derived p-adic heights and p-adic l-functions

Derived P-Adic Heights and P-Adic L-FunctionsAuthor(s): Benjamin HowardSource: American Journal of Mathematics, Vol. 126, No. 6 (Dec., 2004), pp. 1315-1340Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/40067899 .

Accessed: 19/12/2014 21:06

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access toAmerican Journal of Mathematics.

http://www.jstor.org

This content downloaded from 169.230.243.252 on Fri, 19 Dec 2014 21:06:36 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=jhup

http://www.jstor.org/stable/40067899?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


DERIVED P-ADIC HEIGHTS AND P-ADIC L-FUNCTIONS

By Benjamin Howard

Abstract. If £ is an elliptic curve defined over a number field and p is a prime of good ordinary reduction for E, a theorem of Rubin relates the /?-adic height pairing on the p-power Selmer group of E to the first derivative of a cohomologically defined p-adic L-fiinction attached to E. Bertolini and Darmon have defined a sequence of "derived" p-adic heights. In this paper we give an alternative definition of the p-adic height pairing and prove a generalization of Rubin's result, relating the derived heights to higher derivatives of p-adic L-functions. We also relate degeneracies in the derived heights to the failure of the Selmer group of E over a Zp -extension to be "semi-simple" as an Iwasawa module, generalizing results of Perrin-Riou.

0. Introduction and notation. Fix forever a rational prime p > 2. By a coefficient ring we mean a commutative ring which is complete, Noetherian, and local with residue field of characteristic p. Fix a finite set of places £ of a number field F containing all archimedean places and all primes above /?, and let Gx = Gal (Fz/F) where F^ is the maximal extension of F unramified outside of Z. For any coefficient ring /?, denote by Modx(/?) the category of free /?-modules of finite type equipped with continuous, /Minear actions of Gj;. The notation M(k) for any G^-module M means Tate twist, as usual.

Throughout this article we work with a fixed coefficient ring O, which is assumed to be topologically discrete (at least until Section 4), for example O =

Z/pkZ. With F as above let Foo/F be a Zp-extension. If Fn c Foo is the unique subfield with [Fn : F] =pn, we define

T, = Gal (Fn/F) T = Gal (F^/F) An = O[Tn] A = OUT]],

and denote by J the augmentation ideal of A. Let 7 E F be a topological generator and denote by 1 the involution of A induced by 7 h> 7"1. We will also view t as a functor M \-> ML from the category of A-modules to itself, where the underlying group of Ml is the same as that of M but with A acting through A^>A-+Endo(M).

In Section 1 we consider two objects 5, T of Mod^CO) which are assumed to be in Cartier duality: i.e. we assume that there exists a perfect O-bilinear,

Manuscript received January 16, 2003; revised July 8, 2003.

Research partially conducted by the author for the Clay Mathematics Institute.

American Journal of Mathematics 126(2004), 1315-1340.

1315



1316 BENJAMIN HOWARD

Gs-equivariant pairing 5 x T- >O(l). For such objects we define generalized Selmer groups

tfJKF^oo) C KmH\Fn9S) Hlg(F, 7^) C lim//1^, T)

and show that there is a canonical (up to sign) height pairing

whose kernels on either side are the submodules of universal norms in the sense of Definition 1.3.

We continue to work in great generality in Section 2, using the construction of Section 1 to define derived height pairings similar to those of Bertolini and Darmon. A theorem of Rubin [14], Thm. 1, relates the p-adic height pairing to the special values of the first derivatives of certain (cohomologically defined) p-adic L-functions, and we prove similar formulas relating the derived heights to special values of higher derivatives.

In Section 3 we consider the special case where S and T arise from torsion points on an abelian variety with good, ordinary reduction at primes of F above p, and show that our Selmer groups agree with the usual ones (up to a controlled error).

In Section 4 we continue to work with torsion points on an abelian variety A, and explain how degeneracies in the derived heights are reflected in the structure of the Selmer group of A over F^ as an Iwasawa module, generalizing work of Perrin-Riou [12].

The reader is encouraged to begin by reading the results of Section 4, in particular Theorems 4.2 and 4.5, and Corollary 4.3.

1. Construction of the p-adic height pairing. We wish to work with a very general notion of Selmer group, borrowing some notation and conventions from [9].

Definition 1.1. Suppose R is a coefficient ring and M is topological /{-module equipped with a continuous /{-linear action of G%. A Selmer structure, T, on M is a choice of /?-submodule

HxT{Fv,M)dH\Fv,M)

for every v e I. Given a Selmer structure, the associated Selmer module //jr(F, M) is defined to be the kernel of

Hx (Gz, M) -> 0 Hl (Fv, M)/HlT(Fv, M).



DERIVED P- ADIC HEIGHTS AND P- ADIC L-FUNCTIONS 1317

If we are given a Selmer structure T on M and a surjection M - > M', we obtain a Selmer structure on M' (still denoted T) by taking the local condition at any v G X to be the image of

HXT{FV,M)^H\FV,M'\

If instead we are given an injection M1 - M then we define a Selmer structure on M' by taking the preimage of Hjr(Fv, M) under

H\FV,M')^H\FV9M)-

We will refer to this as propagation of Selmer structures. If (7\ tt) belongs to Mods(0), then induction from Gal (Fz/Fn) to Gal (FZ/F)

defines a module (rn,7rn) in Modi(Aw). Explicitly,

Tn = {f: Gz-+T\ f(gx) = 7r(g)f(x) Vg G Gal (Fz/Fn)}.

Here Gx acts by (7rw(g)/)(x) =/(jcg) and An acts by (7/)(x) = ir(<y)f(>y-lx) where

7 is any lift of 7 G Tn to Gx. Define

roo=lim71w riw = limrw

where the direct limit is with respect to the natural inclusions (restriction) and the inverse limit is with respect to the norm operators in An (cores triction). Let ev: Too - T be evaluation at the identity element of G^. By Shapiro's lemma the

composition

Hl(F, Too) =f WiFoo, Too) ̂ //'(Foo, 71)

is an isomorphism, and similarly we have

Hi{F,Tiv/)*]imHi(Fn9T).

An element A G A is said to be distinguished if A 0 mA, where m is the maximal ideal of O.

Lemma 1 .2. Ifgn is any sequence of elements of A which converges to zero and

f G A is distinguished, thenf | gnfar n > 0.

Proof Identify A with a power series ring. The map taking A G A to its remainder upon division by/ is continuous, and so if we write gn = qj + rn with

degree of rn less than that of/, we have rn - 0 with the rn running through a discrete set (recall O is assumed discrete).



1318 BENJAMIN HOWARD

If we fix a topological generator 7 G F and let gw = 7^_11 be the norm element in An, the above lemma shows that m G M is divisible by every distinguished / G A if and only if m G g^Af for every n. This motivates the following:

Definition 1.3. If M is a A-module, we say that m G Af is a universal norm if m G /M for every distinguished / G A.

We denote by c^taut- Gx- >F- >AX the tautological character, and let A{&} denote the ring A, viewed as a module over itself, on which G% acts through u;faut. The unadorned symbol A is always interpreted as A{0}, i.e., with trivial Galois action. For any object M on which A and Gx act we let M{k} = M <8>a A{£}, and we regard the underlying A-modules of M and M{k} as being identified via m h- m ® 1.

Lemma 1.4. For/ G Trt am/ 7 G rm sef /i/(7) = ev(7"1/) G 7\ 77k? map f 1- 5Z7 A*/(7) ® 7 defines an isomorphism in ModsCA^)

r« = r®oA«{-i}.

7n the limit this defines an isomorphism 7iw = T <S>o A{- 1}.

Proof. This amounts to verifying the relations

M7(/(7) = M/CT^V) VnnigVil) = <S) (M/(^taut(g)7))

which are elementary.

Let K denote the ring obtained by localizing A at the prime generated by the maximal ideal of O, i.e., inverting all distinguished elements. Applying Lemma 1.2 one may easily check that K is noncanonically isomorphic to the ring of Laurent series with non-essential singularities in one variable over O. Define P (the module of poles) by exactness of the sequence

0->A^K-^P^0.

For any T in Mod%(O) we tensor the above sequence (over A) with 7iw to obtain

0^7iw^7*-+7W0.

By Lemma 1.4 there are canonical identifications of A-modules and Gx-modules Oiw ̂ A{-1}, OK ̂ K{-1}, and OP * />{-l}.

Lemma 1.5. A choice of topological generator ofT determines an isomorphism

TP ̂ Too.



DERIVED P- ADIC HEIGHTS AND P- ADIC L-FUNCTIONS 1319

If the map associated to the generator 7 is denoted 7;7 and u € Z* then

Proof. The map ?77 is described as follows. Any element of Tp[^pn - 1] may be written as r ® (7^ - I)"1 with r G Tiw. Such an element is sent by 7/7 to the

image of r in Tn = T^^ - 1]. The relation between 7?7 and t]^ follows from

^r^j = u (mod rf ~ 1)A)' D

Lemma 1 .6. Suppose e\ SxT-^O(l)isan O -bilinear, G^-equivariant, perfect pairing of objects in Modx(O). There is an induced G^-equivariant and perfect pairing

en: SnxTn^An(l)

which satisfies

en(Xs, t) = Xen(s, t) = en(s9 \Lt).

One may pass to the limit and then tensor with K to obtain perfect pairings

eui'. Siw x 7iw -> A(l) eK: SKxTK^ K(l).

IfS = T and the pairing on S x T is symmetric (resp. alternating) then the

induced pairings satisfy e.(s9 1) = e.(t, s)L (resp. e.(s, i) = -e.(t, s)L).

Proof Using the identifications of Lemma 1.4, the pairing is defined by en(s,t) = E7Grw M7)®7 where ̂ (7) = Zxern e{^s(x\iit{x^-x)). It is elementary to check that the stated properties hold.

Lemma 1.7. Let T be an object of Modz(£>), and let v be a prime of F not

dividing p. The map

Hlum(Fv,TK)-+Hlum(Fv,TP)

is surjective. If vis finitely decomposed in Fqo, then H\m(Fv, Tp) = 0.

Proof Let X C Gal(F^/F^) be the inertia group of v. Since

TK = T ®zp K{-\] and the restriction of o;taut to J is trivial, (TK)X = T1 ® K.

Similarly, (TP)X = Tx ® P, and so (TK)J surjects onto (Tp)1. Using the fact that

Gal (F"m/Fv) has cohomological dimension one we deduce that the map

Hl(F™/Fv, (TKf) -> H\FT/FV9 (TPf)

is surjective, proving the first claim.



1 320 BENJAMIN HOWARD

If v is finitely decomposed in Fqq, then Lemma 1.5 and Shapiro's lemma allow us to identify

ûnr(^> Tp) *= ̂ unr(^V> ôo) *= £]7 ̂ unr(ôo,w > ̂ H*

where the sum is over places of Foo above v. The pro-p-part of Gal (F"nr/ôo,w) is trivial, and so the right-hand side is zero.

The construction of the pairing of the following theorem, as well as the verification of its properties, is a modification of the construction of the Cassels- Tate pairing as described in [6].

Theorem 1.8. Suppose S and T are objects in Modi(O) and that there is a perfect G^-equivariant pairing S x T-+O(l). Suppose further that we are given Selmer structures T and Q on Sk and Tk, respectively, which are everywhere exact orthogonal complements under the pairing

SKxTK^K(l).

Then there is a canonical pairing

[, ]:Hlr(F9SP)xHlg(F,Tp)^P

whose kernels on the left and right are the images ofHjr(F, Sk) and Hg(F, Tk), and these images are exactly the submodules of universal norms. This pairing satisfies [As, t] = A[s, t] = [s, \Lt]. IfS = T, T = G, and the pairing onT xT is symmetric (resp. alternating) then we also have [s, t] = [t, s]L (resp. [s, t] = - [t, s]L).

Proof Given cocycles s and t representing classes in Hjr(F, SP) and Hg(F, 7», respectively, choose cochains

seCl((h,sK) tec\(h9TK)

whose images under the maps induced by 5a: - Sp and Tk - > 7> are s and t. By the definition of propagation of Selmer structures, there are exact sequences

(1) HXT{FV, Siw) - Hlf(Fv, SK) -> HlT(Fv, SP) -> 0

Hlg(Fv, 7iw) -> Hlg(FVi TK) -+ Hlg(FV9 TP) -. 0

at every ^Gl, and so we may choose semi-local classes

he® HXT(FV, SK) he® Hlg(Fv, TK) vex vex



DERIVED P-ADIC HEIGHTS AND P-ADIC L-FUNCTIONS 1321

which reduce to the semi-localizations

loc2 (s) e 0 Hljr(Fv, SP) locz (0 € 0 Hlg(Fv, 7».

From the fact that s and t are cocycles it follows that the image of ds U di in (^(GziPil)) is trivial, and so also are the images of d(ds\Jt) = dsUdi = d(s\Jdf). Using H3(Gj:,P(1)) = 0 we may therefore choose eo,ei G C2(Gx,P(l)) such that

de0 = ds U ? de\=sUdi

in Z^G^PU)). Writing invi: 0y€l//2(Fi;,JP(l))-^P for the sum of the local invariants, we now define

(2) [s, t] = invz (locz (s) Uh~ locz (co)) = - invz (& U Iocs (?) + locs (ci ))

where the second equality follows from ( loc^ (s) - s^) U ( loc^ (?) - ?z) = 0 in

Q)velHl(Fv,P(l)) and the reciprocity law of class field theory. It is elementary to check that this is independent of the choices made (or see Flach's paper for

essentially the same calculations). Furthermore, it is clear from the construction that the kernels on either side contain the images of Hjt(F,Sk) and Hg(F, 7a:).

For a A-module M, we write Mv = HomA (Af\ P). If M is a topological group on which Gx acts continuously we define

UI'(F, M) = ker ( //'(Gs, M) -+ JJ H!(FV9 M) ] .

The Poitou-Tate nine-term exact sequence provides a perfect pairing

W2(F,Slv,)xUl\F,TP)^P

which defines the right vertical arrow in the exact and commutative diagram

0 > Hlf(F,Sp)°/K > Hlf(F,SP)/K

> UI2(F,5iw)

i i 4(F,rP)v

> Wl(F,TP)V.

Here HlT(F,SP)° denotes HlT(F,SP) intersected with the image of Hl(F,SK) in

Hl(F,SP) and the subscript /K indicates quotient by the image of Hjt(F,Sk) in




Hjr(F, Sp). The top row is extracted from the cohomology of

using exactness of (1). The left vertical arrow is induced by the pairing of the theorem, and a dia-

gram chase shows that to check injectivity of this arrow it suffices to show that

HlT(F,SP)°/K injects into Hxg{F, 7»v. In other words, if s G HlT(F9SP) is in the kernel on the left then we are free to assume that s is chosen to be a cocycle and that e0 = 0. Then we have invs (locz (s) U h) = 0 for every h G (&vel Hlg(Fv, TK) whose image in (&vexHg(Fv,Tp) comes from a global t G Hg(F,Tp). Denote by c the image of loc2(S) in (BveItHl(Fv,SK)/HlT(Fv,SK). It follows from the exactness of (1) that there is a class

d G @H\Fv,Slv,)/Hlf(FV9Slw)

whose image in ®veIiHl(Fv,SK)/Hljr(Fv,SK) is equal to c. Then

invs ( d U Iocs (t)) = inv£ (c U h) = invzOocxC^uFx) = 0

in P for every t G Hxg{F, 7». It follows from Poitou-Tate global duality that d is the image of a global class 6 G /^(G^Siw), and that s - 6 G Hjt(F,Sk) reduces to s G Hjr(F, Sp). This and a similar argument with the roles of S and T reversed show that the kernels on the left and right are contained in the images of

Hlf(F, SK) -+ i/>(F, SP) Hlg(F, TK) -> Hlg(F, 7»

respectively. The image of Hjr(F, Sk) is clearly contained in the universal norms which are clearly contained in the left kernel, and similarly for 7\ and so the kernels on either sides are exactly the universal norms.

The final claim regarding the case S = T follows from the two descriptions of the pairing in (2) and the skew-symmetry of the cup product. Details can be found in Flach's paper.

Lemma 1.9. Let

Ooo{l} = lima{l} = limAw{0},




so that Ooq{ 1 } is just Ooo with Gz acting trivially. For any A-module of finite type, At the map ev: 0oo{l} - > O induces a canonical isomorphism in Mod^C?)

HoniA (A, Ooo{l}) * Homo (A, O).

Proof. This is a special case of Frobenius reciprocity. The inverse map may be described explicitly as follows: for <\> G Homo (A, O) we must have

<t>((lpn - 1)A) = 0 for n > 0, by discreteness of O and continuity of <j>. Taking n

very large we then define O G HomA (A, On{\}) by <S>(a)(g) = <Kv\ant(g) ' a) for

g€Gx.

Keep the assumptions of Theorem 1.8. Fixing a topological generator 7 of F, the isomorphisms 7/7 of Lemma 1.5 determine Selmer structures, still denoted

by T and Q, on 5qo and 7^, and these Selmer structures do not depend on the choice of 7. The composition

(3) P^C?p{1}Ôoo{1}Ô

allows us to construct from the pairing of Theorem 1.8 a pairing

hy. HlT(F, Soo) x HlG(F, T^) -> O

whose kernel (by Lemma 1.9) on either side is exactly the submodule of universal norms.

Lemma 1.10. The above pairing satisfies (a)/i7(AÔ = MÂ'O (b)forue Z*, hîsj) = u~lh7(s,t) (c) ifS = T,Jr = G, and the pairing S x T- (9(1) is symmetric (resp. alter-

nating) then /i7 is alternating (resp. symmetric).

Proof. Let 07: P- O be the composition (3), so that

/i7(5,r) = 07([r/71W,r?71(O]).

The first equality is then immediate from the corresponding property of the pairing [ , ]. For the second property, we compute

hjU(s,t) = ̂ ([^(slrj-Jit)]) = w-2^^^1^),^1^)])

and so it suffices to check 07« = m07, which is clear from the definition. For the third property we must show that (j)(pL) = -0(p) for p G P. If we write p =

:^rry




for some integer n > 0 and A G A, then

( A v= -*

in P, and so under the isomorphism P^Ooo{l} = lim An the action of t on P becomes minus the natural action of i on limAn. For A G An, ev(A*0 = ev(A) and the claim follows.

Part (b) of the lemma implies that the pairing of the following theorem is well defined.

Theorem 1.11. Keep the assumptions of Theorem 1.8 and let J C A be the augmentation ideal There is a canonical pairing

h: H^Soo) x H^F.T^-tJ/J2

defined by h(s, t) = /i7(s, 0(7 - 1) where 7 is any topological generator ofT. This pairing satisfies h(Xs, t) = h(s, \Li) and the kernels on either side are exactly the universal norms. If S = T, T = G, and the pairing T x T-+O(l) is symmetric (resp. alternating) then h is alternating (resp. symmetric).

Proof. All of the claims follow easily from Lemmas 1.9 and 1.10, and the properties of the pairing of Theorem 1.8.

Remark 1.12. Keeping the notation of the thoerem, suppose L is a subfield of F with Foo/L Galois, and assume that the action of Gf on 5 and T extends to an action of Gl. Then for • = Iw, P, K, or 00, the action of A on Hl(F,S.) extends to an action of AL = (9[[Gal (Foo/L)]]. Similarly for every place v of L the action of A on the semi-localization

®H\FW,S.) w\v

extends to an action of AL. If we assume that the local conditions T and Q are stable under the action of AL, then1 AL acts on the associated Selmer groups. The action of Gal (F/L) on T determines a character

lj: Gal(F/L)->Z^

by 0-7(7" l = 7a;(<7) for every a G Gal (Foo/L) and 7 G T. Then for a G Gal (Foo/L) it can be shown that h(s°, f) = u(a) • h(s, t).




2. Derived heights and derivatives of L-functions. Throughout this section we work with fixed objects 5 and T of Mod^(O), and assume that these modules are in Cartier duality. Let T and Q be Selmer structures on Sk and 7>, respectively, which are everywhere exact orthogonal complements under the

pairing of Lemma 1 .6, and let

h: HlT(F, Soo) x Hlg(F, 7^) -> J/J2

be the height pairing of Theorem 1.11. We abbreviate

Ys = //>(F, Soo) YT = Hlg(F, Too).

Let Y = Ys or YT. For any r > 1 set 6r(Y) = Y[Jr]/Y[Jr~xl A choice of

topological generator 7 e F determines an injection

0r,7: 6r(Y)^Y[J]

given by (j)r^{y) = (7 - l)r~1>;» and we denote its image by y(r) C F[7]. This

image is independent of the choice of 7 and defines a decreasing filtration

• • . C F(3) C Y(2) C r(1) = Y[J].

The intersection ny(r) consists of the elements of Y[J] which are universal norms in Y.

Remark 2.1. We define a Selmer structure on 5 by propagating T through the natural inclusion S- Soo, and again denote this by T. This inclusion induces a surjective map

Hlf(F,S)->Ys[J]

whose kernel is bounded (by the inflation-restriction sequence) by the order of the group H^F^/F^iF^S)). Similar remarks hold for T.

Definition 2.2. We define the rth derived height

by the composition

y<r> x ^*^6r(ys) x y^y/y^-JrV//"1.

It is easily checked that this is independent of the choice of 7. Note that /i(1) is

nothing more than the restriction of h to Ys[J] x Yt[J].




If S = T and T = <?, then for any x,y G y5,

AW/ *'y " _ /( " W&Xyrf if * symmetric

*'y " _ [( - I)r+1/z(r)(j,jc)) if e alternating.

The following lemma implies that the kernels of /i(r) on the left and right are

lf+1) and Y{Tr+l\ respectively.

Lemma 2.3. If Ao, Ai G A are distinguished then the kernels on the left and right of the restriction ofh to Y$[\o\ x ^[Ai] are the images of

rs[AoAi]^ys[Ao] lVWAi]^yr[Ai]

Pwo/ Let Z5 and Z^ be the quotients of Ys and y^ by the submodules of universal norms. The natural map

Ys[\o]/\\Ys[\\\o]-+Zs[\o]/\\Zs[\\\o]-^Zs/\\Zs

is an injection, and the height pairing defines an injection

ZS/X\ZS * Uom{ZT[\xlJ/J2)^Hom(YT[\x],J/J2).

This shows that

istAoA^] ̂ l Ys[\q\ -^Hom(yr[Ai],//y2)

is exact. The kernel on the right is computed similarly.

The pairing e\w of Lemma 1.6 induces a perfect pairing

eP: Siwx7>^P(l).

The identification 7> = T^ of Lemma 1.5 and the map (3), both of which depend on a choice of topological generator, induce a perfect pairing

£oo: Siw x Too -+0(1)

which does not depend on the choice of generator. If one views 5iw as the space of 5- valued measures on T, and T^ as the spaces of locally constant T- valued functions on T, then this pairing is integration. Using Lemma 1.9, it can be checked that the local conditions T and Q on 5iw and T^ are everwhere exact




orthogonal complements under the local Tate pairing

induced by e^. Denote by ^rrel the Selmer structure on 5iw whose local conditions at places

of F not dividing p are the same as those of T, but with no condition imposed at primes above /?. At any place v of F we denote

H/j?(Fv, ^iw) = H (Fv, S\vi)/H<F{FV, 5iw),

and we let

hI(fp> ) = ®Hl(FV9 ) v\p

denote the semi-local cohomology at p, where • is either T or / T. Using the fact that the local condition T on 5iw is propagated from a local condition on S# , it is

easy to see that Hl^iF^S^lf] = 0 for any place v of F and any distinguished

For the motivation behind the following definition, see [13], [14], or [15].

Definition 2.4. For any element

z = {zn} e lim//^,(F,5n) S #^,(F,Siw)

define the /?-adic L-function of z, Cz, to be the image of z in

H}jr(Fp, 5iw) = Horn (Hlg(Fp, 7^), O).

Define the order of vanishing of £z, ord (£z), to be the largest power of J by which Cz is divisible in //^(Fp,5iw). Equivalently, ovd(Cz) is the largest integer r such that

A(4(Fp,roc)[/r]) = o.

For any topological generator 7 G F and any r < ord (Cz) we define Der^ (£z) to be the preimage of Cz under the injection

FL l<p(Fp, Siw) -^ H /jr(Fp, 5iw).

Define

£«: HfrFptToo^r/J"1




by C{r)(c) = Der^(£z)(c) • (7 - l)r. Then C{r) is independent of the choice of

7. The restriction of C[r) to Hlg(Fp, T^J] should be thought of as the "special value" of the rth derivative of Cz, and we denote it by A^r).

The following is a higher derivative version of Theorem 1 of [14]. Note that the theorem asserts that a local divisibility (of the p-adic L-function Cz by a

power of /) implies a global divisibility (of zo by a power of J).

Theorem 2.5. Keep notation as above, with r < ord(£z), and propagate the Selmer structure T

' to S via S\w -> S. This is equal to the Selmer structure obtained

by propagation through S - S^. Then (a) \{0) = 0 if and only ifzo G H^iF, S), (b) \{r) = 0 if and only ifr < ord (£z), (c) suppose 0 < r < ord (Cz), then zo G 1^ and for any c G Yj

h(r\zo,c) = \zr\cp),

where cp is the image ofc in Hg(Fp, T^).

Proof Fix a topological generator 7 G F. We have A^0) = 0 if and only if Cz is divisible by /. The first claim now follows from exactness of

H}jr(Fp9 Siw) 7^ H}r(Fp9 Slw) -+ H)T(FP, 5).

For the second, r < ord (Cz) if and only if Der^ (Cz) is divisible by 7 - 1 in

HJpiFpjSiw), and this is equivalent, by local duality, to Der^(£z) vanishing on the /-torsion in Hxg(F, T^).

For the third claim, first suppose zo G Y^ and fix c G Yj\ Let z G

H^iiF.SK) be defined by z = z ® (7 - l)~r and let y denote the image of z in H^(F9Sp). Under the map

H\F9Sp)^Hl(F9Soo)

of Lemma 1.5, (7 - l)r~ly maps to zo. Since we are assuming that zo G Y^r) and c G Y{j\ we may choose s G Hljr(F,SP) and d G Hxg(F, 7» which satisfy

(7 - iy-%(s) = zo (7"1 - iy~lvj(d) = c.

We have (7 - iy~lri^(s - y) = 0, and so

h{r\zo,c) = (7-iy*"1 -Krj^slc) = (7-ir1-/i(r?7(y),c).




Unraveling the definition of h, we find

h(rh(yU) = (7 - 1) • h ([^(T"1 -

Dr"^])

where 07 is the composition (3) and [ , ] is the pairing of Theorem 1.8. Choose a lift, ~dp, of locp (d) to Hlg(Fp, TK\ and let cp = (7-1 - \)r~xdp. We now have

07 ([^(T"1 -

l)r"^]) = <£7(inv/> ((7

- i)r~lyudp))

= ^(inv^zUCT-'-ir^))

= Der;(A)(cp).

Combining all of this gives

/*(r)(zo, c) = (7 - Dr • Der^ (Cz)(cp) = \zr\cp).

We now show by induction on r that zo £ Y^ for 1 < r < ord(Cz). The case r = 1 follows from parts (a) and (b): since 0 < ord(£z) we must have

zo € #jr(F,Soo)[/]. For the inductive step, if zo € l^"^ then we have shown that

*(r-1Wc) = A<r-1)(cp)

for every c G F^"^. But since r - 1 < ord(£z), part (b) of the proposition implies that A^"1^^) = 0 and we conclude that zo is in the kernel on the left of

/j(r-i) jhjg kernel js exactly Y^sr\ and the claim is proven.

3. Selmer groups of ordinary abelian varieties. Let A be an abelian vari-

ety defined over F and let Av be the dual abelian variety. We assume throughout that A has good ordinary reduction at all primes of F above /?, and that the primes of bad reduction are finitely decomposed in Foo. Assume further that p does not

ramify in F, but that all primes of F above p do ramify in F^. We wish to prove the following: for each power pk of p there are generalized Selmer groups

ffJr(Foo,A[/>*]) C Jf^FocAIp*]) H^v(Foo,Av[/]) C H1(Foo,Av[pA:])

such that the inclusion A[pk] ̂ -> A[p°°] induces a map of A-modules

tfkFoo, A[pk]) -> Selpoo (A/Foo)[/]

whose kernel and cokernel are finite and bounded as k varies (and similarly for

Av), where Sel^oo (A/Fqo) is the usual p-power Selmer group associated to A.




These generalized Selmer groups are of the type described in Section 1, and so there is a height pairing

hk: Hjr(Foo,A[/i*]) x Hlfv(Foo,Aw[pk])^Jk/J2k

where Jk denotes the augmentation ideal of (Z/pkZ)[[r]]9 and this pairing enjoys all the properties of that of Theorem 1.11.

For every place of F, fix once and for all an extension to F. At any place v of F above p and for any k we let Filv A[pk] be the kernel of the reduction map

A(Fv)[pk]^A[pk]

where A is the reduction of A at v. Define Filv Aw[pk] similarly. Define gr vA[pk] by exactness of

(4) 0 - Fil, A[pk] ̂ A(Fv)[pk] - gvvA[pk] -+ 0

and similarly for Av. The reduction map on pk -torsion is surjective, and so

grvA[pk]*A[pk].

Lemma 3.1. The submodules ¥'\\v A[pk] and Y\\v Av[pk] are exact orthogonal complements under the Weil pairing.

Proof. The assumption that A has ordinary reduction ensures that Fil^ A[pk] and Filv Av[pk] have exact order pkg, where g = dim(A). As modules for the inertia group Iv of v, each is isomorphic to a product of copies of fi^c, and there are nonontrivial Xv invariant pairings /A x /A -

>/i^.

We set O = Z//Z, S = A[pkl and T = Ay[pk] and use the notation of the first section. Shapiro's lemma and Lemma 1.5 allow us to identify

(5) /^(FocAtp*]) ^ i/k^Soo) ^ Hl(F,SP).

For • = Iw, K, P, or oo, and any place v of F above p, the submodule Fil^ S C S induces a submodule Fil^S, C S. in an obvious way, and similarly with S replaced by T or with Fil^ replaced by gry.

Following Coates and Greenberg [4], we make the following:

Definition 3.2. We define a Selmer structure, T, on S# by setting

hUf r "' s K )=iH^F"^) *vKp hUf r "' s K \xm&ge{H\Fv,Yi\vSK)^H\Fv,SK)) ifv\p

and define Ty on Tk similarly.




The local conditions T and Ty are everywhere exact orthogonal complements under the local Tate pairing induced by the Weil pairing and Lemma 1.6. We use the Selmer group Hjr(F, S» and the identification (5) to define a Selmer group Hjr(Foo,A[pk]), and make the definition for Av similarly.

We must compare these generalized Selmer groups with the usual definitions. Let

Fil<, S = lim ( Fil^, 5^) S = lim 5^

where the limits are over k. Shapiro's lemma identifies Hl(F, S) = Hl(Foo, A[p°°]) and for any place v of F

Hl(Fv,S) * lim0H1(Fn,vv,A[p00])

where the sum is over places w of Fn lying above v .

Definition 3.3. Define the ordinary Selmer structure on S[pk] by

i k _(Hlm{Fv^[pk}) else

Hord(Fv,S[p i k ]) -

|.mage î^ p^g^jj^î^s^ ifv]p

and let /^(F.S) = lim^rd(F,S[/]).

Proposition 3.4. The isomorphism Hl(F,S) = H^FÂip00]) identifies

»id(^S)^Selpoo(A/Foo).

Proof. Let X be the set of primes of F containing all archimedean primes, primes above p, and primes at which A has bad reduction, and let F^ be the maximal extension of F unramified outside Z. Then both Selmer groups are defined as the subgroup of H^F^/FooÂlp00]) of elements which are locally trivial at every v G X not dividing p (this follows from Lemma 1.7 and Proposition 1.6.8 of [16]), and are in the kernel of reduction

HHFÂlp^V^H'iFÂlp™])

at places above p. This description of the image of the Kummer map for w \ p can be found in [4], in particular Proposition 4.3.

Proposition 3.5. There are natural maps

H^Foo, A[pk]) -> <d(F, S[/]) -> <d(F, S)[/]

whose kernels and cokernels are finite and bounded as k varies.




Proof. For the first arrow, let S = A[pk], O = Z/p*Z, and recall the identifications S[pk] = Sqq = Sp. The first arrow becomes the inclusion

(6) Hljr(F,Sp)cHlord(F,SP).

The local conditions defining both Selmer groups are unramified (using Lemma 1 .7 for T) away from p. Above /?, the local condition Hjr(Fv, Sp) is defined as the image of the composition

H{ (Fv, Fil, SK) -> H l (FV9 Fil, SP) -> Hl (FV9 SP),

while the ordinary local condition is defined as the image of the second arrow. It follows that we have injections

HlOTd(F,Sp)/Hlf(F,Sp) - @Hl(Fv,Fi\vSp)/H\Fv,FilvSK) V

-> 0H2(F,,Fil,5iw) V

where the sums are over the primes v of F above p. By local duality and Shapiro's lemma the order of H2{FV,¥'\\VS\VJ) is bounded by

©flV^gr^oo) S 0iiv(LH,)[/>*] v w\p

where T = A v [/?*], the second sum is over all primes of Fqq above p, L^ denotes the residue field of Foo at w, and Av is the reduction of Av at w. This group is finite and bounded as k varies, by the assumption that all primes of F above p ramify in F^.

To control the kernel and cokernel of

(7) HUF, S[/]) - HUF, S)[pk]

we use the exact sequence

0 -> A(FOO)[/7OO]//A(FOO)[/7O°] -, H^F, S[/]) -. H^F, S)[/] -^ 0

The kernel of (7) is bounded by the order of AiF^p00] modulo its maximal divisible subgroup. By the snake lemma, to bound the cokernel of (7) it suffices




to bound the kernel of

0 Hl (Fv, S[pk])/HUFv, S[pk]) - 0 H\FV, S)/HUFv, S) V V

and we compute this kernel term by term in three cases.

Suppose first that v does not divide /?, and that A has good reduction at v. Then the kernel of

(8) H\FV, S[pk])/HUFv, S[pk]) -> H\FV, S)///^, S)

injects into the kernel of Hl(Flm,S[pk])^Hl(Flm,S) which is

H°(Flm, S)/pkH°(F™T9 S) * S/pkS = 0.

If v does not divide p and A has bad reduction at v, then our assumption that v is finitely decomposed in F^ implies that

HlOTd(Fv,S) = HUFv,S[pk]) = 0

by Lemma 1.7, and so we must bound the kernel of the map

Hl(Fv,S[pk])^H\Fv,S).

This kernel is H°(FV, S)/pkH°(Fv, S) which has order bounded by the size of the

quotient of 0 AiFoo^lp00] by its maximal divisible subgroup, where the sum is over primes w of Foo above v.

Lastly, if v \ p it suffices to bound the kernel of

Hl (FV9 ffv S[/]) -> Hx (FV9 grv S).

This kernel is controlled by the order of

f/V,,gr,S)^0A(LJ[/?~] w

modulo its maximal divisible subgroup, where the sum is over primes of F^ above p, L^ is the residue field F^ at w, and A is the reduction of A at v.

4. Semi-simplicity of Iwasawa modules. Set A = Zp[[ Gal (Foo/F)]], let J be the augmentation ideal of A, and denote by /„ C A the kernel of the natural

projection A - > Zp[ Gal (Fn/F)], so that /o = /. Keep A/F as in the preceeding section, so that A has good ordinary reduction at all primes of F above /?, the

primes of bad reduction are finitely decomposed in Foo, P does not ramify in F, and all primes of F above p do ramify in F^.




Let Yk = //jrCFocAt/?*]) be the generalized Selmer group of Section 3, and set Yk(Fn) = Yk[In], We denote by

. . . c y(3) c y(2) c jKl) = yk(F)

the filtration of Section 2. The surjection A[pk+l] -Â[pk] induces a map Y%\ -+ F^r), and we define

1*2 = lim Y(kr) YîFn) = lim r*(Fn)

so that Yj£ defines a decreasing filtration of Yoo(F). Set

Y = Selôo (A/Foo) X = HomZ/, (F, Qp/Zp).

These are cofinitely and finitely generated A-modules, respectively. By Proposi- tions 3.4 and 3.5 we have canonical maps Yk - Y[pk] with kernel and cokernel finite and bounded as k varies, and these induce maps

r<r) s n^r]/n[/r"1]^(^r]/^r~1])[/]

whose kernels and cokernels are again finite and bounded as k varies. The Zp- module Y[Jr]/Y[Jr~l] is cofinitely generated, and so

(9) nmk(yg) = rank (lim(Y[Jr]/Y[Jr-l])[pk])

= corankCytri/nT"1]) = rank(Jr-lX/JrX).

Lemma 4.1. Define SP(A/Fn) = lim Sey (A/Fn). There is a canonical isomorphism

Yoo(Fn) ® Qp i. 5P(A/FW) ® Qp.

Furthermore this isomorphism is semi-integral in the sense that there are integers to, t\, independent ofn, such that p^jiY^Fn)) is contained the lattice generated by SP(A/Fn), andptxj~x{Sp{A/Fn)) is contained in the lattice generated by Yoo(Fn).

Proof. We have maps

Yk(Fn) -> Selôo (A/Foo)[In +pkA] +- Selôo (A/Fn)[pk]

whose kernels and cokernels are finite and bounded as both k and n vary (the second arrow by Mazur's control theorem [8]), and so taking the inverse limit in




k and tensoring with Qp we obtain a semi-integral isomorphism

(10) Y^iFn) ® Qp <* ( lhn Sel^oc (A/Fn)[pk]) ® Qp.

For every k there is a canonical surjection

(11) Sey (A/Fw) -> Selpoo (A/Fn)[p*]

and these maps are compatible, in the obvious sense, as k varies. As k varies the kernels are uniformly bounded by the order of the finite group A(Fn)[p°°], and so passing to the inverse limit over k and tensoring with Qp, we see that the right- hand side of (10) is isomorphic to SP(A/Fn) <g> Qp. Surjectivity of (11) implies that this isomorphism identifies the lattices generated by limSelpoo (A/Fn)[pk] and SP(A/Fn).

^~

The subspace of SP(A/F) ® Qp generated by the image of Y]£ under the

isomorphism

(12) Yoo(F) ® Qp -> SP(A/F) ® Qp

will be denoted S^{A/F\ and we set S^°\A/F) = DrS{pr)(A/F). Let Wr be the one-dimensional Qp-vector space

Wr = (Jr/Jr+l)®Qp.

The derived height pairings of Section 2 are compatible as k varies, and passage to the limit yields the pairing of the following theorem.

Theorem 4.2. There is a filtration

• • • C Sf\A/F) C Sf\A/F) C $\A/F) = SP(A/F) ® Qp

such that dimQpS<r)(A/F) = rank(/r-1X//rX) fanrf similarly for Av ), and a se-

quence of pairings

h<r): %\A/F) x $\AV/F) -» Wr

5«c/i r/iaf tfie iteA7ieZ o« f/te /^ (resp. right) is S^+X\A/F) (resp. ty+l\Ay/F)). The subspace S^°\A/F) is the subspace o/universal norms in the usual sense.

That is, 5^°°\A/F) is the subspace generated by the intersection overn of the image




of corestriction SP(A/Fn) - SP(A/F). Furthermore, if4>: A - > Av is a polarization then /i(r) satisfies

hir\a,(t>(b)) = (-lY+lh(r\b,(j>(a))

foralla9beS$\A/F).

Proof. All of the claims are immediate from the corresponding properties of the derived heights over discrete coefficient rings, together with the equality (9), except for the characterization of S^°\A/F). Suppose x G S^°\A/F). Then for some integer t there is a y G Yoo(F), contained in Yj£ for every r, such that y maps to p'x under (12). Fix a topological generator 7 G F and set gn =

"^J^1 G A. If yk denotes the image of y in Yk(F), we claim that yk is in the image of gn: Yk{Fn)^>Yk(F) for every n. Indeed, yk G (7 - 1)T* for every r, and it follows from Lemma 1 .2 that yk is a universal norm (in the sense of Definition 1.3) in Y^ In particular yk is divisible by every gn. Passing to the limit, we must have that y is in the image of gn: Y^iFn) - Y^F) for every n9 say y = gnZn- If to is as in Lemma 4.1, then the image of ptozn in SP(A/Fn) ® Qp is integral and corestricts to pt+t°x. Hence //+'°jc is a universal norm. The opposite implication is entirely similar.

By the structure theorem for finitely-generated Iwasawa modules, we may fix a pseudo-isomorphism of A-modules

X~Aeoo®M®Mf

such that M' is a torsion A-module with characteristic ideal prime to 7, and M has the form

m * (A/jy* e (A//2)*2 e • • • .

The first statement of the following is due to Perrin-Riou [12].

Corollary 4.3. The integers et satisfy the following properties: (a) the height pairing h^ is nondegenerate if and only ifet = Ofor 1 < i < 00, (b)^0O=dimQ;,5<0O)(A/FX (c) er = dimQp (S£XA/F)/Spl\A/F)) , (d) er = 0 (mod 2) when r is even.

Proof. By Theorem 4.2 we have the equality

dimQp$\A/F) = rankZ/) (Jr~lX/JrX) = er + er+l +er+2 + + eoo

which proves all but the final claim. A choice of polarization of A determines an




isomorphism

$\A/F) * 5<r)(Av/F)

and the induced height pairing on Sfr\A/F) is alternating when r is even, by the last part of Theorem 4.2. This implies that S(pr)(A/F)/S(pr+l)(A/F) is even dimensional.

In the case where Foo is the cylotomic Zp-extension, it is conjectured that h^ is nondegenerate. When F = Q and A is modular it is known by the work of Kato that e^ = 0 (i.e. X is a torsion module), but it is not known that ex = 0 for i> 1.

Now suppose that F is a quadratic imaginary field, F^ is the anticyclotomic Zp-extension, and A = E xqF for some elliptic curve E/q satisfying the "Heegner hypothesis" that all primes of bad reduction are split in F (which, in particular, implies our hypothesis that the primes of bad reduction of A are finitely decom-

posed in Foo). In this situation it is known by the work of Bertolini [2] and Cornut

[5] that £oo = 1, hence Mr) is degenerate for every r. The next best thing one could hope for is that S(p\A/F) is one dimensional, hence equal to Sfi°\A/F)9 but this is still too optimistic. By Remark 1.12, /i(1) satisfies

h<l\xT,yT) = -h<l\x9y)9

where r is complex conjugation. This forces the plus and minus eigencomponents of S{p\A/F) under r to be self-orthogonal, and so if

s+ = dimQp S<pl\A/F)+ s~ = dimQ/7 S^\A/F)',

the kernel of A(1) has dimension at least \s+ - s~\.

Conjecture 4.4. (Bertolini-Darmon, Mazur) In the situation above, the dimension ofSf\A/F) is \s+ - s~\, and the dimension ofS$\A/F) is 1.

Assuming the conjecture, Corollary 4.3 implies that e2 = |s+- s~ | - 1. Mazur's control theorem gives

s+ + s~ = dimQp S(pl\A/F) =l+ex+e2

and so

x - a e (A//)"1 e (A//2)'2 e m'

with e\ - 2min{5+,5~} and Mf having characteristic ideal prime to J.

Returning to the general case, we wish to reformulate Theorem 2.5 in the

present setting. This is merely an exercise in passing from results on A[pk] to




results on the Tate module TP(A), although some caution is needed, owing to our slightly strange choice of Selmer structures on A[pk]. Let £ be the set of places of F consisting of all archimedean places and all primes at which A has bad reduction. Define

HU^/Foo,Tp(A)) = lim Hl(Fz/FniTp(A))

Zoo = lim®v\pH\Fn^Tp(A))

Zoo/ = Km®v\pE(Fn9V)®Zp

where we regard Zoo/ as a A-submodule of Zoo via the Kummer map. Suppose we are given some z G //^(Fx/Fqo, Tp(A)) whose image Cz G Zoo,5 actually lands in rZoQs with r > 0, say Cz = (7- \)ry for some choice of topological generator 7 G F and some y G Zoo,*. Let yo denote the image of y in Hl(Fp, Tp{A))/{A{Fp)®Zp) and define

\{r): Ay{Fp)®Qp^Wr

by \zr\a) = (yo,a) • (7 - l)r, where ( , ) is the local Tate pairing.

Theorem 4.5. With notation and definitions as above, let zo be the image ofz in H\F, Tp(A)). Then zo G S(pr)(A/F) and for any c G S(pr)(Aw/F) we have

hir\zo,c) = \?\cp)

where cp is the image ofc in Av(Fp) <g> Q^.

Proof. Fix some positive integer k. First note that our assumption that all finite primes of X are finitely decomposed in Fqo implies that z is unramified away from /?, by Corollary B.3.5 of [16]. Hence, if z(k) denotes the image of z in lim Hl(Fn,A[pk]) (the limit being over n), we have z(k) G H^(F9Siw) where 5 = A[pk], T is the Selmer structure obtained by propagating the Selmer structure of Definition 3.2 through the injection S\W^SK, and the notation T^ has the same meaning as in Section 2. We claim that the image of z(k) in //^(F,5iw), which we shall denote Cz(k), is divisible by (7- l)r. Indeed, it suffices to show that the natural map Z^-^^iF^Si^) takes Z^/ into HXjr(Fp,S\^\ so that we have a well-defined map Zoo,5 ->//}JF(F,Siw). By Tate local duality, this is equivalent to the condition that the natural map

H\FP, Too) - lim ®v\pH\Fn^ Av[p°°])




takes Hjrv(Fp, Too) into the image of lim @v\p{A(Fn,v) ® Qp/7jp) under the Kum-

mer map, where T = Av[pk] and Ty is the Selmer structure of Definition 3.2. This claim now follows by replacing A by Av in Propositions 3.4 and 3.5 (strictly speaking, these propositions state that there is a natural map on global Selmer

groups //jrv(F, Too)- >Selpoo (Av/Foo), but the proofs proceed by showing that the isomorphisms of local cohomology induced by Shapiro's lemma take the local conditions defining the left-hand side into the local conditions defining the

right-hand side). We have now shown that the order of vanishing of Cz(k) is at least r, and so appealing to Theorem 2.5 we see that zo(k), the image of z(k) under

//1(F,5iw)->H1(F,A[/])-^//1(Foo,A[/])[7],

lies in the submodule Y^ defined at the beginning of this section. Passing to the limit over k we have zo G Y%>9 and therefore zo G S{pr)(A/F).

Now fix some c G Y^r) (the module defined in the same way as Yj£9 but with A and T replaced by Av and JrV). Passing to the limit in Theorem 2.5 we have the equality h{r)(zo,c) = \[r)(cp) in Jr/Jr+\ and extending by Qp-linearity proves the same result for c G 5^r)(Av/F).

In the special case where Sfi\A/F) is one-dimensional and S^°\A/F) is

trivial, the above theorem may be regarded as ap-adic, cohomological formulation of the Birch and Swinnerton-Dyer conjecture. In this situation 36 > 1 such that

If zo ¥ 0, the left-hand side of the equality of the theorem can then be interpreted as a regulator term. By the theorem, Cz must vanish to order 6, and the right-hand side is like the value of the 6th -derivative. This interpretation breaks down when the dimension of S(p\A/F) is greater than one, since the formula only allows one to compute the heights of elements against some fixed element zo, and not all

pairwise heights in a basis. Of course this is only interesting if one can construct an element z for which

we are somehow justified in calling Cz a p-adic L-function, and there are essen-

tially two cases where this is understood: when A/Q is an elliptic curve, Kato has constructed such an element for which Cz is related to the Mazur-Swinnerton-

Dyer /?-adic L-function of £, and when A is a CM elliptic curve the Euler system of elliptic units is related, by work of Yager, to the two- variable p-adic L-function constructed by Katz. For applications of Theorem 4.5 in these special cases, see

[1] for the elliptic unit case and [14, 15] for the case of Kato's Euler system.

Department of Mathematics, Stanford University, Stanford, CA 94305




Current address: Department OF Mathematics, University OF CHICAGO, 5734 S. Unversity Ave., Chicago, IL 60637

REFERENCES

[1] A. Agboola and B. Howard, Anticyclotomic Iwasawa theory of CM elliptic curves, preprint, 2003. [2] M. Bertolini, Selmer groups and Heegner points in anticyclotomic Zp -extensions, Compositio Math. 99

(1995), 153-182. [3] M. Bertolini and H. Darmon, Derived p-adic heights, Amer. J. Math. Ill (1995), 1517-1554. [4] J. Coates and R. Greenberg, Kummer theory for abelian varieties over local fields, Invent. Math. \2A

(1996), 129-174. [5] C. Cornut, Mazur's conjecture on higher Heegner points, Invent. Math. 148 (2002), 495-523. [6] M. Flach, A generalisation of the Cassels-Tate pairing, J. ReineAngew. Math. 412 (1990), 113-127. [7] R. Greenberg, Iwasawa theory for p-adic representations, Algebraic Number Theory, Adv. Stud. Pure

Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 97-137. [8] B. Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18

(1972), 183-266. [9] B. Mazur and K. Rubin, Kolyvagin systems, preprint, 2002.

[10] B. Mazur and J. Tate, Canonical height pairings via biextensions, Arithmetic and Geometry (M. Artin and J. Tate, eds.), vol. 1, Birkhauser, Boston, MA, 1983, pp. 195-238.

[11] J. Nekovaf, Selmer complexes, unpublished manuscript. [12] B. Pernn-Riou, Th6orie d'lwasawa et hauteurs p-adiques, Invent. Math. 109 (1992), 137-185. [13] , p-adic L-functions andp-adic Representations, American Mathematical Society, Providence,

RI, 2000. [14] K. Rubin, Abelian varieties, p-adic heights and derivatives, Algebra and Number Theory, Walter de

Gruyter and Co., Berlin, 1994, pp. 247-266. [15] , Euler systems and modular elliptic curves, Galois Representations in Arithmetic Algebraic

Geometry, London Math. Soc. Lecture Notes, vol. 254, Cambridge Univ. Press, 1998, pp. 351- 367.

[16] , Euler Systems, Princeton University Press, 2000.



derived p-adic heights and p-adic l-functions

Documents