Note on linear relations in Galois cohomology and {\'e}tale $K$-theory of curves

TL;DR

This paper explores the local to global principle in Galois cohomology and étale K-theory of curves, establishing conditions for its validity, providing counterexamples, and extending results to dynamical and Quillen K-theories.

Contribution

It identifies optimal conditions for the local to global principle in étale K-theory of curves and extends the results to dynamical and Quillen K-theories under certain conjectures.

Findings

Conditions for the validity of the local to global principle are optimal.

Counterexamples are provided when conditions do not hold.

Results extend to Quillen K-theory assuming certain conjectures.

Abstract

In this paper we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In \cite{bk13} G. Banaszak and the author obtained the sufficient condition for the validity of the local to global principle for {\'e}tale $K$-theory of a curve . This condition in fact has been established by means of an analysis of the corresponding problem in the Galois cohomology. We show that in some cases this result is the best possible i.e if this condition does not hold we obtain counterexamples. We also give some examples of curves and their Jacobians. Finally, we prove the dynamical version of the local to global principle for {\'e}tale $K$-theory of a curve. The dynamical local to global principle for the groups of Mordell-Weil type has recently been considered by S. Bara{\'n}czuk in \cite{b17}. We show that all our results remain valid for Quillen $K$-theory of ${\cal X}$ if the Bass and Quillen-Lichtenbaum conjectures hold true for ${\cal X}.$

Equations

$$X:y^2+(x^3+x^2+x+1)y=-x^2-x.$$

$$E_2^{p,-q}=H_{cont}^p(Y,{\mathbb{Z}}_l(q/2))\Rightarrow K_{q-p}^{et}(Y),$$

$$E_2^{p,-q}=H_{et}^p(Y,\mathbb{Z}/l^k(q/2))\Rightarrow K_{q-p}^{et}(Y,\mathbb{Z}/l^k),$$

$$0\to H^2(G_{S_l};{\mathbb{Z}}_l(k))\to H_{cts}^2(\mathcal{X};{\mathbb{Z}}_l(k))\to H^1(G_{S_l};T_l(J)(k-1))\to 0,$$

$$K_{2n}^{et}(\mathcal{X})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$h_{l^k}:K_m(\mathcal{X},\mathbb{Z}/l^k\mathbb{Z})\to K_m^{et}(\mathcal{X},\mathbb{Z}/l^k\mathbb{Z}),$$

$$\overline{h_{l^k}}:K_m({\mathcal{X}}_v,\mathbb{Z}/l^k\mathbb{Z})\to K_m^{et}({\mathcal{X}}_v,\mathbb{Z}/l^k\mathbb{Z}).$$

$$\underleftarrow{\lim }{K}_m(\mathcal{X},\mathbb{Z}/l^k\mathbb{Z})=K_m(\mathcal{X})\otimes {\mathbb{Z}}_l.$$

$$h:K_{2n}(\mathcal{X})\otimes {\mathbb{Z}}_l\to K_{2n}^{et}(\mathcal{X})$$

$$\overline{h}:K_{2n}({\mathcal{X}}_v)\otimes {\mathbb{Z}}_l\to K_{2n}^{et}({\mathcal{X}}_v)$$

$$K_{2n}(\mathcal{X})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H_{f,S_l}^1(G_L;{\tilde{T}}_{l,i})=ker(H^1(G_L;{\tilde{T}}_{l,i})\to \prod _{w\notin S_l}{H}^1(G_w;{\tilde{T}}_{l,i})/H_f^1(G_w;{\tilde{T}}_{l,i})).$$

$$J_{f,S_l}({\tilde{T}}_{l,i})=\underset{\genfrac{}{}{0ex}{}{⟶}{L/F}}{\lim }{H}_{f,S_l}^1(G_L;{\tilde{T}}_{l,i}).$$

$$J_{f,S_l}({\tilde{T}}_{l,i}{)}_l\cong {\tilde{V}}_{l,i}/{\tilde{T}}_{l,i}\cong V_l(A_i)/T_l(A_i)(n)\cong A_i[l^{\infty }](n).$$

$$J_{f,S_l}({\tilde{T}}_l)\cong \underset{i=1}{\overset{t}{⨁}}{J}_{f,S_l}({\tilde{T}}_{l,i}).$$

$$i^{\ast }:H^1(G_{F,S_l},T_l(J)(n))\to H^1(G_{L,S_l},T_l(J)(n))$$

$$H^0(G_{F,S_l},V_l(J)/T_l(J)(n))\to H^0(G_{L,S_l},V_l(J)/T_l(J)(n)),$$

$$H^1(g_v,T_l(J_v)(n))\to H^1(g_w,T_l(J_v)(n))$$

$$H^1(G_{F,S_l};T_l(J)(n))\cong H^1(G_{F,S_l};T_l(A)(n))=\prod _{i=1}^t{H}^1(G_{F,S_l};T_l(A_i)(n){)}^{e_i}.$$

$$r_w:H_{f,S_l}^1(G_L;T_l{)}_{tor}\to H^1(g_w;T_l)$$

$${\psi }^{\alpha }:A(F)/\alpha A(F)\to {\mathrm{Hom}}_{G_{\alpha }}(􏰂G_{F_{\alpha }},A[\alpha ])$$

$$A(F)/\alpha A(F)\hookrightarrow 􏰇H^1(F,A[\alpha ])\stackrel{\mathrm{res}}{\to }{H}^1(F_{\alpha },A[\alpha ]{)}^{G_{\alpha }}$$

$$0\to A[\alpha ]\to A(\overline{F})\stackrel{\alpha }{\to }A(\overline{F})\to 0$$

$${\psi }^{\alpha }(x)(\sigma )=\sigma (\frac{x}{\alpha })-\frac{x}{\alpha },$$

$${\psi }^{l^k}:A(F)/l^{k}A(F)\to {\mathrm{Hom}}_{G_{l^k}}(􏰂G_{F_{l^k}},A[l^k]).$$

$$A(F)\otimes {\mathbb{Z}}_l(n)\to {\mathrm{Hom}}_{G_{l^{\infty }}}(G_{F_{l^{\infty }}},T_l(A))\otimes {\mathbb{Z}}_l(n),$$

$$A(F)\otimes {\mathbb{Z}}_l(n)\to \mathrm{Hom}(G_{F_{l^{\infty }},S_l},T_l(A))\otimes {\mathbb{Z}}_l(n)$$

$$A(F)\otimes {\mathbb{Z}}_l(n)\to H^1(G_{F_{l^{\infty }},S_l},T_l(A)(n)).$$

$$A(F)/l^k\hookrightarrow H^1(G_{F,S_l},A[l^k]).$$

Full text

1. Introduction

The local to global type questions are of mathematical interest since the celebrated Hasse principle was proven. For the history of these type of problems in number theory and its extensions to the context of abelian varieties and linear algebraic groups see [FK17].

In [BK13] G. Banaszak and the author proved a sufficient condition for the local to global condition to hold for étale $K$-theory of curves. This was done via analysis of the reduction map in the suitable Galois cohomology. The main result of the current paper is Theorem 1.3. It asserts, for the local to global questions considered there, that if the sufficient condition is not fulfilled then one can produce a counterexample. This gives some evidence that the criterion obtained in [BK13] is the best possible (cf. Remark 1.3).

To state the aforementioned local to global principle for étale $K$-theory of a curve let us recall the following definitions and notations from [BK13]. Let $X/F$ be a smooth, proper and geometrically irreducible curve of genus $g$ defined over a number field $F$ and let $J$ be the Jacobian of $X.$

Definition 1.1**.**

We call a finite field extension $F^{\prime}/F$ an isogeny splitting field of the Jacobian $J$ if $J$ is isogenous over $F^{\prime}$ to the product $A_{1}^{e_{1}}\times\dots\times A_{t}^{e_{t}}$ where $A_{1},\dots,A_{t}$ are pairwise nonisogenous, absolutely simple abelian varieties defined over $F^{\prime}$.

Remark 1.1*.*

Notice that $J$ is an abelian variety and therefore by Poincaré decomposition theorem the isogeny splitting field exists.

Let ${\tilde{V}}_{l,\,i}$ denote $T_{l}(A_{i})(n)\otimes_{\mathbb{Z}_{l}}{\mathbb{Q}}_{l},$ where $T_{l}(A_{i})(n)$ is the $n$-th twist of the $l$-adic Tate module of $A_{i}$ (cf. [T68]).

The main result of [BK13] is the following theorem:

Theorem 1.1**.**

Let $X/F$ be a smooth, proper and geometrically irreducible curve of genus $g.$ Let $F^{\prime}$ be an isogeny splitting field of the Jacobian $J$ and assume that for the corresponding product $A_{1}^{e_{1}}\times\dots\times A_{t}^{e_{t}},$ we have $End_{{\overline{F}}}\,A_{i}={\mathbb{Z}}$ for each $1\leq i\leq t.$ Let $l>2$ be a prime number which is coprime to the polarisation degrees of the abelian varieties $A_{i}.$ Let $S_{l}$ be a set of places of $F$ containing the places of bad reduction, the archimedean places and the primes above $l.$ Let ${\mathcal{X}}$ be a regular and proper model of $X$ over ${\mathcal{O}}_{F,S_{l}}.$ Assume that ${{dim}}_{{\mathbb{Q}}_{l}}{\tilde{V}}_{l,\,i}\,\,\geq e_{i}$ for each $1\leq i\leq t.$ Let $\hat{P}\in K_{2n}^{et}({\mathcal{X}})$ and let $\hat{\Lambda}$ be a finitely generated ${\mathbb{Z}}_{l}$-submodule of $K_{2n}^{et}({\mathcal{X}}).$ If $r_{v}(\hat{P})\in r_{v}(\hat{\Lambda})$ for almost all $v$ of ${\mathcal{O}}_{F,S_{l}}$ then $\hat{P}\in\hat{\Lambda}+K_{2n}^{et}({\mathcal{X}})_{tor}.$

Theorem 1.1 is equivalent with the following (cf. diagram (3.2) and [BK13] especially section 4):

Theorem 1.2**.**

Let $X/F,$ be a curve and $S_{l}$ the set of primes as in Theorem 1.1. Let $F^{\prime}$ be an isogeny splitting field i.e. $J$ is isogenous over $F^{\prime}$ to $A_{1}^{e_{1}}\times\dots\times A_{t}^{e_{t}}$ and for all $1\leq i\leq t$ we have $End_{{\overline{F}}}\,A_{i}={\mathbb{Z}}$ and ${{dim}}_{{\mathbb{Q}}_{l}}{\tilde{V}}_{l,\,i}\,\,\geq e_{i}.$ Let ${\hat{P}}\in H^{1}(G_{S_{l}};{T}_{l}(J)(n))$ and ${\hat{\Lambda}}$ be a finitely generated ${\mathbb{Z}}_{l}$-submodule of $H^{1}(G_{S_{l}};{T}_{l}(J)(n)).$ If $r_{v}(\hat{P})\in r_{v}({\hat{\Lambda}})\subset H^{1}({g}_{v};{T}_{l}(J_{v})(n))$ for almost all $v$ of ${\mathcal{O}}_{F,S_{l}}$ then $\hat{P}\in\hat{\Lambda}+H^{1}(G_{S_{l}};{T}_{l}(J)(n))_{tor}.$

For the definition of $G_{S_{l}}$ and $g_{v}$ see section 3.

We prove the following theorem:

Theorem 1.3**.**

Let $A=A_{1}^{e_{1}}\times\dots\times A_{t}^{e_{t}}$ be an abelian variety such that $End_{{\overline{F}}}\,A_{i}={\mathbb{Z}}$ for all $i.$ Let $S_{l}$ be the set of primes containing archimedean places, primes of bad reduction of $A$ and primes over $l.$ Assume that ${\mathrm{rank}}A_{i}(F)\geq e_{i}$ and ${{dim}}_{{\mathbb{Q}}_{l}}{\tilde{V}}_{l,\,i}\,\,<e_{i}$ for some $1\leq i\leq t.$ Then:

1. (1)

the local to global principle for the map $H^{1}(G_{S_{l}};{T}_{l}(A))\rightarrow H^{1}({g}_{v};{T}_{l}(A_{v}))$ does not hold i.e. there exists $\Lambda\subset H^{1}(G_{S_{l}};{T}_{l}(A))$ and $P\in H^{1}(G_{S_{l}};{T}_{l}(A))$ such that $r_{v}(P)\in r_{v}(\Lambda)$ but $P\notin\Lambda+H^{1}(G_{S_{l}};{T}_{l}(J)(n))_{tor}$ 2. (2)

the local to global principle for the map $H^{1}(G_{F_{{l^{\infty},S_{l}}}};{T}_{l}(A)(n))\rightarrow H^{1}({g}_{v};{T}_{l}(A_{v})(n)),$ where $F_{{l^{\infty},S_{l}}}:={\bigcup}_{n=1}^{\infty}F_{S_{l}}({\mu}_{l^{n}})$ and ${\mu}_{l^{n}}$ is the $l^{n}$-th root of unity, does not hold.

Remark 1.2*.*

Without loss of generality in the proof of Theorem 1.3 we may assume $F^{\prime}=F$. From now on we assume this.

Remark 1.3*.*

The reason that we could not produce a counterexample for an arbitrary Tate twist i.e. for the map $r_{v}:H^{1}(G_{{S_{l}}};{T}_{l}(A)(n))\rightarrow H^{1}({g}_{v};{T}_{l}(A_{v})(n))$ is the difficulty in estimating the rank of $H^{1}(G_{S_{l}};{T}_{l}(A)(n)).$ However, if this rank is sufficiently big then, similar to that given in section 6, proof shows that the corresponding to (1) result holds true. Then abelian varieties from Example 2.6 yield, if the numeric condition of Theorem 1.1 is violated, counterexamples to the local to global principle for even étale $K$-theory of a curve.

In section 2 we give some examples of Jacobians of curves satisfying the assumptions of Theorem 1.1 and examples satisfying assumptions of Theorem 1.3. Sections 3, 4 and 5 contain necessary background from Galois cohomology, intermediate Jacobians and Kummer theory. In section 6 we prove our main theorem (Theorem 1.3). Section 7 is devoted to proof of the dynamical version of the local to global principle. This is done by checking the axioms for the dynamical local to global principle introduced in [B17]. We check these axioms for Galois cohomology (cf. Lemma 7.3) and then pass to the étale $K$-theory. We also show that we can replace étale $K$-theory by Quillen theory of a curve provided Bass and Quillen-Lichtenbaum conjectures hold true.

2. Examples

In this section we give examples of curves whose Jacobian decomposes as $A_{1}^{e_{1}}\times\dots A_{t}^{e_{t}},$ with ${\operatorname{End}}_{\overline{F}}A_{i}={\mathbb{Z}}.$ In general deciding whether the Jacobian of a curve splits is a difficult problem and has vast literature (cf. e.g. [HN65], [ES93]). However, we have the following [CO12], p.589:

Theorem 2.1**.**

Any abelian variety $A$ defined over $\overline{\mathbb{Q}}$ of dimension $1\leq g\leq 3$ is isogenous to $J(C)$, where $C$ is a curve of genus $g$.

Remark 2.1*.*

Genus $1$ case of Theorem 2.1 is trivial, genus $2$ is [W57] Satz 2 and $g=3$ is Theorem 4 of [OU73].

Example 2.2*.*

Let $A/F$ be principally polarized abelian surface with ${\operatorname{End}_{{\overline{F}}}}A={\mathbb{Z}}$ then by Theorem 2.1 there exists a curve $X$ defined over $F^{{}^{\prime}}\subset{\overline{Q}}$ such J(X) is isogenous to $A.$

Example 2.3*.*

Let $B=A\times E$, where $A$ is as in Example 2.2 and $E$ is an elliptic curve without complex multiplication then there exists a curve $X$ such that J(X) is isogenous to $B.$

A special case of Example 2.2 is given in the following (see [BPP18] section 7.1 p. 39):

Example 2.4*.*

Let $X/{\mathbb{Q}}$ be a smooth projective curve given by the following equation

[TABLE]

Then J(X) is a principally polarized abelian isotypical surface over ${\mathbb{Q}}$ of conductor 277.

Remark 2.5*.*

Since the ${\mathbf{j}}$ invariant of a curve with complex multiplication is an algebraic integer (cf. [C89], Theorem 11.1 ), in Example 2.3 it is enough to pick an elliptic curve defined over ${\mathbb{Q}}$ whose ${\mathbf{j}}$ invariant is non-integral rational number.

The following lemma guarantees that there are many simple abelian surfaces $A$ with ${\operatorname{End}}_{\overline{F}}A={\mathbb{Z}}$

Lemma 2.2**.**

([BPP18], Lemma 4.1.1) Let $A$ be a simple, semistable abelian surface over ${\mathbb{Q}}$ with non-square conductor then $A$ is isotypical, i.e. ${\operatorname{End}}_{\overline{\mathbb{Q}}}A={\mathbb{Z}}$.

Now we can give examples of abelian varieties satisfying the assumptions of Theorem 1.3

Example 2.6*.*

In Example 2.3 take $A$ to be any simple abelian surface $A$ over ${\mathbb{Q}}$ with ${\operatorname{End}}A={\mathbb{Z}}$ and $E/{\mathbb{Q}}$ to be any elliptic curve without complex multiplication such that ${\mathrm{rank}}E(\mathbb{Q})>2.$ One specific choice of that sort will be $A=J(X)$ where $X$ is a curve from Example 2.4 and $E$ the elliptic curve (listed in Cremona tables) of conductor 5077 with the coefficients $[a_{1},a_{2},a_{3}.a_{4},a_{6}]=[0,0,1,-7,6]$ and the Mordell-Weil group over ${\mathbb{Q}}$ of rank 3. One readily verifies that the $j$-invariant is not a rational integer, so this curve has no CM (cf. Remark 2.5).

3. necessary results concerning étale $K$-theory of curves and cohomology

We start with the definition of continuous cohomology [J88] and [DF85].

Definition 3.1**.**

Let $Y$ be a scheme over $\mathbb{Z}[\frac{1}{l}]$ and let $({\mathcal{F}}_{m})$ be a projective system of ${\mathbb{Z}}/{l^{m}}$ - étale sheaves ${\mathcal{F}}_{m}$. The functor $({\mathcal{F}}_{m})\rightarrow\varprojlim H^{0}(Y,\,({\mathcal{F}}_{m}))$ is left exact and its $i$-th right derived functor is by definition the continuous cohomology group $H^{i}_{cts}(Y,\,({\mathcal{F}}_{m})).$

The étale K-theory spaces and groups were defined in [DF85] by W. Dwyer and E. Friedlander. In [BGK99] using the spectral sequences of [DF85]:

[TABLE]

[TABLE]

it was shown that one has the following exact sequence connecting continuous cohomology of $\mathcal{X}$ and the Galois cohomology of the Galois group of the maximal unramified outside $S_{l}$ extension of $F:$

[TABLE]

as well as existence of the following commutative diagram:

[TABLE]

where $g_{v}:=Gal({\bar{k_{v}}}/k_{v}).$ The right hand vertical arrow in the diagram (3.2) is an isomorphism whereas the left vertical arrow is an epimorphism with finite kernel (cf. [BK13], [BGK99], [BG08]).

One has the following Dwyer-Friedlander homomorphisms, for odd prime $l$ and $k\geq 1$, connecting Quillen $K$-theory with the étale $K$-theory with coefficients (cf. [DF85] ):

[TABLE]

[TABLE]

In the sequel we assume the following:

Conjecture 3.1**.**

(Bass conjecture) The Quillen $K$-theory groups $K_{m}({\mathcal{X}})$ are finitely generated for $m>0$.

Assuming Conjecture 3.1 we obtain the following equality:

[TABLE]

The Dwyer-Friedlander homorphisms (3.3) and (3.4) induce the following homomorphisms:

[TABLE]

[TABLE]

The maps (3.6) and (3.7) are isomorphism if we assume that the Quillen-Lichtenbaum conjecture holds true for $\mathcal{X}.$ Thus we obtain the following commutative diagram:

[TABLE]

where the map $h^{{}^{\prime}}$ (resp. ${\overline{h^{{}^{\prime}}}}$ ) is the composition of the natural homomorphism $K_{2n}({\mathcal{X}})\rightarrow K_{2n}({\mathcal{X}})\otimes{\mathbb{Z}}_{l}$ (resp. $K_{2n}({\mathcal{X}}_{v})\rightarrow K_{2n}({\mathcal{X}}_{v})\otimes{\mathbb{Z}}_{l}$) with $h$ (resp. $\overline{h}$).

Concatenation of diagrams (3.2) and (3.8) yields the following commutative diagram:

[TABLE]

with the vertical maps having finite kernels.

4. Intermediate Jacobian

In this section to ease notation we denote ${\tilde{T}}_{l,i}:=T_{l}(A_{i})(n).$ Let $L$ be a finite extension of $F.$ For $w\notin S_{l},$ let $G_{w}:=G({\overline{L}_{w}}/L_{w})$ be the absolute Galois group of the completion $L_{w}$ of $L$ at $w.$ Let $k_{w}$ be the residue field at $w.$ Put $H^{1}_{f}(G_{w};{\tilde{T}}_{l,\,i})=i_{w}^{-1}H^{1}_{f}(G_{w};{\tilde{V}}_{l,\,i}).$ Here $i_{w}:H^{1}(G_{w};{\tilde{T}}_{l,\,i})\rightarrow H^{1}(G_{w};{\tilde{V}}_{l,\,i})$ and $H^{1}_{f}(G_{w};{\tilde{V}}_{l,\,i})=ker(\,res:H^{1}(G_{w};{\tilde{V}}_{l,\,i})\rightarrow H^{1}(I_{w};{\tilde{V}}_{l,\,i}))$ where $I_{w}\subset G_{w}$ is the inertia subgroup and $res$ is the restriction map. Let

[TABLE]

We define the intermediate Jacobian (see [BGK05], [BGK03]):

[TABLE]

In [BGK05], [BGK03], in the more general situation of any free ${\mathbb{Z}}_{l}$-module of finite rank $T_{l},$ we made the following:

Assumptions 4.1**.**

For any finite extension $L/F$ and any place $w\in{\mathcal{O}}_{L}$ such that $w\notin S_{l}\,,$ we have $T_{l}^{Fr_{w}}=0.$

Remark 4.1*.*

We have $H^{1}_{f,S_{l}}(G_{L};{\tilde{T}}_{l,\,i})\cong H^{1}(G_{L,S_{l}};{\tilde{T}}_{l,\,i})$ and the assumption 4.1 is satisfied (cf. [BGK05], p.5 ).

By [BGK03, Prop. 2.14] we have the following isomorphisms:

[TABLE]

Let ${\tilde{T}}_{l}:=\bigoplus_{i=1}^{t}\,{\tilde{T}}_{l,\,i}.$ Since each ${\tilde{T}}_{l,\,i}$ is a free ${\mathbb{Z}}_{l}(n)$-module we have the following natural isomorphism:

[TABLE]

For a finite extension $L/F$ the following map corresponding to these used in the direct systems (4.2):

[TABLE]

is an injection. One readily verifies, using transfer in the Galois cohomology, that the map (4.5) has $l$-torsion kernel.. But the $l$-torsion part of (4.5) is just

[TABLE]

which is clearly injective. Similarly, the reduction of the map (4.5)

[TABLE]

is injective for any finite extension $L/F$ and any prime $w$ of $\mathcal{O}_{L}$ over $v,\,v\notin S_{l}.$ Notice that by Definition 1.1 we have

[TABLE]

Let us also recall the following:

Lemma 4.2**.**

( [BGK03], Lemma 2.13) For any finite extension $L/F$ and any prime $w\notin S_{l}$ in ${\mathcal{O}}_{L}$ the natural map

[TABLE]

is an imbedding.

5. Kummer theory

Kummer theory in the context of abelian varieties was developed in [R79]. In this section we collect necessary facts which will be useful in section 7.

Definition 5.1**.**

Let $A$ be an abelian variety over a number field $F$. Let ${\mathcal{R}}=\operatorname{End}_{F}A$. For $\alpha\in{\mathcal{R}}$ we set $F_{\alpha}=F(A[\alpha])$ and $G_{\alpha}=Gal(F_{\alpha}/F)$. The Kummer map:

[TABLE]

is defined as the composition:

[TABLE]

with the first map a coboundary map for the $G_{F}$-cohomology of the Kummer sequence:

[TABLE]

and the second map the restriction to $F_{\alpha}$.

Explicitly, the map (5.1) is given by the following formula

[TABLE]

where $\frac{x}{\alpha}$ is a fixed ”$\alpha$-root” of $x$ i.e. an element $y\in A({\overline{F}})$ such that ${\alpha}y=x.$

We are interested in Kummer maps where $l$ is a rational prime and ${\alpha}=l^{k}\in{\mathbb{Z}}\hookrightarrow{\mathcal{R}}.$ Thus we have the the family of Kummer maps:

[TABLE]

The maps (5.5) are compatible with the natural maps induced by multiplication by $l$. Therefore taking the inverse limit of both sides and twisting it with ${\mathbb{Z}}_{l}(n)$ yields the following map:

[TABLE]

where $F_{l^{\infty}}={\bigcup_{k}}F_{l^{k}}$ and $G_{F_{l^{\infty}}}=Gal({\overline{F}}/F_{l^{\infty}})$. From the map (5.6) by restriction of the Kummer maps to $G_{F_{l^{\infty}},S_{l}}$ we obtain the map:

[TABLE]

Notice that ${\operatorname{Hom}}(G_{F_{l^{\infty}},S_{l}},T_{l}(A))\cong{H^{1}}(G_{F_{l^{\infty}},S_{l}},T_{l}(A))$ and $H^{0}(G_{F_{l^{\infty}},S_{l}},{\mathbb{Z}}_{l}(n))={\mathbb{Z}}_{l}(n).$ Composition with the cup product yields the map:

[TABLE]

Obviously, this map factors through $A(F_{l^{\infty}})\otimes{\mathbb{Z}}_{l}(n)$ We will also need the coboundary map in the $G_{F,S_{l}}$-cohomology of the Kummer sequence (5.3) with $\alpha$ equal to multiplication by $l^{k}$:

[TABLE]

Passing to the inverse limits in (5.9) we obtain an injection:

[TABLE]

Remark 5.1*.*

Notice that we could pass to $G_{F,S_{l}}$-cohomology because of the choice of $S_{l}$ which guarantees that multiplication by $l$ on $A(F)$ is étale (cf. Remark 4.1 and discussion on p. 148 of [BGK03] ).

We have the following commutative diagram:

[TABLE]

In the diagram (5.11) horizontal arrows are injections by (5.10) and the left vertical arrow is an obvious injection.

We also have the following:

Lemma 5.1**.**

The restriction homomorphism composed with the natural injection $H^{1}(G_{F,S_{l}},T_{l}(A)(n))\rightarrow H^{1}(G_{F_{l^{\infty}},S_{l}},T_{l}(A)(n))$ is a map with finite kernel.

Proof.

By definitions of $S_{l}$ and the field $F_{l^{\infty},S_{l}}$ we see that $G_{F_{l^{\infty},S_{l}}}$ acts trivially on $T_{l}(A)(n)$ and therefore $H^{1}(G_{F_{l^{\infty}},S_{l}},T_{l}(A)(n))=Hom(G_{F_{l^{\infty}},S_{l}},T_{l}(A)(n)).$ We have the Leray spectral sequence in Galois cohomology (cf [Mi80] ) with $E_{2}^{p,q}=H^{p}(G(F_{l^{\infty}}/F);H^{q}(G_{F_{l^{\infty}},S_{l}},T_{l}(A)(n)))$ converging to $H^{p+q}(G_{F,S_{l}},T_{l}(A)(n)).$ The inflation - restriction sequence has the following form:

[TABLE]

[TABLE]

But

[TABLE]

[TABLE]

Notice that the groups $H^{1}(G(F_{l^{\infty}}/F),T_{l}(A)(n))$ are finite ( cf. [S71], Corollaire on p.734 ). The assertion of the lemma follows. ∎

Remark 5.2*.*

The Corollaire to Théoréme 2 in [S71] asserts that for any $k\geq 0$ the groups $H^{k}(G(F_{l^{\infty}}/F),T_{l}(A))$ are finite $l$-groups. However, the same proof shows that for any $k$ the groups $H^{k}(G(F_{l^{\infty}}/F),T_{l}(A)(n))$ are finite $l$-groups.

Corollary 5.2**.**

In particular the restriction map composed with the natural injection $H^{1}(G_{F,S_{l}},T_{l}(A))\rightarrow H^{1}(G_{F_{l^{\infty}},S_{l}},T_{l}(A))$ has finite kernel. Therefore if $Q_{j}\otimes 1\in A(F)\otimes{\mathbb{Z}}_{l}(n),j=1,\dots r$ are linearly independent elements over ${\mathbb{Z}}_{l}$ then their images by the left verical arrow in the diagram (6.2) are also linearly independent.

6. Proof of Theorem 1.3

Notice that for any abelian variety we have the following commutative diagram ( cf. [BGK03] diagram (6.10) ):

[TABLE]

From the diagram (6.1) tensoring with ${\mathbb{Z}}_{l}(n)$ we obtain the following commutative diagram:

[TABLE]

where $w|v$ are primes of ${\mathcal{O}}_{F_{l^{\infty}},S_{l}}.$ By Corollary 5.2 the image in $H^{1}(G_{F_{l^{\infty}},S_{l}},T_{l}(A)(n))$ of any non-torsion element $Q\otimes 1\in A(F)\otimes{\mathbb{Z}}_{l}(n)$ is a non-torsion element. Therefore by the commutativity of the diagram (6.2) the reduction of this image in $H^{1}(g_{w},T_{l}(A_{v})(n))$ comes from the reduction $red_{v}(Q\otimes 1)$. The proof of Theorem 1.3 (1) is essentially the same as that of Theorem 1.3 (2). For (1) we use the diagram (6.1) whereas for the point (2) the diagram (6.2). Therefore we will give a proof of Theorem 1.3 (2). In what follows we consider elements of $H^{1}(G_{F_{l^{\infty}},S_{l}},T_{l}(A)(n))$ coming from $A(F)\otimes{\mathbb{Z}}_{l}(n)$.

Our proof is a generalization of the counterexample to local - global principle for abelian varieties constructed by P. Jossen and A. Perucca in [JP10] and later extended to the context of $t$-modules in [BoK18]. Because of (4.8) and the choice of $F$ we may assume that $J=A^{e}$ where $A$ is a geometrically simple abelian variety with ${\mathrm{End}}A={\mathbb{Z}}$ such that ${d=\dim\tilde{V}}_{l},e=d+1$ and ${\mathrm{rank}}A(F)\geq e.$

Let $v\in{\mathcal{O}_{F,S_{l}}}$ be a prime of good reduction for $A.$

Pick non-torsion points $Q_{1},\dots Q_{e}$ linearly independent over ${\mathbb{Z}}.$ Then $Q_{1}\otimes 1,\dots,Q_{e}\otimes 1$ in $A(F)\otimes{\mathbb{Z}}_{l}(n)$ and their images $P_{1},\dots P_{e}\in H^{1}(G_{F_{l^{\infty}},S_{l}},T_{l}(A)(n))$ are linearly independent over ${\mathbb{Z}}_{l}.$ Let $\overline{P}_{i}\in H^{1}({g}_{w};{T}_{l}(A_{v})(n))$ be the reduction of $P_{i}$ ${\operatorname{mod}}\,\,{w.}$ Consider the reductions ${{\bar{Q}}_{1}},\dots,{{\bar{Q}}_{e}}\in A_{v}(k_{v})_{l}.$

Let

[TABLE]

and $\overline{P}=[\overline{P}_{1}\,,\dots,\overline{P}_{e}]^{T},$ $\overline{P}_{i}\in H^{1}({g}_{w};{T}_{l}(A_{v})(n))$ be the reduction ${\operatorname{mod}}\,\,{w}$ of $P.$ Notice that $P\notin{\Lambda}+H^{1}(G_{F_{l^{\infty}},S_{l}},T_{l}(A)(n))_{tor}$ since $P_{1},\dots,P_{e}$ are ${\mathbb{Z}}_{l}$-linearly independent. We will find a matrix $M\in{\mathrm{Mat}}_{e\times e}({\mathbb{Z}}_{l})$ of trace zero such that $\overline{P}=M{\overline{P}}.$ This will show that ${\overline{P}}\in{\overline{\Lambda}}.$ We will give two constructions of such a matrix.

First construction

Since the group $Y=({\overline{P}}_{1},\dots,{\overline{P}}_{e})\subset H^{1}({g}_{w};{T}_{l}(A_{v})(n))$ is finite there exist ${\alpha}_{1},\dots,{\alpha}_{e}\in{\mathbb{Z}}$ with ${\alpha}_{i}$ minimal positive such that

[TABLE]

We will show that $D={\gcd({{\alpha}_{1},\dots\alpha_{e}}})=1.$ Assume opposite. Choose a rational prime $p$ that divides $D.$ This means, by our choice of ${\alpha}_{1},\dots,{\alpha}_{e},$ that $p$ divides coefficients of any yielding zero linear combination of points ${\overline{P}}_{1},\dots,{\overline{P}}_{e}\in H^{1}({g}_{w};{T}_{l}(A_{v})(n)).$ In particular, we see that $p$ divides the orders of ${\overline{P}}_{i},i=1,\dots e.$ Since $H^{1}({g}_{w};{T}_{l}(A_{v})(n))$ is $l$-torsion the only possibility is $p=l.$

Let ${\alpha}_{1}=x_{1}l$ and

[TABLE]

be a linear relation.

According to [ST68] (cf. also [L57] ), if $l$ is prime to the characteristic of the field of definition $k$ of $A$ then over the separable closure $k^{s}$ of $k$ we have $A(k^{s})[l]=({\mathbb{Z}}/l{\mathbb{Z}})^{2g}$ where $A(k^{s})[l]$ denotes subgroup of $A(k^{s})$ of points of order dividing $l$. Therefore, $A(k)[l]$ is isomorphic to one of the following groups $\{0,({\mathbb{Z}}/l{\mathbb{Z}})^{i},\,\,\,i=1,\dots,2g\}$ and thus is again a ${\mathbb{Z}}/l{\mathbb{Z}}$-vector space. Since $v\in{\mathcal{O}}_{F,S_{l}}$ we get $A(k_{v})[l]\cong({\mathbb{Z}}/l{\mathbb{Z}})^{d}.$ Consider $x_{1}l{\bar{Q}}_{1}+\dots+{x_{e}}l{\bar{Q}}_{e}=0$ in $A_{v}(k_{v})_{l}$

Then

[TABLE]

where ${\tilde{T}}\in\bar{\Lambda}\cap A(k_{v})[l].$ Now, since ${\dim}_{{\mathbb{Z}}/l{\mathbb{Z}}}A(k_{v})[l]=d$ we see that ${\tilde{T}}$ is a linear combination of say ${\bar{Q}}_{2},\dots,{\bar{Q}}_{e}$ and this gives the desired contradiction (cf. diagram 6.2).

Hence, there exist $a_{1},\dots,a_{e}\in{\mathbb{Z}}$ such that

[TABLE]

Put $m_{i,i}=1-a_{i}{\alpha}_{i}.$ Then $m_{1,1}+\dots+m_{e,e}=0$ and

[TABLE]

Therefore ${\overline{P}}\in{\overline{\Lambda}}.$ We view the matrix M in (6.6) as the matrix with the ${\mathbb{Z}}_{l}$ coefficients.

Second construction

We start with the following general group-theoretic lemma:

Lemma 6.1**.**

Assume $G=(U_{1},\dots U_{\mu})$ is an abelian $l$-torsion group generated by ${\mu}$ elements. Then any nontrivial subgroup $H=(V_{1},\dots V_{\nu})$ of $G$ can be generated by at most ${\mu}$ elements. Moreover if ${\nu}>{\mu}$ then one can express ${\nu}-{\mu}$ generators say $V_{i_{1}},\dots V_{i_{{\nu}-{\mu}}}$ as ${\mathbb{Z}}$-linear combination of the remaining generators.

Proof.

Assume that a subgroup $H$ of the group $G$ is generated by ${\nu}>{\mu}$ elements. Consider $G$ as a ${\mathbb{Z}}$-module. Then $G\cong{\mathbb{Z}}/{l^{n_{1}}}{\mathbb{Z}}\oplus\dots\oplus{\mathbb{Z}}/{l^{n_{\mu}}}{\mathbb{Z}}$ where $n_{1}\leq n_{2}\leq\dots\leq n_{\mu}.$ Pick elements $S_{i}=(0,\dots,0,1,0,\dots,0)$ (1 on the $i$-th place) and expand $V_{i}=\sum{\beta}_{j,i}^{(1)}S_{j},i=1,\dots,{\nu},j=1,\dots\mu$ in the ${\mathbb{Z}}$-basis $\{S_{i}\}$ of $G.$ In the columns of the matrix $({\beta}^{(1)}_{j,i})\in M_{{\mu}\times{\nu}}$ we have the coefficients of the expansion of $V_{i}.$ In the first row we pick a nonzero coefficient with the smallest $l$-valuation. Without loss of generality assume this element is in the first column. Subtract from each column appropriate multiple of the first column to obtain zeros for $i>1$ in the first row. This is possible since every element in the first row is of the form ${\beta}_{1,i}^{(1)}=u_{i}\cdot l^{k_{i}},$ where $u_{i}\in{\mathbb{Z}}/{l^{n_{1}}}{\mathbb{Z}}$ is a unit and $k_{i}<n_{1}.$ We obtain a new ${\mu}\times{\nu}$ matrix ${\beta}_{j,i}^{(2)}$ with one nonzero entry in the first row, namely ${\beta}^{(2)}_{1,1}={\beta}^{(1)}_{1,1}$. Now, for $i\geq 2$ the $i$-th column contains expansion of the point $V_{i}-c_{i,1}^{(1)}V_{1}$ where $c_{i,1}^{(1)}$ is the coefficient ( viewed in ${\mathbb{Z}}$) by which we multiply the first column to obtain $0\in{\mathbb{Z}}/{l^{n_{1}}}{\mathbb{Z}}$ in the place $(1,i).$ Continue this process with the second row i.e. pick an element ${\beta}_{j,i}^{(2)},\,i\geq 2$ with the smallest $l$-valuation, transpose columns and enumerate points correspondingly so that the element with the smallest $l$-valuation is at the place $(2,2).$ Obtain zeros for columns with the index $i>2$. Continue the process until we obtain the lower diagonal matrix. Notice that at the $k$-th stage for $i\geq k$ the $i$-th column contains the expansion of the element $V^{(k)}_{i}-{\sum}_{s<i}c_{i,s}^{(k)}V^{(k)}_{s}$ in the basis $\{S_{j},\,j=1,\dots,{\mu}\}.$ Assume that at the $m$-th stage we obtain zeros in the columns $i>r$ where $r\leq{\mu}.$ Let $C=\{V^{(m)}_{r+1},\dots,V^{(m)}_{{\nu}}\}$ be the elements corresponding to the zero columns. Thus any element from $C$ is a linear combination of the generators not belonging to $C.$ Since our possible transposition at the $k$-th stage involved transposition of the $k$-th and $i$-th columns with $i>k$ we see that the coefficient at each $V^{(m)}_{r+1},\dots,V^{(m)}_{{\nu}}$ is one. Thus for $i\in C$ we have $V^{(m)}_{i}={\sum}_{s\notin C}c_{i,s}^{(m)}V^{(m)}_{s}$ ∎

Now we can apply Lemma 6.1 to $G=\operatorname{Im}(A_{v}(k_{v})_{l}\otimes{\mathbb{Z}}_{l}(n))\subset H^{1}({g}_{w};{T}_{l}(A_{v})(n))$ (cf. diagram (6.2)) and $H=({\bar{P}}_{1},\dots,{\bar{P}}_{e}).$ Since $G$ is a finite $l$-group generated by $d$ elements and $e=d+1$ we may assume without loss of generality that ${\bar{P}}_{1}={\sum}_{i=2}^{e}c_{i}{\bar{P}}_{i}.$ Then we can choose

[TABLE]

Obviously ${\overline{P}}=M{\overline{P}}$ and therefore ${\overline{P}}\in{\overline{\Lambda}}$ but $P\notin{\Lambda}+H^{1}(G_{F_{l^{\infty}},S_{l}},T_{l}(A)(n))_{tor}.$

7. Local to global principle and dynamical systems

S. Barańczuk in [B17] considers abelian groups satisfying the following two axioms.

Assumptions 7.1**.**

Let $B$ be an abelian group such that there are homomorphisms $r_{v}:B\rightarrow B_{v}$ for an infinite family $v$, whose targets $B_{v}$ are finite abelian groups.

1. (1)

Let $l$ be a prime number, $(k_{1},\dots,k_{m})$ a sequence of nonnegative integers. If $P_{1},\dots,P_{m}\in B$ are points linearly independent over ${\mathbb{Z}},$ then there is a family of primes $v$ in $F$ such that $l^{k_{i}}||{\operatorname{ord}}_{v}P_{i}$ if $k_{i}>0$ and $l\nmid{\operatorname{ord}}_{v}P_{i}$ if $k_{i}=0.$ 2. (2)

For almost all $v$ the map $B_{\text{\rm tors}}\rightarrow B_{v}$ is injective.

Here ${\operatorname{ord}}_{v}P$ is the order of a reduced point $P\operatorname{mod}v$. Under the Assumptions 7.1, S. Barańczuk was able to prove the following dynamical version of the local lo global principle

Theorem 7.2**.**

( [B17]) Let $\Lambda$ be a subgroup of $B$ and $P\in B$ be a point of infinite order and $\phi$ be a natural number. Then the following are equivalent:

1. i)

For almost every $v$

[TABLE] 2. ii)

$O_{\phi}(P)\cap({\Lambda})\neq\emptyset$.

Here $O_{\phi}(P)=\{{\phi}^{n}(P):n\geq 0\}.$ We have the following lemma:

Lemma 7.3**.**

Let $A=A_{1}^{e_{1}}\times\dots\times A_{t}^{e_{t}},$ be an abelian variety such that ${\operatorname{End}}A_{i}={\mathbb{Z}}$ for $i=1,\dots,t.$ Then $B=H^{1}(G_{F,S_{l}};{T}_{l}(A)(n))$ and $B_{v}=H^{1}({g}_{v};{T}_{l}(A_{v})(n))$ for $v\in{\mathcal{O}}_{v}$ fulfil Assumptions 7.1.

Proof.

Assumption (1) of 7.1 is a specialization to ${\mathcal{R}}_{i}={\mathbb{Z}}$ of Corollary 3.5 of [BK13]. Assumption (2) is an assertion of Lemma 4.2. ∎

Theorem 7.4**.**

Let $X/F$ be a smooth, proper and geometrically irreducible curve of genus $g.$ Let $F^{\prime}$ be an isogeny splitting field of the Jacobian $J$ and assume that for the corresponding product $A_{1}^{e_{1}}\times\dots\times A_{t}^{e_{t}},$ we have $End_{{\overline{F}}}\,A_{i}={\mathbb{Z}}$ for each $1\leq i\leq t.$ Let $l>2$ be a prime number which is coprime to the polarisation degrees of the abelian varieties $A_{i}.$ Let $S_{l}$ be a set of places of $F$ containing the places of bad reduction, the archimedean places and the primes above $l.$ Let ${\mathcal{X}}$ be a regular and proper model of $X$ over ${\mathcal{O}}_{F,S_{l}}.$ Let $\hat{P}\in K_{2n}^{et}({\mathcal{X}})$ be a point of infinite order and let $\hat{\Lambda}$ be a finitely generated ${\mathbb{Z}}_{l}$-submodule of $K_{2n}^{et}({\mathcal{X}}).$ Let $w$ be a natural number and $O_{w}=\{w^{k}P:k\geq 0\}$ Then the following are equivalent

1. (1)

For almost every prime ${v}\in{\mathcal{O}}_{F,S_{l}}$

[TABLE] 2. (2)

$O_{w}(P)\cap{\Lambda}^{\prime}\neq\emptyset$* where ${\Lambda}^{\prime}$ is a subgroup of $K_{2n}^{et}({\mathcal{X}})$ generated by $\Lambda$ and the finite kernel of the map $K_{2n}^{et}({\mathcal{X}})\rightarrow H^{1}(G_{S_{l}};{T}_{l}(J)(n)).$*

Proof.

The proof follows from Lemma 7.3 , commutativity of the diagram (3.2). ∎

Remark 7.1*.*

If the Bass Conjecture and Quillen-Lichtenbaum Conjecture hold true for ${\mathcal{X}}$ then we obtain ( using diagram (3.9) instead of (3.2)  ) the corresponding to Theorem 7.4 statement for Quillen $K$-theory of ${\mathcal{X}}.$

Acknowledgments We would like to thank G. Banaszak for helpful discussions and D. Ulmer for drawing our attention to [HN65] and [OU73]. We would also like to thank the referee whose remarks helped to improve the manuscript.