The Functional Equation

Therefore, the (non-exact) restriction of p3.39qto integral structures fits into the diagram 0 ^//Gp^gapOq ^{exp //}J{pO_Zpq ^//Jp{O^red

Fpq ^//0

Gp^gappq

Ă

Jˆppq

oo log

ι

OO

Gp^gapp^rq

Ă oo ^– // ˆJpp^rq

Ă

Since, moreover, Gp^gapp^kq M

Gp^gapp^k`1q – Jˆpp^kq

MJˆpp^k`1q for all k ě 1, the trivialization factor associated to (3.39) does not depend onr and agrees with the trivialization factor of

0 ÝÑ Jˆppq ÝÑ^ι J{pO_Zpq ÝÑ Jp{O^red

Fpq ÝÑ 0 (3.40)

up to a factor of#Gp^gapOq{Gp^gappq “ pś

p|pNpq^g – which equalsś

p|ppNpq^ppa since we assume Z to be smooth. So, it suffices to show that (3.40) is exact.

First, let t“ rt₀ :t₁:¨ ¨ ¨:t_Ns PJ{pO_Zpq be an integral point that is congruent to O modulo p. TheXj “XjpT0 :¨ ¨ ¨:Tnq are rational functions in theTi that vanish at O. Therefore we have x_j :“X_jptq Pp. We then haveιpx_jqj “tproving exactness at the middle component.

Let nowZn“X ˆSpecZ{pⁿ. By the Theorem on Formal Functions the categories of line bundles on X andX are equivalent. So, one has JpO_Zpq “Pic⁰X “ lim

ÐÝÝPic⁰Z_n. Therefore it suffices to show that each Pic⁰Znsurjects onto JpO_Fpq “Pic⁰Z. This in turn follows from the long exact sequence associated to

1 ÝÑ 1`pO_Zn ÝÑ G^Z_mⁿ ÝÑ G^Z_m ÝÑ 1

sinceH²pZ,1`pO_Znq “0 becauseZ is one-dimensional. Note that exactness of (3.40) in the special case of an elliptic surface is precisely [32] Ch. VII, Prop. 2.1 & 2.2.

and consequently

CpX, nq “ ppn´1q!¨ pn´2q!q^mpg´1q. Proof. Letně2. Corollary 2.35 unfolds to

1ApXq^{2´n2 p}L^˚₈pH¹pXq,2´nq¹Λ8pX,2´nq “ ˘¹ApXq^{n2 p}L^˚₈pH¹pXq, nq¹Λ8pX, nq¹CpX, nq.

By virtue of Theorem 3.13 and Lemma 2.34 this simplifies to

1CpX, nq “ ˘¹ApXq^1´n¨

pL^˚₈pH¹pXq,2´nq

pL^˚₈pH¹pXq, nq ¨

1Λ₈pX,2´nq

1Λ8pX, nq

“ ˘¹ApXq^1´n¨ ˆ

p2πq^2pn´1qΓ^˚p2´nq Γ^˚pnq

˙_mg˜

p2πiq^2mgpn´1q

1ApXq^n´1

¸_´1

“ ˘

ˆΓ^˚p2´nq Γ^˚pnq

˙mg

.

Similarly, forně1we obtain from the well-known functional equation FE(S) CpS, nq “ ˘ApSq^12´n¨Γ^˚

Rp1´nq^rΓ^˚

Cp1´nq^s Γ^˚

Rpnq^rΓ^˚

Cpnq^s ¨Λ₈pS,1´nq Λ₈pS, nq

“ ˘p#D_Fq^12´n

˜ π^n´12

`₁

2

˘´n

Γ^˚p^1´n₂ q Γ^˚pⁿ₂q

¸rˆ

p2πq^2n´1Γ^˚p1´nq Γ^˚pnq

˙s

1

2^p´1qn´1rp2πq^mn´rn´sp#D_Fq^12´n

“ ˘2^´p^n´12q^r2^nr

˜Γ^˚p^1´n₂ q Γ^˚pⁿ₂q

¸r

pn´1q!^´2s 2^p´1qn´1rp2πq^m2´rn´s

“ ˘2p<sup>´n`1</sup>2`nq^rp2πq^´p´1q

n

2 r2^p´1qnr

˜Γ^˚p^1´n₂ q Γ^˚pⁿ₂q

¸r

1 pn´1q!^2s

“ ˘2^p1´nqr

ˆ 2^n´1 pn´1q!π^p´1q

n 2

˙r

1 pn´1q!^2s

“ pn´1q!^´m.

Note that the factor p1{2q^´n arises from the relation between leading Taylor coefficients Γ^˚

Rpkq “ p1{2q^{ords“kΓRpsq}π^´k{2Γ^˚pk{2q.(3.32) concludes the proof.

In particular,CpS, nq “ pn´1q!^´m also holds for ně1 when assumingTC(S,n) instead of the conjectureC_EPpQppnqq fromp-adic Hodge Theory.

A simplified version of FE(X) for integersn“s. Flach and Morin provide a reformu- lation of FE(X) for integer arguments in terms of a newly defined quantityx8pX, nq PR^ą0, satisfying

x₈pX, nq² “ Λ₈pX,2´nq

Λ₈pX, nq . (3.42)

The definition of x₈pX, nq² willnot incorporate the full fundamental line, mirroring the fact that the quotient on the right hand side is easier to evaluate than each term separately. We will review it here and use it to computex₈pX, nqindependently of Theorem 3.13. Set

Ξ8pX{Z, nq :“det_ZRΓWpX₈,Zpnqq b det^´1

Z RΓddRpX{Zq{Fⁿ b det^´1

Z RΓWpX₈,Zp2´nqq b det_ZRΓddRpX{Zq{F^2´n.

(3.43) This definition is made to have an isomorphism

φ: ∆pX{Z, nq bΞ8pX{Z, nqÝÑ^– ∆pX{Z,2´nq.

Observe that the distinguished triangle

RΓ_dRpX₈{Rq{Fⁿr´1s ÝÑ RΓDpX{_R,Rpnqq ÝÑ RΓ_WpX₈,Zpnqq_R ÝÑ (3.44) together with the duality (2.23) for Deligne cohomology gives a trivialization

Ξ₈pX{Z, nq_R » det_RRΓDpX{_R,Rpnqq b det^´1

R RΓDpX{_R,Rp2´nqq

» det_RRHompRΓ_DpX{_R,Rp2´nqq,Rr´3sq b det^´1_R RΓ_DpX{_R,Rp2´nqq

» det_RRΓDpX{_R,Rp2´nqq b det^´1

R RΓDpX{_R,Rp2´nqq

» R.

(3.45)

We denote it byξ₈pX, nq:RÝÑ^– Ξ₈pX{Z, nq_R and definex²₈pX, nq PR^ą0 via ξ8pX, nqpZq “x²₈pX, nq ¨Ξ8pX{Z, nq

as an equality of lattices inΞ₈pX{Z, nq_R. Clearlyx₈pX,2´nq “x₈pX, nq^´1. One has Proposition 3.25. (cf. [8] Prop. 5.28, Cor. 5.30)The diagram

∆pX{Z, nq bΞ8pX{Z, nq bR ^{φbR //}∆pX{Z,2´nq bR

RbR

λ8pX,nq bξ8pX,nq

OO

R

λ8pX,2´nq

OO

commutes. Therefore – when assuming TC(X,n) for all integers n – the functional equation FE(X) holds for all integers s“n if and only if for alln

ApXqⁿ² ¨ζ^˚pX₈, nq ¨CpX, nq

x₈pX, nq “ ˘ApXq^2´n2 ¨ζ^˚pX₈,2´nq ¨CpX,2´nq

x₈pX,2´nq . (3.46) Motivic decomposition of Ξ₈pX{Z, nq. By the work of last chapter all determinants in (3.43) admit motivic decompositions. So, we obtain further decompositions

Ξ₈pX{Z, nq “Ξ₈pS{Z, nq b¹Ξ₈pX{Z, nq^´1bΞ₈pS{Z, n´1q and

x₈pX, nq “x₈pS, nq ¨¹x₈pX, nq^´1¨x₈pS, n´1q.

Proposition 3.26. Let n be any integer. One has

x²₈pS, nq “2^p´1qnrp2πq^rn`s´mnp#D_Fq^n´12.

Moreover, when assuming the technical condition RP(X) (or the formula (2.39)), one has

1x²₈pX, nq “

1ApXq^n´1 p2πq^2mgpn´1q and consequently

x²₈pX, nq “

´

p2πq^2mpg´1qApXq

¯n´1

.

Proof. It suffices to consider n ě1 since x₈pS,1´nq “ x₈pS, nq^´1 and x₈pX,2´nq “ x₈pX, nq^´1. The second quasi-isomorphism of (3.45) is due to Poincaré Duality which holds integrally; the third and the fourth follow directly from the determinant formalism. Therefore the trivialization factors x²₈pX, nq andx²₈pS, nqarise fully from a comparison of the integral structures of the complexes in (3.44) (and its analogue forS). The motivic degree1 part of its associated long exact sequence is

0 ÝÑ H_W,8^1,n pXq_R ÝÑ H_dR^1,npXq_R ÝÑ H_D^2,npXq ÝÑ 0, which is exact integrally with respect to Bⁿ. Consequently, for ně2,

1x²₈pX, nq “ ¹t^pnq_ddRpXq ¨detMBⁿ_ddR,Bⁿpidq

“ 1

p2πq^2mgpn´1q ¨¹t^pnq_ddRpXq ¨detMBⁿ_ddR,B_dRpidq by Lemma 3.4

“

1ApXq^n´1

p2πq^2mgpn´1q by (3.13).

The 2-torsion ofRΓWpX₈,Zpnqqdoes not contribute due to Corollary A.9.

ForS we begin noting that by Corollary A.16(i) one has for alln χpRΓ_WpS₈,Zpnqqq “ź

iPZ

´

#TorH_W,8^i,n pSq

¯p´1qⁱ

“2^{p´1qn n`n2 r}. The long exact sequence associated to the analogue of (3.44) for S becomes

0 ÝÑ H_W,8^0,n pSq_R ÝÑ H_dR^0,npSq_R ÝÑ H_D^1,npSq ÝÑ 0.

The implied integral structures are obvious. The outer cohomology groups contribute factors of p2πiq^{´npnr`sq</sup> andp2πiq<sup>´pn´1qpnr`sq} respectively to x²₈pS, nq. The relative determinant between the lattices H_ddR^0,n pS{Zq and H⁰pS,O_Fq “ O_F equals ApSq^n´1 “ p#D_Fq^n´1. A remaining factor of ?

#DF results from the comparison ofO_F with the integral lattice of the Minkowski spaceF_R. Therefore one obtains

x₈pS, nq² “ 2^p´1q1´n

1´n`1´n

2 r

2^{p´1qn n`n2 r}

p2πiq^´npnr`sq´pn´1qpnr`sqp#D_Fq^n´1a

#D_F

“2^p´1qnrp2πq^rn`s´mnp#DFq^n´12

as claimed. Finally, we combine this to x²₈pX, nq “ x₈pS, nqx₈pS, n´1q

1x₈pX, nq

“

ˆ |DF| p2πq^m

˙_2pn´1q

¨

˜

p2πq^2mgpn´1q

1ApXq^n´1

¸

“

´

p2πq^2mpg´1qApXq

¯_n´1 .

In particular, (3.42) holds for every motivic degree component separately. So, we can rederive the formula for the correction factor (3.41) from (3.46).