Integral Structures

and using the vanishing order part of Lemma 2.34 verifies ord_s“nLpH¹pXq, sq “ mg for nď0. The last statement follows since (3.8) is known forX being an elliptic curve overQ

by virtue of the Modularity Theorem. l

does not necessarily generate the lattice H¹pXpCq,Zq ĂH¹pXpCq,Rq since (3.9) is generally not an equality¹. Next, define

δ^`;n_vj “

#

p2πiqⁿδ_vj^` for n even

p2πiqⁿδ_vj^´ for n odd and δ_vj^´;n“

#

p2πiqⁿδ_vj^` for n odd p2πiqⁿδ_vj^´ for n even and letB^˘;n:“ tδ_vj^˘,nuvj ĂH¹pXpCq,Cq.

B^{`;n</sup> is a basis of H<sub>W,8</sub><sup>1,n</sup> pXq<sub>R</sub> “ H<sup>1</sup>pXpCq,Rpnqq<sup>GR</sup> that generates the integral lattice H<sub>W,8</sub><sup>1,n</sup> pXq Ă H<sub>W,8</sub><sup>1,n</sup> pXq<sub>R</sub>. Moreover, the set B<sup>`,n´1} is for n ě 2 a basis of H_D^2,npXq “ H¹pXpCq,Rpn´1qq^GR and for nď 1 it is a basis of H_D^1,n´1pXq “ H¹pXpCq,Rpn´1qq^GR. Their integral structures shall be theZ-lattices generated by B^`;n´1. Further, for nď0 we endow Hc^2,npXq with an integral structure viaHc^2,npXq –H_D^1,npXq. Finally, for any fixed integer nwe giveH¹pXpCq,Cq^GR an integral structure via

H¹pXpCq,Cq^GR “H¹pXpCq,Rpnqq^GR ‘H¹pXpCq,Rpn´1qq^GR i.e. Bⁿ

C:“B^{`;n</sup>YB<sup>`;n´1} is its integral basis.

Integral structures coming from H²pX,Zpnqq forně2. For each ně2 we fix a set of generators

Cⁿ“ tc_vjⁿ |v|8,1ďjďgu

of the image of H^2,npXq in H²pX,Zpnqq_R. Since H^i,npXq_R – H_W^i,npXq_R for these n, the set Cⁿ also determines an integral structure on H_W^i,npXq_R. Further, (2.34) shows that H_W^3,2´npXq_R–HompH^2,npXq,Rq forně2. We writec_vj^2´nPH_W^3,2´npXq_R for the cycle class corresponding toc_vj² under this isomorphism. C^2´n:“ tc_vj^2´n|v|8,1ďjďgu is a basis of the integral structure ofH_W^i,npXq_R determined byH_W^i,npXq since the duality (2.34) is a duality of integral lattices.

If there is no risk of confusion we will also use δ_vj^`;‚ andc_vj^‚ to refer to integral elements of Hc^2,npXqfor nď0 andH_W^2,npXq, H_W^3,2´npXq for ně2respectively.

Integral structures for de Rham cohomology. One has the decomposition H_dR¹ pXpCq{Cq^GR –H¹pX,O_Xq_R‘H⁰pX, ωXq_R.

Serre duality for coherent sheaves provides a perfect pairing

^:H¹pX,O_Xq ˆH⁰pX, ωXq ÝÑ H¹pX, ωXq –Z (3.11)

1E.g. ifX is an elliptic curve overQso that one may writeXpCq –C{Λ, the matrixB describing the base change fromH¹pXpCq,ZqtoBis given byB“ p¹₁qorB“`_{1 1}

1´1

˘respectively, depending on whether complex conjugation acts on a basis of the latticeΛby`₁

´1

˘orp₁¹q.

of free abelian groups. Let B_dR¹⁰ “ tωvjuvj be a basis of the image of H¹pX,O_Xq ^´bÝÑ^R1 H¹pX,O_Xq_Rand let B_dR⁰¹ :“ pB¹⁰_dRq^˚ :“ tη_vjuvj be the basis ofH⁰pX, ωXqdual to B_dR¹⁰. We will use

B_dR :“B¹⁰_dRYB_dR⁰¹ “ tω_vj, η_vjuvj.

A subtlety needs to be taken into account for the choice of bases of derived de Rham coho- mology groups. LetBrⁿ_ddR and Br^p2q;n_ddR be bases of the images of H_ddR^1,n pX{Zq andH_ddR^2,npX{Zq inH_dR¹ pX₈{RqandH_dR² pX₈{Rqunder base change to R. Similarly, let B_dR^p2q be a basis of the image of H⁰pS, ω_Fq^´bÝÑ^R1H_dR² pX₈{Rq. After splitting off the motivic degree 0 component (2.47) unravels to

t^pnq_ddRpXq t^pnq_ddRpSq ¨

detM

Br^p2q;n_ddR,B^p2q_dR

`id_H2pX8{Rq

˘ detM_Brn

ddR,B_dR

`id_H1pX₈{Rq

˘ “

ˆApXq ApSq

˙_n´1

. (3.12)

Due to the ad-hoc Definition 2.31(i) the motivic degree 1and 2 parts of the quotient (2.47) can only be expressed in terms of differently defined integral bases ofH_dRⁱ pX₈{Rq fori“1,2 – which we will denote B_ddRⁿ ,B^p2q,n_ddR. They have to be chosen in such a way that (2.47) holds for each motivic degree component separately. Concretely, we let B_ddR^p2q;n be any basis of H_dR² pX₈{Rq, satisfying

t^pn´1q_ddR pSq ¨detM_Bp2q;n

ddR,B^p2q_dR

`id_H2pX₈{Rq

˘“ det_ZRΓ_ddRpS{Zq{F^n´1 det_ZRΓpS, ω_Fq

and then choose Bⁿ_ddR to be such a basis of H_dR¹ pX₈{Rq that (3.12) remains valid when replacing Br_ddRⁿ ,Br^p2q,n_ddR withBⁿ_ddR,B^p2q,n_ddR. Now the 1-part of (2.47) is

1t^pnq_ddRpXq ¨detMBⁿ_ddR,BdR

´ id_H1

dRpX₈{Rq

¯

“ ¹ApXq^n´1. (3.13) Obviously, if we had a duality result of the kind (2.48) one could chooseB_ddRⁿ “Brⁿ_ddR and B^p2q;n_dR “ Br^p2q;n_dR . In any case, the difference will not concern us in the remainder of this thesis.

The Period Isomorphism. Let

Φ :H¹pXpCq,CqÝÑ^– H_dR¹ pXpCq{Cq

be the period isomorphism and writeΦ^GR :H¹pXpCq,Cq^GR ÑH_dR¹ pX₈{Rqfor its restriction to the G_R-invariant part.

Lemma 3.4. One has

detM_B¹_,BdRpΦ^GRq “1.

Consequently, for all integers n,

detMBⁿ,B_dRpΦ^GRq “ p2πiq^2mgpn´1q. (3.14)

Proof. Most of the work towards the above identity is hidden in the definitions of B¹ “ p2πiqB^´YB^` andB_dR “B¹⁰_dRYB⁰¹_dR as self-dual bases. It suffices to show that

detM_B`YB^´,B_dR⁰¹YB_dR¹⁰pΦ^GRq “ p2πiq^´mg.

In other words, we need to calculate the quantity cPR^ˆ{t˘1u for which ľ2gm

Φ^GR : ľ2gm

H¹pXpCq,Cq^GR ÝÑ

ľ2gm

H_dR¹ pX₈{Rq (3.15) acts as

ľ

v|8

1ďjďg

´

δ_vj^` ^δ_vj^´

¯

ÞÝÑ c¨ ľ

v|8

1ďjďg

pω_vj^η_vjq. (3.16)

By the Poincaré and Serre dualities (3.10) and (3.11) δ_vj^` ^δ_vj^´ andω_vj^η_vj are generators of (thev-component of)H²pXpCq,Zq and H²pX, ω^‚_X{Zq, i.e. they correspond to classes in H²pXpCq,Cq^GR andH_dR² pX₈{Rqrepresented by a point. But it is well-known that the period isomorphism on second cohomology

Φ :H²pXpCq,Cq^GR ÝÑ H²pX₈{Rq

is just multiplication withp2πiq^´1(for eachv) with respect to point class bases. Consequently, one has c“ p2πiq^´mg.

Finally, note that the period isomorphism restricts to a map Φ¹⁰: H¹pXpCq,Rp1qq^GR ÝÑ^– H¹pX,OXq_R. We define

ΩpXq:“detM_B`;1,B_dR¹⁰pΦ¹⁰q.

The duality isomorphism h_BpX,nq. Let h^piq_BpX,nq : H^i,npXq_R ÝÑ^– H_c^4´i,2´npXq^˚ be the isomorphism induced by the conjectural perfect pairing (2.29). It is related to the Beilinson regulator map ρ₂ :H^2,npXq_RÑH_D^2,npXqas follows.

Lemma/Definition 3.5. Let ně2. Then one has

detM_Cn,B^{`;2´n</sup>ph<sup>p2q</sup><sub>BpX,nq</sub>q “detM<sub>C</sub>n,B<sup>`;n´1}pρ₂q.

We write RⁿpXq for the above determinants and call it the n-th regulator of X. Proof. Theh^piq_BpX,nq fit into a commutative diagram (cf. [8] Rmk. 2.6)

// H^i,nc pXq ^//

– ph^p4´iq_BpX,2´nqq^˚

H^i,npXq_R ^{ρ ////}

– h^piq_BpX,nq

H_D^i,npXq ^//

– h^i,n_D

Hc^i`1,npXq ^//

– ph^p3´iq_BpX,2´nqq^˚

ρ^˚ //H^4´i,2´npXq^˚_R ^//H^4´i,2´nc pXq^{˚ //}H^3´i,2´n_D pXq^{˚ ρ}

˚ //H^3´i,2´npXq^˚_R ^//

with exact rows coming from (2.27) and where h^i,n_D :H_D^i,npXq –H_D^3´i,2´npXqis the isomor- phism induced by the perfect pairing (2.23) of Deligne cohomology groups. Specializing to i“2 yields the commutative square

H^2,npXq_R ^ρ_–^{2 //}

– h^p2q_BpX,nq

H_D^2,npXq

– h^2,n_D

Hc^2,2´npXq^{˚ – //}H_D^1,2´npXq^˚

(3.17)

(2.23) simplifies for i “ 2 to the restriction of the Poincaré duality pairing of algebraic topology

H¹pXpCq,Rpn´1qq ˆH¹pXpCq,Rp2´nqq ÝÑ H²pXpCq,Rp1qq.

to itsG_R-equivariant part. Since Poincaré duality also holds integrallyh^2,n_D does not contribute to the determinant of the upper right decompositionh^2,n_D ˝ρ2:H^2,npXq_RÑH_D^1,2´npXq^˚ of the square. Finally, sinceHc^2,2´npXq derives its integral structure from the bottom map of (3.17) the claim follows from taking determinants in (3.17).