The Regulator RpX q

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).

It shows that the motivic decomposition of Hr^2,1pXq is given by Hr^2,1pXq “cokerpρ₁q ‘

ˆPic⁰X Cl_F

˙

R

‘ R. (3.21)

Hr^2n,npXq is isomorphic to then-th Arakelov Chow group CHⁿpXq_R (cf. [8] Prop. 2.11) and the pairingB(X,n) translates into a perfect pairing

Hr^2n,npXq ˆHr^4´2n,2´npXq ÝÑ R. (3.22)

Remark 3.6. Flach’s and Morin’s identification Hr^2n,npXq – CHⁿpXq_R results from an application of the5-Lemma and thus depends on the choice of a splittingH^2,1pXq_RÑHr^2,1pXq of (3.20). However, the identifications (3.19) and the decomposition (3.21) into motivic degree components provide one such splitting, i.e. we haveHr^2n,npXq –CHⁿpXq_R canonically.

From now on we will assume the following enhanced version ofB(X,n).

Conjecture 3.7 (B(X,n)). Conjecture B(X,n) holds and the perfect pairing CHⁿpXq_RˆCH^2´npXq_R ÝÑ R

obtained from (3.22) via the canonical identifications Hr^2n,npXq –CHⁿpXq_R coincides with the Arakelov Intersection Pairing x´,´yAr.

For arithmetic surfaces x´,´yAr is only non-trivial ifn‰0,1,2. Moreover, it only involves information from motivic degree 1 ifn “ 1. We now make the Arakelov pairing explicit for n “ 1 and compare it to the intersection pairing x´,´yX from algebraic geometry, following [15].

Proposition 3.8. (Arakelov, Hriljac)

(i) (3.21) is an orthogonal decomposition of Hr^2,1pXq, i.e. one has Hr^2,1pXq “cokerpρ1q K

ˆPic⁰X Cl_F

˙

R

K R

andx´,´yAr is defined on each summand separately. x´,´yAr is negative definite on pPic⁰X{Cl_Fq_R (cf. [15] Thm. 3.4, Prop. 3.3).

(ii) Let D, D¹ PPic⁰X be fibral divisor classes with support in the special fiber X_p. Then xD, D¹yAr “logNp¨ xD, D¹yX

(cf. [15] def. ofpD¨Eq_v in Sec. 2).

(iii) There is a unique linear splitting

Pic⁰X ÝÑ pPic⁰Xq_Q, P ÞÑP

of the natural projection Pic⁰X ÑPic⁰X such that the image is orthogonal to all fibral divisor classes inPic⁰X (cf. [15] Thm. 1.3).

(iv) Fix an isomorphism φ :Pic⁰X ÝÑ^– JacX. X has a divisor such that its associated canonical height h satisfies for all P PPic⁰X (cf. [15] Thm. 3.1)

xP,PyAr “ ´hpφpPqq.

Definitions of RpXq, RpXq and c_ppXq. Let P be a basis of the image of ¹H^2,1pXq in

1H^2,1pXq_R– pPic⁰X{Cl_Fq_R. P also defines an integral basis onHc^2,1pXq due to Hc^2,1pXq – H^2,1pXq_R. LetP^˚ be the basis of ¹H^2,1pXq^˚_R dual toP. We define theregulator RpXq of X to be

RpXq:“det`

xP,P¹yAr

˘

P,P¹PP “detM_P,P^˚

´

ph^p2q_BpX,1qq^˚

¯ .

Next, fix a basis P “ tP_iu_{1ďiďrk Pic}⁰_X of the image of Pic⁰X in pPic⁰Xq_R. The regulator RpXq of the generic fiberX equals

RpXq “det`

xP,P¹yAr

˘

P,P¹PP

since the Arakelov Intersection Pairing is by Proposition 3.8(iv) the same as the Neron-Tate height pairing on Pic⁰X.

Now, fix a prime p. Let J ÑS_p denote the Neron model of the Jacobian J “JacX_Fp of the generic fiber of the local surface X_Op overS_p“SpecO_p. Let J˜“J_p denote the special fiber ofJ and let J˜⁰ be its identity component. We also write J⁰ ĂJ for the subgroup scheme with generic fiberJ and special fiber equal toJ˜⁰. We define

c_ppXq:“

#

^JpFpq

J⁰pF_pq “ # ˜Jpkppqq

# ˜J⁰pkppqq.

Decomposition of RpXq. Fix a prime pof O. Recall the notationsdppqand n_jppq from Lemma 2.2. Also, let tC_j^pu1ďjďdppq be the reduced irreducible components of X_p and let m_jppq be the multiplicity of C_j^p inX_p. The sections:SÑX provides a rational point on one component – sayC_dppq^p – ofX_p. ThusC_dppq^p must be simple and cannot decompose further over any algebraic extension of kppq, i.e. m_dppqppq “n_dppqppq “1. We conclude that the set of classesD^p :“ trC_j^ps P pΛ_pq_Ru_1ďjădppq is a basis of the image ofΛ_p insidepΛ_pq_R.

Lemma 3.9. (Raynaud;Bosch,Liu) Fix a prime p of O. The sequence CH⁰pX_pq

α

ÝÝÝÝÑ CH⁰pX_pq

β

ÝÝÝÝÑ Z

C_j^p ÞÑ

dppq

ÿ

i“1

xC_i^p, C_j^pyX¨C_i^p

C_j^p ÞÑ mjppq

(3.23)

is a chain complex and one has

#

^Kerβ

Imα “c_ppXq ¨

dppq

ź

j“1

n_jppq.

Proof. This is [5] Theorem 1.11 applied to the abelian varietyA“J. Indeed, the right-most term in Thm 1.11qdZ{d¹Zvanishes sinceX_phas a component satisfyingm_dppqppq “n_dppqppq “ 1. Moreover, the geometric multiplicitiesej of C_j^p inC_j^p (cf. [6] Def. 9.1.3) occurring in [5]

Rmk. 1.12 equal 1 since the base change of any reduced curve over a perfect field to its algebraic closure remains reduced (see e.g. [26] example (6.1.7)).

Remark 3.10. Raynaud has shown the analogue of Lemma 3.9 for algebraically closed residue fields (cf. [26] Prop. 8.12); Bosch and Liu extended it to more general residue fields. As part of his proof Raynaud has shown that, in the casekppq “kppq, the Picard-scheme Pic⁰_X

p

is isomorphic to the group of components J˜{J˜⁰ of J – a finite étale group scheme that only depends on the generic fiberX. This should serve as intuition for why#^Ker_Imα^β does not depend on the special fiber X_p beyond the values of the njppq.

Proposition 3.11. One has

RpXq “ ˘ 1 p#Tor Pic⁰Xq²

ˆ

#

^{Tor Pic0X}

Cl_F

˙2

¨RpXq ¨ź

p

¨

˝plogNpq^dppq´1

dppq

ź

j“1

njppq

˛

‚cppXq

“ ˘ 1

p#Tor Pic⁰Xq² ˆ

#

^{Tor Pic0X}

Cl_F

˙2

¨RpXq ¨Π^˚pX,1q ¨ź

p

c_ppXq.

Proof. (3.3) gives a short exact sequence of real vectorspaces 0 ÝÑ

ˆ à

p

Λ_p

˙

R

ÝÑ

ˆPic⁰X Cl_F

˙

R

ÝÑ `

Pic⁰X˘

R ÝÑ 0. (3.24)

Due to the splitting provided by Proposition 3.8(iii) the above sequence yields an orthogonal decomposition

ˆPic⁰X Cl_F

˙

R

– ˆ

à

p

Λ_p

˙

R

K `

Pic⁰X˘

R (3.25)

and we may regardP¹ :“PYŤ

pD^p as a further (R-)basis of ¹H^2,1pXq_R. (3.25) gives RpXq “`

detM_P,P¹pidq˘2

¨det`

xP,P¹yAr

˘

P,P¹PP¹

“`

detM_P,P¹pidq˘2

¨det`

xP,P¹yAr

˘

P,P¹PP¨ ź

p

det

´

xrC_i^ps,rC_j^psy

¯

1ďi,jădppq

“RpXq ¨`

detM_P,P¹pidq˘₂

¨ ź

p

plogNpq^dppq´1det

´

xC_i^p, C_j^pyX

¯

1ďi,jădppq

(3.26)

where the last equation uses Proposition 3.8(ii). We evaluate the remaining factors separately.

First, since (3.3) is an integral exact sequence,detM_P,P¹pidqmeasures the discrepancy in torsion between Pic⁰X{Cl_F and its surrounding terms in (3.3), i.e. one has

detM_P,P¹

´ id1

H^2,1pXq_R

¯

“ 1

#Tor Pic⁰X ¨

#

^{Tor Pic0X}

Cl_F .

Finally, recall the sequence (3.23) of the previous Lemma. Sincem_dppq“1the set D^p may also be viewed as a basis of Kerβ. Besides,α is represented by the full intersection matrix

´

xC_i^p, C_j^pyX

¯

1ďi,jďdppq. It follows that det

´

xC_i^p, C_j^pyX

¯

1ďi,jădppq “

#

^Kerβ

Imα. Lemma 3.9 completes the proof.