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ALGEBRAIC CYCLES, MODULAR FORMSAND EULER SYSTEMS

TOM WESTON

Fix a squarefree integer N and let f be a newform of weight 2 for Γ0(N); weassume that f does not have complex multiplication. It was shown in [14] and[15] that for a set of primes l of density 1 the naive deformation theory of themod l Galois representation associated to f is unobstructed (in the sense that theuniversal deformation ring is a power series ring over the Witt vectors). In [31] thesemethods were modified to obtain results on the deformation problems studied byTaylor-Wiles. In this paper we extend the results of Flach and Mazur to the caseof newforms f of weight κ ≥ 2 for Γ1(N).

We now state our results more precisely. Fix l > max5, κ+1, let f be as aboveand let H be the associated l-adic representation: H is a free module of rank 2 overa certain Hecke algebra A, which itself is a finite, flat, local, Gorenstein Zl-algebra.Let T be the Tate twist End0

AH(1) of the module of trace zero endomorphismsof H. Using techniques of Flach we construct a collection of cohomology classescp in H1(Q, T ) with tightly controlled ramification. With some mild additionalhypotheses, applying the methods of Kolyvagin to these classes yields a certainannihilator η ∈ A of the Selmer group H1

f (Q, T ∗) of the Cartier dual of T . ThisSelmer group is dual to the differentials ΩR⊗RA, where R is the universal minimallyramified deformation ring of the residual representation of H. In the case that ηis a unit this then implies that both R and A are isomorphic to the ring of Wittvectors over the residue field of A.

In the general case, following Mazur we show that our construction yields aderivation from A to the Selmer group H1

f (Q, T/ηT ); it follows by a formal argu-ment that the natural surjection R A induces an isomorphism ΩR⊗RA ∼= ΩA.Although not the strongest possible result, this does provide a great deal of infor-mation on the structure of the ring R. (It is possible that any such map R Amust be an isomorphism, although as far as I know this question remains open.)We also show that the isomorphism ΩR⊗RA ∼= ΩA is characterized by the fact that

ΩA ∼= ΩR⊗RA ∼= HomZl

(H1f (Q, T ∗),Ql/Zl

)identifies the differential of the trace of a geometric Frobenius at p acting on Hwith −12 times the dual of cp under the Bockstein pairing

H1f (Q, T/ηT )⊗H1

f (Q, T ∗)→ Ql/Zl.

(The factor of 12 appears as the weight of the modular form ∆ used to definethe classes cp.) This non-obvious congruence exhibits a certain naturality of ourconstruction which is not otherwise apparent and answers the question [31], p. 98.

In order to explain the methods of the construction of the classes cp we considera more general situation. Let X be a smooth projective variety over a number field

Partially supported by an NSF graduate research fellowship.

1

2 TOM WESTON

F . Set V = H2met (XFac ,Ql/Zl(m)) for some m. The Selmer group H1

f (F, V ) isthe subgroup of the Galois cohomology group H1(F, V ) consisting of cohomologyclasses which are everywhere locally “minimally ramified”. This Selmer group oftenhas an interesting arithmetic interpretation (as with the deformation theory caseconsidered above) and is connected to special values of L-functions by the Bloch-Kato conjectures.

The most successful methods to date for understanding Selmer groups involveKolyvagin’s method of Euler systems: one uses certain collections of classes in theGalois cohomology of the Cartier dual of V to produce an annihilator (or even abound) for H1

f (F, V ). Most of these constructions rely on realizing appropriateelements of motivic cohomology in Galois cohomology. The map we use is an Abel-Jacobi map

H2m+1M

(X,Q(m+ 1)

)→ H1

(F,H2m

et (XFac ,Ql(m+ 1))).

(We actually use a formulation via coniveau spectral sequences so that we can workintegrally.) Our key result, Theorem 3.1.1, is a reciprocity law in the sense of Katodescribing this Abel-Jacobi map locally as the composition of a certain divisor mapand the cycle class map. This generalizes [15], Lemma 3 and is precisely the sortof result needed to check that classes in the image of the Abel-Jacobi map form anEuler system.

The first four sections of this paper can be regarded as the generalization tohigher dimensions of [31]. In Section 1 we give our version of standard material onSelmer groups. Very little in this section is new; our treatment is a synthesis of[17], [31] and [37]. Our notion of a “partial geometric Euler system” is weak butwell-suited to applications. We also describe Mazur’s notion of a cohesive Flachsystem and give the applications to deformation theory.

The next three sections are concerned with the production of cohesive Flachsystems. Section 2 discusses the global behavior of the Abel-Jacobi map. Wereview its construction and investigate its relations with algebraic correspondences.The main result is the Leibniz relation Proposition 2.3.3; this is used to constructthe derivation to Galois cohomology.

Section 3 begins with the statement and proof of the reciprocity law for theAbel-Jacobi map. We then explain how these results can be used, in the presenceof appropriate geometric data, to construct partial geometric Euler systems andcohesive Flach systems.

Section 4 is concerned with exhibiting this data for modular curves and Kuga-Sato varieties; this yields our applications to modular forms. The key idea is theproduction of certain modular units on Hecke correspondences.

In Section 5 we give a proof of our congruence description of the map ΩR⊗RA→ΩA. Our proof is quite computational; it would be useful to have a more conceptualargument. We hope to give applications of this result in a later paper. We includea short appendix on linear algebra over Gorenstein rings and bilateral derivations.

We should note that stronger results on the deformation theory of Galois repre-sentations associated to modular forms have been obtained by Diamond, Flach andGuo. Given this, we will not feel a need to maintain the absolute most general hy-potheses and we will make additional (not terribly restrictive) hypotheses as needed.We refer to their paper [8] for the applications to the Bloch-Kato conjecture.

This paper is a shortened version of my Ph.D. thesis. I would like to thank mythesis advisor Barry Mazur for leading me towards these problems and for all of

ALGEBRAIC CYCLES, MODULAR FORMS AND EULER SYSTEMS 3

the ideas and insights he provided. Matthew Emerton and Richard Taylor readearlier versions of this paper and their comments were extremely helpful (as weremany other conversations with them). I would also like to thank Brian Conrad,Fred Diamond, Mark Dickinson, Matthias Flach, Robert Pollack and Karl Rubinfor helpful conversations.

If K is a field we write Ks (resp. Kac) for a separable (resp. algebraic) closure ofK. (We will tend to write Ks even when over a perfect field to emphasize that ourperfect hypotheses are often excessive.) We set GK = Gal(Ks/K). All cohomologyin this paper is either Galois or etale; for Galois cohomology we always use coho-mology with continuous cochains as in [37], Appendix B. We write Hi(L/K, T ) forHi(Gal(L/K), T ) and Hi(K,T ) for Hi(Gal(Ks/K), T ). By a local (resp. global)field we mean a finite extension of Qp or Fp((t)) for some p (resp. Q or Fp(t) forsome p). If F is a number field and v is a place of F , we write Fr(v) ∈ GF for a geo-metric Frobenius element at v. If χ is a character of GF of conductor prime to v, wewrite χ(v) for χ(Fr(v)). If H is a Galois representation, we simply write EndH(1)(rather than the cumbersome (EndH)(1)) for the Tate twist of the endomorphismsof H.

1. Selmer groups and geometric Euler systems

1.1. Local cohomology groups.

1.1.1. l-adic Galois modules. Fix a prime l and let A be a finite, flat, local Zl-algebra to be fixed throughout the discussion. Let K be a field. By a finitely gen-erated (resp. discrete) l-adic GK-module we mean an A-module T which is finitelygenerated over Zl (resp. torsion of finite corank), endowed with an A-linear actionof GK which is continuous for the l-adic (resp. discrete) topology on T . The l-adicGK-modules form a category in the obvious way; morphisms are assumed to beA-linear, continuous and compatible with GK-actions.

1.1.2. Local finite/singular structures. We now restrict to the case of a local fieldK with residue field k of characteristic p; we allow p = l only if K itself hascharacteristic zero. Let Kur denote the maximal unramified extension of K andlet IK = Gal(Ks/Kur) denote the inertia group of K. We say that an l-adicGK-module T is unramified if IK acts trivially on T . In any case we define theunramified subgroup H1

ur(K,T ) of H1(K,T ) by

H1ur(K,T ) = ker

(H1(K,T )→ H1(Kur, T )

).

By [37], Lemma 1.3.2, H1ur(K,T ) identifies with H1(k, T IK ) via inflation.

Definition 1.1.1. A local finite/singular structure S on T consists of a choice ofA-submodule H1

f,S(K,T ) ⊆ H1(K,T ). A structured GK-module (T,S) is a pair ofan l-adic GK-module T and a finite/singular structure S on T .

We callH1f,S(K,T ) the finite subgroup; we define the singular quotientH1

s,S(K,T )to be H1(K,T )/H1

f,S(K,T ). We will often omit the finite/singular structure S fromthe notation if it is clear from context. We write cs for the image of c ∈ H1(K,T )in H1

s,S(K,T ).We call the structure with H1

f,S(K,T ) = H1ur(K,T ) the unramified structure.

When p 6= l we say that a structured GK-module (T,S) is unramified if T is

4 TOM WESTON

unramified and S is the unramified structure. In this case we have

(1.1.1) H1s,S(K,T ) ∼= T (−1)Gk ,

as follows from the fact that the maximal pro-l quotient of IK is isomorphic toZl(1) as a Gk-module.

In the case l 6= p the only non-trivial structures we will consider is the minimallyramified structure. For this we assume that T is free over Zl (resp. l-divisible). SetV = T ⊗ZlQl (resp. V = lim←− T [ln]⊗ZlQl). The minimally ramified structure on Tis the pullback (resp. pushforward) of H1

ur(K,V ) via the natural map T → V (resp.V → T ). If T is unramified one checks easily that this agrees with the unramifiedstructure.

In the case that l = p we will usually consider only the crystalline structure givenby the procedure of the preceeding paragraph with

H1f (K,V ) = ker

(H1(K,V )→ H1(K,V ⊗Ql

Bcris)).

1.1.3. Induced finite/singular structures. A morphism f : (T,S)→ (T ′,S ′) of struc-tured GK-modules is a morphism f : T → T ′ of l-adic GK-modules such thatf∗H

1f,S(K,T ) ⊆ H1

f,S′(K,T′). If (T,S) is a structured GK-module and i : T ′ → T

(resp. j : T → T ′′) is a map of l-adic GK-modules, we defined the induced fi-nite/singular structure i∗S on T ′ (resp. j∗S on T ′′) to be the full inverse image(resp. pushforward) of H1

f,S(K,T ). The map i (resp. j) is then a map of structuredmodules.

Let 0 → T ′ → T → T ′′ → 0 be a sequence of structured GK-modules. We saythat this is sequence is exact (as a sequence of structured GK-modules) if it is exactas a sequence of A-modules and if the structures on T ′ and T ′′ are induced fromthe structure on T . A diagram chase and cohomological dimension arguments showthat one then obtains exact sequences

(1.1.2) 0→ H0(K,T ′)→ H0(K,T )→ H0(K,T ′′)→H1f (K,T ′)→ H1

f (K,T )→ H1f (K,T ′′)→ 0

and

(1.1.3) 0→ H1s (K,T ′)→ H1

s (K,T )→ H1s (K,T ′′)→

H2(K,T ′)→ H2(K,T )→ H2(K,T ′′)→ 0.

1.1.4. Cartier dual structures. The category of l-adic GK-modules has a natural in-volution T 7→ T ∗ where T ∗ = HomZl(T, µl∞(Ks)) with A-module structure inducedfrom T and adjoint Galois action. Tate local duality yields a perfect pairing

〈·, ·〉K : H1(K,T )⊗ZlH1(K,T ∗)→ Ql/Zl.

If p 6= l and T is unramified, one also has that H1ur(K,T ) and H1

ur(K,T∗) are exact

orthogonal complements; see [37], Chapter 1, Section 4.We extend Cartier duality to the category of structured GK-modules by letting

the finite subgroup H1f,S∗(K,T

∗) on T ∗ be the exact annihilator of H1f,S(K,T )

under the Tate pairing. This respects minimally ramified structures for l 6= p; by[1], Proposition 3.8 it also respects crystalline structures if T ⊗ZlQl is deRham.


1.1.5. Archimedean structures. We briefly consider the archimedean case. Let Kdenote either R or C and let T be an l-adic GK-module. The cohomology groupH1(K,T ) is trivial, so that there is only one choice for the finite/singular structure,unless K = R and l = 2. We refer to [37], Remark 1.3.7 for the natural choices inthis case.

1.2. Global cohomology groups.

1.2.1. Global finite/singular structures. Let F be a global field. For every placev of F we fix now and forever embeddings Fs → Fv,s; these induce injectionsGFv → GF . Let kv denote the residue field of Fv and let Iv = Gal(Fv,s/Fv,ur)denote the inertia group of Fv.

Let A be a Zl-algebra as before; we now write m for its maximal ideal and kfor its residue field, contrary to our earlier notation. We assume that l does notequal the characteristic of F . We modify the definition of an l-adic GF -module toinclude only those which are unramified at almost all places of F . Let Σl (resp.Σ∞) denote the set of places of F above l (resp. the set of archimedean places);both are empty if F has positive characteristic.Definition 1.2.1. A global finite/singular structure S on an l-adic GF -module Tconsists of choices of local finite/singular structures H1

f,S(Fv, T ) for every place vof F such that (T,S) is unramified as a stuctured GFv -module for almost all v.

The structures considered in [1], [17] and [37] are those which are minimallyramified away from Σl. The definitions of morphisms, induced structures, exactsequences and Cartier duals extend to structured GF -modules by considering eachplace of F individually.

1.2.2. Selmer groups. Let (T,S) be a structured GF -module. For every place vthere is a canonical restriction map resv : H1(F, T ) → H1(Fv, T ). We often writeresv(c) = cv.Definition 1.2.2. The Selmer group H1

f,S(F, T ) of (T,S) is defined by

H1f,S(F, T ) = ker

(H1(F, T )→

∏v

H1s,S(Fv, T )

).

See [37], Chapter 1, Section 6 for interpretations of Selmer groups in terms ofideal class groups, global units and rational points on abelian varieties.

Let Σ be a set of places which contains Σl, Σ∞ and all places where T is ramified.Let S be the finite/singular structure which is unramified away from Σ and withH1f,S(Fv, T ) = H1(Fv, T ) for v ∈ Σ. By [43], Proposition 6 inflation induces an

isomorphism H1f,S(F, T ) ∼= H1(FΣ/F, T ) where FΣ is the maximal extension of

F unramified outside Σ. It follows from this and [37], Proposition B.2.7 that if(T,S) is a finite (resp. finitely generated, resp. discrete) structured GF -module,then H1

f,S(F, T ) is finite (resp. finitely generated over Zl, resp. of finite corank).Let 0 → T ′ → T → T ′′ → 0 be an exact sequence of structured GF -modules.

The local exact sequences (1.1.2) yield an exact sequence

0→ H0(F, T ′)→ H0(F, T )→ H0(F, T ′′)→H1f (F, T ′)→ H1

f (F, T )→ H1f (F, T ′′).

As an immediate consequence we have the following useful lemma.

6 TOM WESTON

Lemma 1.2.3. Let (T,S) be a structured GF -module and let α ∈ A be such that(αT )GF = (T/αT )GF = 0. Let T [α] have the induced structure. Then H1

f (F, T [α])injects into H1

f (F, T ) and under this identification it identifies with H1f (F, T )[α].

1.2.3. Global pairings. Let (T,S) be a structured GF -module. We recall now twoglobal pairings. We begin with the Kolyvagin pairing

〈·, ·〉F :(⊕vH1s,S(Fv, T )

)⊗ZlH

1f,S∗(F, T

∗)→ Ql/Zl

given by〈(cv), d〉F =

∑v

〈cv, dv〉v

where 〈·, ·〉v is the Tate pairing for Fv. We define the compactly supported cohomol-ogy H1

c (F, T ) to be the A-submodule of H1(F, T ) of classes which lie in H1f (Fv, T )

for almost all v. (Note that H1c (F, T ) = H1(F, T ) if T is discrete.) The basic fact

about the Kolyvagin pairing is the following.Proposition 1.2.4. Let T be a structured GF -module. Then the image of H1

c (F, T )in ⊕H1

s (Fv, T ) is orthogonal to H1f (F, T ∗) under the Kolyvagin pairing.

Proof. This follows from global class field theory; see [30], Section 12.

Now let0→ T ′

α−→ Tβ−→ T ′′ → 0

be an exact sequence of finite structured GF -modules. There is then a Bocksteinpairing

·, ·α,β : H1f (F, T ′′)⊗H1

f (F, T ′∗)→ Ql/Zl.We give the definition of x′′, yα,β only in the case that x′′ is the image of x ∈H1(F, T ). Since x′′ ∈ H1

f (F, T ′′), it follows from (1.1.3) that for each v the singularrestriction xv,s is the image of x′v ∈ H1

s (Fv, T ′); in fact, we have x′v = 0 for almostall v. We define

x′′, yα,β = 〈(x′v), y〉F .By Proposition 1.2.4 this is independent of the choice of x. See [17], Chapitre 2,Section 1.4 for the general definition.

1.3. Partial geometric Euler systems.

1.3.1. Definitions. Let A and F be as above. Let (T,S) be a structuredGF -module.If C is an A-submodule of H1(F, T ) and v is a places of F , we write Cv,s for theimage of C in H1

s (Fv, T ).Definition 1.3.1. Let L be a (possibly infinite) set of places of F and let η be anideal of A. A partial (geometric) Euler system Cvv∈L of depth η for (T,S) is anassignment of A-submodules Cv ⊆ H1(F, T ) for each v ∈ L such that

• Cvw,s = 0 for all places w 6= v;• H1

s (Fv, T )/Cvv,s is killed by η.(That is, Cv is supported in the singular quotients only at v, and here it containsηH1

s (Fv, T ).) If in addition Cv vanishes in H1s (Fv, T/ηT ) for all v ∈ L, we say

that the partial Euler system has strict depth η. In this case the image of each Cv

in H1(F, T/ηT ) lies in H1f (F, T/ηT ) and we define the Euler module Φ to be the

A-submodule of H1f (F, T/ηT ) generated by the images of the Cv for all v ∈ L.


For a set of places L, define

H1L(F, T ) = ker

(H1(F, T )→

∏v∈L

H1(Fv, T )

).

The basic utility of a partial Euler system comes from the next lemma.Lemma 1.3.2. Let (T,S) be a finitely generated structured GF -module. Assumethat T admits a partial Euler system Cvv∈L of depth η. Then

ηH1f (F, T ∗) ⊆ H1

L(F, T ∗).

Proof. Fix v ∈ L. By Proposition 1.2.4 the Kolyvagin pairing is trivial on the imageof Cv. However, by the definition of Cv the restriction of the Kolyvagin pairing toCv “coincides” with the restriction

〈·, ·〉v : Cvv⊗Zl im(H1f (F, T ∗)→ H1

f (Fv, T ∗))→ Ql/Zl

of the Tate local pairing at v. The lemma follows from this and the duality ofH1s (Fv, T ) and H1

f (Fv, T ∗).

1.3.2. Annihilation theorems. Let T be a finite l-adic GF -module with splitting fieldF ′; set ∆ = Gal(F ′/F ). For τ ∈ ∆, define Lτ to be the set of non-archimedeanplaces of F which are unramified in F ′ and such that Fr(v) is conjugate to τ on F ′.We say that τ acts on T as a scalar if its action factors through the natural mapA× → AutA T . The next lemma is a variation due to Flach and Mazur of ideas ofKolyvagin.Lemma 1.3.3. Let T be a finite l-adic GF -module with l 6= 2. Suppose that T [m]is absolutely irreducible as a k[GF ]-module and that τ ∈ ∆ acts on T as a non-scalar involution. Then for any set of places L cofinite in Lτ we have H1

L(F, T ) ⊆H1(∆, T ).

Proof. By inflation-restriction it suffices to show that the image of c ∈ H1L(F, T ) in

Hom∆(GabF ′ , T ) is trivial. Let ψ : Gab

F ′ → T be this ∆-equivariant homomorphism.Let F ′′ be the fixed field of kerψ and set Γ = Gal(F ′′/F ′). Fix a lifting τ ∈Gal(F ′′/Q) of τ .

Fix now a g ∈ Γ. By the Tchebatorev density theorem there exist infinitely manyunramified places v′′ of F ′′ such that FrF ′′/F (v′′) = τ g. Let v′ (resp. v) denote therestriction of v′′ to F ′ (resp. F ). We have FrF ′/F (v′) = τ , so that v ∈ Lτ . We picksuch a place v′′ so that v ∈ L. Thus by assumption ψ|Gal(F ′′

v′′/F′v′ )

= 0. Equivalently,ψ(FrF ′′/F ′(v′′)) = 0.

Since τ has order 2, F ′v′/Fv is of degree 2 and FrF ′′/F ′(v′′) = (τ g)2. Thusψ(τ gτg) = 0. Since τ is an involution and ∆ acts on Gab

F ′ by conjugation, thismeans that ψ(τg · g) = 0. By ∆-equivariance we conclude that τψ(g) = −ψ(g).

Thus the A-span Ψ of the image of ψ is a GF -stable submodule of T τ=−1,It follows that Ψ[m] ⊆ T [m]τ=−1. As τ is non-scalar, T [m]τ=−1 6= T [m]; thusΨ[m] 6= T [m]. Since T [m] is absolutely irreducible this implies that Ψ[m] = 0, andthus that Ψ = 0. We conclude that ψ = 0, which completes the proof.

Lemmas 1.3.2 and 1.3.3 yield the following result. Let δ be the A-annihilator ofH1(∆, T ∗).Proposition 1.3.4. Let (T,S) be a finite structured GF -module. Suppose that:

• T ∗[m] is absolutely irreducible as a GF -module over k;

8 TOM WESTON

• There is a non-scalar involution τ ∈ ∆;• T admits a partial Euler system Cvv∈L of depth η for some set of placesL cofinite in Lτ .

Then δη annihilates the Selmer group H1f (F, T ∗).

There is a version of Proposition 1.3.4 for finitely generated T , but it involvessome additional complications which will not come up in our applications.

1.4. Flach systems.

1.4.1. Review of deformation theory. We continue with our previous notation. Wenow require that F is a number field with at least one real embedding; fix a complexconjugation τ ∈ GF . We also assume that l is unramified in F/Q. We assume nowthat l > 2, and we require that A is Gorenstein; in particular, HomZl(A,Zl) is freeof rank 1 as an A-module. Fix a Gorenstein trace tr : A→ Zl. (See Appendix A.1.)Let W (k) denote the ring of Witt vectors for k.Definition 1.4.1. Let H be an l-adic GF -module which is free of rank 2 as anA-module. Let Σ be the set of places at which H is ramified together with Σl andΣ∞. We say that H is minimally ramified if the following conditions are satisfied:

• H⊗Ak is absolutely irreducible;• For every v ∈ Σ−Σl ∪Σ∞, the image of the inertia group Iv in AutAH ∼=

GL2(A) is conjugate to the subgroup ( 1 a0 1 ) ; a ∈ A;

• For some κ < l and each v ∈ Σl, H is crystalline of weight κ at v inthe following sense: H arises via the Tate module functor from a stronglydivisible lattice D in H0(Fv, Bcris⊗ZlH) such that the filtration F iD satisfies

rankOFv FiD =

2 i ≤ 0;1 1 ≤ i ≤ κ− 1;0 κ ≤ i;

(see [16] and [1], Section 4);• The determinant character χ : GF → A× of H factors through W (k)× and

satisfies χ(τ) = −1;• The W (k)-algebra A is generated by the traces of Fr(v) acting on the inertia

coinvariants HIv for all v /∈ Σl.For each v /∈ Σl we define the Hecke operator Tv ∈ A to be the trace of Fr(v)

acting on HIv ; note that this is well-defined since HIv is always free for such vby our assumptions. For v /∈ Σ, the characteristic polynomial of Fr(v) on H isx2 − Tvx+ χ(v).

We are interested in deformations of H⊗Ak as a Galois module which satisfy thesame local conditions as H. Specifically, let C denote the category of inverse limitsof artinian local rings with residue field k. Let H ′ be a lifting of H⊗Ak over a ringB of C; that is, H ′ is a free B-module of rank 2 with H ′⊗Bk ∼= H⊗Ak. FollowingDiamond, we say that H ′ is minimally ramified if:

• H ′ is unramified away from Σ;• For every v ∈ Σ − Σl, the image of Iv acting on H ′ is conjugate to a

subgroup of ( 1 b0 1 ) ; b ∈ B;

• H ′ is crystalline at every place v ∈ Σl (in the sense of Fontaine-Laffailleabove) and the filtration on the associated Dieudonne module D satisfiesF 0D = D and FκD = 0;


• H ′ has determinant χ.We let D : C → Sets be the corresponding deformation functor, so that D(B) is theset of isomorphism classes of pairs (H ′, α′) of minimally ramified liftings of H⊗Akto B and isomorphisms α′ : H ′⊗Bk ∼= H⊗Ak.

By [32] the functor D is representable; we let R be the universal deformation ring.Let π : R → A be the map corresponding to the minimally ramified deformationH. Let Tv be the trace of Fr(v) acting on the inertia coinvariants of the universaldeformation; we have π(Tv) = Tv, from which it follows that π is surjective.

In order to study the ring R via our given deformation H we need to define asecond deformation functor. Let C′ be the category of inverse limits of local artinianrings, with residue field k, equipped with a local homomorphism to A (inducingthe identity map on k). We define a deformation functor DA : C′ → Sets byletting DA(f : B → A) be the inverse image of (H, id) ∈ D(A) under the mapf∗ : D(B) → D(A). That is, DA classifies the deformations which are “congruentto H”. [32], Section 20, Proposition 4 shows that DA is represented by π : R→ Afor the same π and R as above.

1.4.2. Tangent spaces and Selmer groups. For any ideal a of A let Aa be the ringA[ε]/(aε, ε2). There is a well-known canonical isomorphism

(1.4.1) DA(Aa) ∼= HomA(ΩR⊗RA,A/a)

where ΩR is the module of differentials of R over W (k) (or equivalently over Zl);see [32], Section 17.

Set T = End0AH(1). T is a free A-module of rank 3; the composition of the usual

trace pairing with the Gorenstein trace tr yields an isomorphism

(1.4.2) T ∗ ∼= End0AH⊗ZlQl/Zl.

We give T the finite/singular structure S which is minimally ramified away from Σland crystalline at all places of Σl; the Cartier dual structure S∗ on T ∗ has the samedescription. For every ideal a of A we give T/aT and T ∗[a] the induced structures.We now have a second isomorphism

(1.4.3) DA(Aa) ∼= H1f (F, T ∗[a]);

see [32], Sections 21, 22, 24, 29 and [1], Lemma 4.5 for details.Combining (1.4.1) and (1.4.3) and passing to the limit yields

(1.4.4) H1f (F, T ∗) ∼= HomA

(ΩR⊗RA,A⊗ZlQl/Zl

).

Postcomposition with the Gorenstein trace yields an isomorphism

(1.4.5) H1f (F, T ∗) ∼= HomZl

(ΩR⊗RA,Ql/Zl

)which is fundamental to what follows.

1.4.3. Singular quotients. Let F ′ be the splitting field of H⊗Ak. By assumptionτ acts on H as a non-scalar involution; it follows that it acts in the same way onT and T ∗. Let L denote the set of non-archimedean places of F , unramified withFrobenius conjugate to τ on F ′.

Note that for v ∈ L the characteristic polynomial of Fr(v) acting on H is con-gruent to x2 − 1 modulo m. In particular, χ(v) ≡ −1 (mod m) and we have afactorization

x2 − Tvx+ χ(v) ≡ (x− 1)(x+ 1) (mod m).

10 TOM WESTON

By Hensel’s lemma this lifts to a factorization

x2 − Tvx+ χ(v) = (x− α)(x− β)

with α, β ∈ A and α ≡ −β ≡ 1 (mod m). An argument using Nakayama’s lemmaand a dimension count now yields the following.Lemma 1.4.2. Let v be a place of L. There is a direct sum decomposition (de-pending on v) H = Hα ⊕ Hβ, where Hα (resp. Hβ) is free of rank 1 over A andFr(v) acts on it as the scalar α (resp. β).Corollary 1.4.3. For all places v ∈ L and all ideals a of A, the singular quotientH1s (Fv, T/aT ) is a free A/a-module of rank 1.

Proof. By (1.1.1) we haveH1s (Fv, T ) ∼= T (−1)Gkv ∼= (End0

AH)Gkv . Gkv is generatedby Fr(v), and Fr(v) acts on End0

AH (with respect to the basis of Lemma 1.4.2) asconjugation by

(α 00 β

). This sends a matrix

(a bc −a

)∈ End0

AH to(a α2

χ(v)bβ2

χ(v)c −a

).

Note that α2 and β2 are not equal to χ(v) (or even congruent to it modulo m)since χ(v) ≡ −1 (mod m). It follows that (End0

AH)Gkv is free of rank 1 over A,generated by

(1 00 −1

). The result for a 6= 0 is proven in the same way.

1.4.4. Flach systems. We now introduce a slightly refined version of a partial geo-metric Euler system. We assume that T ⊗A k is absolutely irreducible and thatH1(F (T ∗[a])/F, T ∗[a]) = 0 for all ideals a of A. By [44], Proposition III.5.1 thissecond condition holds if #k 6= 5 and GF → AutAH is surjective; see also [14],Lemma 1.2 and [15], Section 4, Remark 1..Definition 1.4.4. Let η be a non-zero divisor in A. A Flach system cvv∈L ofdepth η for T is a partial Euler system Cvv∈L of strict depth ηA such that eachCv is a cyclic A-module, generated by cv ∈ H1(F, T ).

Let cvv∈L be a Flach system for T and let Φ denote its Euler module. ByCorollary 1.4.3 we see that this Flach system induces (by pushforward) a partialEuler system of depth η for T/aT for every ideal a of finite index in A. Proposi-tion 1.3.4 and our vanishing assumption above now show that ηH1

f (F, T ∗[a]) = 0 forevery a; thus ηH1

f (F, T ∗) = 0. Since T⊗Ak is absolutely irreducible the hypothesesof Lemma 1.2.3 are satisfied and we conclude that H1

f (F, T ∗) = H1f

(F, T ∗[η]

).

Consider now the Bockstein pairing

(1.4.6) ·, ·η : H1f (F, T/ηT )⊗ZlH

1f (F, T ∗)→ Ql/Zl

associated to the exact sequence

0→ T/ηTη→ T/η2T → T/ηT → 0.

(One checks easily that each factor of T/ηT receives the same induced finite/singularstructure.)Lemma 1.4.5. The restriction of ·, ·η to Φ on the left is right non-degenerate.In particular it induces an injection

(1.4.7) H1f (F, T ∗) → HomZl(Φ,Ql/Zl).


Proof. Let c ∈ Φ be the image of cv for some v ∈ L. Fix y ∈ H1f (F, T ∗). We

compute the Bockstein pairing c, yη. Proceeding as in Section 1.2.3, we firstmust lift c to H1(F, T/η2T ). The image of cv yields such a lift. cvw,s is 0 for w 6= v,and it follows that

c, yη =⟨

1η cvv,s, yv

⟩v.

Here 1η cvv,s denotes the element of H1

s (Fv, T/ηT ) which maps to the class cvv,s ∈H1s (Fv, T/η2T ) under multiplication by η.By definition of a Flach system, cvv,s generates ηH1

s (Fv, T/η2T ) as an A-module.Thus 1

η cvv,s generates H1

s (Fv, T/ηT ) as an A-module. In particular, since the Tatepairing is perfect we conclude that c, yη = 0 only when yv = 0 in H1

f (Fv, T ∗).Thus if y is orthogonal to all of Φ, then we have yv = 0 for all v ∈ L. By

Lemma 1.3.3 and our assumptions we then have y = 0. This completes the proof.

(1.4.7) and (1.4.5) yield a surjection Φ ΩR⊗RA. In particular ΩR⊗RA isη-torsion. If η is a unit this implies that ΩR⊗RA = 0; an easy argument then showsthat W (k) ∼= R ∼= A. In order to obtain strong results when η is not a unit we willneed to impose more structure on our Flach system.

1.4.5. Cohesive Flach systems. We continue with the assumptions of the previoussection.

Definition 1.4.6. A cohesive Flach system of depth η for T is a collection of classescv ∈ H1(F, T ) for v /∈ Σl ∪ Σ∞ such that:

• cvv∈L is a Flach system of depth η;• cv vanishes in H1

s (Fw, T ) for all w 6= v;• cv vanishes in H1

s (Fv, T/ηT );• The map Θ : A → H1(F, T/ηT ) sending each Tv to cv is a well-defined

continuous derivation.

(Note that the second two conditions are redundant for v ∈ L.)

Note that since the Tv generate A the image of Θ is contained in H1f (F, T/ηT ).

Note also that Θ is automatically W (k)-linear since W (k) is unramified over Zl.Θ induces an A-linear map ΩA → H1

f (F, T/ηT ). This yields a surjection ΩA im Θ. Of course, there is an injection Φ → im Θ. We also have the surjectionΦ ΩR⊗RA of the previous section. Lastly, since π : R→ A is surjective we havea surjection ΩR⊗RA ΩA. The existence of these four maps and [28], Theorem2.4 imply that all four are isomorphisms. We define the Flach automorphism Ξ :ΩA

'−→ ΩA to be the appropriate composition

ΩA im Θ = Φ ΩR⊗RA ΩA.

Returning to the isomorphism ΩR⊗RA'−→ ΩA, we see that the surjection

π : R→ A induces an isomorphism on differentials. Such a π is called an evolution.If A is a complete intersection, then such a π is necessarily an isomorphism. As faras we know no examples of non-trivial evolutions are known in our setting. See [9]for examples of non-trivial evolutions in positive characteristic.

12 TOM WESTON

1.4.6. Cohesive Flach systems of Eichler-Shimura type. One can give an explicitdescription of the map Ξ introduced above for certain cohesive Flach systems. Wefirst fix some notation. Fix a place v /∈ Σ and let λ0 be a uniformizer of Fv. Fix acompatible system λ = (λn) of l-power roots of λ0. Fix also a generator ζ = (ζn)of Zl(1). These choices determine a generator σ of Gal(Fv,ur(λ)/Fv,ur) ∼= Zl(1) byrequiring σ(λn) = ζnλn for all n. The isomorphism (1.1.1) is given explicitly by

H1s (Fv, T ) ∼= HomGkv

(Gal(Fv,ur(λ)/Fv,ur), T

).

Let Fr(v) denote the endomorphism of H induced by Frobenius at v. We definethe verschiebung Ver(v) to be the endomorphism χ(v) Fr(v)−1.Definition 1.4.7. A cohesive Flach system cv is said to be of Eichler-Shimuratype of weight 2w if for each v /∈ Σ the class cvv,s ∈ H1

s (Fv, T ) is given by

cvv,s : Gal(Fv,ur(λ)/Fv,ur

)→ T

τ j 7→ wjη(Fr(v)−Ver(v)

)⊗ζ.

One checks immediately that this is a well-defined cocycle with values in T andthat this definition is independent of the choices of λ and ζ. Our main result oncohesive Flach systems is the following, which is proved in Section 5.Theorem 1.4.8. Let cv be a cohesive Flach system of Eichler-Shimura type ofdepth η and weight 2w for T . Assume also that H1

s (Fv, T ) = 0 for v ∈ Σ − Σl.Then the Flach automorphism Ξ : ΩA → ΩA is multiplication by 2w.

Theorem 1.4.8 can be restated in the following terms. Consider the pairing onΩA induced from the Bockstein pairing, the cohesive Flach system, (1.4.5) and theisomorphism ΩR⊗RA ∼= ΩA:

H1f (F, T/ηT )⊗ZlH

1f (F, T ∗) // Ql/Zl

ΩA ⊗Zl HomZl(ΩA,Ql/Zl)

OO OO

Theorem 1.4.8 says that this induced pairing is precisely 2w times the canonicalduality pairing.

1.5. Additional applications to deformation theory. A simple generalizationof [14], Section 3 yields the following application to deformation theory. We willonly use it in the situation of the previous section; nevertheless, as the general caseis no more difficult we relax our hypotheses. Let F be a global field of characteristicdifferent from l. Let A be a finite, flat, local, Gorenstein Zl-algebra with residuefield k and let H be an l-adic GF -module which is unramified away from a set ofplaces Σ (containing Σl and Σ∞) and which is free of finite rank as an A-module.We assume that H⊗ZlQl is deRham over Fv for each v ∈ Σl. Assume also thatthe determinant character of H factors through W (k)×. Set T = End0

AH(1) (resp.T ′ = End0

AH) with finite/singular structure S (resp. S ′) minimally ramified awayfrom Σl and crystalline at Σl. We will also use the finite/singular structure S0 onT which is unramified away from Σ and with H1

f,S0(Fv, T ) = 0 for v ∈ Σ. We have

T ′ ⊗Zl Ql/Zl = T ∗. Let GF,Σ = Gal(FΣ/F ).Proposition 1.5.1. Assume that H0(Fv, T ⊗A k) = 0 for all v ∈ Σ − Σ∞; thatT ′ ⊗Zl Ql is critical in the sense of Deligne; and that H1

f,S∗(F, T∗) = 0. Then the


deformation problem (with fixed determinant) associated to H⊗Ak as a k[GF,Σ]-module is unobstructed.

Proof. By [29], Section 1.6 it suffices to show that H2(GF,Σ, T ′⊗Ak) = 0. By thePoitou-Tate exact sequence∏

v∈Σ−Σ∞

H0(Fv, T ⊗Ak)→ H2(GF,Σ, T ′⊗Ak)∨ → H1f,S0

(F, T⊗Ak)

and our assumptions it suffices to show that H1f,S0

(F, T ⊗A k) = 0. This groupinjects into H1

f,S(F, T⊗ZlQl/Zl), and we will show that this Selmer group vanishes.We begin with the exact sequence (as in [14], Section 1)

0→ π∗H1f,S′(F, T

′⊗ZlQl)→ H1f,S∗(F, T

∗)→X(F, T ∗)→ 0.

Here π : T ′⊗ZlQl → T ∗ is the natural map and we are using the obvious notion ofthe Selmer group of T ′⊗ZlQl. By assumption we thus have H1

f,S′(F, T′⊗ZlQl) =

X(F, T ∗) = 0. By [13] and [12], Corollary 1.5 (using that T ′⊗Zl Ql is critical),it follows that H1

f,S(F, T⊗ZlQl) = X(F, T ⊗Zl Ql/Zl) = 0. Thus H1f,S(F, T⊗Zl

Ql/Zl) = 0, as claimed.

2. The Abel-Jacobi map

2.1. Coniveau spectral sequences.

2.1.1. Etale cohomology. Let X be a scheme of finite Krull dimension, let Y be aclosed subscheme of X and let F be a torsion etale sheaf on X. As in [21], Section10.1 and [3], Section 1 there is a spectral sequence

(2.1.1) Ep,q1,Y (X,F) = ⊕x∈Xp∩Y

Hp+qx (X,F)⇒ Hp+q

Y (X,F)

coming from filtration by codimension of support and excision. Here Xp denotesthe points of X of codimension p and

Hix(X,F) def= lim−→

Z(x

Hi¯x−Z(X − Z,F).

It is clear from its construction that (2.1.1) is contravariant for flat morphisms,covariant for finite flat morphisms and that these functorialities are compatible withedge maps when they exist. When Y = X we write (2.1.1) simply as Ep,q1 (X,F).

There is a localization sequence when U = X−Y is dense in X. Specifically, foreach q there is a short exact sequence of complexes

0→ E•q1,Y (X,F)→ E•q1 (X,F)→ E•q1 (U,F)→ 0

which yields a long exact sequence

· · · → Ep,q2,Y (X,F)→ Ep,q2 (X,F)→ Ep,q2 (U,F)→ Ep+1,q2,Y (X,F)→ · · · .

The boundary maps of this exact sequence are compatible with edge maps whenthey exist; see [15], Proposition 3.

14 TOM WESTON

2.1.2. Purity. Assume now that X is a regular scheme of finite type over a perfectfield (resp. smooth over a discrete valuation ring with perfect residue field). Leti : Y → X be a regular closed subscheme of X (resp. of the closed fiber of X) ofconstant codimension c and let F be a locally constant torsion sheaf on X of orderinvertible in OX . In this situation one knows that Grothendieck’s purity conjectureholds: there are functorial isomorphisms

Rji!F ∼=

0 j 6= 2c;i∗F(−c) j = 2c;

where i!F is the sheaf of sections of F supported on Y . In particular, one then has

Hj+2cY (X,F) ∼= Hj

(Y, i∗F(−c)

)for all j. See [24], Expose 16, Section 3 (resp. [35], Lemma 2.1) for details. (One canextend much of what we will say to local and global fields of positive characteristicwith some mild hypotheses on the sheaf F ; see [41], Corollary 3.7 for the relevantpurity statements. We do not treat this except to say that, assuming that purityholds, the rest of our arguments go through.)

Return now to the closed immersion i : Y → X; note that we also know purityfor the inclusion of every regular closed subscheme of Y into X. In this situation onecan use purity, the existence of an open locus of regularity [23], Corollary 6.12.6 andthe compatibility of etale cohomology with direct limits to obtain an isomorphism

Hix(X,F) ∼= Hi−2p

(Spec k(x),F(−p)

)for any x ∈ Xp ∩ Y . In particular,

(2.1.2) Ep,q1,Y (X,F) ∼= ⊕x∈Xp∩Y

Hq−p(Spec k(x),F(−p)).

2.1.3. K-theory. There is an analogous spectral sequence in K-theory; see [15],Sections 5.1 and 5.2 and [34], Section 7, Theorem 5.4. Here we require that Xis a regular noetherian scheme of finite Krull dimension and that Y is a closedsubscheme of X; the spectral sequence is

(2.1.3) Ep,q1,Y (X) = ⊕x∈Xp∩Y

K−p−qk(x)⇒ K−p−q,YX.

It is contravariant for flat morphisms, covariant for finite flat morphisms and hasa localization sequence as in Section 2.1.1; all of these are compatible with edgemaps when they exist.

Note that

Ep−1,−p1,Y (X) = ⊕

x∈Xp−1∩Yk(x)×; Ep,−p1,Y (X) = ⊕

x∈Xp∩YZ.

By [34], Proposition 5.14 and Remark 5.17 and [20] the spectral sequence differentialbetween these terms is nothing other than the divisor map. In particular Ep,−p2 (X)identifies with the codimension p Chow group CHpX and Ep,−p−1

2 (X) is a quotientof the abelian group of formal sums

∑(Zi, fi) of pairs of codimension p cycles Zi

and rational functions fi on Zi such that∑

divZi fi = 0. Alternately, by [25],Lemma 6.12.4 we may write Ep,−p2 (X)⊗Z Q and Ep,−p−1

2 (X)⊗Z Q as the motiviccohomology groups H2p

M(X,Q(p)) and H2p+1M (X,Q(p+ 1)).


2.1.4. Chern characters. The coniveau spectral sequences in K-theory and etale co-homology are connected by the Chern characters of [19]; see also [27], Chapter 3.Specifically, let X be a regular noetherian scheme of finite Krull dimension n andlet Y be a closed subscheme of X. For any i and for any N invertible on X andrelatively prime to p! there is a map of spectral sequences

Ep,qr,Y (X)→ Ep,q+2ir,Y

(X,Z/NZ(i)

)which is compatible with all functorialities and localization sequences. For the casep = −q, there is a commutative diagram

(2.1.4) Ep,−p1,Y (X) // Ep,p1,Y

(X,Z/NZ(p)

)

⊕x∈Xp∩Y

Z // ⊕x∈Xp∩Y

Z/NZ

with the bottom map the obvious one. (At the second stage this becomes thestatement that the cycle class map is a special case of the Chern character.)

2.2. The Abel-Jacobi map. We are now in a position to define the Abel-Jacobimap to Galois cohomology. Let F be a perfect field and let X be a smooth separatedF -scheme of finite type and dimension n. Fix m, 0 ≤ m ≤ n and an integer Nrelatively prime to m! and the characteristic of F . Let F denote the constant sheafZ/NZ on X.

We begin with the Chern character

(2.2.1) Em,−m−12 (X)→ Em,m+1

2

(X,F(m+ 1)

).

Using the expression (2.1.2) we see that there is an edge map

(2.2.2) Em,m+12

(X,F(m+ 1)

)→ H2m+1

(X,F(m+ 1)

).

Consider now the Leray spectral sequence for u : X → SpecF :

Hp(SpecF,Rqu∗F(m+ 1)

)⇒ Hp+q

(X,F(m+ 1)

).

We define H2m+1(X,F(m+ 1))0 to be the kernel of the edge map

H2m+1(X,F(m+ 1)

)→ H0

(SpecF,R2m+1u∗F(m+ 1)

).

(Of course, H2m+1(X,F(m + 1))0 and H2m+1(X,F(m + 1)) coincide wheneverH2m+1(XFs ,F(m+ 1)) has no GF -invariants.) There is an edge map

H2m+1(X,F(m+ 1)

)0→ H1

(F,R2mu∗F(m+ 1)

).

Let Em,m+12 (X)0,F be the inverse image of H2m+1(X,F(m+ 1))0 under (2.2.2)

and (2.2.1). The Abel-Jacobi map is the map

σm : Em,−m−12 (X)0,F → H1

(SpecF,R2mu∗F(m+ 1)

).

We will usually identify the range with H1(F,H2m(XFs ,F(m+ 1))).The Abel-Jacobi map is functorial for change of base field; for flat morphisms of

relative dimension 0; and (covariantly) for finite flat morphisms. All of these followeasily from what we have said so far and compatibilities of Leray spectral sequenceswith edge maps.

16 TOM WESTON

It is clear that the Abel-Jacobi map is compatible with change in N . In partic-ular, we can pass to the limit to obtain an l-adic Abel-Jacobi map

σm : Em,−m−12 (X)0,Zl → H1

(F,H2m(XFs ,Zl(m+ 1))

).

Note that by the Weil conjectures H2m+1(XFs ,Zl(m+1)) has no GF -invariants (sothat Em,−m−1

2 (X)0,Zl = Em,−m−12 (X)) if it is torsion-free, F is a local field (and

X has good reduction) or a global field and X is projective.

2.3. The Leibniz relation.

2.3.1. Algebraic correspondences. Let F be a perfect field and let X and Y besmooth proper varieties of dimension n over F . We will be forced to work with afairly restrictive notion of algebraic correspondences.Definition 2.3.1. An irreducible correspondence from X to Y is an integral closedsubscheme α → X×FY such that the projections α→ X and α→ Y are finite andfaithfully flat. A correspondence is a formal sum (with integer or Zl-coefficients asappropriate) of irreducible correspondences.

We use algebraic correspondences to define maps in K-theory and etale coho-mology in the usual way: for any irreducible correspondence α → X×FY , we definemaps

α∗ : Hi(X,Zl)→ Hi(α,Zl)→ Hi(Y,Zl)

α∗ : Hi(Y,Zl)→ Hi(α,Zl)→ Hi(X,Zl)as the composition of pullback and trace maps on etale cohomology. In the sameway we obtain analogous maps of coniveau spectral sequences in etale cohomologyand K-theory. We can also apply the construction to αFs → XFs×Fs YFs ; in thiscase αFs∗ and α∗Fs

commute with the action of GF since α is defined over F . Inparticular, one obtains maps on the Galois cohomology of the etale cohomologygroups of XFs and YFs . All of these definitions extend immediately to generalcorrespondences by linearity.

The functorialities of the Abel-Jacobi map show that it is compatible with alge-braic correspondences in the sense that there is a commutative diagram

Em,−m−12 (X)0,Zl

α∗ //

σm

Em,−m−12 (Y )0,Zl

σm

H1(F,H2m(XFs ,Zl(m+ 1))

) αFs∗ // H1(F,H2m(YFs ,Zl(m+ 1))

)and similarly for α∗.

We say that a variety X is cohomologically torsion-free at l if all of the groupsHi(XFs ,Zl) are torsion-free. Let α → X×Y and β → X ′×Y ′ be irreducible corre-spondences and assume that X,X ′, Y, Y ′ are all cohomologically torsion-free at l.One then has Kunneth projections which are compatible with algebraic correspon-dences:

Hi+j(XFs×X ′Fs,Zl) //

(α×β)Fs∗

Hi(XFs ,Zl)⊗ZlHj(X ′Fs

,Zl)

αFs∗⊗βFs∗

Hi+j(YFs×Y ′Fs,Zl) // Hi(YFs ,Zl)⊗ZlH

j(Y ′Fs,Zl)

and similarly for contravariant maps.


2.3.2. Markings. Assume that for every F -variety X of dimension n we are given aZariski sheaf LX which is invertible on the smooth locus of X; further assume thatthis assignment is functorial in the sense that for a map f : X → Y over SpecFthere is an induced map f∗LY → LX . (For example, if n = 1 one can take LXto be any fixed pluricanonical sheaf Ω⊗wX/F .) We define an L-marking ωX on anF -variety X to be a non-zero rational section of LX ; a marked variety is a varietyX together with an L-marking ωX .

Let X and Y be smooth proper marked varieties of dimension n over SpecF . Letα → X×Y be an irreducible correspondence. We define a rational function fα onα as the ratio of the pullbacks of ωX and ωY to Lα. We may view the pair (α, fα)as an element of the coniveau spectral sequence En,−n−1

1 (X×Y ); if α =∑miαi

by this we mean the element∑

(αi, fmiαi ). We say that α is admissible for the givenmarkings if the Weil divisor of fα is trivial on α; in this case (α, fα) defines anelement of En,−n−1

2 (X×Y ).

2.3.3. Composition of algebraic correspondences. Let X,Y, Z be smooth proper va-rieties of dimension n over F . Let α → X×Y and β → Y ×Z be irreduciblecorrespondences. Under certain circumstances we can define a composition β αas a correspondence from X to Z.

Let Γ denote the scheme-theoretic intersection of α×Z and X×β in X×Y ×Z.An easy argument with tangent spaces shows that each irreducible component Γiof Γ is generically reduced. Let γi → X×Z be the scheme-theoretic image of Γiunder the map X×Y ×Z → X×Z. Checking on the level of geometric pointsand using [22], Proposition 4.4.2 one can show that each γi is finite and surjectiveover X and Z. If in addition each γi is flat over X and Z, we define β α tobe the correspondence

∑[k(Γi) : k(γi)]γi. We extend this definition to arbitrary

correspondences by distributivity, at least when all subproducts are defined.Lemma 2.3.2. Suppose that X,Y, Z are marked varieties and α and β are admis-sible. If β α is defined, then it is admissible as well.

Proof. This is straightforward from the definitions; one does need to use the factthat if π : X → Y is a finite surjective morphism, then π∗π

∗ is injective on cycles.

2.3.4. The Leibniz relation. Let X,Y, Z be smooth proper marked varieties of di-mension n over F and let α → X×Y and β → Y ×Z be admissible correspondencessuch that γ = β α is defined (and thus, by Lemma 2.3.2, admissible). We assumefurther that X,Y, Z are cohomologically torsion-free at l. It was observed by Mazurand Beilinson in the case n = 1 that there is a remarkable relation between theAbel-Jacobi classes

σX×Y (α, fα) ∈ H1(F,H2n(XFs×YFs ,Zl(n+ 1))

);

σY×Z(β, fβ) ∈ H1(F,H2n(YFs×ZFs ,Zl(n+ 1))

);

σX×Z(γ, fγ) ∈ H1(F,H2n(XFs×ZFs ,Zl(n+ 1))

).

We now prove their Leibniz relation for arbitrary n.We first need some notation. Let ∆Z → Z×Z be the diagonal viewed as an

algebraic correspondence from Z to Z; both ∆Z∗ and ∆∗Z are the identity map onetale cohomology and K-theory. We view α×∆Z as a correspondence from X×Zto Y ×Z and ∆X×β as a correspondence from X×Y to X×Z.

18 TOM WESTON

Proposition 2.3.3. With the above hypotheses and notation we have

σX×Z(γ, fγ) = (α×∆Z)∗σY×Z(β, fβ) + (∆X×β)∗σX×Y (α, fα).

Proof. By linearity we may assume that α and β are irreducible. Since the Abel-Jacobi map is compatible with algebraic correspondences it suffices to prove that

(γ, fγ) = (α×∆Z)∗(β, fβ) + (∆X×β)∗(α, fα)

in En,−n−11 (X×Z).

Consider first (α×∆Z)∗(β, fβ). The “cycle” part of this is obtained as follows:one pulls back and pushes forward β → Y ×Z in the diagram

α×∆Z

$$JJJJJJJJJ

zzttttttttt

Y ×Z X×Z

Let β′ be the image of β under the map id×∆Z : Y ×Z → Y ×Z×Z. Pulling backβ to α×∆Z is the same as forming the scheme-theoretic intersection

(2.3.1) (X×β′) ∩ (α×∆Z) → X×Y ×Z×Z.The projection from here to X×Z factors through X×Y ×Z; here the image of(2.3.1) is just the intersection of X×β and α×Z. In particular, the image of β inX×Z is nothing other than β α = γ.

Since fβ “is” ωY /ωZ (we will systematically omit pullback maps from our no-tation for the remainder of this proof), tracing through the maps we see that theinduced rational function on an irreducible component γi of γ is

Nk(Γi)/k(γi)ωY

ωmiZ

where Γi is the irreducible component of Γ = (α×Z) ∩ (X×β) surjecting onto γiand mi = [k(Γi) : k(γi)]. That is, writing γ =

∑miγi as a sum of irreducible

correspondences, we have

(2.3.2) (α×∆Z)∗(β, fβ) =∑(

γi,Nk(Γi)/k(γi)ωY

ωmiZ

).

Similarly, we have

(2.3.3) (∆X×β)∗(α, fα) =∑(

γi,ωmiX

Nk(Γi)/k(γi)ωY

).

Adding (2.3.2) and (2.3.3) in En,−n−11 (X×Z) yields precisely (γ, fγ), as claimed.

2.4. Derivations to Galois cohomology.

2.4.1. Algebras of correspondences. Let X be a smooth proper marked variety ofdimension n over F ; we assume that X is cohomologically torsion-free at l andthat En,−n−1

2 (X)0,Zl = En,−n−12 (X). (This second condition is redundant if F

is a global field.) By an algebra of correspondences A0 on X we will mean a setof correspondences from X to X which is closed under addition and composition(in particular, we assume that every product is defined) and contains ∆X ; A0 isnaturally a ring with identity ∆X . We say that A0 is admissible if every elementof A0 is admissible (with respect to the fixed marking on X).


Set V = Hn(XFs ,Zl) and A = A0⊗Z Zl; A admits two maps to EndZlV , onegiven by α 7→ α∗ and the other by α 7→ α∗. We let B∗ and B∗ denote the imagesof these maps.

Assuming that A0 is admissible, we have a map

σ : A → En,−n−12 (X×X)⊗ZZl → H1

(F, V ⊗ZlV (n+ 1)

)given as the composition of α 7→ (α, fα), the Abel-Jacobi map and the Kunneth pro-jection. By Proposition 2.3.3 we have

σ(βα) = (α∗⊗ 1)σ(β) + (1⊗ β∗)σ(α).

2.4.2. Bilateral derivations. We assume now that A is commutative. We can usethe Poincare pairing ϕ : V⊗ZlV (n)→ Zl to identify V (n) with V † = HomZl(V,Zl)by v 7→ ϕ(·, v). Identifying V ⊗ZlV

† with EndZlV , we can view σ as a map

σ′ : A → H1(F,EndZlV (1)

).

However, the Poincare pairing satisfies ϕ(v, α∗v′) = ϕ(α∗v, v′) and ϕ(v, α∗v′) =ϕ(α∗v, v′). Thus the isomorphism V (n) ∼= V † interchanges the actions of B∗ andB∗; in other words, σ′ satisfies

σ′(βα) = (α∗⊗ 1)σ′(β) + (1⊗ β∗)σ′(α)

where we view EndZlV as a B∗⊗ZlB∗-module via (α∗⊗ β∗)f(v) = α∗f(β∗v). Thus

σ′ is an A-bilateral derivation when V is viewed as A-module via α 7→ α∗.

2.4.3. Derivations. Choose now a maximal ideal m of B∗ and set A = B∗m, H =V ⊗B∗A. We have natural maps i : H → V and j : V H.

We say that m is dualizing if A is reduced and Gorenstein and H is free of rank 2over A. (This last hypotheses will not be relevant until later sections.) Given suchan m, fix a Gorenstein trace tr : A → Zl with congruence element η ∈ A. Definean A-bilateral derivation

D : A → H1(F,EndZlH(1)

)to be the composition of σ′ with the map on cohomology induced by the mapEndZl V → EndZlH given by f 7→ jfi. Define an A-derivation

∂ : A → H1(F,EndAH(1)

)as the composition of D with the map htr : EndZlH → EndAH of Appendix A.1.2.

Let I denote the kernel of A → A. By Lemma A.2.1 the maps above restrict toA-module homomorphisms

D : I/I2 → H1(F,EndZlH(1)

)δ

∂ : I/I2 → H1(F,EndAH(1)

).

By Lemma A.2.3 our choice of Gorenstein trace tr yields an A-linear Galois equi-variant isomorphism (EndZlH)δ ∼= EndAH. If we assume that the hypotheses ofLemma A.2.2 are satisfied (as is the case if H⊗Ak is absolutely irreducible), we canuse this to view D as a map

D : I/I2 → H1(F,EndAH(1)

)

20 TOM WESTON

which by Lemma A.2.3 satisfies ηD = ∂. That is, there is a commutative diagram

ID //

H1(F,EndAH(1)

)η

A ∂ // H1(F,EndAH(1)

)This induces a derivation on cokernels

Θ : A→ H1(F,EndA(H/ηH)(1)

).

In Section 3.2 we will use this derivation to construct cohesive Flach systems.It is worth mentioning an argument which is often implicit in the proof that a

maximal ideal m is dualizing. For this we ignore Galois actions; in particular, we fixan isomorphism Zl ∼= Zl(1) so that we can regard ϕ as a pairing ϕ : V ⊗ZlV → Zl.Suppose that we are given a Zl-linear automorphism w of V such that w(α∗v) =α∗w(v) and w(α∗v) = α∗w(v) for all α ∈ A and v ∈ V . Suppose also that we knowthat H = Vm is free of rank 2 over A = B∗m. The pairing ϕ = ϕ (id⊗w) on Vis now a perfect B∗-pairing. Localizing yields a perfect A-pairing H⊗ZlH → Zl.By Section A.1.2 it follows that A is Gorenstein. In the case of modular formsthe automorphism w is given by an Atkin-Lehner involution w∗ζ . In fact, one cankeep track of the Galois twist given by w∗ζ ; in this way one can rewrite the aboveconstruction to take values in a twist of the symmetric square of H. This point ofview is closer to that of Flach and Mazur.

3. Construction of Euler systems

3.1. A reciprocity law for the Abel-Jacobi map. The Abel-Jacobi map pro-duces Galois cohomology classes from appropriate pairs of algebraic cycles andrational functions; in order to show that such classes form a partial Euler systemwe will need some control over the local behavior of these classes. This is ac-complished in the next result which relates this local behavior to divisor maps inpositive characteristic; it is an explicit reciprocity law in the sense of Kato and isformally quite similar to [40], Corollary 3.2.4.

Let R be a complete local ring with perfect residue field k of characteristicp ≥ 0 and perfect fraction field K. (As before our arguments go through for any Rsatisfying appropriate purity hypotheses.) Let X be a smooth proper scheme overSpecR of relative dimension n; let X denote the generic fiber of X. Fix an integerm and let F denote either the constant sheaf Z/NZ or Zl (for N, l relatively primeto m!p). Let V denote the GK-module H2m(XKs ,F(m + 1)); it is unramified bysmooth base change and we give it the unramified finite/singular structure. Wehave an Abel-Jacobi map

σm : Em,−m−12 (X)0,F → H1(K,V ).

For Z ⊆ X let Z denote the closure of Z in X. Let

divk : Em,−m−12 (X)→ CHmXk

be the map sending a pair (Z, f) to the divisor of f on Z; it is supported on Zk.(That divk is well-defined will become apparent in the proof below. Alternately,divk can be viewed as a localization map in motivic cohomology.)


Theorem 3.1.1. With the notation above there is a commutative diagram

Em,−m−12 (X)0,F

divk //

σm

CHmXk

s

H1(K,V )

H1s (K,V ) ' // H2m

(Xks ,F(m)

)GkHere s is the cycle class map in etale cohomology, and the bottom isomorphism isthat of (1.1.1) and smooth base change.

Somewhat more “motivically”, one can write this diagram over Q as

H2m+1M

(X,Q(m+ 1)

)//

ch

H2mM(Xk,Q(m)

)ch

H2m+1et

(X,Ql(m+ 1)

)

H2met

(Xks ,Ql(m)

)Gk'

H1(K,H2m

et (XKs ,Ql(m+ 1)))

// H1s

(K,H2m

et (XKs ,Ql(m+ 1)))

We use the formulation in terms of coniveau spectral sequences so that we can workintegrally.

Proof. Assume first that F = Z/NZ. We show that the map divk relates the Abel-Jacobi map for X to a relative Abel-Jacobi map for the pair X,Xk. This relativeAbel-Jacobi map will be identified with the cycle class map via purity.

We claim that there is a commutative diagram

(3.1.1) Em,−m−12 (X)0,F

δ1 // Em+1,−m−12,Xk

(X)

a1

Em,m+12

(X,F(m+ 1)

)0

δ2 //

Em+1,m+12,Xk

(X,F(m+ 1)

)a2

H2m+1(X,F(m+ 1)

)0

δ3 // H2m+2Xk

(X,F(m+ 1)

)a3

H1(

SpecK,R2muK∗F(m+ 1)) δ4 // H2

Spec k

(SpecR,R2mu∗F(m+ 1)

)where u : X → SpecR, uK : X → SpecK are the structure maps. Here theleft-hand vertical maps are those in the definition of the Abel-Jacobi map.

The map a1 is a relative Chern character and δ1 and δ2 are the boundary mapsof the localization sequences for the coniveau spectral sequences as in Section 2.1.1.We already commented on the commutativity of this square in Section 2.1.4. Themap a2 is an edge map which exists by purity for closed subschemes of Xk in X.

22 TOM WESTON

The map δ3 is the boundary map in etale cohomology for the pair X,Xk. Thissquare commutes by Section 2.1.1.

The map a3 is an edge map of a Leray spectral sequence (with supports); itsexistence uses purity for the pair SpecR, Spec k. The map δ4 is a boundary mapin etale cohomology combined with a base change map. The commutativity of thissquare is a fairly straightforward and general fact of homological algebra; see [44],Proposition A.7.1. This completes the proof of the commutativity of (3.1.1).

We claim that the right-hand side of (3.1.1) identifies via purity with the sequence

(3.1.2) CHmXk = Em,−m2 (Xk)a′1−→ Em,m2

(Xk,F(m)

) a′2−→

H2m(Xk,F(m)

) a′3−→ H0(Spec k,R2muk∗F(m)

).

Here uk : Xk → Spec k is the structure map, a′1 is a Chern character and the nexttwo are the obvious edge maps.

To see this, note first that from the construction of the coniveau spectral sequencein K-theory as in [34], Section 7 there is a canonical isomorphism of spectral se-quences Ep,qr (Xk) '−→ Ep+1,q−1

r,Xk(X) which for r = 1 coincides with the obvious

identification of the corresponding expressions (2.1.3). There is an analogous iso-morphism of spectral sequences Ep,qr (Xk,F(m)) '−→ Ep+1,q+1

r,Xk(X,F(m+1)) coming

from purity and the construction of the coniveau spectral sequence; for r = 1 itcoincides with the obvious identifications of (2.1.2). The identification of a1 with a′1now follows from these identifications and the description of the Chern character in(2.1.4). The identifications of a2 with a′2 and a3 with a′3 are straightforward fromthe definition of the purity isomorphism (see for example [19], pp. 205–207) andeasy compatibilities of edge maps. That the composition (3.1.2) is the cycle mapfollows easily from (2.1.4). (All of this is really just an integral version of standardcomputations of motivic cohomology.)

It remains to identify the maps

(3.1.3) Em,−m−12 (X) δ1−→ Em+1,−m−1

2,Xk(X) '←− Em,−m2 (Xk) = CHmXk

(3.1.4) H1(K,V ) = H1(SpecK,R2muK∗F(m+ 1)

) δ4−→

H2Spec k

(SpecR,R2mu∗F(m+ 1)

) '←− H0(Spec k,R2muk∗F(m)

) ∼= H1s (K,V )

with divk and the singular restriction map, respectively. For (3.1.3) this followsfrom the definition of δ1 as a boundary map of the localization sequence and thedescription of the spectral sequence differential as the divisor map. The desireddescription of (3.1.4) is straightforward from the definition of purity via cup productwith the fundamental class; we omit the details.

For the case of F = Zl it suffices to observe that everything above is obviouslycompatible with change in N .

3.2. Construction of cohesive Flach systems.

3.2.1. Divisorial liftings. Let F be a number field and let S be an open subschemeof the spectrum of the ring of integers of F . Let X be a smooth proper scheme ofrelative dimension n over S with generic fiber X. Fix a closed point v ∈ S andlet Z be a codimension m cycle of the closed fiber Xkv . We say that a finite set(Zi, fi) of pairs of codimension m cycles Zi on X and rational functions fi on Zi


is a divisorial lifting of Z if∑

divZi fi = Z; here Zi is the closure of Zi in X and Zis considered as a vertical cycle of codimension m+ 1 on X.

Note that a divisorial lifting of Z yields an element∑

(Zi, fi) of Em,−m−12 (X)

such that

divkw(∑

(Zi, fi))

=

Z w = v;0 w ∈ S − v.

By Theorem 3.1.1 such an element will yield a Galois cohomology class which isunramified at w ∈ S − v and locally at v looks like the cycle class of Z.

3.2.2. Partial Euler systems. Let F , S and X be as above. Fix an integer m and aprime l > m such that Σl ⊆ S; set V = H2m(XFs ,Zl(m+ 1)). We assume that Vis torsion-free.

Fix a Zl-algebra A of scalars and let T be a torsion-free l-adic GF -module ad-mitting a map V → T with finite cokernel. (V itself need not have any structureof A-module.) We will consider the Abel-Jacobi map

σ : Em,−m−12 (X)→ H1(F, V )→ H1(F, T ).

V is unramified at all places of S−Σl by smooth base change; thus T is unramifiedat all such places as well. We let S denote the finite/singular structure on T with

H1f,S(Fv, T ) =

H1

ur(Fv, T ) v ∈ S − Σl;H1f,cris(Fv, T ) v ∈ Σl;

H1(Fv, T ) v /∈ S.Definition 3.2.1. Let v be a closed point of S − Σl and let η be an element ofA. We say that a collection of codimension m cycles Z1, . . . , Zr on Xkv generate Twith depth η if the A-submodule of T (−1)Gkv generated by the cycle classes of theZi contains ηT (−1)Gkv .Lemma 3.2.2. Let Z1, . . . , Zr be cycles on Xkv which generate T with depth η.Assume that each of the Zi admits a divisorial lifting to X. Then there is an A-submodule C of H1(F, T ) such that Cw,s = 0 for w ∈ S − v and such that Cv,shas depth η in H1

s (Fv, T ).

Proof. Let (Zij , fij) be a divisorial lifting of Zi and define C to be the A-submodule of H1(F, T ) generated by the σ(

∑(Zij , fij)). That C has the desired

properties at w ∈ S−Σl follows immediately from the definition of a divisorial lift-ing and Theorem 3.1.1. The fact that C has trivial singular restriction at w ∈ Σlfollows from [33], Theorem 3.1.

Proposition 3.2.3. Let L be a set of closed points of S−Σl. Assume that for eachv ∈ L there is a set of codimension m cycles on Xkv which generate T with depth ηand which admit divisorial liftings. Then there is a partial Euler system Cvv∈Lof depth η for (T,S).

Proof. This is immediate from Lemma 3.2.2.

Of course, if one can choose L appropriately above, then Proposition 1.3.4 yieldsan annihilator of the Selmer group of T .

In the special case of a product variety X×X one can often use powers of thegraph of Frobenius as generators. This is essentially the point of view we will takein our construction of cohesive Flach systems.

24 TOM WESTON

3.2.3. Construction of cohesive Flach systems: preparation. Assume now that F isas in Section 1.4 and let S and X be as above. Fix a prime l > n with Σl ⊂ S andsuch that X is cohomologically torsion-free at l. Set V = Hn(XFs ,Zl).

Let A = A0⊗Zl be a commutative l-adic algebra of correspondences on X.Let B∗ and B∗ denote the images of A → EndZlV . Assume also that we have adualizing maximal ideal m of B∗; set A = B∗m and H = V ⊗B∗ A. We assume thatH is minimally ramified in the sense of Section 1.4.1. (Note that H is automaticallyunramified away from the set Σ consisting of Σl and the places not lying in S. His also crystalline at every place v of Σl by [10].) A is Gorenstein by assumption;fix a Gorenstein trace tr with congruence element η.

Let T denote End0AH(1) endowed with the finite/singular structure as in Sec-

tion 1.4.2. Consider the Galois equivariant map

V ⊗ZlV (n) ∼= V ⊗ZlV† ∼= EndZlV → EndZlH → EndAH End0

AH

as in Section 2.4.3. Since by the Kunneth theorem V ⊗Zl V (n) is a quotient ofH2n(XFs×XFs ,Zl(n)), we see that in this way we obtain a map

(3.2.1) H2n(XFs×XFs ,Zl(n+ 1)

)→ T.

3.2.4. Construction of cohesive Flach systems: generators. For a place v of S−Σl,by a diamond operator for v we mean an automorphism 〈v〉 of X such that j 〈v〉∗ i =χ(v)ε(v)n as an automorphism of H; here χ is the determinant character of H, εis the cyclotomic character and i, j are as in Section 2.4.3. Assuming that suchdiamond operators exist, we let Γv be the graph of the Frobenius morphism Fr(v)on Xkv and let Γ′v be the image of Fr(v)×〈v〉 : Xkv → Xkv×Xkv . Let L denote theset of places of F with Frobenius conjugate to complex conjugation on H⊗Ak.

Lemma 3.2.4. Fix a place v ∈ L and integers a, b such that l does not divide a−b.Then aΓv + bΓ′v generates T with depth η via (3.2.1).

Proof. By Lemma 1.4.3 we know that T (−1)Gkv ∼= (End0AH)Gkv is a free A-module

of rank 1 generated by the matrix(

1 00 −1

)(for an appropriate choice of basis of H).

We must compute the image of the cycle class of aΓv + bΓ′v in T (−1). The imageof Γv in EndZlV is precisely the geometric Frobenius endomorphism of V by [18],pp. 155–156 and Chapter II, Section 4. This maps to the Frobenius endomorphismFr(v) of H in EndZlH. Since Frobenius is A-linear, by Section A.1.2 this maps toη times the Frobenius morphism in EndAH.

By [18], pp. 155–156, the image of Γ′v in EndZlV is 〈v〉∗ Fr(v)adj where Fr(v)adj

is the Poincare adjoint of Fr(v), characterized by ϕ(Fr(v)adjx, y) = ϕ(x,Fr(v)y).Since ϕ is Galois equivariant, one computes that Fr(v)adj is just ε(v)−n Fr(v)−1.Thus 〈v〉∗ Fr(v)adj = χ(v) Fr(v)−1. We therefore conclude as above that Γ′v mapsto ηχ(v) Fr(v)−1 in EndAH.

Choose a basis of H with respect to which Fr(v) is given by a matrix(α 00 β

). Since

H has determinant χ, χ(v) Fr(v)−1 is given by(β 00 α

). It follows that aΓv + bΓ′v

maps to12 (a− b)(α− β)η

(1 00 −1

)in End0

AH. The lemma follows.


3.2.5. Construction of cohesive Flach systems: conclusion. We are now in a posi-tion to prove our main result on the construction of cohesive Flach systems. Recallthat if ω is a marking on X and α ∈ A0, we write fα for the induced rationalfunction on α. We continue to assume that H is minimally ramified.Proposition 3.2.5. Let ω be an admissible marking for the algebra of correspon-dences A on X. Assume that for all v ∈ S −Σl there is a correspondence Tv ∈ A0

and integers av, bv such that:• (Tv, fTv ) is a divisorial lifting of avΓv + bvΓ′v;• l does not divide av − bv;• T ∗v agrees with the Hecke operator Tv = trace Fr(v) on H;

and that for v /∈ S there is a correspondence Tv ∈ A0 with T ∗v = Tv. Assumefurther that:

• T⊗Ak is absolutely irreducible;• H1(F (T ∗[a])/F, T ∗[a]) = 0 for every ideal a of finite index in A;• H1

s (Fv, T ) = 0 for every v /∈ S.Then T admits a cohesive Flach system of depth η. If the differences av − bv are aconstant c independent of v, then this cohesive Flach system is of Eichler-Shimuratype of weight c.

Proof. The classes cv are defined to be σ(Tv, fTv ), with σ the map

A → En,−n−12 (X×X)⊗ZZl → H1

(F,H2n(XFs×XFs ,Zl(n+ 1))

)→ H1(F, T ).

That cvv∈L is a Flach system of depth η and that each cv vanishes in H1s (Fw, T )

for w 6= v is immediate from the assumptions and Lemmas 3.2.2 and 3.2.4. Thateach cv vanishes in H1

s (Fv, T/ηT ) is proved in the same way as Lemma 3.2.4. Thefact that the map Θ : A→ H1(F, T/ηT ) sending Tv to cv is a derivation was provedin Section 2.4.3. The last statement is clear from the proof of Lemma 3.2.4.

4. Applications to modular forms

4.1. The modular curve X1(N).

4.1.1. Definitions. It is a fairly straightforward exercise to apply Proposition 3.2.5in the case of classical modular forms of squarefree level. (The squarefree assump-tion simplifies a few points; it is probably not essential.) Fix a squarefree integerN ≥ 5 and let X1(N) → Spec Z[ 1

N ] be the modular curve of level N as in [7]; it isa fine moduli scheme for generalized elliptic curves with sections of exact order Non fibers (which are required to meet all irreducible components of fibers which areNeron polygons). X1(N) → Spec Z[ 1

N ] is proper, smooth, geometrically connectedand of relative dimension 1. The maps 〈d〉 : (E,P ) 7→ (E, dP ) on moduli realize(Z/NZ)× as a group of automorphisms of X1(N).

For any prime number p we let X1(N ; p) be the modular curve classifying triples(E,P,C) of elliptic curves, points of exact order N and cyclic subgroups of orderp; one requires that C contains no non-trivial multiple of P in the case that pdivides N . X1(N ; p) is a proper Z[ 1

N ]-scheme of relative dimension 1 and is smoothover Z[ 1

Np ]. It admits two natural degeneracy maps jp, j′p : X1(N ; p) ⇒ X1(N);on moduli the first sends a triple (E,P,C) to (E,P ) and the second sends it to(E/C,P ).

26 TOM WESTON

4.1.2. Hecke correspondences. We define the pth Hecke correspondence Tp to be thescheme-theoretic image of the map

jp× j′p : X1(N ; p)→ X1(N)×Spec Z[ 1N ] X1(N).

For p not dividing N the fiber Tp,Fp is reduced and has two irreducible components:the graph of Frobenius (which we write as Γp) and the image of Fr×〈p〉 (which wewrite as Γ′p).

The modular curve X1(N) is the generic fiber of X1(N); it is a smooth projectivecurve over Q. Write Tp for the generic fiber of the Hecke correspondence Tp. Onecan more generally define Hecke correspondences Tn for any n; see [36], Chapter2, Sections 1–3, for example. Each Tn is a divisor on X1(N)×X1(N), and thusis Cohen-Macaulay. The projections Tn ⇒ X1(N) are visibly quasi-finite, properand surjective, so that by [23], Proposition 15.4.2 and [22], Proposition 4.4.2 theyare finite and faithfully flat. In particular the Tn are correspondences in our sense.Regarding the diamond operators as correspondences via their graphs, it followsfrom the relations [26], Chapter 7, Theorem 2.1 (which already hold on the level ofmoduli) that the Hecke correspondences Tp and the diamond operators generate acommutative algebra of correspondences T1(N).

4.1.3. Galois representations. We fix a prime l ≥ 5 not dividing N and let V =H1(X1(N)Qac ,Zl). Let B∗ and B∗ denote the images of T1(N) in EndZlV as usual.By [6], Lemma 4.1 we can omit Tl from the generating set of B∗ and B∗.

Let m be a non-Eisenstein maximal ideal of B∗ associated to a newform of levelN ; see [6], Chapter 4 and [42], Theorem 3.4 for definitions. Assume also that mdoes not contain l. Then by the above references m is dualizing and H ⊗A k isabsolutely irreducible. In particular A = B∗m is Gorenstein. Fix a Gorenstein tracetr : A→ Zl with congruence element η. Set H = V ⊗B∗A and T = End0

AH(1).

4.1.4. The modular unit ∆. For any curve X over Q we set LX = Ω⊗6X/Q. Following

Flach and Mazur, we use the modular form ∆ as a marking on the curve X1(N).In particular, ∆ is a global section of LX1(N).Lemma 4.1.1. T1(N) is an admissible algebra of correspondences for X1(N) withthe L-marking ∆.

Proof. It is clear that the diamond operators are admissible for ∆, so that byLemma 2.3.2 we need only check that the divisor of fp = j∗p∆/j∗p

′∆ vanishes on Tpfor each p. It suffices to work on the level of geometric points. We first computethe divisor of ∆ on X1(N). For this we recall the combinatorics of the cuspsof X1(N): for each d dividing N there are φ(N)/2 distinct Γ1(N)-structures onNeron d-gons, so that the cusps of X1(N) are indexed as cd,i(N) with d dividing Nand i ∈ (Z/NZ)×/ ± 1. A cusp cd,i(N) has ramification degree d over the uniquecusp c of X(1); since the divisor of ∆ on X(1) is c, it follows that the divisor of ∆on X1(N) is

∑dcd,i(N).

Assume now that p does not divide N . There are two cusps cd,i(N ; p) andcdp,i(N ; p) of X1(N ; p) lying over cd,i(N) under jp and j′p; under jp (resp. j′p) thefirst is unramified (resp. ramified of degree p) while the second is ramified of degreep (resp. unramified). Thus the divisor of fp on X1(N ; p) is

∑d(1− p)(cd,i(N ; p)−

cdp,i(N ; p)). Since both cd,i(N ; p) and cdp,i(N ; p) map to cd,i(N)×cd,i(N) underjp× j′p, it follows that the divisor of fp on Tp is trivial.


The argument is similar for p dividing N . Here the behavior of the cusps is morecomplicated: there are three cusps cd,p,i(N ; p), cdp,1,i(N ; p), cdp,p,i(N ; p) lying overthe pair of cusps cd,i(N) and cdp,i(N). Under jp (resp. j′p) the cusp cd,p,i(N ; p)(resp. cdp,1,i(N ; p)) is ramified of degree p over cd,i(N), while cdp,1,i(N ; p) (resp.cd,p,i(N ; p)) is unramified over cdp,i(N). In both cases cdp,p,i(N ; p) is ramified ofdegree p − 1 over cdp,i(N). It follows easily that fp has trivial divisor already onX1(N ; p), and thus on Tp as well. This completes the proof.

If one follows through the calculations in the second case above, it becomes clearthat the only cusp forms for which T1(N) is admissible are multiples of powers of∆.

4.1.5. Divisors in positive characteristic. We now prove the following key lemma.Lemma 4.1.2. For all p not dividing N the pair (Tp, fp) is a divisorial lifting of6(Γ′p − Γp) on X1(N)×X1(N)→ Spec Z[ 1

N ].

Proof. By Lemma 4.1.1 we know that the divisor of fp on Tp has no horizontalcomponent. In characteristics not dividing Np the analysis is identical to that incharacteristic 0, so that the divisor of fp on Tp is supported in characteristic p.

We compute on Tp,Fp = Γp+Γ′p. Since the divisor of fp on Tp,Fp is of codimensonzero, it suffices to compute it generically on each irreducible component. Recall thatΓp is the scheme-theoretic image of the map

id×Fr : X1(N)Fp → X1(N)Fp×X1(N)Fp ;

on this irreducible component fp is simply π∗1∆/π∗2∆ where π1, π2 are the twoprojections. In particular, π1 = id is an isomorphism so that the divisor of π∗1∆has no generic support on Γp. However, π2 = Fr is purely inseparable; since ∆ isin the sixth power of the canonical sheaf, it follows that π∗2∆ vanishes to order 6on Γp. Thus the divisor of fp on Γp is −6Γp. In the same way one computes thatthe divisor of fp on Γ′p is 6Γ′p (using that 〈p〉 is an isomorphism). This completesthe proof.

4.1.6. The cohesive Flach system. We are now in a position to prove our maintheorem. Recall that N ≥ 5 is squarefree and l ≥ 5 is a prime not dividing N . A isthe localization of the “contravariant” image of T1(N) in EndZlH

1(X1(N)Qac ,Zl)at a non-Eisenstein maximal ideal m of B∗ corresponding to a newform and H =H1(X1(N)Qac ,Zl)m. A is Gorenstein and we have fixed a Gorenstein trace tr withcongruence element η. H is minimally ramified of weight 2 by [6], Lemma 3.27 and[38]. Set T = End0

AH(1) endowed with the usual finite/singular structure.Theorem 4.1.3. Let H be a modular Galois representation of weight 2 as above.Assume that T⊗Ak is absolutely irreducible and that H1(Q(T ∗[a])/Q, T ∗[a]) van-ishes for all ideals a of finite index in A (this holds in particular if GF → AutAHis surjective). Then T admits a cohesive Flach system of Eichler-Shimura type ofdepth η and weight −12.

Proof. We need to check the hypotheses of Proposition 3.2.5 for the algebra ofcorrespondences T1(N) with the marking ∆. That the 〈p〉 are diamond operatorsin our sense follows from [6], Theorem 3.1. The first two conditions on (Tp, fp)follow from Lemma 4.1.2. That T ∗p agrees with the trace of Fr(p) on H is also[6], Theorem 3.1. It remains to check that H1

s (Qp, T ) = 0 for every p dividing N .By [2], Theoreme A, as a GQp-module, H is either ordinary or a direct sum of an

28 TOM WESTON

unramified character and a tamely ramified character. The fact that H1s (Qp, T ) = 0

is straightforward in the second case; for the first case see [44], Lemma I.5.2 or [14],Lemma 2.10. This completes the proof.

We should note that it is not necessary to use [33] (as in Lemmaa 3.2.2) tocheck the crystalline condition in this theorem; one can easily modify the originalmethods of [14] to deal with it.

4.2. Kuga-Sato varieties.

4.2.1. Universal elliptic curves. Fix a squarefree integer N and let Y1(N)→ Z[ 1N ]

be the open complement of the cusps in X1(N)→ Z[ 1N ]. Let f : E1(N)→ Y1(N) be

the universal elliptic curve and let f : E1(N)→ X1(N) be the universal generalizedelliptic curve; the first is a smooth open subscheme of the second, which is properbut not smooth over Z[ 1

N ]. For κ > 0 we let Eκ1 (N) (resp. Eκ1 (N)) denote the κ-foldfiber product of E1(N) over Y1(N) (resp. E1(N) over X1(N)). Eκ1 (N) → Z[ 1

N ] issmooth but not proper, while Eκ1 (N)→ Z[ 1

N ] is proper but not smooth. Eκ1 (N) hasa proper smooth model Eκ1 (N) which contains Eκ1 (N) as a dense open subscheme;see [39], Section 2 and [4], Section 4.3. We use the obvious notation for the genericfibers of each of these Z[ 1

N ]-schemes.The Galois representation associated to a modular form of weight κ + 2 ≥ 3 is

initially a localization of

H1(Y1(N)Qac ,SymκR1fQac∗Zl

) def=

im(H1c (Y1(N)Qac ,SymκR1fQac∗Zl)→ H1(Y1(N)Qac ,SymκR1fQac∗Zl)

).

It can also be realized in the cohomology group Hκ+1(Eκ1 (N)Qac ,Zl), which is itselfa quotient of Hκ+1(Eκ1 (N)Qac ,Zl). This is the group in which we will work.

4.2.2. Hecke correspondences. Let Y1(N ; p) → Z[ 1N ] be the open complement of

the cusps in X1(N ; p)→ Z[ 1N ] and let E1(N ; p)→ Y1(N ; p) be the universal elliptic

curve. There are two natural degeneracy maps jp, j′p : E1(N ; p) ⇒ E1(N) and wedefine the Hecke correspondence Tκp to be the scheme-theoretic image of

jκp× j′pκ : Eκ1 (N ; p)→ Eκ1 (N)×Z[ 1N ] Eκ1 (N).

We let Tκp (resp. Tκp) be the closure of Tκp in Eκ1 (N)×Eκ1 (N) (resp. Eκ1 (N)×Eκ1 (N)).Tκp is also the strict transform of Tκp under the blow-up Eκ1 (N)× Eκ1 (N)→ Eκ1 (N)×Eκ1 (N). We write T κp , T κp and T κp for the generic fibers of these correspondences;that they are correspondences in our sense is shown in [39], Section 4. There arealso diamond operators 〈d〉κ on Eκ1 (N) and Eκ1 (N). The diamond operators and theT κp generate an algebra T1(N)κ of correspondences on Eκ1 (N) which is isomorphicto the algebra of correspondences on Eκ1 (N) generated by the diamond operatorsand the T κp .

For p not dividng N , the characteristic p fiber Tκp,Fp is reduced and has twoirreducible components Γκp , Γκp

′; the first is the image of id×Fr and the second isthe image of Fr×〈p〉κ. See [4], Theorem 5.3.3.1.


4.2.3. Galois representations. We follow the presentation of [4] via the integraltheory of [11]. Fix a prime l > max5, κ+ 1 not dividing N . Let V be the Galoismodule H1(Y1(N)Qac ,SymκR1fQac∗Zl). By [4], Section 5.4 and [39], Proposition4.1.1, V is a subquotient of Hκ+1(Eκ1 (N)Qac ,Zl); thus it is a subquotient of V0 =Hκ+1(Eκ1 (N)Qac ,Zl) as well.

We must assume that Eκ1 (N) is cohomologically torsion-free at l and that V isa direct summand of V0 as a Zl-module; both of these conditions hold for almostall l. We will apply the procedure of Proposition 3.2.5 directly to V rather than toV0; it is straightforward to check that this is valid as V is stable under all relevantoperations.

Let B∗ and B∗ denote the images of T1(N)κ in EndZlV . Let m be a non-Eisenstein maximal ideal of B∗ associated to a newform of weight κ+ 2 as in [11],Theorem 2.1; such an m is dualizing. In particular, H = V ⊗B∗A is free of rank 2over the Gorenstein ring A = B∗m and by [11], Theorem 3.38 H⊗Ak is absolutelyirreducible. Fix a Gorenstein trace tr : A → Zl with congruence element η. SetT = End0

AH(1) with the usual finite/singular structure.

4.2.4. Admissible markings. We have maps Eκ1 (N) → X1(N) and T κp → Tp; weassign sheaves L• to Eκ1 (N) and T κp as the pullback of the sixth power of thecanonical sheaf on X1(N) and Tp via these maps. In particular, ∆ is a globalsection of LEκ1 (N).

Lemma 4.2.1. T1(N)κ is an admissible algebra of correspondences for Eκ1 (N)with the L-marking ∆.

Proof. This is clear for the diamond operators. By Lemma 4.1.1 we know that fp =fTp has trivial divisor on each Tp. It follows that fκp = fT κp has trivial divisor on T κp .We need to check that this divisor does not become non-trivial as Eκ1 (N)×Eκ1 (N) isblown-up to Eκ1 (N)×Eκ1 (N). This is straightforward from the explicit descriptionof the resolution process as in [39], Section 2; one finds that the singularities of T κpcoming from the singularities of Eκ1 (N ; p) are resolved into a single κ-cycle. (Thesingularity of T κp coming from its embedding into Eκ1 (N)× Eκ1 (N) is not resolved.)Since the divisor of fκp on T κp must be supported on this κ-cycle, it follows that itis still trivial.

Lemma 4.2.2. For all p not dividing N the pair (T κp , fκp ) is a divisorial lifting ofthe cycle 6(Γκp

′ − Γκp) on Eκ1 (N)×Eκ1 (N)→ Z[ 1N ].

Proof. The proof of this is identical to that of Lemma 4.1.2; one also uses thedescription of T κp given in the proof of Lemma 4.2.1.

4.2.5. The cohesive Flach system. Recall that we have assumed that N is square-free; that l > max5, κ + 1; that Eκ1 (N) is cohomologically torsion-free at l; andthat V is a direct summand of V0. H is minimally ramified by [2] and [38].

Theorem 4.2.3. Let H be a modular Galois representation of weight κ+2 as aboveand set T = End0

AH(1). Assume that T⊗Ak is absolutely irreducible and that thecohomology group H1(Q(T ∗[a])/Q, T ∗[a]) vanishes for all ideals a of finite index inA. Then T admits a cohesive Flach system of depth η and weight −12.

30 TOM WESTON

Proof. The proof is essentially the same as that of Theorem 4.1.3, using our extrahypotheses to pass all constructions to V .

Of course, assuming the hypotheses above we now obtain immediate applicationsto deformation theory as in Section 1.4.5 and Section 1.4.6. If η is a unit, then onecan also apply Proposition 1.5.1.

5. Flach systems of Eichler-Shimura type

In this section we give the proof of Theorem 1.4.8. Since A is generated bythe Tv with v /∈ Σl and H1

s (Fv, T ) = 0 for v ∈ Σ − Σl it suffices to show thatΞ(∂Tv) = 2w∂Tv for v /∈ Σ.

5.1. The map on differentials. We begin by recalling the details of the construc-tion of the map Ξ. Fix a power ln of l such that η divides ln in A; such a powerexists since η is a non zero-divisor. We have

H1f (F, T ∗) = H1

f

(F, T ∗[η]

)= H1

f

(F, T ∗[ln]

).

We can and will work at finite levels since everything is ln-torsion. For any Z/lnZ-module M we write M∨ for its Pontrjagin dual HomZl(M,Z/lnZ).

The map Ξ : ΩA → ΩA is defined to be the composition

ΩAξ1−→ H1

f (F, T/ηT )ξ2−→ H1

f (F, T ∗[ln])∨ξ3−→ H1

f

(F,End0

A(H/lnH))∨ ξ4−→

HomA(ΩR⊗RA,A/lnA)∨ξ5−→ HomZl(ΩR⊗RA,Z/lnZ)∨

ξ6−→ ΩR⊗RAξ7−→ ΩA

ξ1 is the A-linear map induced by Θ; we have ξ1(∂Tv) = cv. ξ2 is induced bythe Bockstein pairing (1.4.6) and we computed in the proof of Lemma 1.4.5 thatξ2(cv)(κ) =

⟨1η cvv,s, κv

⟩v. ξ3 comes from (1.4.2); ξ4 is the dual of the isomorphism

of ln-torsion in (1.4.4); ξ5 is induced by the Gorenstein trace tr; ξ6 is the doubleduality isomorphism; and ξ7 is induced by π : R A.

Note that the cohesive Flach system enters only into the very first map ξ1;the remaining maps are at most dependent on the choice of Gorenstein trace tr,although the composite does not depend on that choice.

5.2. The Tate pairing. In order to explicitly compute the map Ξ we will need towork with the Tate pairing. Let M be a finite GFv -module of exponent m and letM∗ = HomZ(M,µm) be its Cartier dual. Recall that the Tate pairing is the map

H1(Fv,M)⊗H1(Fv,M∗)→ Q/Z

defined as the composition of

H1(Fv,M)⊗H1(Fv,M∗)cup−→ H2(Fv,M⊗ZM

∗) Cartier−→ H2(Fv, µm) '−→

H2(L/Fv, L×) val−→ H2(L/Fv,Z) δ←− H1(L/Fv,Q/Z) eval−→ Q/Z.

Here L is the unique unramified extension of Fv of degree m. The unlabeled iso-morphism comes from local class field theory. The map δ is the boundary map forthe exact sequence 0→ Z→ Q→ Q/Z→ 0; it is an isomorphism.

In our computation we will compute as far as H2(L/Fv, L×); we now give theinverse image in here of e

m ∈ Q/Z so that we have something to compare with.Let · : Q/Z → Q be the map sending x ∈ Q/Z to the unique x ∈ Q such that


0 ≤ x < 1 and x ≡ x (mod Z). Using the definition of δ one finds that the cocycleCe ∈ H2(L/Fv, L×) given by

(5.2.1) Ce(Fr(v)i,Fr(v)j) = λ e(i+j)m − eim−

ejm

0

maps to em under eval δ−1 val.

5.3. A special case.

5.3.1. Additional hypotheses. In this section we compute Ξ(∂Tv) with some addi-tional simplifying hypotheses; this computation will still contain most of the contentof the general case, but it is significantly simpler algebraically.

We make two assumptions. First, assume that the action of Fr(v) on H isdiagonal with respect to a fixed basis x, y; that is, Fr(v) acts on H by a matrix( s 0

0 t ) with s, t ∈ A. In particular, st = χ(v) and s + t = Tv. It follows from thedefinition of a cohesive Flach system of Eichler-Shimura type that in this case thecocycle cvv,s is given by

cvv,s : Gal(F urv (λ)/F ur

v

)→ T

τ j 7→ wjη(s− t)(

1 00 −1

)⊗ζn(5.3.1)

with the notation of Section 1.4.6.The second simplifying assumption is that the map π : R→ A is an isomorphism;

that is, A is the universal minimally ramified deformation ring of H⊗Ak and H isthe universal deformation. Of course, we will identify R with A via π.

5.3.2. Preliminaries. To compute Ξ(∂Tv), we begin by computing the image of ∂Tvin HomA(ΩA, A/lnA)∨. So let ω : ΩA → A/lnA be a fixed map; we will computeits image in Z/lnZ under ξ4 ξ3 ξ2 ξ1(∂Tv).

5.3.3. Making explicit the definition of ξ4 (see [44], Section V.1) one computesthat this is the same as the image under ξ3 ξ2 ξ1(∂Tv) of the cohomology classrepresented by the cocycle κ′ : GF → End0

A(H/lnH) given by

κ′(σ) = 1ad−bc

(dω(∂a)− bω(∂c) dω(∂b)− bω(∂d)aω(∂c)− cω(∂a) aω(∂d)− cω(∂b)

).

Here ρA(σ) =(a bc d

)∈ GL2(A) with ρA : GF → GL2(A) the Galois representation

on H with respect to the basis x, y

5.3.4. Using the definition of ξ3, this is just the image under ξ2 ξ1(∂Tv) ∈H1f (F, T ∗[ln])∨ of the cohomology class represented by the cocycle

κ : GF → T ∗[ln] = HomZl

(End0

A(H/lnH)(1), µln)

= HomZl

(End0

A(H/lnH),Z/lnZ)

explicitly given as

(5.3.2) κ(σ)(α βγ −α

)= tr

(1

ad−bc(αdω(∂a)− αbω(∂c) + γdω(∂b)− γbω(∂d)+

βaω(∂c)− βcω(∂a)− αaω(∂d) + αcω(∂b))).

32 TOM WESTON

5.3.5. Using the definition of ξ2 and its explicit expression for cv = ξ1(∂Tv), wefind that the desired element of Z/lnZ is the value of the Tate pairing

⟨1η cvv,s, κv

⟩v;

here we are identifying the image 1lnZ/Z with Z/lnZ. It remains to compute this.

5.3.6. The Tate pairing : preliminaries. To begin with, note that κv factors throughGal(F ur

v /Fv), as it is unramified at v; thus we need only concern ourselves with κ

evaluated at powers of Fr(v). Using the fact that ρA(Fr(v)i) =(si 00 ti

)which has

determinant χ(v)i, we find that (5.3.2) simplifies to

(5.3.3) κ(Fr(v)i)(α βγ −α

)= tr

(χ(v)−iα(tiω(∂si)− siω(∂ti))

).

Next, note that

si∂ti = siti−1i∂t = χ(v)i−1si∂t; ti∂si = χ(v)i−1ti∂s.

We therefore can write (5.3.3) as

κ(Fr(v)i)(α βγ −α

)= tr

(iχ(v)−1α (tω(∂s)− sω(∂t))

).

Setting K = Fv(H/lnH) (so that K/Fv is unramified), we see that κv factorsthrough Gal(K/Fv).

For cv, we computed in (5.3.1) that


v )→ T/lnT

τ j 7→ wjη(s− t)(

1 00 −1

)⊗ζn

Since τ ln

goes to 0 under this map, cv factors through Gal(F urv (λn)/F ur

v ).In order to compute the Tate pairing of κ and cv we first must descend cv

to a cocycle over Fv. We can do this over the field K(λn) as follows: let G =Gal(K(λn)/Fv). Denote by ϕ the element of G which acts as Frobenius on K andfixes λn, and denote by τ the element of G which is the identity on K and sendsλn to ζnλn. Then ϕ and τ generate G with the relations

ϕ[K:Fv ] = τ ln

= 1, τϕ = ϕτ ε(v).

One checks that 1η cvv,s is represented by the cocycle

θv : G→ T/lnT

ϕiτ j 7→ wε(v)ij(s− t)(

1 00 −1

)⊗ζn.

Via inflation we can represent κv by the cocycle G→ T ∗[ln] given by

(5.3.4) κv(ϕiτ j)(α βγ −α

)= tr

(iχ(v)−1α(tω(∂s)− sω(∂t))

).

5.3.7. The Tate pairing : cup product. We now compute the Tate pairing. The firststep is to form the cup product θv ∪κv. By the formula for the cup product we seethat θv ∪ κv sends the pair (ϕiτ j , ϕi

′τ j′) ∈ G×G to

θv(ϕiτ j)⊗ κϕiτj

v (ϕi′τ j′) ∈ T/lnT ⊗Zl T

∗[ln].

Under Cartier duality this maps to the cocycle C ∈ H2(G,µln) given by

C(ϕiτ j , ϕi′τ j′) = κϕ

iτj

v (ϕi′τ j′)(wε(v)ij(s− t)

(1 00 −1

))ζn.


Recall that κv(ϕi′τ j′) is a map End0

A(H/lnH) → Z/lnZ. In particular, ϕiτ j actstrivially on both the domain and the range. Thus by the definition of the adjointGalois action we find that

C(ϕiτ j , ϕi′τ j′) = tr

(i′χ(v)−1wε(v)ij(s− t)(tω(∂s)− sω(∂t))

)ζn.

If we let C ′ : G×G→ µln be the cocycle C ′(ϕiτ j , ϕi′τ j′) = ε(v)ii′jζn, then we

conclude that ω maps to

(5.3.5) tr(ω(wχ(v)−1(t∂s− s∂t))

)I

where I is the image of C ′ under the invariant map

H2(K(λn)/Fv, µln)→ Z/lnZ.

5.3.8. At this point, thankfully, we get the maps ξ5 and ξ6 for free. Specifically,suppose that we began with ω0 : ΩA → Z/lnZ and wished to compute its image inZ/lnZ under ξ5 · · · ξ1(∂Tv). This is the same as the image under ξ4 · · · ξ1(∂Tv) ofthe unique ω : ΩA → A/lnA such that tr ω = ω0. By (5.3.5) this is visibly just

(5.3.6) ω0

(wχ(v)−1(s− t)(t∂s− s∂t)

)I.

Similarly, in HomZl(ΩA,Z/lnZ)∨ the element (5.3.6) is just the evaluation at

(5.3.7) wχ(v)−1(s− t)(t∂s− s∂t)I

map, so that (5.3.7) is the final image of ∂Tv in ΩA. It remains, then, to computeI and to simplify our expression.

5.3.9. Computation of the invariant. We begin by computing I; it is the image ofC ′ under the maps

H2(K(λn)/Fv, µln)→ H2(L/Fv, L×)→ Z/lnZ,

where L is the unique unramified extension of Fv of degree ln. We first need tomodify C ′ by a coboundary to get it to factor through Gal(L/Fv) and to take valuesin L×. We can do this using the cochain f : G→ K(λn)× given by f(ϕiτ j) = λ

〈i〉n

where 〈i〉 is the unique integer in 0, 1, . . . , ln − 1 which is congruent to i moduloln. The coboundary formula states that

C ′∂f(ϕiτ j , ϕi′τ j′) =

C ′(ϕiτ j , ϕi′τ j′)f(ϕiτ jϕi

′τ j′)

ϕiτjf(ϕi′τ j′)f(ϕiτ j)

= λ〈i+i′〉−〈i〉−〈i′〉

n .

This is simply the inflation to H2(K(λn)/Fv,K(λn)×) of the cocycle C1 of (5.2.1).Since C1 was defined to map to 1 under the invariant map, we see that C ′ does aswell. Thus I = 1.

5.3.10. Differentials and Hecke operators. We conclude by (5.3.7) that

Ξ(∂Tv) = wχ(v)−1(s− t)(t∂s− s∂t) ∈ ΩA.

It remains to simplify this expression. Using that st = χ(v) we find that

(s− t)(t∂s− s∂t) = 2χ(v) (∂s+ ∂t) .

Thus we conclude that Ξ(∂Tv) = 2w∂(s+ t) = 2w∂Tv. This completes the proof ofTheorem 1.4.8 in this case.

34 TOM WESTON

5.4. A matrix computation. The key to removing both of the assumptions of theprevious computation is the following matrix lemma; the proof is an unenlighteninginduction and we omit it.Lemma 5.4.1. Let R be a ring, S an R-algebra and M an S-module. Let ∂ :S → M be an R-linear derivation. Let T =

(a bc d

)∈ GL2(S) be a matrix with

determinant δ = ad− bc and trace t = a+ d. Assume that δ lies in the image of Rin S. Let e be a positive integer, and write T e = (A B

C D ) ∈ GL2(S). Then

2eδe∂t = (−2bC + aD − dD)∂A+ (2cD + aC − dC)∂B+

(2bA− aB + dB)∂C + (−2cB − aA+ dA)∂D

5.5. Computation of Ξ in the non-diagonal case. We now explain how tocompute Ξ(∂Tv) when Fr(v) is not necessarily diagonal. We continue to assume thatπ is an isomorphism. This computation is fundamentally the same as the previousspecial case, just a bit messier and with the simple computation of Section 5.3.10replaced by the more elaborate computation of Lemma 5.4.1.

The complication is that Fr(v) no longer acts diagonally. Write ρA(Fr(v)) =(a bc d

), ρA(Fr(v)i) =

(ai bici di

)for some fixed basis x, y of H. Note that a + d = Tv,

ad− bc = χ(v) and aidi − bici = χ(v)i. The formula for the cocycle κ is exactly ascomputed in (5.3.2), replacing a, b, c, d with ai, bi, ci, di for σ = Fr(v)i. The Flachclass is now


v )→ T

τ j 7→ wjη(a−d 2b2c d−a

)⊗ζn.

In order to compute the Tate pairing of 1η cvv,s and κv we first must lift 1

η cvv,s to

H1(Fv, T/lnT ). In fact, the lifting

θv : Gal(K(λn)/Fv)→T/lnTϕiτ j 7→wε(v)ij

(a−d 2b2c d−a

)⊗ζn

still works.We now compute the cup product of κv and θv as cohomology classes for G.

Writing C = θv ∪ κv, one finds that

C(ϕiτ j , ϕi′τ j′) =

tr(wε(v)ijχ(v)−i

′(

(−2bci′ + adi′ − ddi′)ω(∂ai′) + (2cdi′ + aci′ − dci′)ω(∂bi′)+

(2bai′ − abi′ + dbi′)ω(∂ci′) + (−2cbi′ − aai′ + dai′)ω(∂di′)))ζn.

Applying Lemma 5.4.1, we conclude that

C(ϕiτ j , ϕi′τ j′) = 2wε(v)ii′j tr

(ω(∂Tv)

)ζn

and from here the computation is identical to the earlier case; we conclude thatΞ(∂Tv) = 2w∂Tv.

5.6. Computation of Ξ in the general case. We now remove the assumptionthat π is an isomorphism. The computation in this case is essentially the sameas in the previous case. First, recall that universality of R means that, fixing auniversal deformation ρR : GF → GL2(R), there is some basis of H with respect towhich ρA = πρR. We can conjugate in GL2(A) from this basis to our fixed basis


x, y; since π is surjective (and R is local) we can lift this conjugation to GL2(R).That is, we can conjugate ρR so as to assume that ρA = πρR where ρA is now therepresentation on our fixed basis x, y of H.

To compute Ξ this time, we begin with ω : ΩR⊗RA → A/lnA and computeits image in Z/lnZ. Proceeding as before, we find that this is the image underthe Tate pairing of two cocycles κv : G → T ∗[ln] and θv : G → T/lnT . WritingρR(Fr(v)i) =

(ai bici di

), ρA(Fr(v)i) =

(ai bici di

)we find that

κv(ϕiτ j)(α βγ −α

)tr(χ(v)−i

(αdiω(∂ai)− αbiω(∂ci) + γdiω(∂bi)−

γbiω(∂di) + βaiω(∂ci)− βciω(∂ci)− αaiω(∂di) + αciω(∂bi))).

The cocycle θv is given by

θv(ϕiτ j) = wε(v)ij(a−d 2b2c d−a

)⊗ζn

where ρA(Fr(v)) =(a bc d

)as before.

From these expressions the computation works out exactly as in the previouscase, with ∂ai, ∂bi, ∂ci, ∂di replaced by ∂ai, ∂bi, ∂ci, ∂di respectively. Lemma 5.4.1applies to show that

C(ϕiτ j , ϕi′τ j′) = 2wε(v)ii′j tr

(ω(∂a+ ∂d)

)ζn,

where ρR(Fr(v)) =(a bc d

). From here the computation is as before, with the fact

that ξ7(∂a) = ∂a and ξ7(∂d) = ∂d showing that Ξ is still multiplication by 2w.This completes the proof of Theorem 1.4.8.

Appendix A. Linear algebra

In this appendix we give a quick summary of the basic constructions we willneed with Gorenstein rings. We give no proofs; all of the results we state arestraightforward.

A.1. Gorenstein rings.

A.1.1. Definitions. Let A be a finite, flat, local Zl-algebra with maximal ideal mand residue field k. A is Gorenstein if the following equivalent conditions hold:

• Ext1A(k,A) ∼= k;

• dimk(A/lA)[m] = 1;• HomZl(A,Zl) is free of rank 1 as an A-module.

See [42], Section 1. Note that local complete intersections are Gorenstein; in par-ticular, this includes rings of the form Zl[x]/f(x) for f(x) monic.

By a Gorenstein trace we mean an A-basis of HomZl(A,Zl). Fix a Gorensteintrace tr and consider the ring A⊗ZlA which we regard as an A-algebra via multipli-cation on the right factor. tr⊗1 is an A⊗ZlA-generator of the free rank 1 moduleHomA(A⊗ZlA,A). Thus we can write the diagonal map ∆ : A⊗ZlA→ A as ι(tr⊗1)for a unique ι ∈ A⊗ZlA; we define the congruence element ηtr ∈ A of tr to be ∆(ι).For u ∈ A× we have ηu tr = u−1ηtr so that the congruence ideal ηtrA is independentof the choice of tr. One also checks that ηtr is a non-zero divisor if and only if A isreduced.

36 TOM WESTON

A.1.2. Duality. Let M be a finitely generated free A-module. It follows immedi-ately from the definition of a Gorenstein trace that the map f 7→ tr f sets upan isomorphism HomA(M,A) ∼= HomZl(M,Zl). If M is any finitely generated A-module, on taking a resolution by free A-modules one finds that f 7→ tr f yieldsan isomorphism

HomA(M,A⊗ZlQl/Zl) ∼= HomZl(M,Ql/Zl).

We return now to the case that M is free. We define a map htr : EndZlM →EndAM as the composition

EndZlM∼= HomZl(M,Zl)⊗ZlM

∼= HomA(M,A)⊗ZlM

HomA(M,A)⊗AM ∼= EndAM.

Here the second isomorphism is induced by tr as above. The composition of theinclusion EndAM → EndZlM with htr is multiplication by ηtr.

Consider now a free A-module H of rank 2. It is straightforward to check that thecondition that A is Gorenstein is equivalent to the existence of a perfect A-pairingψ : H⊗ZlH → Zl (one such pairing is given by ψ(ax + by, cx + dy) = tr(ad − bc)for an A-basis x, y of H).

A.2. Bilateral derivations. We review the (very) basic theory of bilateral deriva-tions developed in [31].

A.2.1. Definitions. Let A be a commutative Zl-algebra and let M be an A⊗ZlA-module. A bilateral derivation from A to M is a Zl-linear map D : A → M suchthat

D(βα) = (α⊗ 1)D(β) + (1⊗ β)D(α)

for all α, β ∈ A. The fundamental example is the map δ : A → A⊗ZlA given byδ(α) = α⊗ 1− 1⊗ α; note that the image lies in the kernel I of the diagonal map∆. One can show that δ : A → I is the universal bilateral derivation, although wewill not use this.

Define Mδ to be the set of m ∈ M such that δ(α)m = 0 for all α ∈ A; it iscanonically an A-module via ∆. The next two lemmas are straightforward.Lemma A.2.1. Let D : A → M be a bilateral derivation and let a be an ideal ofA such that 1⊗ a and a⊗ 1 annihilate M . Then the restriction of D to a yields anA-module homomorphism D : a/a2 →Mδ.

If M is an A-module, we give EndZlM the structure of A⊗ZlA-module by(α⊗ β)f(m) = αf(βm) for f ∈ EndZlM .Lemma A.2.2. Let M be a free A-module of finite rank with a continuous A-linear action of some group G. Assume also that every Jordan-Holder constituentof EndZlM has trivial G-invariants. Then there is a canonical isomorphism

H1(G,EndZlM)δ ∼= H1(G, (EndZlM)δ

).

A.2.2. Bilateral derivations from Gorenstein rings. Let A be a finite, flat, local, re-duced, Gorenstein Zl-algebra. Fix a Gorenstein trace tr and let η be the associatedcongruence element; η is a non-zero divisor since A is reduced. Let M and N befree A-modules of finite rank; HomZl(M,N) is an A⊗ZlA-module in the obviousway.


Lemma A.2.3. There is a unique A-module isomorphism ν : HomZl(M,N)δ'−→

HomA(M,N) fitting into a commutative diagram

HomZl(M,N)δ //

ν

HomZl(M,N)

HomA(M,N) η// HomA(M,N)

Proof. The uniqueness of such a ν is clear. To define it it suffices to consider thecase M = N = A; we leave this to the reader.

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Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109

E-mail address: [email protected]

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