisfies 
should be positive [Reference Delaunay11].
More generally, for any abelian variety
$A/\mathbb {Q}$
, the group 
parameterizes A-torsors which violate the Hasse principle. Like the 1-dimensional case of elliptic curves, there are few examples with 
for large primes p, beyond examples where
$A = \mathrm {Res}^F_{\mathbb {Q}} B$
is the Weil restriction of an abelian variety B over a number field F with 
(see, for example, [Reference Clark and Sharif9, Reference Kloosterman18, Reference Kloosterman and Schaefer19]). However, the second author and Weiss [Reference Shnidman and Weiss37] recently showed that for every prime p, there exist absolutely simple abelian varieties A over
$\mathbb {Q}$
with 
. They prove such A exist among the quadratic twists of quotients of modular Jacobians
$J_0(N)$
with prime level
$N \equiv 1\ \pmod {p}$
, but the proof does not yield explicit examples.
1.1 Results
Our first main result is an explicit construction of A-torsors X which violate the Hasse principle. In our examples, both A and X have very simple equations. To state the theorem, recall the p-th power character
$\left (\frac {q}{\ell }\right )_p$
, which satisfies
$\left (\frac {q}{\ell }\right )_p = 1$
if and only if q is a p-th power in
$\mathbb {Q}_{\ell }^\times $
.
Theorem 1.1. Let
$p> 5$
be a prime and let
$u,v$
be integers not divisible by
$3$
. Let U be the set of primes dividing
$3puv(u-3v)$
. Let
$t \geq 2$
, and let
$k = p_1 p_2 \cdots p_t$
, where each
$p_i$
is a prime not in U satisfying
-
(1)
$\left (\frac {p_i}{p_j}\right )_p = 1$
, for all
$i \neq j$
in
$\{1,\ldots , t\}$
, -
(2)
$\left (\frac {p_i}{q}\right )_p = 1$
, for all
$i \in \{1, \ldots , t\}$
and all
$q \in U$
, -
(3)
$\left (\frac {q}{p_i}\right )_p = 1$
, for all
$i \in \{1, \ldots , t\}$
and all
$q \in U\backslash \{3\}$
, -
(4)
$\left (\frac {3}{p_i}\right )_p \neq 1$
, for all
$i \in \{1, \ldots , t\}$
.
Let
$g = p-1$
and consider the variety
$\tilde A \subset \mathbb {A}_{\mathbb {Q}}^{2g+1}$
defined by the equations
$$\begin{align*}y_i^p = x_i(x_i - 3uk)(x_i - 9vk),\mbox{ for } i = 1,\ldots ,g, \mbox{ and } z^p = \prod_{i = 1}^g x_i(x_i - 3uk).\end{align*}$$
The symmetric group
$S_g$
acts on
$\tilde A$
, and the quotient
$\tilde A/S_g$
is birational to a unique g-dimensional abelian variety A over
$\mathbb {Q}$
. Let
$I \subset \{1, \ldots , t\}$
be a proper nonempty subset, and let
$q = \prod _{i \in I} p_i$
. Let
$\tilde X \subset \mathbb {A}_{\mathbb {Q}}^{2g+1}$
be defined by the equations
$($
with
$i = 1,\ldots , g)$
$$\begin{align*}y_i^p = x_i(x_i - 3uk)(x_i - 9vk) \mbox{ and } qz^p = \prod_{i = 1}^g x_i(x_i - 3uk).\end{align*}$$
Then
$\tilde X/S_{g}$
is birational to an A-torsor X that violates the Hasse principle, and the class of X in 
has order p.
Remark 1.2. Both A and X are
$\mu _p$
-covers of the Jacobian J of the genus
$p-1$
superelliptic curve
$C \colon y^p = x(x-3uk)(x-9vk)$
. Since J is birational to the symmetric power
$C^g/S_g$
, the
$\mu _p$
-covers can be seen from the equations above as well.
Using the Cebotarev density theorem, we show in Proposition 6.1 that there exist primes
$p_1,\ldots , p_t$
satisfying the hypotheses of Theorem 1.1. Here is an example with
$p = 29$
.
Example 1.3. Let
$\tilde X \subset \mathbb {A}_{\mathbb {Q}}^{28} \times \mathbb {A}_{\mathbb {Q}}^{28} \times \mathbb {A}_{\mathbb {Q}}^1$
be the variety defined by the
$28$
equations
$$\begin{align*}y_i^{29} = x_i(x_i - 3\cdot 386029093\cdot545622299) (x_i +9\cdot386029093\cdot545622299)\end{align*}$$
for
$i = 1, \ldots , 28$
, as well as the additional equation
$$\begin{align*}386029093 z^{29} = \prod_{i = 1}^{28} x_i(x_i - 3\cdot 386029093\cdot545622299).\end{align*}$$
Then
$\tilde X/S_{g}$
is birational to a torsor X for a
$28$
-dimensional abelian variety A over
$\mathbb {Q}$
. Moreover, X violates the Hasse principle and represents an order
$29$
element of 
.
Remark 1.4. As a point of comparison, work of Radičević [Reference Radičević32] gives a method to compute equations for order p torsors in the Tate-Shafarevich group of an elliptic curve E over
$\mathbb {Q}$
. Even for
$p = 11$
, the equations for these torsors are not so easy for humans to write down. As p grows, the computations quickly become intractable even for computers.
Since the hypotheses of Theorem 1.1 are always met, this gives a second proof of [Reference Shnidman and Weiss37, Thm. 1], and moreover gives explicit examples for any prime p. Moreover, the flexibility of the index set I allows us to prove our second main result, that 
can be arbitrarily large.
Theorem 1.5. For every prime p and every integer
$k \geq 1$
, there exists an absolutely simple abelian variety A over
$\mathbb {Q}$
with 
.
The cases
$p = 2,3,5$
not covered by Theorem 1.1 were proven by Bölling [Reference Bölling3], Cassels [Reference Cassels6] and Fisher [Reference Fisher12], respectively. Indeed, it was previously known that the p-part of the Tate-Shafarevich group of absolutely simple abelian varieties over
$\mathbb {Q}$
can be arbitrarily large only for certain small primes p. Our examples are special since they arise as
$\mu _p$
-covers of a specific type of Jacobian, so we leave open the question of existence of order p elements in 
for ‘generic’ abelian varieties over
$\mathbb {Q}$
(i.e., those such that the Mumford-Tate group is
$\mathrm {GSp}_{2g}$
and
$A[p]$
is irreducible as a
$\mathrm {Gal}(\bar {\mathbb {Q}}/\mathbb {Q})$
-module). In both this paper and [Reference Shnidman and Weiss37], the abelian varieties are such that
$\mathrm {rk}\, \mathrm {End}(A_{\bar {\mathbb {Q}}}) = \dim A$
and
$A[p]$
is reducible.
Since we can control the dimension of our examples, we also conclude the following:
Corollary 1.6. Suppose
$g = p-1$
for some prime
$p \geq 7$
. Then the Tate-Shafarevich groups of absolutely simple abelian varieties A over
$\mathbb {Q}$
of dimension g can be arbitrarily large. More precisely, the groups 
can be arbitrarily large.
Our construction generalizes in an obvious way to any global field. We work over
$\mathbb {Q}$
because it is the most interesting case and to keep the notation simple. The restriction
$p \neq 5$
in our results is related to some quirky numerology (see Proposition A
$(iii)$
in the Appendix) that could probably be removed by tweaking the construction slightly.
1.2 Previous work
Previous work on elliptic curves ([Reference Bhargava, Klagsbrun, Lemke Oliver and Shnidman2, Reference Bölling3, Reference Cassels6, Reference Fisher12, Reference Kloosterman18, Reference Kloosterman and Schaefer19, Reference Kramer20, Reference Lemmermeyer22, Reference Lemmermeyer and Mollin23, Reference Matsuno26]) has found arbitrarily large p-torsion part of the Tate-Shafarevich group for
$p \leqslant 7$
and
$p=13$
. In higher dimension, Creutz [Reference Creutz10] has shown that for any principally polarized abelian variety A over a number field K, the p-torsion in the Tate-Shafarevich group can be arbitrarily large over a field extension L of degree which is bounded in terms of p and the dimension of A, generalizing work of Clark and Sharif [Reference Clark and Sharif9]. In higher dimension over
$\mathbb {Q}$
, the first author [Reference Flynn14] has recently shown that the 2-torsion subgroup of Tate-Shafarevich groups of absolutely simple Jacobians of genus
$2$
curves over
$\mathbb {Q}$
can be arbitrarily large, and then in [Reference Flynn15] that the
$2$
-torsion of the Tate-Shafarevich groups of absolutely simple Jacobians of curves of any genus over
$\mathbb {Q}$
can be arbitrarily large. With Bruin, the authors recently showed in [Reference Bruin, Flynn and Shnidman5] that 
can be arbitrarily large among certain abelian surfaces
$A/\mathbb {Q}$
. Many of these works make use of Jacobians with an isogeny to another Jacobian, comparing the bound obtained using isogeny-descent against that of a complete p-descent.
1.3 Approach
Our method makes use of Jacobians with two independent
$\mathbb {Q}$
-rational p-torsion points, so we also make (implicit) use of isogenies. However, instead of bounding the Mordell-Weil rank, we construct locally soluble torsors and show directly that they have no rational points. Since our method does not require knowledge of L-functions nor any information related to the rank of
$A(\mathbb {Q})$
, it is more widely applicable. Our technique is similar in spirit to that of Cassels in [Reference Cassels6] who used the Cassels-Tate pairing to show that the
$3$
-part of the Tate-Shafarevich group of elliptic curves can be arbitrarily large. However, our approach is more direct. In the appendix by Tom Fisher, an alternative interpretation of our proof is given in terms of an appropriate Cassels-Tate pairing.
We construct our torsors purely geometrically, as
$\mu _p$
-covers. In fact, we avoid the use of Galois cohomology in this paper, as a way of emphasizing the geometry. Experts will see that the proof can be interpreted cohomologically using standard descent techniques [Reference Cassels and Flynn8, Reference Schaefer34], but the geometric point of view is the most direct way to understand the construction and will perhaps be more accessible to those less familiar with Selmer groups (though we do assume familiarity with the basics of abelian varieties).
1.4 Outline of proof
In Section 2 and 3, we prove some preliminary material on
$\mu _p$
-covers and
$\mu _p$
-descent. Most of this will be well known to experts, but we have customized the discussion to our needs and made it fairly self-contained. In Section 4, we specialize the discussion to
$\mu _p$
-covers of Jacobians of superelliptic curves. In Section 5, we prove Theorem 1.1. We must show that the torsors have local points everywhere and yet have no rational points. For most primes
$\ell $
, it is easy to see that the torsors have
$\mathbb {Q}_{\ell }$
-points using the fact that almost all of the primes in the set
$\{p_1,\cdots , p_t\} \cup U$
are p-th powers modulo each other. The subtle case is where
$\ell = p_i$
, and in this case, we construct points explicitly using the torsion points
$D_0 = (0,0) - \infty $
and
$D_1= (3uk,0) - \infty $
on J. The more interesting argument is the proof that the torsors have no global points. For this, we first show that the two global torsion divisors
$D_0$
and
$D_1$
generate a certain quotient of
$J(\mathbb {Q}_{p_i})$
, for each
$p_i$
. The presence of the powers of
$3$
in the model of the curve, and the fact that
$3$
is not a p-th power locally, then ‘glues together’ the localizations of the torsors in a certain way that makes it impossible for them to have a global point unless the parameter q is divisible by either all or none of the primes
$p_1, \ldots , p_t$
. The particular choice of the prime
$3$
here is not special (we could replace it by
$5$
or
$7$
, etc.), but the presence of this ‘gluing prime’ plays the crucial role in the argument.
In Section 6, we deduce Theorem 1.5 from Theorem 1.1. First, we use a Cebotarev argument to show that given p, the set U, and any
$t \geq 1$
, there exist primes
$p_1, \ldots , p_t$
satisfying the conditions of Theorem 1.1. Second, the flexibility in the choice of q allows us to generate a subgroup of
$\mathbb {F}_p$
-rank at least
$t - 1$
in 
. Finally, we use a theorem of Masser to show that for
$100\%$
of integers
$u,v$
not divisible by
$3$
, the corresponding abelian variety is geometrically simple. In the appendix, Tom Fisher recasts our proof in terms of a Cassels-Tate pairing.
2
$\mu _p$
-covers
2.1 Classifying
$\mu _p$
-covers
Let X be a proper variety over a field F. Let
$\mu _p$
be the F-group scheme of p-th roots of unity. A
$\mu _p$
-cover of Y (or more formally, a
$\mu _p$
-torsor over Y in the fppf topology) is a Y-scheme X together with a
$\mu _p$
-action that is simply transitive on fibers over Y. The
$\mu _p$
-covers of Y form a category
$\mathcal {M}_p(Y)$
whose morphisms are
$\mu _p$
-equivariant isomorphisms. The following proposition gives a concrete way to think about
$\mu _p$
-covers.
Proposition 2.1. There is an equivalence of categories between
$\mathcal {M}_p(Y)$
and the category of pairs
$(\mathcal {L}, \eta )$
where
$\mathcal {L}$
is an invertible sheaf on Y and
$\eta \colon \mathcal {L}^{\otimes p} \simeq \mathcal {O}_Y$
is an isomorphism. Here, the morphisms
$(\mathcal {L}, \eta ) \to (\mathcal {L}', \eta ')$
are isomorphisms
$g \colon \mathcal {L} \to \mathcal {L}'$
such that
$\eta ' \circ g^{\otimes p} = \eta $
.
Proof. This is well known (see [Reference Arsie and Vistoli1] or [Reference Mumford29, pg. 71]), so we just describe the functors in both directions. If
$\pi \colon X \to Y$
is a
$\mu _p$
-cover, then there is a
$\mathbb {Z}/p\mathbb {Z}$
-grading on the
$\mathcal {O}_Y$
-module
$$\begin{align*}\pi_*\mathcal{O}_X = \mathcal{O}_Y \oplus \bigoplus_{i = 1}^{p-1} \mathcal{L}_i\end{align*}$$
where each
$\mathcal {L}_i$
is the invertible subsheaf of
$\pi _*\mathcal {O}_X$
on which
$\mu _p$
acts by
$\zeta \cdot s = \zeta ^i s$
. The algebra structure of
$\pi _*\mathcal {O}_X$
gives isomorphisms
$\mathcal {L}_i \otimes \mathcal {L}_j \simeq \mathcal {L}_{i + j}$
, where indices are to be taken modulo p and where
$\mathcal {L}_0 = \mathcal {O}_Y$
. Thus, we obtain an isomorphism
$\mathcal {L}_1^{\otimes p} \simeq \mathcal {O}_Y$
. Conversely, starting with a pair
$(\mathcal {L},\eta )$
, we can define a sheaf of
$\mathcal {O}_Y$
-algebras
$\mathcal {O}_Y \oplus \bigoplus _{i = 1}^{p-1} \mathcal {L}^i$
using the given isomorphism
$\eta $
to define the multiplication
$\mathcal {L}^i \otimes \mathcal {L}^j \simeq \mathcal {L}^{i+j} \simeq \mathcal {L}^{i+j - p}$
on the factors with
$i + j \geq p$
. The relative spectrum of this sheaf over Y is then naturally endowed with a
$\mu _p$
-action making it a
$\mu _p$
-cover.
Remark 2.2. If
$Y = \mathrm {Spec} \, F$
, this recovers Kummer theory.
2.2
$\mu _p$
-covers of abelian varieties
Let us now specialize to the case where Y is an abelian variety over a field F of characteristic not p. We will think of a
$\mu _p$
-cover
$\pi \colon X \to Y$
in terms of the corresponding pair
$(\mathcal {L}, \eta )$
. The isomorphism class of
$\mathcal {L}$
is a well-defined element of
$\mathrm {Pic}(Y) = \mathrm {Pic}_Y(F)$
, called the Steinitz class of
$\pi $
. The existence of
$\eta $
means that
$\mathcal {L}$
is p-torsion, so that
$\mathcal {L} \in \widehat {Y}[p](F)$
, where
$\widehat {Y} = \mathrm {Pic}^0_Y \subset \mathrm {Pic}_Y$
is the dual abelian variety parameterizing algebraically trivial line bundles on Y.
From one
$\mu _p$
-cover
$(\mathcal {L}, \eta )$
, we may construct many more, simply by scaling
$\eta \colon \mathcal {L}^{\otimes p} \to \mathcal {O}_Y$
by any
$r \in F^*$
. Two
$\mu _p$
-covers
$(\mathcal {L}, r\eta )$
and
$(\mathcal {L}, s\eta )$
are isomorphic if and only if
$r/s\in F^{*p}$
. More generally, given two
$\mu _p$
-covers
$(\mathcal {L},\eta )$
and
$(\mathcal {L}', \eta ')$
, the tensor product
$(\mathcal {L} \otimes \mathcal {L}', \eta \otimes \eta ')$
is another. Let
$H^1(Y, \mu _p)$
denote the set of isomorphism classes of
$\mu _p$
-covers of Y.
Proposition 2.3. The set
$H^1(Y, \mu _p)$
is naturally an abelian group and sits in a short exact sequence
$$\begin{align*}0 \to F^*/F^{*p} \to H^1(Y, \mu_p) \to \widehat{Y}[p](F) \to 0.\end{align*}$$
Proof. This follows from Proposition 2.1 and the discussion above.
Remark 2.4. We use the notation
$H^1(Y, \mu _p)$
since the étale cohomology group
$H^1_{\mathrm {et}}(Y, \mu _p)$
is also in bijection with isomorphism classes of
$\mu _p$
-covers. From this point of view, one obtains Proposition 2.3 by applying the long exact sequence in cohomology to the short sequence of sheaves
$0 \to \mu _p \to \mathbb {G}_m \to \mathbb {G}_m \to 0$
.
Lemma 2.5. The
$\mu _p$
-cover
$\pi \colon X \to Y$
corresponding to
$(\mathcal {L}, \eta )$
is geometrically connected if and only if
$\mathcal {L}\not \simeq \mathcal {O}_X$
.
Proof. If
$\mathcal {L} \simeq \mathcal {O}_X$
, then
$\eta $
is scalar multiplication by some
$r \in F^\times $
. In this case, X is isomorphic to
$Y \times _F F(\sqrt [p]{r})$
as an F-scheme, which is not geometrically connected. Conversely, if X is not geometrically connected, then the
$\mu _p$
-cover
$X_{\bar F} \to Y_{\bar F}$
induces an isomorphism on connected components, forcing
$X_{\bar F}$
to be isomorphic to the trivial
$\mu _p$
-torsor
$Y_{\bar F} \times _{\bar F} \mu _p$
. It follows that
$\pi $
is in
$\ker (H^1(Y, \mu _p) \to H^1(Y_{\bar F}, \mu _p)) \simeq F^*/F^{*p}$
, and hence has trivial Steinitz class.
Suppose now that
$\pi \colon X \to Y$
is a geometrically connected
$\mu _p$
-cover corresponding to
$(\mathcal {L}, \eta )$
, so that
$\mathcal {L} \not \simeq \mathcal {O}_Y$
. Since every connected finite étale cover of the abelian variety
$Y_{\bar F}$
is itself an abelian variety [Reference Mumford29, §18], X becomes an abelian variety over the algebraic closure
$\overline {F}$
. It follows that X is a torsor for a certain abelian variety, which we will now identify.
Let
$ \widehat {\psi } \colon \widehat {Y} \to \widehat {Y}/\langle \mathcal {L} \rangle $
be the degree p isogeny obtained by modding out by
$\mathcal {L}$
. Let
$\psi \colon A_{\mathcal {L}} \to Y$
be the dual isogeny, which is also of degree p. Then
$\psi $
can itself be given the structure of
$\mu _p$
-cover. Indeed, we have
$$\begin{align*}\ker(\psi) \simeq \widehat{\ker(\widehat{\psi})} \simeq \widehat{\mathbb{Z}/p\mathbb{Z}} = \mathrm{Hom}(\mathbb{Z}/p\mathbb{Z}, \mathbb{G}_m) \simeq \mu_p.\end{align*}$$
Note that there are
$p-1$
different isomorphisms
$\ker (\psi ) \simeq \mu _p$
, corresponding to the different
$\mathbb {Z}/p\mathbb {Z}$
-gradings we can put on
$\psi _*\mathcal {O}_{A_{\mathcal {L}}}$
. Exactly one of them will have the property that the corresponding
$\mu _p$
-cover has Steinitz class
$\mathcal {L}_1 \subset \psi _*\mathcal {O}_{A_{\mathcal {L}}}$
isomorphic to
$\mathcal {L}$
. We choose this
$\mu _p$
-cover structure for
$\psi $
.
Lemma 2.6. Let
$\pi \colon X \to Y$
be a
$\mu _p$
-cover with nontrivial Steinitz class
$\mathcal {L} \in \widehat {Y}[p](F)$
. Then
$\pi $
is a twist of the
$\mu _p$
-cover
$\psi \colon A_{\mathcal {L}} \to Y$
and X is a torsor for
$A_{\mathcal {L}}$
.
Proof. If
$\psi \colon A_{\mathcal {L}} \to Y$
corresponds to
$(\mathcal {L}, \eta )$
, then
$\pi \colon X \to Y$
corresponds to
$(\mathcal {L}, s \eta )$
for some scalar
$s \in F^*$
. Over
$\bar {F}$
there is an isomorphism
$\rho \colon (A_{\mathcal {L}})_{\bar {F}} \to X_{\bar {F}}$
of
$\mu _p$
-covers, which satisfies
$$\begin{align*}\rho^g(P) = \sqrt[p]{s}^g/\sqrt[p]{s} + \rho(P)\end{align*}$$
for all
$g \in \mathrm {Gal}(\bar F/F)$
and
$P \in A_{\mathcal {L}}(\bar {F})$
; here,
$\sqrt [p]{s}^g/\sqrt [p]{s} \in \mu _p$
and
$+$
is the torsor action. The torsor structure
$A_{\mathcal {L}} \times X \to X$
is given by
$(P, Q) \mapsto \rho (P + \rho ^{-1}(Q))$
. Using the formula for
$\rho ^g$
, we see that this torsor is indeed defined over F.
We have seen that for each nonzero
$\mathcal {L} \in \widehat {Y}[p](F)$
, there is, in fact, a distinguished
$\mu _p$
-cover with Steinitz class
$\mathcal {L}$
– namely, the cover
$A_{\mathcal {L}} \to Y$
. This means there must be a distinguished isomorphism
$\eta \colon \mathcal {L}^p \simeq \mathcal {O}_Y$
. We will describe this isomorphism
$\eta $
in Lemma 3.6, in the context of rational points. For simplicity, we will specialize to the case where Y is a Jacobian, and in particular principally polarized (so that
$\widehat {Y} \simeq Y$
). However, most of what we prove can be generalized to arbitrary abelian varieties in a straightforward way.
2.3
$\mu _p$
-covers of Jacobians
Let C be a smooth projective geometrically integral curve over F, and let
$J = \mathrm {Pic}^0(C)$
be its Jacobian. Let g be the genus of C, and hence also the dimension of the abelian variety J. Let
$D \in J[p](F)$
be a divisor class of order p. Let
$J \to J/\langle D \rangle $
be the quotient and let
$\psi \colon A_D\to \widehat {J}$
be the corresponding dual isogeny, where
$A_D$
is the dual of
$J/\langle D \rangle $
. Then
$\psi $
is a
$\mu _p$
-cover of
$\widehat {J}$
corresponding to a pair
$(\mathcal {L}, \eta )$
, as in the previous section.
Remark 2.7. As before, we may choose the
$\mu _p$
-cover structure on
$\psi $
so that
$\mathcal {L} \in \mathrm {Pic}^0(\widehat {J})(F)$
is mapped to D under the isomorphism
$\hat {\hat J} \simeq J$
.
From now on, we identify J and
$\widehat {J}$
via the principal polarization
$\lambda \colon J \to \widehat J$
coming from the theta divisor of the curve C. To make this explicit, we assume that C contains a rational point
$\infty \in C(F)$
. The theta divisor
$\Theta \subset J$
is the subvariety of degree 0 divisor classes of the form
$E - (g-1)\infty $
, where E is an effective divisor of degree
$g -1$
. The isomorphism
$J \to \widehat {J}$
sends P to
$t_P^*\mathcal {O}_J(\Theta ) \otimes \mathcal {O}_J(\Theta )^{-1}$
, where
$t_P \colon J \to J$
is translation by P. We can also describe
$\lambda (P)$
as the line bundle on J associated to the divisor
$[\Theta - P] - [\Theta ]$
.
After making the identification
$J \simeq \widehat {J}$
, we may view
$\psi $
as a
$\mu _p$
-cover of J and
$\eta $
as an isomorphism
$\mathcal {L}^{\otimes p} \to \mathcal {O}_J$
. By Proposition 2.3, we have the exact sequence
$$\begin{align*}0 \to F^*/F^{*p} \to H^1(J, \mu_p) \to J[p](F) \to 0. \end{align*}$$
3
$\mu _p$
-descent
We continue with our assumptions on
$J = \mathrm {Pic}^0(C)$
. We have seen that to each
$D \in J[p](F)$
of order p, there is a corresponding
$\mu _p$
-cover
$\psi \colon A_D \to J$
giving rise to the data
$(\mathcal {L}, \eta )$
. These particular
$\mu _p$
-covers are by construction abelian varieties, but general
$\mu _p$
-covers corresponding to pairs
$(\mathcal {L}, r\eta )$
, for
$r \in F^\times $
, may only be torsors for abelian varieties. We characterize those which are abelian varieties, or equivalently, those which have rational points.
3.1 Descent over general fields
Fix
$D \in J[p](F)$
and
$(\mathcal {L}, \eta )$
, as above. Given
$P \in J(F)$
, we may consider the
$\mu _p$
-cover
$\psi _P = t_P \circ \psi \colon A_D \to J$
, where
$t_P \colon J \to J$
is translation by P. The
$\mu _p$
-cover
$\psi _P$
is endowed with the same
$\mu _p$
-action as
$\psi $
, but different structure map to J. Since the Steinitz class is in
$\mathrm {Pic}^0(J)$
, it is invariant under translation, and hence,
$\psi _P$
and
$\psi $
have isomorphic Steinitz classes. If
$\psi _P = (\mathcal {L}', \eta ')$
, then we can choose an isomorphism
$\mathcal {L}' \simeq \mathcal {L}$
, and under this isomorphism, we have
$\eta ' = r_P \eta $
for some
$r_P \in F^*$
. Any other choice of isomorphism
$\mathcal {L}' \simeq \mathcal {L}$
differs by a scalar, so the element
$r_P$
is well defined up to
$F^{*p}$
.
Lemma 3.1. The map
$P \mapsto r_P$
induces an injective map
$\partial ^D \colon J(F)/\psi (A_D(F)) \to F^*/F^{*p}$
.
Proof. Note that
$r_P\in F^{*p}$
if and only if
$\psi _P$
is isomorphic as a
$\mu _p$
-cover to
$\psi $
. But any isomorphism of
$\mu _p$
-covers induces an isomorphism of
$A_D$
-torsors, and hence must be given by translation by Q for some
$Q \in A_D(F)$
. Translation by Q gives an isomorphism between these two
$\mu _p$
-covers if and only if
$P = \psi (Q)$
.
For completeness, we state the following result, connecting the map
$\partial ^D$
to a boundary map in Galois cohomology:
Lemma 3.2. The map
$\partial ^D$
is the boundary map
$J(F) \to H^1(F, \mu _p) \simeq F^*/F^{*p}$
in the long exact sequence in group cohomology for the short exact sequence of
$\mathrm {Gal}(\bar F/F)$
-modules
$$\begin{align*}0 \to \mu_p \to A_D(\bar F) \stackrel{\psi}{\longrightarrow} J(\bar F) \to 0. \end{align*}$$
Proof. The boundary map
$J(F) \to H^1(F, \mu _p)$
sends
$P \in J(F)$
to the cocycle
$c \colon \mathrm {Gal}(\bar F/F) \to \mu _p \simeq A_D[\psi ]$
given by
$g \mapsto Q^g - Q$
, where
$Q \in A_D$
is such that
$\psi (Q) = P$
. We must show that this cocycle agrees with the cocycle
$g \mapsto \sqrt [p]{r}^g/\sqrt [p]{r}$
, where
$r = r_P$
. From the proof of Lemma 2.6, we see that the
$A_D$
-torsors
$(\mathcal {L}, r\eta )$
and
$(\mathcal {L}, \eta )$
are isomorphic (over
$\bar F$
) via translation by Q. By the explicit formula given there, this exactly means that
$Q^g - Q$
is equal to the element
$\sqrt [p]{r}^g/\sqrt [p]{r} \in \mu _p$
.
Lemma 3.3. The image of
$\partial ^D$
is the set of
$r \in F^*/F^{*p}$
such that the
$\mu _p$
-cover
$(\mathcal {L}, r\eta )$
has a rational point.
Proof. Every torsor in the image clearly has a rational point since it is isomorphic to
$A_D$
as a variety. Conversely, if a
$\mu _p$
-cover
$X \to J$
of the form
$(\mathcal {L}, r\eta )$
has a rational point, then the underlying
$A_D$
-torsor is isomorphic to the trivial
$A_D$
-torsor up to translation by a point P. Hence,
$\partial ^D(-P) = r$
.
Remark 3.4. It follows that for a
$\mu _p$
-cover
$\pi \colon X \to J$
with Steinitz class
$\mathcal {L}$
, X is isomorphic to
$A_D$
(as varieties) if and only if
$\pi $
corresponds to
$(\mathcal {L}, r\eta )$
, with r in the image of
$\partial ^D$
.
The following lemma is immediate from the definitions and can be used to give an explicit formula for the homomorphism
$\partial ^D$
.
Lemma 3.5. Let
$F(J)$
be the function field of J and view
$\eta ^{-1} \colon \mathcal {O}_J \to \mathcal {L}^p$
as a global section of
$\mathcal {L}^p$
. Fix an embedding of
$\mathcal {L}$
as a subsheaf of
$F(J)$
, so that
$\eta ^{-1}$
is a nonzero element f of
$F(J)$
. Let Q be such that Q and
$Q + P$
are in a domain of definition for f. Then
$\partial ^D(P) = r_P = f(P+Q)/f(Q)$
, up to p-th powers.
Thinking of
$\eta ^{-1}$
as a function on J allows us to distinguish the unique
$\mu _p$
-cover
$(\mathcal {L}, \eta )$
corresponding to
$\psi \colon A_D \to J$
among all
$\mu _p$
-covers with Steinitz class
$\mathcal {L}$
, as promised.
Lemma 3.6. The
$\mu _p$
-cover corresponding to
$(\mathcal {L}, \eta )$
, which is isomorphic to the
$\mu _p$
-cover
$A_{\mathcal {L}} = A_D \to J$
, is characterized among all
$\mu _p$
-covers with Steinitz class
$\mathcal {L}$
by the fact that the value
$f(0_J)$
of the function
$f = \eta ^{-1} \in F(J)$
at
$0_J$
is a p-th power in
$F^*$
. (Here we assume that
$\mathcal {L}$
is chosen within its isomorphism class so that
$f(0_J) \in F^\times $
.)
Proof. The
$\mu _p$
-cover
$A_D \to J$
is distinguished among
$\mu _p$
-covers with Steinitz class
$\mathcal {L}$
by the fact that the fiber above
$0$
has a rational point. Indeed, if
$\pi \colon X \to J$
is a
$\mu _p$
-cover of type
$(\mathcal {L}, r\eta )$
with a rational point
$Q \in X(F)$
above
$0 \in J(F)$
, then
$\pi = \psi _P$
for some
$P \in J(F)$
, and
$\pi ^{-1}(0) = \psi ^{-1}(-P)$
. It follows that
$P \in \psi (A_D(F))$
, and hence, r is a p-th power, or in other words,
$\pi $
is isomorphic to
$\psi $
as
$\mu _p$
-covers.
However, the pullback of the
$\mu _p$
-cover
$(\mathcal {L}, \eta )$
on J to
$\mathrm {Spec}\, F$
, via the inclusion
$\{0_J\} \hookrightarrow J$
, is
$\mathrm {Spec}\, k[z]/(z^p - h)$
where
$h = f(0_J)$
. This has an F-rational point if and only if h is a p-th power.
Let
$\mathrm {Sym}^g C = C^g/S_g$
be the g-th symmetric power of C. Points of
$\mathrm {Sym}^g C$
correspond to effective degree g divisors E on C. Recall that the map
$\mathrm {Sym}^g C \to J$
sending
$E \mapsto E - g\infty $
is birational [Reference Milne, Cornell and Silverman27, Thm. 5.1], and hence induces an isomorphism of function fields
$F(\mathrm {Sym}^g C) \simeq F(J)$
.
Lemma 3.7. Suppose
$pD = \mathrm {div}(\tilde f)$
for some
$\tilde f \in F(C)$
. Then
$\mathcal {L}\simeq \mathcal {O}_J(\tilde D)$
for a divisor
$\tilde D$
on J such that
$p\tilde D = \mathrm {div}(f)$
, where
$f \in F(J) \simeq F(\mathrm {Sym}^g C)$
is the rational function
$f(\sum _{i = 1}^g (x_i,y_i) - g\infty ) = \prod _{i = 1}^g \tilde f(x_i,y_i)$
.
Proof. Assume, for simplicity, that
$D = \infty - Q$
for some
$Q \in C(F)$
. Under the polarization
$J \to \widehat {J}$
, the point D gets sent to the divisor
$[\Theta - D] - [\Theta ]$
. Note that
$$\begin{align*}\Theta - D = \{E + Q - g\infty \colon E \, \mbox{effective of degree } g-1\}\end{align*}$$
is the locus of poles of the function f. Similarly,
$\Theta $
is the zero locus. Taking into account multiplicities, the divisor of f is
$p[\Theta - D] - p[\Theta ]$
, as claimed. The general case where
$D = \sum _j (\infty - Q_j)$
is similar.
Finally, we will use a generalization of the map
$\partial $
and Lemma 3.1. Let
$H = \{D_1, \ldots , D_m\} \subset J[p](F)$
be a subset of
$\mathbb {F}_p$
-linearly independent elements. For each
$i = 1, \ldots , m$
, let
$\psi _i \colon A_i \to J$
be the
$\mu _p$
-covers corresponding to
$D_i$
. Let
$A_H = \widehat {J/\langle H\rangle }$
and let
$\psi _H \colon A_H \to J$
be the isogeny dual to
$J \to J/\langle H\rangle $
. Then we have a homomorphism
$$\begin{align*}\tilde \partial^H \colon J(F) \longrightarrow \prod_{i = 1}^m F^*/F^{*p}\end{align*}$$
sending P to
$(\partial ^{D_1}(P), \ldots , \partial ^{D_m}(P))$
.
Lemma 3.8. The map
$\tilde \partial ^H$
induces an injection
$\partial ^H \colon J(F)/\psi _H(A_H(F)) \hookrightarrow \bigoplus _{i = 1}^m F^*/F^{*p}$
.
Proof. We prove this in the case
$m = 2$
, which is the only case we will use. The general case follows by an inductive argument. Suppose
$\tilde \partial ^{D_1}(P) = 0$
and
$\tilde \partial ^{D_2}(P) = 0$
, so that
$P = \psi _i(Q_i)$
for some
$Q_i \in A_i(F)$
by Lemma 3.1. Let
$g_i \colon A_H \to A_i$
be the natural maps, of degree p; note that
$\psi _1 g_1 = \psi _2 g_2$
. The fiber diagram
$$ \begin{align} \begin{array}{ccc} A_H &\longrightarrow&A_2 \\ \downarrow & &\downarrow \\ A_1& \longrightarrow&J\\ \end{array} \end{align} $$
shows that there is a unique point Q in
$A_H(\bar F)$
such that
$g_i(Q) = Q_i$
for
$i = 1,2$
. The uniqueness of Q implies that it is
$\mathrm {Gal}(\bar F/F)$
-stable, and so we have
$P = \psi _H(Q)$
with
$Q \in A_H(F)$
. This shows that
$\partial ^H$
is injective.
3.2 Descent over global fields
Suppose now that C is a curve over
$\mathbb {Q}$
. The preceding discussion applies for
$F =\mathbb {Q}$
, but also for
$F =\mathbb {Q}_{\ell }$
for any prime
$\ell \leq \infty $
. Having fixed
$D \in J[p](\mathbb {Q})$
, let
$$\begin{align*}\mathrm{Sel}(A_D) \subset \mathbb{Q}^*/\mathbb{Q}^{*p} \end{align*}$$
be the subgroup of classes r with the property that for every prime
$\ell $
, the class of r in
$\mathbb {Q}_{\ell }^*/\mathbb {Q}_{\ell }^{*p}$
is in the image of
$\partial ^D \colon J(\mathbb {Q}_{\ell })/\psi (A_D(\mathbb {Q}_{\ell })) \to \mathbb {Q}_{\ell }^*/\mathbb {Q}_{\ell }^{*p}$
. In other words, an element of
$\mathrm {Sel}(A_D)$
is a
$\mu _p$
-cover
$X \to J$
with Steinitz class D and such that
$X(\mathbb {Q}_{\ell }) \neq \emptyset $
for every prime
$\ell $
.
Recall that if A is an abelian variety over
$\mathbb {Q}$
, then 
is the group of A-torsors which are trivial over
$\mathbb {Q}_{\ell }$
, for all primes
$\ell \leq \infty $
.
Proposition 3.9. Let 
be the Tate-Shafarevich group of
$A_D$
. There is an exact sequence
where 
is the kernel of the map 
induced by
$\psi $
.
Proof. The map 
sends the
$\mu _p$
-cover
$X \to J$
to the underlying
$A_D$
-torsor (c.f. Lemma 2.6). The cocycle
$c \colon \mathrm {Gal}(\bar F/F) \to A_D(\bar F)$
with
$c(g) = \sqrt [p]{s}^g/\sqrt [p]{s}$
determines this torsor and has image in
$\mu _p \simeq \ker (\psi )$
, so the cocycle indeed becomes trivial in 
. The exactness of the sequence in the middle follows from Lemma 3.3. The exactness on the right can be proved using direct geometric arguments but is most easily seen using Lemma 3.2. Since we will not actually use the exactness on the right, we omit the proof.
The group
$\mathrm {Sel}(A_D)$
is isomorphic to the usual Selmer group
$$\begin{align*}\mathrm{Sel}_\psi(A_D) \subset H^1(F, A_D[\psi]) \simeq H^1(F, \mu_p) \simeq F^*/F^{*p}.\end{align*}$$
In particular, it is finite. This can also be seen from the following proposition.
Proposition 3.10. Suppose
$\ell \neq p$
is a prime of good reduction for J. Then the image of
$\partial ^D \colon J(\mathbb {Q}_{\ell }) \to \mathbb {Q}_{\ell }^*/\mathbb {Q}_{\ell }^{*p}$
is the subgroup
$\mathbb {Z}_{\ell }^*/\mathbb {Z}_{\ell }^{*p}$
.
Proof. This well-known fact follows from [Reference Česnavičius7, Prop. 2.7(d)] if we grant Lemma 3.2, but we will give a more geometric proof in the spirit of this paper. The classes in
$\mathbb {Z}_{\ell }^*/\mathbb {Z}_{\ell }^{*p}$
represent
$\mu _p$
-covers
$X \to J$
which are trivialized by an unramified field extension; hence, the corresponding
$A_D$
-torsor X is also trivialized by an unramified field extension. Since
$A_D$
has good reduction at
$\ell $
, the torsor X has a Néron model
$\mathcal {X}$
over
$\mathbb {Z}_{\ell }$
, which is a torsor for the Néron model
$\mathcal {A}$
of
$A_D$
[Reference Bosch4, Reference Cassels6.5. Cor. 4]. Since any torsor for a smooth proper group scheme over
$\mathbb {Z}_{\ell }$
has a
$\mathbb {Z}_{\ell }$
-point, it follows that such classes are in the image of
$\partial ^D$
by Lemma 3.3. Conversely, any element r in the image of
$\partial ^D$
corresponds to a
$\mu _p$
-cover (and
$A_D$
-torsor) X which is abstractly isomorphic to
$A_D$
, and hence has good reduction over
$\mathbb {Q}_{\ell }$
. By the Néron mapping property, X extends to a
$\mu _p$
-cover and even an
$\mathcal {A}$
-torsor over
$\mathbb {Z}_{\ell }$
. It follows that
$r \in \mathbb {Z}_{\ell }^*$
, since we can interpret this scalar as an automorphism of a line bundle on an abelian scheme
$\mathcal {J}$
over
$\mathbb {Z}_{\ell }$
(well-defined up to p-th powers).
We will also consider more general Selmer groups. Given a subset
$H = \{D_1, \ldots , D_m\} \subset J[p](F)$
of linearly independent elements, we can define an analogous Selmer group
$\mathrm {Sel}(A_H) \subset \prod _{i = 1}^m F^*/F^{*p}$
which sits in an exact sequence
and which is isomorphic to the usual Selmer group
$\mathrm {Sel}_{\psi _H}(A_H)$
.
4 Jacobians of curves of the form
$y^p = x(x-e_1)(x-e_2)$
4.1 A special family of curves
Let
$p> 5$
be a prime and let
$e_0, e_1,e_2$
be distinct integers. Let C be the smooth projective curve over
$\mathbb {Q}$
with affine model
$$ \begin{align} y^p = (x-e_0)(x-e_1)(x-e_2). \end{align} $$
There is no loss in generality in assuming
$e_0 = 0$
, so we will do so. The affine model is itself smooth, and its complement in C is a single rational point we call
$\infty $
. The genus of C is
$g = p-1$
. For more details on such curves, see [Reference Poonen and Schaefer30].
Let J be the Jacobian of C. Note that
$J(\mathbb {Q})$
has p-torsion of rank at least
$2$
, generated by the three divisor classes
$D_i = [(e_i,0) - \infty ]$
, for
$i \in \{0,1,2\}$
. The equality of divisors
$D_0 + D_1 + D_2 = \mathrm {div}(y)$
means that
$D_0 + D_1 + D_2 = 0$
in J. Let
$D = D_0 + D_1$
and define the abelian varieties
$$ \begin{align} {\widehat{A}} = J/\langle D_0, D_1 \rangle \end{align} $$
$$ \begin{align} {\widehat{B}} = J/\langle D \rangle \end{align} $$
and the corresponding quotient isogenies
${\widehat \phi } : J \rightarrow {\widehat {A}}$
and
${\widehat \psi } \colon J \to {\widehat {B}}$
. As before, we identify J with its dual via the canonical principal polarization, so that we may write
$\phi : A \to J$
and
$\psi \colon B \to J$
for the dual isogenies. We also define
$A_{D_i}$
to be the dual of
$J/\langle D_i\rangle $
for
$i = 0,1,2$
with isogenies
$\psi _i \colon A_{D_i} \to J$
. We have
$B \simeq A_{D_2}$
since
$D = -D_2$
.
Let
$H = \langle D_0, D_1\rangle $
, and define the map
$$ \begin{align} \begin{aligned} \partial^H : &J(\mathbb{Q})/\phi\bigl( A(\mathbb{Q}) \bigr) \longrightarrow {\mathbb{Q}}^*/\mathbb{Q}^{*p} \times {\mathbb{Q}}^*/\mathbb{Q}^{*p},\\ &\left[ \sum_{j=1}^g (x_j,y_j) - g \cdot \infty \right] \mapsto \Bigl( \prod_{j=1}^g x_j,\ \prod_{j=1}^g (x_j - e_1) \Bigr) \end{aligned} \end{align} $$
as in Lemmas 3.5 and 3.7 of Section 2.3. In the above definition, each
$x_j,y_j \in \overline {\mathbb {Q}}$
, the divisor
$\sum _{j=1}^g (x_j,y_j) - g \cdot \infty $
is Galois stable, and the left-hand side is its divisor class. That such representatives exist follows from the fact that
$C(\mathbb {Q}) \neq \emptyset $
[Reference Schaefer34, Prop. 2.7]. The above description of
$\partial ^H$
applies whenever it makes sense – that is, when all
$x_j$
and
$x_j - e_1$
are nonzero. Every class in
$J(\mathbb {Q})/\phi \bigl ( A(\mathbb {Q}) \bigr )$
can be represented by such a divisor [Reference Lang21, pg.166].
For
$i \in \{1,2,3\}$
, we have similar homomorphisms
$$ \begin{align} \begin{aligned} \partial^{D_i} :&J(\mathbb{Q})/\psi_i\bigl( A_{D_i}(\mathbb{Q}) \bigr) \longrightarrow {\mathbb{Q}}^*/\mathbb{Q}^{*p},\\ &\left[ \sum_{j=1}^g (x_j,y_j) - g \cdot \infty \right] \mapsto \prod_{j=1}^g \Bigl( x_j - e_i \Bigr), \end{aligned} \end{align} $$
as well as the homomorphism
$$ \begin{align} \begin{aligned} \partial^{D} :&J(\mathbb{Q})/\psi\bigl( B(\mathbb{Q}) \bigr) \longrightarrow {\mathbb{Q}}^*/\mathbb{Q}^{*p},\\ &\left[ \sum_{j=1}^g (x_j,y_j) - g \cdot \infty \right] \mapsto \prod_{j=1}^g \Bigl( x_j(x_j - e_1) \Bigr). \end{aligned} \end{align} $$
As before, the description of these maps is for representative divisors for which it makes sense. However, note that by the equation for the curve, we have
$\partial ^{D_0} \cdot \partial ^{D_1} \cdot \partial ^{D_2} = 1$
. This allows us to describe the maps
$\partial ^{D_i}$
even on points where the formula above is not well defined. For example,
Lemma 4.1. We have
$$\begin{align*}\partial^H(\left[(0,0)-\infty\right]) = [ e_1^{-1}e_2^{-1}, -e_1 ]\end{align*}$$
and
$$\begin{align*}\partial^H( \left[(e_1,0)-\infty\right]) = [e_1, e_1^{-1}(e_1 - e_2)^{-1} ].\end{align*}$$
In the next section, we will make critical use of the following commutative diagram
$$ \begin{align} \begin{array}{ccc} J(\mathbb{Q})/\phi\bigl( A(\mathbb{Q}) \bigr) &\overset{\partial^H}{\longrightarrow}&{\mathbb{Q}}^*/\mathbb{Q}^{*p}\times {\mathbb{Q}}^*/\mathbb{Q}^{*p}\\ \downarrow & &\downarrow \\ J(\mathbb{Q})/\psi\bigl(B(\mathbb{Q})\bigr)&\overset{\partial^D} {\longrightarrow}&{\mathbb{Q}}^*/\mathbb{Q}^{*p},\\ \end{array} \end{align} $$
whose right vertical map is
$[r_1,r_2] \mapsto r_1r_2$
.
4.2 Models
There are simple birational models for the
$\mu _p$
-covers of J with a given Steinitz class. For concreteness, assume that the Steinitz class is
$D = D_0 + D_1$
. The distinguished
$\mu _p$
-cover with this Steinitz class is the cover
$B \to J$
. A birational model for J is given by the equations
$$\begin{align*}y_i^p = x_i(x_i - e_1)(x_i - e_2)\end{align*}$$
for
$i = 1,\ldots , g$
, modulo the action of
$S_g$
. By Lemmas 3.5, 3.6 and 3.7, a birational model for B is given by the same g equations above along with the additional equation
$$\begin{align*}z^p = \prod_{i = 1}^g x_i(x_i - e_1),\end{align*}$$
all modulo the action of
$S_g$
. Similarly, if
$r \in \mathbb {Q}^\times $
, then the
$\mu _p$
-cover corresponding to
$(\mathcal {L}, r\eta )$
is given by the same equations, with the last one twisted by r:
$$\begin{align*}y_i^p = x_i(x_i - e_1)(x_i - e_2)\end{align*}$$
for
$i = 1,\ldots , g$
and
$$\begin{align*}rz^p = \prod_{i = 1}^g x_i(x_i - e_1), \end{align*}$$
again modulo the action of
$S_g$
.
5 Arbitrarily large p-torsion part of the Tate-Shafarevich group
We wish to produce examples of arbitrarily large p-torsion subgroups in the Tate-Shafarevich group 
, by finding elements of
$\mathrm {Sel}(B)$
which can be shown to violate the Hasse principle by using
$\mathrm {Sel}(A)$
. We will choose fairly generic curves of the form
$y^p = x(x-e_1)(x-e_2)$
, but then we will twist them in a carefully chosen way to produce our examples.
Let
$$\begin{align*}C = C_{u,v} : y^p = x(x-3u)(x-9v), \end{align*}$$
where
$p> 5$
is prime and where
$u,v \in \mathbb {Z}$
are not divisible by
$3$
. Let J be the Jacobian of C and let A be the isogenous abelian variety, as in the previous section.
Lemma 5.1. Suppose q is a prime such that
$q \equiv 1\ \pmod p$
. Then
$J(\mathbb {Q}_q)/\phi \bigl (A(\mathbb {Q}_q)\bigr )$
has order
$p^2$
.
Proof. The congruence condition on q implies
$\mathbb {Q}_q^*$
contains a primitive p-th root of unity
$\zeta $
. Over any field containing
$\zeta $
, the automorphism
$(x,y) \mapsto (x, \zeta y)$
of C induces a ring embedding
$\iota \colon \mathbb {Z}[\zeta ] \hookrightarrow \mathrm {End}(J)$
. The degree of
$\iota (\alpha )$
is equal to
$\mathrm {Nm}(\alpha )^2 = \#(\mathbb {Z}[\zeta ]/\alpha )^2$
. Indeed, the degree function restricted to
$\mathbb {Z}[\zeta ]$
is a power of the norm [Reference Mumford29, §19], and we have
$\deg ([n]) = n^{2g} = n^{2[\mathbb {Q}(\zeta ) \colon \mathbb {Q}]}$
, so it is the square of the norm in this case. The kernel of
${\widehat \phi } \colon J \to {\widehat {A}}$
is then equal to the kernel of the endomorphism
$1 - \iota (\zeta )$
. It follows that
${\widehat \phi }$
agrees with
$1 - \iota (\zeta )$
up to post-composition with an automorphism (since they have the same degree and one factors through the other); hence, the abelian varieties A and
$\widehat {A}$
are isomorphic to J (over any field containing
$\zeta $
, and in particular over
$\mathbb {Q}_{q}$
). However, we have [Reference Shnidman36, Cor. 3.2]
$$\begin{align*}\dfrac{\#J( \mathbb{Q}_{q} )/ \phi\bigl(A(\mathbb{Q}_{q})\bigr)}{\#A( \mathbb{Q}_{q} )[\phi]} = c_{q}(J)/c_{q}(A),\end{align*}$$
where the right-hand side is the ratio of Tamagawa numbers over
$\mathbb {Q}_{q}$
. Since
$J \simeq A$
over
$\mathbb {Q}_{q}$
, this ratio is 1. We also have
$\#A(\mathbb {Q}_{q})[\phi ] = p^2$
, which shows that
$\#J( \mathbb {Q}_{q} )/ \phi \bigl (A(\mathbb {Q}_{q})\bigr ) = p^2$
.
For each integer k, we will consider the curve
$$\begin{align*}C_k = C_{u,v,k} \colon y^p = x(x-3uk)(x-9vk).\end{align*}$$
Another model for
$C_k$
is
$k^{-3} y^p = x(x-3u)(x-9v)$
, which shows that
$C_k$
is a
$\mu _p$
-twist of the original curve
$C = C_{u,v}$
. Let
$J_k$
,
$A_k$
and
$B_k$
be the corresponding abelian varieties for the curve
$C_k$
. These are
$\mu _p$
-twists of
$J,A$
and B, respectively.
For two primes q and
$\ell $
, set
$\left (\frac {q}{\ell }\right )_p = 1$
if and only if q is a p-th power in
$\mathbb {Q}_{\ell }^\times $
. Recall the exact sequence
from Proposition 3.9.
Proposition 5.2. Let U be the set of primes that divide
$3puv(u-3v)$
. Suppose k is a product of distinct primes
$k = p_1 p_2 \cdot \ldots \cdot p_t$
, where
$t\geq 2$
and each prime
$p_i$
is not in U, and satisfies
-
(1)
$\left (\frac {p_i}{p_j}\right )_p = 1$
, for all
$i \neq j$
in
$\{1,\ldots , t\}$
, -
(2)
$\left (\frac {p_i}{q}\right )_p = 1$
, for all
$i \in \{1, \ldots , t\}$
and all
$q \in U$
, -
(3)
$\left (\frac {q}{p_i}\right )_p = 1$
, for all
$i \in \{1, \ldots , t\}$
and all
$q \in U\backslash \{3\}$
, -
(4)
$\left (\frac {3}{p_i}\right )_p \neq 1$
, for all
$i \in \{1, \ldots , t\}$
.
Then, for all i, we have
$p_i \in \mathrm {Sel}(B_k)$
but
$p_i \notin \partial ^D(J_k(\mathbb {Q}))$
. More generally, if
$q = \prod _{i \in I} p_i^{a_i}$
, where I is any nonempty proper subset of
$ \{1,\ldots , t\}$
and
$1 \leq a_i \leq p-1$
, then
$q \in \mathrm {Sel}(B_k)$
but
$q \notin \partial ^D(J_k(\mathbb {Q}))$
.
Note that condition
$(4)$
implies that
$p_i \equiv 1\ \pmod {p}$
for all i.
Proof. By Lemma 4.1, we have
$$\begin{align*}\partial^H([(0,0)-\infty]) = [ 3^{-3} u^{-1} v^{-1} k^{-2}, -3 u k ]\end{align*}$$
and
$$\begin{align*}\partial^H( [(3uk,0)-\infty]) = [ 3uk, 3^{-2}u^{-1}(u-3v)^{-1}k^{-2} ].\end{align*}$$
For
$i = 1,\ldots , t$
, the images of these two elements in
$\mathbb {Q}_{p_i}^*/(\mathbb {Q}_{p_i}^*)^p \times \mathbb {Q}_{p_i}^*/(\mathbb {Q}_{p_i}^*)^p$
are
$[3^{-3}p_i^{-2},3p_i]$
and
$[3p_i,3^{-2}p_i^{-2}]$
, respectively, by our assumptions on the
$p_i$
. These two elements are linearly independent, and hence generate all of
$\partial ^H\bigl ( J_k(\mathbb {Q}_{p_i}) /\phi (A_k(\mathbb {Q}_{p_i})) \bigr )$
, by Lemma 5.1.
Let
$[r_1,r_2]$
be in
$\partial ^H\bigl ( J_k(\mathbb {Q})/\phi (A_k(\mathbb {Q})) \bigr )$
. We shall consider elements of
$\mathbb {Q}^*/\mathbb {Q}^{*p}$
as pth-power-free integers. Note that by Proposition 3.10, the integers
$r_1$
and
$r_2$
can only be divisible by primes in the set
$\{p_1, \ldots , p_t\} \cup U$
.
If there is no i such that
$\mathrm {ord}_{p_i}(r_1) \equiv \mathrm {ord}_{p_i}(1/r_2)\ \pmod p$
, then
$\mathrm {ord}_{p_i}(r_1r_2) \not \equiv 0\ \pmod {p}$
for every
$p_i$
, and so
$r_1 r_2$
cannot be of the form
$\prod _{i \in I} p_i^{a_i}$
, for a nonempty proper subset of indices I, and any tuple of exponents
$a_i$
with
$1 \leq a_i \leq p-1$
.
Suppose there exists i such that
$\mathrm {ord}_{p_i}(r_1) \equiv \mathrm {ord}_{p_i}(1/r_2)\ \pmod p$
. Since
$[3^{-3}p_i^{-2},3p_i]$
and
$[3p_i,3^{-2}p_i^{-2}]$
generate all of
$\partial ^H\bigl ( J_k(\mathbb {Q}_{p_i})/ \phi (A_k(\mathbb {Q}_{p_i})) \bigr )$
, considering only the
$p_i$
-adic valuation, we have
$$\begin{align*}[r_1,r_2] \equiv [p_i^{-2a},p_i^a] \cdot [p_i^b,p_i^{-2b}] \equiv [p_i^{b-2a}, p_i^{a-2b}]\end{align*}$$
for integers a and b. So
$$\begin{align*}r_1r_2 \equiv p_i^{b-2a + a - 2b} = p_i^{-b-a}\end{align*}$$
and we must have
$b \equiv -a$
(mod
$p_i$
). In other words, the image of
$[r_1,r_2]$
in
$\mathbb {Q}_{p_i}^*/(\mathbb {Q}_{p_i}^*)^p \times \mathbb {Q}_{p_i}^*/(\mathbb {Q}_{p_i}^*)^p$
must be a power of the quotient of these generators. Thus, in
$\mathbb {Q}_{p_i}^*/(\mathbb {Q}_{p_i}^*)^p \times \mathbb {Q}_{p_i}^*/(\mathbb {Q}_{p_i}^*)^p$
, we have
$$\begin{align*}[r_1,r_2] = [ 3^{-4} p_i^{-3}, 3^3 p_i^3 ]^m = [3^{-4m} p_i^{-3m},3^{3m} p_i^{3m}],\end{align*}$$
for some
$0 \leq m < p$
. Thus, in
$\mathbb {Q}_{p_i}^*/(\mathbb {Q}_{p_i}^*)^p$
, we have
$$\begin{align*}r_1^3r_2^4 = 3^{-12m + 12m} p_i^{-9m+12m} =p_i^{3m}.\end{align*}$$
Since
$3$
is not a p-th power modulo
$p_i$
, this implies that
$3$
divides
$r_1^3$
and
$1/r_2^4$
to the same power. Since, for any j, the elements
$[ 3^{-3} p_j^{-2}, 3 p_j ]$
and
$[ 3p_j, 3^{-2}p_j^{-2} ]$
generate all of
$\partial ^H(( J_k(\mathbb {Q}_{p_j})/\phi (A_k(\mathbb {Q}_{p_j}))))$
, and since we have already shown that
$3$
divides
$r_1^3$
and
$1/r_2^4$
to the same power, we deduce that the image of
$[r_1,r_2]$
in
$\mathbb {Q}_{p_j}^*/(\mathbb {Q}_{p_j}^*)^p \times \mathbb {Q}_{p_j}^*/(\mathbb {Q}_{p_j}^*)^p$
must be a power of the quotient of these generators, and so
$p_j$
must divide
$r_1$
and
$1/r_2$
to the same power. This applies for all j; hence, no
$p_j$
will appear in
$r_1 r_2$
. So again we have that
$r_1 r_2$
cannot be of the form
$\prod _{i \in I} p_i^{a_i}$
with
$0 < \#I < t$
.
Since
$\partial ^H \bigl ( J_k(\mathbb {Q})/\phi (A_k(\mathbb {Q})) \bigr )$
maps surjectively onto
$\partial ^D \bigl ( J_k(\mathbb {Q})/\psi (B_k(\mathbb {Q})) \bigr )$
, it follows that each element of the form
$\prod _{i \in I} p_i^{a_i}$
with
$0 < \#I < t$
and
$0 < a_i < p$
is not in
$\partial ^D \bigl ( J_k(\mathbb {Q})/\psi (B_k(\mathbb {Q})) \bigr )$
.
However, we show that each
$p_i$
is in
$\partial ^D ( J_k(\mathbb {Q}_{\ell })/\psi (B_k(\mathbb {Q}_{\ell })) )$
for every prime
$\ell $
, as follows. First, note that for every prime
$\ell \in U \cup \{p_1, \ldots , p_t\}$
except
$p_i$
itself,
$p_i$
is a pth power in
$\mathbb {Q}_{\ell }^*$
, and so
$p_i$
is in
$\partial ^D\bigl ( J_k(\mathbb {Q}_{\ell })/\psi (B_k(\mathbb {Q}_{\ell })) \bigr )$
by virtue of being
$\partial ^D$
of the identity. If
$\ell \notin U \cup \{p_1, \ldots , p_t\}$
, we have
$p_i \in \partial ^D ( J_k(\mathbb {Q}_{\ell })/\psi (B_k(\mathbb {Q}_{\ell })) )$
, by Proposition 3.10. For the remaining case
$\ell = p_i$
, note that
$\partial ^D( [(0,0)-\infty ] )= [ -3^{-2} v^{-1} k^{-1} ]$
and
$\partial ^D([(3uk,0)-\infty ]) = [ 3^{-1} (u-3v)^{-1} k^{-1} ]$
, and so the first divided by the square of the second gives
$p_i$
, since all other factors are pth powers in
$\mathbb {Q}_{p_i}$
. Hence,
$p_i$
is in the image everywhere locally.
As a corollary, we deduce the first theorem from the introduction.
Proof of Theorem 1.1.
The Theorem follows from Lemma 3.3 and Proposition 5.2, together with the discussion of models in Section 4.2.
As a second corollary, we have the following:
Corollary 5.3. Let
$C_{u,v,k}$
be as in Proposition 5.2. Then 
.
Proof. Let
$B_k = B_{u,v,k}$
. We have the exact sequence
Moreover, we have seen that any element of the form
$\prod _{i \in I} p_i^{a_i}$
, for some proper subset
$I \subset \{1,\ldots , t\}$
is contained in
$\mathrm {Sel}(B_k)$
but does not lie in the subgroup
$J_k(\mathbb {Q})/\psi (B_k(\mathbb {Q})$
. Thus, these elements must map nontrivially to 
. In fact, we see that the intersection of
$J_k(\mathbb {Q})/\psi (B_k(\mathbb {Q}))$
with the subgroup of
$\mathrm {Sel}(B_k)$
generated by the elements
$p_1, \ldots , p_t$
is at most one dimensional as an
$\mathbb {F}_p$
-vector space. Indeed, any two linearly independent elements in this subgroup can be scaled so that they are divisible exactly once by
$p_1$
, and hence, their ratio is nonzero and not divisible by
$p_1$
, which would be a contradiction. It follows that the image of the subgroup
$\langle p_1, \ldots , p_t\rangle $
in 
has dimension at least
$t - 1$
. Since
$\deg (\psi ) = p$
, we have 
, which finishes the proof.
6 Proof of Theorem 1.5
To deduce Theorem 1.5 from the results of the previous section, we need two extra ingredients.
Proposition 6.1. For any
$u,v$
as above, and for any
$t \geq 0$
, there are primes
$p_1,\dots , p_t$
satisfying the conditions of Proposition 5.2.
Proof. Let
$K = \mathbb {Q}(\zeta ) = \mathbb {Q}(\mu _p)$
. We prove this by induction on t. If
$t = 0$
, then there is nothing to prove.
Now let
$t> 0$
and suppose we have found primes
$p_1,\cdots , p_{t-1}$
satisfying the conditions. Let
$k = p_1 \cdots p_{t-1}$
. Let N be the product of the primes dividing
$puv(u-3v)k$
, and let
$\zeta _{pN}$
be a primitive
$pN$
-th root of unity. Let L be the compositum inside
$\bar {\mathbb {Q}}$
of
$\mathbb {Q}(\zeta _{pN})$
with all of the fields
$\mathbb {Q}(\sqrt [p]{q})$
, with q a prime dividing N. Then L is an abelian extension of K and a Galois extension of
$\mathbb {Q}$
. Finally, let
$E = \mathbb {Q}(\sqrt [p]{3})$
and let
$F =EL$
be the compositum of E and L, which is again a Galois extension of
$\mathbb {Q}$
.
Note that the fields E and L are linearly disjoint over
$\mathbb {Q}$
. Indeed,
$E/\mathbb {Q}$
is totally ramified at
$3$
, whereas L is unramified at
$3$
. Thus, we have an exact sequence
$$\begin{align*}0 \to (\mathbb{Z}/p\mathbb{Z}) \to \mathrm{Gal}(F/\mathbb{Q}) \to \mathrm{Gal}(L/\mathbb{Q}) \to 0.\end{align*}$$
By the Cebotarev density theorem, there exists a prime
$p_t$
(in fact, infinitely many such primes) whose Frobenius conjugacy class in
$\mathrm {Gal}(F/\mathbb {Q})$
is not trivial but restricts to the trivial class in
$\mathrm {Gal}(L/\mathbb {Q})$
. Let us check that
$p_t$
satisfies all the desired properties.
By construction,
$p_t$
splits completely in any subfield of L. In particular,
$p_t$
splits completely in
$\mathbb {Q}(\zeta _{pN})$
and hence is a p-th power in
$\mathbb {Q}_q^*$
for any prime q dividing
$3pN$
. For
$q \nmid 3p$
, this is because
$p_t \equiv 1\ \pmod q$
, and hence,
$p_t$
is a p-th power in
$\mathbb {Q}_q^*$
by Hensel’s lemma. For
$q = p$
, this is because
$p^2 \mid pN$
, and hence
$p_t \equiv 1\ \pmod {p^2}$
, and hence is a p-th power, again by Hensel’s lemma. For
$q = 3$
, this is because every unit in
$\mathbb {Z}_3$
is a p-th power. Similarly,
$p_t$
splits completely in
$\mathbb {Q}(\sqrt [p]{q})$
for all
$q \mid N$
, so all such primes q are p-th powers modulo
$p_t$
.
Finally, we check that 3 is not a p-th power in
$\mathbb {Q}_{p_t}^*$
. It is enough to show that the prime
$p_t$
does not have a degree 1 prime above it in E. If it did, then because
$\zeta \in \mathbb {Q}_{p_t}^*$
, once the polynomial
$x^p - 3$
has one root in
$\mathbb {Q}_{p_t}^*$
, it necessarily has all of its roots in
$\mathbb {Q}_{p_t}^*$
. Therefore,
$p_t$
would split completely in E. Since
$p_t$
splits completely in L, this would mean that
$p_t$
splits completely in
$F = EL$
. But by construction, the Frobenius at
$p_t$
is nontrivial, so
$p_t$
does not split completely.
It remains to show that, for each prime p, there exist examples for which
$J_{u,v,k}$
is absolutely simple, and hence
$B_{u,v,k}$
as well.
Lemma 6.2. For each
$p> 5$
, there exist
$u,v \in \mathbb {Z}$
as in Proposition 5.2 such that
$B_{u,v,k}$
is absolutely simple for all k.
Proof. The Jacobian of the curve
$y^p = x(x-1)(x-t)$
over
$\mathbb {Q}(t)$
is absolutely simple since there is a value of
$t\in \mathbb {C}$
– namely,
$t = \frac {1}{2} + \frac {\sqrt {3}}{2} i$
– which makes the curve isomorphic over
$\mathbb {C}$
to
$y^p = x^3 - 1$
, and the Jacobian of the latter is known to be absolutely simple [Reference Hazama16, Reference Jedrzejak17]. By a result of Masser [Reference Masser25], for an abelian variety over
$\mathbb {Q}(t)$
, the geometric endomorphism ring for 100% of specializations of
$t \in \mathbb {Q}$
(ordered by height) is the same as the generic geometric endomorphism ring. Since the generic abelian variety is geometrically simple, this endomorphism ring is a division ring, and hence,
$100\%$
of specializations are simple as well. But for
$t = a/b \in \mathbb {Q}$
, a positive proportion has
$3$
exactly dividing a and
$3$
not dividing b. So, there are many curves
$y^p = x(x-1)(x-3 v/u)$
with
$u,v \in \mathbb {Z}$
, not divisible by
$3$
, with absolutely simple Jacobian. This is a twist of the curve
$\mathcal {C}_{u,v,1} : y^p = x(x-3u)(x-3^2v)$
, so there are curves of this form with absolutely simple Jacobian, giving that
$B_{u,v,1}$
is also absolutely simple. Since each
$B_{u,v,k}$
is a twist of
$B_{u,v,1}$
, it follows that
$B_{u,v,k}$
is absolutely simple for all k.
Appendix: The Cassels-Tate pairing for p-coverings of Jacobians
by Tom Fisher
The purpose of this appendix is to interpret the proof of Proposition 5.2 in terms of a certain Cassels-Tate pairing. Let
$J/\mathbb {Q}$
be a Jacobian, and identify
$J = \widehat {J}$
in the usual way.
Let
$p> 5$
be a prime. Suppose that
$J(\mathbb {Q})$
contains subgroups
$\mathbb {Z}/p\mathbb {Z}$
and
$(\mathbb {Z}/p\mathbb {Z})^2$
that we take to be the kernels of isogenies
$\widehat {\psi } : J \to \widehat {B}$
and
$\widehat {\phi } : J \to \widehat {A}$
. We further suppose that
$\ker \widehat {\psi } \subset \ker \widehat {\phi }$
, so that
$\widehat {\phi }$
factors via
$\widehat {\psi }$
to give a commutative diagram
There is then a commutative diagram
where
$\iota (s) = (s,s^{-1})$
and
$\pi (r_1,r_2) = r_1 r_2$
. We now give the Weil pairing definition of the Cassels-Tate pairing (see [Reference Milne28, Chapter 1, Proposition 6.9], [Reference Poonen and Stoll31, Section 12.2] or [Reference Fisher13]) simplified by the fact that
$\pi $
has an obvious section given by
$r \mapsto (r,1)$
. The Cassels-Tate pairing
$$\begin{align*}\langle~,~\rangle_{\operatorname{CT}} : S^{(\psi)}(B/\mathbb{Q}) \times S^{(\widehat{\nu})}(\widehat{ B}/\mathbb{Q}) \to \mathbb{Q}/\mathbb{Z} \end{align*}$$
is defined as follows. We start with
$$\begin{align*}r \in S^{(\psi)}(B/\mathbb{Q}) \subset \mathbb{Q}^*/(\mathbb{Q}^*)^p \quad \text{ and } \quad s \in S^{(\widehat{\nu})} (\widehat{B}/\mathbb{Q}) \subset H^1(\mathbb{Q},\mathbb{Z}/p\mathbb{Z}). \end{align*}$$
For each prime
$\ell $
, we pick
$P_{\ell } \in J(\mathbb {Q}_{\ell })$
with
$\delta _{\psi ,\ell }(P_{\ell }) \equiv r\ \mod {(\mathbb {Q}_{\ell }^*)^p}$
. Then
$\delta _{\phi ,\ell }(P_{\ell }) = (r \xi _{\ell },\xi _{\ell }^{-1})$
for some
$\xi _{\ell } \in \mathbb {Q}_{\ell }^*/(\mathbb {Q}_{\ell }^*)^p$
. We define
$$\begin{align*}\langle r, s \rangle_{\operatorname{CT}} = \sum_{\ell} (\xi_{\ell},\operatorname{res}_{\ell} s)_{\ell}, \end{align*}$$
where
$$ \begin{align} (~,~)_{\ell} : H^1(\mathbb{Q}_{\ell},\mu_p) \times H^1(\mathbb{Q}_{\ell},\mathbb{Z}/p\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z} \end{align} $$
is the local pairing given by cup product and the local invariant map. Since the sum of the local invariants of an element in
$H^2(\mathbb {Q}, \mu _p)$
is 0, we have
$\langle r,s\rangle _{\operatorname {CT}}= 0$
for all
$r \in \delta _\psi (J(\mathbb {Q}))$
.
Proposition A. With the notation and assumptions of Proposition 5.2 (noting that
$\delta _\phi $
,
$\delta _\psi $
and
$S^{(\psi )}(B/\mathbb {Q})$
are there called
$\partial ^H$
,
$\partial ^D$
and
${\operatorname {Sel}}(B_k)$
), we have
-
(i)
$p_i \in S^{(\psi )}(B/\mathbb {Q}) \subset \mathbb {Q}^*/(\mathbb {Q}^*)^p$
for all
$1 \leqslant i \leqslant t$
. -
(ii) Let
$\chi _i \in H^1(\mathbb {Q},\mathbb {Z}/p\mathbb {Z}) = \operatorname {Hom}(\operatorname {Gal}({\overline {\mathbb Q}}/\mathbb {Q}),\mathbb {Z}/p\mathbb {Z})$
be the unique continuous character that factors via
$\operatorname {Gal}(\mathbb {Q}(\zeta _{p_i})/\mathbb {Q})$
and satisfies
$\chi _i(\operatorname {Frob}_3) = 1$
(this is possible by assumption (4)). Then for all
$$\begin{align*}\chi_i - \chi_j \in S^{(\widehat{\nu})}(\widehat{B}/\mathbb{Q}) \subset H^1(\mathbb{Q},\mathbb{Z}/p\mathbb{Z}) \end{align*}$$
$1 \leqslant i,j, \leqslant t$
.
-
(iii) For
$a_1,\ldots ,a_t,b_1, \ldots , b_t \in \{0,1, \ldots ,p-1\}$
with
$\sum b_j \equiv 0\ \pmod {p}$
, we have In particular, since
$$\begin{align*}\left\langle \prod_{i=1}^t p_i^{a_i} , \sum_{j=1}^t b_j \chi_j \right\rangle_{\operatorname{CT}} = \frac{-5}{p} \sum_{i=1}^t a_i b_i. \end{align*}$$
$p> 5$
, if
$q = \prod _{i=1}^t p_i^{a_i}$
, and the
$a_i \in \{0,1, \ldots ,p-1\}$
are not all equal, then
$q \not \in \delta _\psi (J(\mathbb {Q}))$
.
Proof. (i) See the final paragraph of the proof of Proposition 5.2.
(ii) The restriction
$\operatorname {res}_{\ell }(\chi _i)$
is unramified for all
$\ell \not = p_i$
, and trivial for all
$\ell \in U \setminus \{3\}$
by assumption (3). So we only need to check the local conditions at
$p_1, \ldots , p_t$
and
$3$
. By Lemmas 4.1 and 5.1, we know that
$\operatorname {im} \delta _{\phi ,p_i}$
has order
$p^2$
, and the natural map
$\operatorname {im} \delta _{\phi ,p_i} \to \operatorname {im} \delta _{\psi ,p_i}$
is an isomorphism. Chasing around (the local analogue of) the above diagram shows that
$\operatorname {im} \delta _{\nu ,p_i}$
is trivial. It follows by Tate local duality that
$\operatorname {im} \delta _{\widehat {\nu },p_i}$
is all of
$H^1(\mathbb {Q}_{p_i},\mathbb {Z}/p\mathbb {Z})$
. In other words, in the definition of
$S^{(\widehat {\nu })} (\widehat {B}/\mathbb {Q})$
, there are no local conditions at
$p_1, \ldots , p_t$
. It is the local condition at
$3$
that forces us to take differences.
(iii) We take
$r = p_i$
in the above description of the pairing. The only primes
$\ell $
that can contribute to the pairing are those in
$\{p_1, \ldots ,p_t\} \cup U$
. For all such primes
$\ell \not = p_i$
we have that r is trivial in
$\mathbb {Q}_{\ell }^\times /(\mathbb {Q}_{\ell }^\times )^p$
by assumptions (1) and (2). Taking
$P_{\ell } = 0$
and
$\xi _{\ell } = 1$
, we see that these primes make no contribution to the pairing. It remains to compute the contribution at
$\ell = p_i$
. By Lemma 4.1, we have
$$\begin{align*}\delta_{\phi,\ell}(P_{\ell}) = ((3^{-3} p_i^{-2})^a(3p_i)^b,(3p_i)^a (3^{-2}p_i^{-2})^b) \end{align*}$$
for some
$a,b \in \mathbb {Z}/p\mathbb {Z}$
. We know that
$(r_1,r_2) \mapsto r_1 r_2$
maps this to
$r = p_i \in \mathbb {Q}_{p_i}^*/(\mathbb {Q}_{p_i}^*)^p$
, and that
$\mathbb {Q}_{p_i}^*/(\mathbb {Q}_{p_i}^*)^p$
is generated by
$3$
and
$p_i$
. Therefore,
$-2a-b \equiv 0\ \pmod {p}$
and
$-a-b \equiv 1\ \pmod {p}$
. We solve these to give
$a = 1$
and
$b=-2$
. Therefore,
$$\begin{align*}\delta_{\phi,\ell}(P_{\ell}) = ( 3^{-5} p_i^{-4}, 3^{5}p_i^{5} ). \end{align*}$$
and
$\xi _{\ell } = 3^{-5} p_i^{-5}$
. Therefore,
$$\begin{align*}\langle p_i, \sum b_j \chi_j \rangle_{\operatorname{CT}} = \sum b_j ((3p_i)^{-5}, \operatorname{res}_{p_i} \chi_j)_{p_i} = \frac{-5 b_i}{p}. \end{align*}$$
Notice that, since
$p_i \equiv 1\ \pmod {p}$
, evaluating the local pairing (6.1) reduces to a computation of Hilbert symbols. The formula in the statement of the proposition follows by linearity in the first argument.
Acknowledgements
The authors thank Michael Stoll for his comments and for organizing Rational Points 2022, where they began working together on this problem. The second author was supported by the Israel Science Foundation (grant No. 2301/20). The authors also thank Ariyan Javanpeykar, Jef Laga and Ariel Weiss for comments on an earlier draft. Finally, we thank Tom Fisher for his suggestions and for letting us include his appendix.
Competing interest
The authors have no competing interest to declare.
. 
for abelian varieties. 
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