Pairings over local fields

Background on Pairings

6.3 Pairings over local fields

LetP ∈A(K). There exists a pointQ∈A K

such that[n]Q=P. Define δ(P) :GK → A

K [n]

σ → σ·Q−Q.

We easily check thatδ(P)is a1-cocycle with image inA K

[n]and that another choice ofQwith [n]Q = P changes this cocycle by a coboundary and so we get a well defined map fromA(K) toH¹

G_K, A K

[n]

. Another immediate check shows that the kernel of this map is exactly [n]A(K). This explains the first part of the Kummer sequence.

We now use the injection ofA K

[n]intoA K

to interpret cocycles with values inA K

[n]

as cocycles with values inA K

. Going to the quotient modulo coboundaries gives the mapα.

Since the arguments of the induced cocycles are points of ordernit follows that the image ofαis contained in the subgroup ofH¹

GK, A K

, which is annihilated by the map “multiplication by n.”

We can check, either directly or by using properties of cohomology, thatαis surjective and that the kernel ofαis equal to the image ofδ.

Next we use thatA K

[n]is self-dual as aGK-module under the Weil pairing, which we denote byW_n. We obtain a cup product

∪:H¹ G_K, A

K [n]

×H¹

G_K, A K

[n]

→H²

G_K, K^∗ [n]

in the following way:

Representζ1, ζ2∈H¹ GK, A

K [n]

by cocyclesc1, c2. Thenζ1∪ζ2is the cohomology class of the2-cocycle

c:G_K×G_K →K^∗ given by

c(σ1, σ2) :=Wn

c1(σ1), c2(σ2) . This map∪is bilinear.

We can apply this to a pointP ∈ A(K)and a cohomology classγ ∈ H¹ GK, A

K [n]to define theTate pairing

·,·T,n:A(K)/nA(K)×H¹ GK, A

K

[n]→H²

GK, K^∗ [n]

by

P+nA(K), γT,n=δ

P+nA(K)

∪α⁻¹(γ).

It is routine to check that·,·T,nis well defined and bilinear.

Remark 6.2 The Tate pairing relates three very interesting groups occurring in Arithmetic Geom- etry: the Mordell–Weil group ofA, the first cohomology group of A, which can be interpreted as group of principally homogeneous spaces overA, andH²

GK, K^∗

, theBrauer groupof the ground fieldK, which can be interpreted as group of classes of central simple algebras with center Kwith the class of full matrix groups as neutral element.

6.3 Pairings over local fields

We now assume thatKis alocal field(e.g., ap-adic field) with finite residue fieldFq.

118 Ch. 6 Background on Pairings

6.3.1 The local Tate pairing

We have the beautiful result of Tate [TAT1958]:

Theorem 6.3 LetAbe a principally polarized abelian variety overK, e.g.,A=JC. Let

·,·T,n:A(K)/nA(K)×H¹ GK, A

K

[n]→H²

GK, K^∗ [n]

be the Tate pairing as defined above. Then·,·T,nis a nondegenerateZ-bilinear map.

It is thus certainly worthwhile to study in more detail the groups that are involved.

To simplify the situation we shall assume thatAhas good reduction, i.e., we find equations for Awith coefficients in the ring of integers ofKwhose reductions modulo the valuation ideal ofK again define an abelian variety overFq. This situation is typical for the applications that we have in mind. In fact we shall begin with an abelian variety overFqand then lift it to an abelian variety overK. This motivates a change of notation:A→A˜and the reduction ofA˜is now denoted byA.

Let us consider the first group occurring in Tate duality. Using Hensel’s lemma we get A(K)/n˜ A(K)˜ A(Fq)/nA(Fq).

Remark 6.4 If we assume thatn = is a prime and thatA(Fq)has no points of order² then A(Fq)/A(Fq)is isomorphic toA(Fq)[]in a natural way.

We now come to the discussion ofH¹ GK, A

K

[n]. Since unramified extensions ofKdo not split elements in this group we can use a well-known inflation-restriction sequence to change our base field fromKto the maximal unramified extensionK^urofK, compute the cohomology group over this larger field, and look for elements that are invariant under the Galois group ofK^ur/K which is topologically generated by (a canonical lift of) the Frobenius automorphismφ_q ofFq. Note that this automorphism acts both onG

K/K^ur

and onA[n] =A(K^ur)[n].

LetK_tamebe the unique cyclic extension ofK^ur of degreen(which has to be fully ramified).

We obtain

Proposition 6.5 The first cohomology groupH¹

GK, A K

[n]is equal to the group of elements in

Hom

G(Ktame/K^ur), A[n]

,

which are invariant under the natural action of the Frobenius automorphism.

After fixing a generatorτofG(K_tame/K^ur)we can identify ψ∈Hom(G(K_tame/K^ur), A[n]) with

ψ(τ) =:P_τ ∈A[n]

and henceHom(G(K_tame/K^ur), A[n])withA[n].

Warning: The identification ofHom(G(K_tame/K^ur), A[n])withA[n]is, in general, not com- patible with Galois actions. Here the cyclotomic character becomes important: overK(µ_n)we can realize a ramified cyclic extensionKnof degreenby choosing ann-th roottof a uniformizing ele- mentπofK. Sinceτmapsttoζntfor somen-th root of unityζnand the Frobenius automorphism φqmapsζntoζ_n^q we deduce thatφqoperates onτby conjugation, sendingτtoτ^−q.

Corollary 6.6 The first cohomology groupH¹ GK, A

K

[n] can be identified with the sub- groupA0of points inA[n]defined by

A0={P ∈A[n]|φq(P) = [q]P}.

§ 6.3 Pairings over local fields 119

We summarize what we have found up to now in the case wherenis a prime.

Proposition 6.7 LetK be a local field with residue fieldFq, letAbe an abelian variety defined overFq, letbe a prime number not dividingq, and assume thatA(Fq)contains no elements of order². DefineA₀={P ∈A[]|φ_q(P) = [q]P}.

The Tate pairing induces a nondegenerate pairing

·,·T,:A(Fq)[]×A0→Br(K)[].

Corollary 6.8 Under the assumptions of the propositions we get thatA0is (as an abelian group) isomorphic toA(Fq)[].

Example 6.9 Assume thatA[](Fq)is cyclic of orderand generated byP. 1. Assume that|(q−1).ThenA0=A[](Fq), andP, P_Fq,= 0.

2. Assume that q−1. Thenφ withφ(τ) = P is not in H¹ GK, A

K []. In particular ”P, PT,” is not defined. Of course we can extend the ground field until the pairing over these larger fields permits the argument(P, P). But the value will then necessarily be equal to0.

Example 6.10 IfA[](Fq)is not cyclic and |q−1then for all pointsP, Q ∈A[](Fq)we can formP, QT,but it is not clear whether there is aP withP, PT,= 0.

We now come to the discussion of the Brauer group. First one has

Theorem 6.11 The Brauer group ofKis (canonically) isomorphic toQ/Z.

More precisely, there is a map, the invariant map invK, such that for alln ∈ Nwe have an isomorphism

inv: Br(K)[n]→Z/nZ.

Thus computations inBr(K)boil down to the computation of the invariant map. A further study of the theory of local fields shows that this is closely related to the computation of the discrete logarithm inF^∗q (see [NGU 2001]).

This becomes more obvious if we replaceFq byFq(µn) = Fq^k withk minimal such thatn | (q^k−1). PutK1=K(µn)and letKnbe a cyclic ramified extension of degreenofK1. The value of the Tate pairing is then inH²(G(K_n/K), K_n^∗)[n], and elementary computations with cohomology groups yield that this group is isomorphic (canonically after the choice ofτ) toF^∗_qk/

F^∗_qk

n

. Proposition 6.12 Letnbe equal to a prime number, and letkbe as above. Assume thatA(Fq) contains no points of order².

The Tate pairing induces a pairing

·,·T,:A[](Fq)×A[](Fq^k)→F^∗q^k/ F^∗q^k

which is nondegenerate on the left, i.e., ifP, QT,= 0for allQ∈A[](Fq^k)thenP = 0.

6.3.2 The Lichtenbaum pairing on Jacobian varieties

In Proposition 6.12 we have described a pairing that can, in principle, be used to transfer the dis- crete logarithm fromA[](Fq)toF^∗_qk/

F^∗_qk

. However, looking at the conditions formulated in Section 1.5.2 we see that one crucial ingredient is missing: we must be able to compute the pairing very fast.

120 Ch. 6 Background on Pairings

The next two sections are devoted to this goal in the case that we are interested in.

We come back from the general theory of abelian varieties to the special theory of Jacobian varietiesJCof projective curvesCof genusgover finite ground fieldsFq.

We want to use the duality theory over local fields and so we liftCto a curveCof genusgover the local fieldKwith corresponding abelian varietyJ_Ce. We remark that the reduction ofJ_CeisJ_C. The central part of the construction of the Tate pairing was the construction of a2-cocycle from G²_K into K^∗. For a pointP ∈ J_Ce(K)and an element γ ∈ H¹

GK, J_Ce K

[n]we choose a 1-cocyclecinα⁻¹(γ)and define

c(σ₁, σ₂) =W_n

δ(P)(σ₁), c(σ₂)

whereδandαare the maps defined in Section 6.3.1 andWnis the Weil pairing.

In his paper [LIC1969] Lichtenbaum used sequences of divisor groups of curves to define a pairing in the following way.

PutC^__=C×K. We have discussed the group of divisor classes of degree0ofC^__together with the action ofG_K on this group in Section 4.4.4. As a consequence we get the exact sequence of G−K-modules (see 6.1)

1→PrincC^__→Div⁰_C^__ →Pic⁰_C^__→0.

We can apply cohomology theory to this sequence and obtain a map δ₁:H¹(G_K,Pic⁰_C^__)→H²(G_K,Princ_C^__) which associates toγ∈H¹(GK,Pic⁰_C^__)a2-cocycle fromG²_KtoPrincC^__.

In other words, givenγwe find for each pair(σ1, σ2)∈GKa functionfσ₁,σ₂ ∈K(C)^__ such that the class of(fσ₁,σ₂; (σ1, σ2)∈GK)is equal toδ1(γ).

Definition 6.13 The notations are as above. Letc∈Pic⁰_Cebe aK-rational divisor class of degree0 with divisorD∈c.

The Lichtenbaum pairing

·,·L: Pic⁰_Ce×H¹(G_K,Pic⁰_C^__)→H²(G_K, K^∗)

maps(c, γ)to the class inH²(GK, K^∗)of the cocycle G²_K →K^∗ given by

(σ1, σ2)→fσ₁,σ₂(D).

(HereD has to be chosen such that it is prime to the set of poles and zeroes offσ1,σ2, which is always possible.)

Since we have seen thatJC

K

= Pic⁰_C^__we can compare Tate’s pairing with·,·L. It is shown in [LIC1969, pp. 126-127], that the two pairings are the “same” in the following sense.

Proposition 6.14 For all natural numbersndenote by·,·L,n the pairing induced by·,·L on Pic⁰_e

C/nPic⁰_e

C×H¹(G_K,Pic⁰_C^__)[n].

Then·,·L,nis equal (up to sign) to the Tate pairing·,·T,napplied to the abelian varietyJ_Ce. In fact, Lichtenbaum uses this result to prove nondegeneracy of his pairing for a local fieldK.

The importance of Lichtenbaum’s result for our purposes is that we have a description of the Tate pairing related to Jacobian varieties that only uses objects directly defined by the curveC. In particular the Weil pairing has completely disappeared.

§ 6.3 Pairings over local fields 121

We can now use the general considerations given in Section 6.3.1 and come to the final version of the pairing in the situation that we are interested in.

Thus we come back to a curveCdefined overFq, choose a lifting toCoverK, and compute the groups occurring in the pairing. As a result we shall obtain a pairing that only involves the curveC itself.

The first group can be identified withPic⁰_C/nPic⁰_C. To avoid trivial cases we shall assume that this group is nontrivial, i.e., thatPic⁰_Ccontains elements of ordern.

The groupH¹(GK,Pic⁰_C^__)[n]can be identified with the subgroupJ0of points inPic⁰_C^__[n]defined by

J₀=

c∈Pic⁰_C^__[n]|φ_q(c) = [q]c .

A first application of these results is that we can describe the Lichtenbaum pairing by Galois coho- mology groups related to a finite extension ofK. We first enlargeKto a fieldK1that is unramified and such that over its residue field all elements ofJ0are rational. AutomaticallyK1contains the n-th roots of unity. It follows that there exists a cyclic ramified extensionKnof degreenofK1.

The image of the pairing·,·L,n will be contained inH²(G(Kn/K1), K_n^∗). Let us fix a gen- eratorτ of the Galois group ofKn/K1. We identifyγinH¹(GK,Pic⁰_C^__)[n]with the class of the cocycleζfromG(Kn/K1)given by

ζ(τⁱ) = [i]˜c for 0in−1, where˜c∈P ic⁰_e

C×K_n[n]is a lift ofc∈J0.

The image ofγunder the mapδ₁ will be a2-cocycle mapping fromτ × τtoPrinc_Ce×K

n

which we must describe. Choose a divisorD∈˜crational overK1. ThusiD∈ζ τⁱ

. By definition

δ1(γ) τⁱ, τ^j

=τⁱjD−ri+jD+jD

for0i, jn−1, whereri+jis the smallest nonnegative residue ofi+jmodulon.

Sinceτ D =Dit follows thatδ₁(γ) τⁱ, τ^j

= 0fori+j < nandδ₁(γ) τⁱ, τ^j

= nDfor i+j n.SinceDis a divisor of degree0in a class of ordernthe divisornDis the divisor of a functionf_D(which is defined overK₁).

Now choosec˜1∈Pic⁰_Ceand a divisorE∈c˜1such thatEis prime toD.

Then

c˜1+nPic⁰_Ce, γL,n

τⁱ, τ^j

= 1 ifi+j < nand

c˜1+nPic⁰_Ce, γL,n

τⁱ, τ^j

=fD(E) ifi+jn.

Thanks to this result we can immediately identify the element in the Brauer group ofK1which corresponds to˜c1+nPic⁰_Ce, γL,n: it is the cyclic algebra split byKncorresponding to the class of the pair(τ, fD(E))(cf. [NGU 2001]). If we fixτ the class is uniquely determined by the norm classfD(E) N_Kn/K1(K_n^∗).

SinceKn/K1is totally ramified we can computeN_Kn/K1(K_n^∗)to be (canonically) isomorphic toF^∗_qk/

F^∗_qk

n

.

Hence˜c1+nPic⁰_Ce, γL,nis uniquely determined by the image offD(E)in the residue fieldFq^k

modulo F^∗_qk

n

.

We can obtain this image directly: choosingc1 ∈ Pic⁰_C ,E ∈ c1,c ∈ J0,D ∈ c, andfD a function onCdefined overFq^kwith no zeroes and poles in divisors ofE, we have

c˜1+nPic⁰_Ce, γL,n=fD(E) F^∗_qk

n

.

122 Ch. 6 Background on Pairings