Special Values of the Zeta Function of an Arithmetic Surface

We study the special value conjecture for the Zeta function of a proper regular arithmetic scheme X introduced by Flach and Morin in the case n=1. We compute the correction factor C(X,1) left unspecified in the original statement of the Flach-Morin Conjecture, thereby developing some results on the eh-topology introduced by Geisser. We then specialize further to the case where X is an arithmetic surface and show that the conjecture of Flach and Morin is equivalent to the Birch and Swinnerton-Dyer Conjecture.


Introduction
Let X be a regular scheme of dimension d, proper over Spec(Z). In [5][Conj. 5.11, 5.12] the first author and Morin have formulated a conjecture on the vanishing order and the leading Taylor coefficient of the Zeta-function ζ(X , s) of X at integer arguments s = n ∈ Z in terms of what we call Weil-Arakelov cohomology complexes. More specifically, our conjecture involves a certain invertible Z-module ∆(X /Z, n) := det Z RΓ W,c (X , Z(n)) ⊗ Z det Z RΓ(X Zar , LΩ <n X /Z ) which can be attached to the arithmetic scheme X under various standard assumption (finite generation of étale motivic cohomology in a certain range being the most important). Under these assumptions one has a natural trivialization (1) λ ∞ : R ∼ − → ∆(X /Z, n) ⊗ Z R and we conjecture λ ∞ (ζ * (X , n) −1 · C(X , n) · Z) = ∆(X /Z, n) which determines the leading coefficient ζ * (X , n) ∈ R up to sign (all identities in this paper involving leading coefficients should be understood up to sign). Here C(X , n) ∈ Q × is a certain correction factor, defined as a product over its p-primary parts and where the definition of each p-primary part involves p-adic Hodge theory. It is easy to see that C(X , n) = 1 for n ≤ 0 and one also has C(X , n) = 1 for X of characteristic p. In [6][Rem. 5.2] it was then suggested that in fact C(X , n) has the simple form for any n ∈ Z and any X . This formula is corroborated by the computation of C(Spec(O F ), n) in [5][Prop. 5.34] for a number field F all of whose completions F v are absolutely abelian.
In the present article we focus on the case n = 1 and then specialize further to arithmetic surfaces. We give more evidence for formula (2) by proving that it holds for n = 1, i.e. that C(X , 1) = 1, for arbitrary X . If X is connected, flat over Spec(Z) and of dimension d = 2 we call X an arithmetic surface. Denoting by F the algebraic closure of Q in the function field of X , the structural morphism X → Spec(Z) factors through a morphism (3) f : X → Spec(O F ) =: S with f * O X = O S . We show that all the assumptions entering into our conjecture are satisfied for n = 1 if X has finite Brauer group, or equivalently the Jacobian J F of X F has finite Tate where Pic 0 (X ) is the kernel of the degree map on Pic(X ), R(X ) is the regulator of a certain intersection pairing on Pic 0 (X ) and Ω(X ) is the determinant of the period isomorphism between the finitely generated abelian groups H 1 (X (C), 2πi·Z) G R and H 1 (X , O X ). The integer δ is the index of X F , i.e. the g.c.d. of the degrees of all closed points. Furthermore, Φ v = J F (F v )/J F (F v ) 0 is the group of components of the group of F v -rational points of J F and δ ′ v , resp. δ v , are the period, resp. index of X Fv over F v . The group Br(X ) coincides with Br(X ) if F has no real places and differs from Br(X ) by a 2-torsion group in general. The group Br(X ) is naturally self-dual and the quantity # Br(X )δ 2 will also arise as the cardinality of a naturally self-dual group in our computation (one could call this group the H 1 -part of the Brauer group, see Lemma 8 below).
We then prove that our conjecture (5) and (6) is equivalent to the Birch and Swinnerton-Dyer conjecture for the Jacobian of X F , provided that our intersection pairing agrees with the Arakelov intersection pairing. This result was shown if f is smooth in [5][Thm. 5.27] but only rather indirectly via compatibility of both conjectures with the Tamagawa number conjecture. Here we give a direct proof without assumptions on f (such as existence of a section which would simplify the proof considerably). If X is a smooth projective surface over a finite field, fibred over a curve f : X → S, the equivalence of the Birch and Swinnerton-Dyer formula for the Jacobian of the generic fibre with a special value conjectures for the Zeta function ζ(H 2 abs , s) of the absolute H 2 -motive of X has been much studied in the literature, going back to the article of Tate [30], see also [10], [20], [21]. At least for suitable choices of f , the two functions ζ(H 1 , s) and ζ(H 2 abs , s) only differ by a very simple Euler factor (see Remark 2 below), and it seems likely that our methods will also give a simplification of the arguments in [10]. To the best of our knowledge, our special value conjecture is the first such for arithmetic surfaces of characteristic zero, and in this context one of course does not have an analogue of H 2 abs . Combined with the analytic class number formula for the Dedekind Zeta function ζ F (s) at s = 1 and s = 0 formula (6) becomes (7) ζ * (X , 1) = 2 r1 (2π) r2 Ω(X ) |D F | · (# Pic 0 (X )) 2 Since there are now examples of elliptic curves E/Q for which X(E) is known to be finite and the Birch and Swinnerton-Dyer conjecture is completely proven [32][Thm. 9.3], [18][Thm. 1.2] our conjectures (5), (6) and (7) hold for any regular model X of any principal homogenous space of any elliptic curve isogenous over Q to such an E. We give a brief summary of each section below. In section 2 we prove that C(X , 1) = 1 for general X . Since Z(1) = G m [−1] the proof involves elementary properties of the sheaf G m in the étale topology. However, the p-part of C(X , 1) is defined in [5][Def. 5.6] via the eh-topology on schemes over F p [8], and our proof eventually reduces to a curious statement, Prop. 2.2 below, which in some sense says that the difference between étale and eh-cohomology is the same for the sheaves G m and O. In order to prove Prop. 2.2 we develop some results on the eh-topology which might be of independent interest.
In section 3 we prove, for an arithmetic surface X , Conjecture AV(X , 1) of [5][Conj. 6.23] using results of Saito [26]. This conjecture is necessary to define the Weil-étale complexes in terms of which our special value conjecture is formulated.
In section 4 we define Weil-étale complexes associated to the relative H 1 -motive of f and then translate our special value conjecture for ζ(H 1 , s) from the formulation in terms of a fundamental line to the explicit form (6) given above.
Finally, in section 5 we prove the equivalence to the Birch and Swinnerton-Dyer conjecture. The two key ingredients are a formula due to Geisser [9] relating the cardinalities of Br(X ) and X(J F ) and Lemma 15 about the Arakelov intersection pairing in exact sequences. We give a different proof of Geisser's formula in order to generalize it from totally imaginary to arbitrary number fields F .

The correction factor for n = 1
We recall some definitions from [5]. For any scheme Z and integer n we denote by Z(n) = z n (−, 2n − •) Bloch's higher Chow complex, viewed as a complex of sheaves on the small étale site of Z. For any prime number p we set and For X regular, proper over Spec(Z) and n ∈ Z we consider the derived de Rham complex modulo the Hodge filtration LΩ * X /Z /Fil n (see [17][VIII.2.1]) as a complex of abelian sheaves on the Zariski site of X . We obtain a perfect complex of abelian groups RΓ dR (X /Z)/F n := RΓ(X Zar , LΩ * X /Z /Fil n ) and its base change RΓ dR (X A /A)/F n to any ring A. and . Fix a prime number p and let X be regular, proper and flat over Spec(Z). The following is [5][Conj.5.5].
Conjecture 1. D p (X , n) There is an exact triangle of complexes of Q p -vector spaces Conjecture 1 gives an isomorphism λ p = λ p (X , n) : We then set (9) χ(X Fp , O, n) := i≤n,j where Ω i is the sheaf of Kaehler differentials on Sm /F p . Under R(F p , d) and for smooth Z/F p one has an isomorphism [8][Thm. 4.7] and Proposition 2.1. Let X be regular of dimension d, proper and flat over Spec(Z), and assume R(F p , d−1). Then Conjecture D p (X , 1) holds and one has C(X , 1) = 1.
Proof. Fix a prime number p. Then the base change X Zp is again a regular scheme.
on Sm /F p . One has an exact sequence of coherent sheaves and an exact sequence of abelian sheaves Here 1+I is just our notation for the kernel, the sections of this sheaf over an étale U → X Zp need not coincide with 1 + I(U ). Passing to p-adic completions and shifting by [−1] we obtain an exact triangle For n = 1 one has an isomorphism LΩ * X /Z /Fil 1 ∼ = O X and (9) specializes to (11) The outer terms in (10) are respectively computed by Lemma 1 and Proposition 2.2 below. Hence for an arithmetic surface we obtain the exact triangle of Conjecture D p (X , 1) after scalar extension to Q p . Moreover, Lemma 1 and Proposition 2.2 show that inside det Qp RΓ(X Zp,et , Q p (1)), i.e. we have d p (X , 1) = p −χ(X Fp ,O) and therefore c p (X , 1) = 1. This finishes the proof of Prop. 2.1.

Lemma 1.
There is an isomorphism is the analogue of (11) for the étale (or Zariski) topology.
Proof. Let (X, O X ) be the formal completion of X Zp at the ideal I = (p). The underlying topological space of X is X Fp and we denote by i : X → X Zp the natural morphism of ringed spaces and étale topoi. We have an exact sequence on X et Lemma 2. For any ν ≥ 1 the natural morphism i * (1 + I) → 1 + I X induces an isomorphism on X et Proof. Note that (14) is meant in the derived sense, so we have to investigate the kernel and cokernel of multiplication by p ν separately. For each (connected) étale Spec(A) → X Zp the map on kernels is whereÂ denotes the p-adic completion. Now any element of µ p ν (A) lies in the integral closure of Z p in A which coincides with the integral closure of Z p in the fraction field K of A, as A is regular. Hence µ p ν (A) is contained in the algebraic closure L of Q p in K. But L/Q p is finite as X Zp → Spec(Z p ) is of finite type. We conclude that µ p ν (A) is contained in O L which is already p-adically complete, i.e. we have µ p ν (Â) = µ p ν (A). Hence the map (15) is also an isomorphism (in fact both sides vanish unless p = 2). Concerning the cokernel of p ν we first note that for each étale Spec(A) → X Zp we have (1 + I X )(Â) = 1 + I X (Â) since the inverse of any element 1 + p · x is given by a geometric series. Moreover, the usual power series of the exponential and the logarithm induce an isomorphism (16) log : . In particular we conclude that any element in 1 + p ν+2 ·Â has a p ν -th root in 1 + p ·Â. Given 1 + x ∈ 1 + p ·Â and ν ≥ 1 we can write with x 0 ∈ I(A) = p · A and x 1 ∈Â and we can also assume that 1 + x 0 ∈ A × since inverting 1 + x 0 does not remove any points of X Fp = X. Hence we find 1 + x ∈ (1 + I)(A) · (1 + I X (Â)) p ν , i.e. that (14) is a surjection on cokernels of multiplication by p ν .
To show that it is also an injection we need to consider an element 1 + x 0 ∈ (1 + I)(A) that becomes a p ν -th power in 1 + I X (B) for some étale neighborhood Spec(B) → Spec(A) of any given point p ∈ Spec(A/(p)) and show that 1 + x 0 ∈ (1 + I(B ′ )) p ν in some étale neighborhood Spec(B ′ ) → Spec(B) of p. Since 1 + x 0 ∈ (1 + I X (B)) p ν we have 1 + x 0 = 1 + p ν+1 · y 0 . Pick a prime q of B above p. If y 0 ∈ q we can write 1 + p ν+1 · y 0 = 1 + p ν+1 · (y 0 + 1) 1 + p ν+1 · (1 + p ν+1 · y 0 ) −1 where both y 0 + 1 and (1 + p ν+1 · y 0 ) −1 do not lie in q. Hence we can assume y 0 / ∈ q and in fact y 0 ∈ B × . We then adjoin an element z satisfying the integral equation is étale and surjective. Indeed it is clearly étale at all primes q ′ ∈ Spec(B) with so inverting z removes all primes where q ′ is ramified. Moreover, there is a prime B[z, 1 z ] · q ′ above q ′ with a finite separable residue field extension (in fact the same residue field if p is odd).
Since since 1 + I X is already p-adically complete. Consider the diagram with exact rows where the vertical isomorphism is induced by the logarithm (16) for k large enough (in fact k ≥ 2). We have isomorphisms of abelian sheaves is a perfect complex of F p -modules since X is proper. It follows that the maps α and α add are isomorphisms after tensoring with Q p . Finally recall the theorem of formal functions [15][Thm. III.11.1] The isomorphism (12) is then the composition of the scalar extensions to Q p of (17), α, log, α add and (19). The above diagram and (18) show that under this isomorphism we have

2.1.
Results on the eh-topology. To prepare for the proof of Prop. 2.2 below we develop some results on the eh-topology which might be of independent interest. We first recall the notion of seminormalization of a scheme.
Definition 1. The seminormalization X sn → X of a scheme X is an initial object in the full subcategory of schemes over X consisting of universal homeomorphisms Z → X which induce isomorphisms on all residue fields.
By [29][Lemma 28.45.7] the seminormalization always exists and X sn is a seminormal scheme, meaning that for any affine open U the ring A = O(U ) is a seminormal ring, i.e. if x 2 = y 3 for some x, y ∈ A then there is a ∈ A with x = a 3 and y = a 2 . Any seminormal scheme is reduced [29][Lemma 28.45.5], hence we have a factorization X sn → X red → X by the universal property of either the reduction or the seminormalization. By [29][28.45.7] the seminormalization is also the final object in the category of seminormal schemes above X, i.e. (20) Hom(Z, X sn ) for any seminormal scheme Z. Any normal scheme is seminormal and hence the normalization X n → X of, say, a Noetherian scheme X [29][Sec. 28.52] factors through the seminormalization X n → X sn by (20). If X is moreover a Nagata scheme [29][Def. 27.13.1], for example if X is of finite type over a field, then X n → X is a finite morphism by [29][Lemma 28.52.10]. It follows that X sn → X is finite since X is Noetherian.
Lemma 3. For any scheme X the seminormalization X sn → X induces an equivalence of étale topoi where X ∼ denotes the eh sheaf associated to the presheaf represented by X. Hence for any (abelian) eh-sheaf F we have Proof. The isomorphism (21) follows from the fact that X sn → X is a universal homeomorphism [13][Exp. IX, 4.10] [14][II, Exp. VIII, 1.1]. Since X sn → X is finite surjective and induces an isomorphism on residue fields it is an eh-cover by [8][Lemma 2.2]. Since it is also a monomorphism it becomes an isomorphism in and we can assume that X is seminormal. We follow the arguments of [31] [3.2]. Since X is reduced, any surjection X ′ → X (for example the disjoint union of the schemes in an eh-cover) is an epimorphism in the category of schemes. Hence the separated presheaf associated to Hom(−, Y ) still has value Hom(X, Y ) on seminormal X, and the map is injective. Any element f ∼ of the right hand side is represented by a family of By [8][Prop. 2.3] every eh-cover has a refinement of the form for each x ∈ X since there is a point y ∈ X ′ with isomorphic residue field. Since X is seminormal, p 0 is an isomorphism and we find that f ∼ is represented by a morphism f ∈ Hom(X, Y ).

Corollary 1. We have
the category of representable sheaves in the eh-topology is equivalent to the category of seminormal schemes.
Proof. This follows from Lemma 4 together with (20), or by applying Lemma 4 to both Y sn and Y together with Y sn,∼ ∼ = Y ∼ .
Proof. Since every scheme has a cover by smooth schemes in the eh-topology un- By [8][Cor.2.6] every eh-cover of a smooth scheme has a refinement by a cover consisting of smooth schemes. Hence the eh-sheafification process again leads to isomorphic groups on both sides.
Next we discuss comparison results between étale and eh-cohomology. Consider the morphism of topoi p = p X : (Sch d /F p ) ∼ eh /X → X et and the natural transformation α ′ : F | X → Rp * F where F | X := p * F denotes restriction to the small étale site. The functor F → F | X extends to complexes but does not preserve quasi-isomorphisms. We obtain a natural transformation on the category of abelian sheaves on (Sch d /F p ) eh . Both α ′ and α are also contravariantly functorial in X.
Then there exists a natural transformation which coincides with (23) for proper X and is compatible with exact localization triangles for open/closed decompositions U ֒→ X ←֓ Z.
Proof. Let j : X →X be an open embedding into a proper F p -schemeX with closed complement i : Z ֒→X. Choose an injective resolution which arises from functoriality of α ′ for the morphism i, is realized by a diagram of maps of complexes of injectives with all maps unique up to homotopy and commuting up to homotopy. Note here that p * and i * preserve injective objects. Taking is an isomorphism for all X.
Proof. The proof is a standard induction over the dimension of X, see [8][Lemma 2.7].
We remark that The transformation α c is then also an isomorphism for all X if F is constructible. However, constructibility is a much stronger assumption than being torsion and will not hold for the p-primary torsion sheaves of interest below.
For non-torsion sheaves, even if étale and eh cohomology agree on smooth schemes X, one cannot expect an isomorphism for general, even normal schemes, as the example [8][Prop. 8.2] shows. The following Lemma (which is not needed in the remainder of the paper) allows to prove such an identification between étale and eh cohomology in some very restricted circumstances.
Proof. The functoriality of p in X and diagram (27) induce a commutative diagram of exact triangles Here exactness at the last term was assumed and exactness at the remaining terms follows from the fact that F is an eh sheaf, and the pullback of (27) to any étale U → X is again an abstract blowup square. We also use that f, i, g are finite so Since Z ′ , Z, X ′ are smooth the maps α 2 , α 3 are quasi-isomorphisms, and hence so is α 1 by the Five Lemma.
Corollary 3. Let X be a seminormal curve over F p . Then we have isomorphisms (27) with X ′ = X n , the normalization, and Z = X sing , the singular locus. Both Z and Z ′ are finite unions of closed points, hence smooth, and X n is smooth since X is a curve. The map G m → i ′ * G m is surjective since i ′ is a closed embedding, and applying f * gives a surjection since f * is exact. So all conditions of Lemma 7 are satisfied. The proof Corollary 4. Let X be an arbitrary curve over F p . Then we have isomorphisms Proof. Combine (22) and Corollary 3.
We finally come back to the proof of Prop. 2.1. For X in Sch d /F p we have a natural map Consider the induced map on p-adic completions and denote by C the [-1]-shift of its mapping cone, so that there is an exact triangle Similarly, we have an exact triangle For a bounded complex K with finite cohomology we set For example, if X is proper, the terms in (29) are perfect complexes of F p -vector spaces by [8][Cor. 4.8] and we have where χ(X, O), resp. χ et (X, O), is defined as in (11), resp. (13), with X Fp replaced by X. Proposition 2.2. Let X → Spec(F p ) be proper of dimension d and assume R(d, F p ). Then C is a bounded complex with finite cohomology and (30) χ(C) = χ(C add ).
In particular for a proper arithmetic scheme X we have an isomorphism the assumption of Cor. 2 is satisfied for G m /p ν by [8][Thm. 4.3], and we deduce that the p-adic completion of α in (28) is an isomorphism for proper X. Moreover, by Lemma 4 we have G ∼ m | X = σ * G m where σ : X sn → X is the seminormalization. So we obtain an exact triangle Since O is a p-torsion sheaf an analogous argument gives an exact triangle where J is the nilradical, as X sn is reduced. There is an analogous sequence of abelian sheaves Both J and 1 + J have finite filtrations J k , resp. 1 + J k , with isomorphic subquotients J k /J k+1 ∼ = (1 + J k )/(1 + J k+1 ); x → 1 + x, hence (31) χ(RΓ(X et , J)) = χ(RΓ(X et , 1 + J)).
To prove (30) it then suffices to show χ(RΓ(X et , K)) = χ(RΓ(X et , K mult )). This in turn will follow from the more general statement that inducing isomorphisms on all residue fields with X reduced. We prove this by induction on the dimension d of X. First we can assume that X ′ is reduced, using (31) for the nilradical of X ′ . If d = 0 both X and X ′ are finite unions of spectra of finite fields and σ ′ is an isomorphism. In general, let Z ֒→ X be the singular locus, a proper closed subset of X, with its reduced scheme structure, and let be the pullback under σ ′ . Then Z ′ → Z is again a universal homeomorphism inducing isomorphisms on all residue fields with Z reduced, to which the induction assumption applies. Since X \ Z is smooth, hence seminormal, the restriction of σ ′ to X ′ \ Z ′ has a section, hence is an isomorphism as X ′ is reduced. It follows that O X ′ /O X is supported on Z (we omit σ ′ * since σ ′ is a homeomorphism). The morphism σ ′ is finite as the seminormalization factors through it. Hence O X ′ /O X is coherent and there exists r such that M r · O X ′ /O X = 0 where M denotes the ideal sheaf of Z. Setting M ′ = MO X ′ we have an exact sequence of coherent sheaves On the multiplicative side we have an exact sequence of abelian sheaves on X et supported in Z and an isomorphism of subquotients.
Note here that all sections 1 + x ∈ 1 + M ′ /M are invertible, since x r+1 ∈ M. We conclude that and together with the induction assumption we obtain (32).

Artin-Verdier duality
A key ingredient in our construction of Weil-étale complexes is Artin-Verdier duality with torsion coefficients, in the form of Conjecture AV(X , n) introduced in [ is a perfect pairing of finite abelian groups for any i ∈ Z and any positive integer m.
This conjecture is known for X smooth proper over a number ring, and for regular proper X as long as n ≤ 0 or n ≥ d. Therefore, if X is an arithmetic surface, the only unresolved case is n = 1 which we shall prove using results of Saito in [26].
Let X be the formal completion of X at the ideal sheaf I = (p) where we view the structure sheaf O X as a sheaf on X Fp,et . By [26][Thm. 4.13] the map is an isomorphism for i = 0, 1. The groups H i (X Fp , G m,X ) can be computed following the computation of H i (X Fp , i * G m ) in [26][Prop. 4.6 (2)], and the result is the same for i = 0, 1. More precisely, the rings O x , O η , O λ of loc. cit. get replaced by the corresponding local rings of X, which are again Henselian local with unchanged residue field. Since G m is a smooth group scheme, its cohomology in degrees ≥ 1 coincides with that of the residue field [22][III. 3.11]. It follows then from [26][Prop. 4.6 (1), (2)] that (35) is an isomorphism for i = 3 and has cokernel the uniquely divisible group (Ẑ/Z) r for i = 3 where r is the number of irreducible components of X Fp . Hence is an isomorphism for all i. On the other hand by Lemma 2 we have an isomorphism i * G m /p ν ∼ = G m,X /p ν and hence an isomorphism for all i. Now consider the duality map on localization triangles where we have shown the outer vertical maps to be quasi-isomorphisms. It follows that the middle vertical map is a quasi-isomorphism which finishes the proof of Prop where K /X is the complex constructed by Spiess in [28] and the middle isomorphism is [33][Thm. 3.8]. The pairing (37), combined with the map (38) and taken modulo p ν , can also be characterized as the unique pairing constructed by the method of Sato in [27]. Even though Sato assumes X to have semistable reduction, his arguments work in our situation where X is a relative curve and n = m = 1. We summarize the properties we need in the following Proposition.

Isolating the H 1 -part
The formulation of the special value conjecture [5][Conj 5.12] for ζ(X , s) at s = 1 involves the fundamental line and the exact triangle [5][ (5)] which induces the trivialization (1) of the determinant of ∆(X /Z, 1) R . For each complex in (42), we shall define in this section a corresponding complex for the relative H 1 -motive of the morphism f : X → S and obtain a corresponding description of ζ(H 1 , s) at s = 1. In the absence of a suitable triangulated category of motivic complexes DM with a motivic t-structure, we isolate the relative H 1 -motive in an ad-hoc way in the derived category of étale sheaves on S. More precisely, by [12][Cor. 3.2] one has R i f * G m = 0 for i ≥ 2 and is the relative Picard functor studied for example in [25]. One has a truncation triangle and we define a complex of étale sheaves P 0 by the exact triangle where the degree map is discussed for example in [12][Sec. 4]. The complex P 0 serves as a substitute for the relative H 1 -motive and we will define Weil-étale and Weil-Arakelov complexes associated to it according to the following table. The first column refers to definitions made in [5]. In particular, X and S denote the Artin-Verdier compactifications of X and S, respectively. If for example f has a section the complexes in the right hand column are direct summands of those in the left hand column. In general the exact triangles (43) and (44) will induce corresponding exact triangles for the Weil-étale complexes associated to Z(1), Z, P and P 0 on S.
The precise definition of Weil-étale modifications will be recalled in the proof of the following Lemma. Lemma 8. If Br(X ) is finite then RΓ W (S, P 0 ) is a perfect complex of abelian groups satisfying a duality

Its cohomology is given by
where H 1 (S, P 0 ) = H 1 (S et , P 0 ) is a finite abelian group of cardinality # Br(X )δ 2 and Br(X ) is defined by the exact sequence Br(X Fv ).
In particular, Br(X ) coincides with Br(X ) if F has no real places and is a subgroup of Br(X ) of co-exponent 2 in general.
Proof. According to their definition in [5][App. A] the Artin-Verdier étale topoi of X and S fit into a commutative diagram of morphisms of topoi Applying Rf * to the defining exact triangle [5][App. A, Cor. 6.8] of the complex Z(1) X we obtain a commutative diagram of exact triangles (47) where the middle vertical triangle is induced by (43) and the left, resp. right, vertical triangle defines P S , resp. K. Since Rπ * (2πiZ), resp. Rπ S * (2πiZ), is a complex of sheaves supported on X (R), resp. S(R), with stalk H j (G R , 2πiZ) = Z/2Z in odd degrees j, and vanishing in even degrees, it follows that the top two complexes in the right hand column are supported in degrees ≥ 3. This gives exactness of the columns in the diagram which is the map of long exact cohomology sequences induced by the left two columns in (47). Here we use the isomorphism (see the proof of Lemma 11 below) and similarly for S ∞ . We deduce that Br(S) := H 3 (S, Z(1)) = 0, hence and that H 3 (X , Z(1)) coincides with the group Br(X ) defined in (45). The continuation of the top long exact sequence gives where the vertical duality isomorphisms follow from [5][Prop. 3.4] for both S and X , taking into account the compatibility of dualities in Corollary 5. Since we have a commutative diagram with surjective vertical map, the cokernel of both degree maps is Z/δZ where δ is the g.c.d. of the degrees of all divisors on X F , i.e. the index of X F . Hence an exact sequence We can similarly extend (44) to the Artin-Verdier compactification. The composite map in Sh(G R , S(C)) clearly vanishes and the commutativity of the right hand square in (46) then shows that the degree map on the middle row of (47) factors through the lower row, i.e. we obtain a commutative diagram with exact rows and columns (50) where the lower row is the defining exact triangle [5][App. A, Cor. 6.8] of the complex Z S , and the left hand column defines P 0,S . Note that in fact Z S = Z. The long exact cohomology sequence of the left hand column is and from (49) that H 1 (S, P 0 ) has cardinality # Br(X )δ 2 . In order to compute H i (S, P 0 ) in degrees ≥ 2 we prove a torsion duality for P 0 /p ν for any prime p, the isomorphism AV(S, P 0 ) in diagram (55) below. Following the proof of [5][Prop. 3.4

] this then implies
and its cohomology is computed by the following Lemma.
Lemma 9. One has Proof. The topological space X ∞ = X (C)/G R is a 2-manifold (with boundary X (R)) and hence has sheaf-cohomological dimension ≤ 2. The complex .23] is concentrated in degrees 0 and 1 and R 1 π * (2πi)Z is supported on the closed subset X (R), a union of circles, hence of cohomological dimension 1. It follows that RΓ W (X ∞ , Z(1)) is concentrated in degrees ≤ 2.
The two triangles (56) and (57) are direct sums over the infinite places v ∈ S ∞ and we denote the respective direct summands by an index v. If v is a complex place or a real place with X (F v ) = ∅ then f ∞,v has a section and the exact triangle [5][(47)] splits into a direct sum of its H i -parts for i = 0, 1, 2. The last term being concentrated in degrees ≥ 3 we find for the H 1 -part. The group H 1 (G R , H 1 (X (F v ⊗ R C), (2πi)Z)) is isomorphic to Φ v as can be verified easily by taking G R -cohomology of the exponential sequence for the abelian variety J F (F v ⊗ R C).
If v is a real place with X (F v ) = ∅ an analysis of the spectral sequence Here one uses the fact that the end term is concentrated in degrees ≤ 2 and that for i ≥ 1 To show (59) first note that X (F v ) = ∅ if and only if δ v = 2, otherwise δ v = 1. Since δ ′ v | δ v this shows (59) at places where X (F v ) = ∅. If X (F v ) = ∅ one has #Φ v = 1 or 2 according to whether the genus g of X is even or odd by [11][Prop. 3.3] and one also has δ ′ v = 1 or 2 according to whether g is even or odd by [19], [24][p. 1126]. This shows (59) at places where X (F v ) = ∅.
The complex RΓ W,c (S, P 0 ) is defined by an exact triangle RΓ W,c (S, P 0 ) → RΓ W (S, P 0 ) → RΓ W (S ∞ , P 0 ) → and its cohomology, or at least the determinant of its cohomology, can easily be computed by combining Lemmas 8 and 9. The H 1 -part of the exact triangle (42) is an exact triangle The fundamental line of the H 1 -part on X (C) and the intersection pairing One easily derives formula (6) in the introduction. Note here that the determinant Ω(X ) of (61) with respect to the natural Z-structures on both sides includes possible torsion in H 1 (X , O X ). This torsion subgroup is nontrivial if and only if f is not cohomologically flat in dimension 0.

Comparison to the Birch and Swinnerton-Dyer Conjecture
The comparison of our conjecture (6) with the Birch and Swinnerton-Dyer formula involves two key ingredients. The first is a precise formula relating the cardinalities of Br(X ) and X(J F ) and the second a Lemma about the behavior of the Arakelov intersection pairing in exact sequences, Lemma 15 below. The essential ideas for the comparison of Br(X ) and X(J F ) can already be found in Grothendieck's article [12] and his results imply that Br(X ) ∼ = X(J F ) if, for example, f has a section and F is totally imaginary. The general case is considerably more complicated and has been studied by a number of authors until it was recently settled by Geisser [9]. Unfortunately, Geisser's formula still has the condition that F is totally imaginary, so we give here a generalization of his result without this condition and with a different proof. What makes both proofs eventually possible are the duality results of Saito in [26].
For any place v of F we denote by δ v , resp. δ ′ v , the index, resp. period of X Fv over F v , i.e. the cardinalities of the cokernel of Pic(X Fv ) where the equality holds since H 0 (S, P 0 ) = Pic 0 (X )/ Pic(O F ) → Pic 0 (X F ) is surjective.
where the product is over all places v of F and Br(X ) was defined in Lemma 8.
Note that η * P 0,S is the sheaf represented by the Jacobian J F and H 0 (P 0 ) = η * J F the sheaf represented by the Neron model of J F over S.

Lemma 10.
The complex E is a sum of skyscraper sheaves in degrees 0 and 1.

One has
where Σ f is the set of finite places of F where f is not smooth, and Proof. The restriction of the middle row of (65) to S is a short exact sequence of sheaves concentrated in degree 0 and has been analyzed by Grothendieck [12][(4.10 bis)]. The restriction of E to S is a sum over v ∈ Σ f of skyscraper sheaves E v placed in degree 0. Viewing E v as a G κ(v) -module there is an exact sequence where C v is the set of irreducible components of the fibre X κ(v) and κ(v) i denotes the algebraic closure of κ(v) in the function field of the component X κ(v),i indexed by i ∈ C v . By [12][ (4.25)] The middle row in (65), the fact that η * η * P S is concentrated in degree 0 and Lemma 11 below imply H i (E) = 0 for i ≥ 2 and an exact sequence We saw already that the restriction of H 1 (E) to S is zero. If is an open subscheme over which f is smooth we haveη * η * P = P and hence u S, * ∞ η * η * P S = u S, * ∞ φ S * j * P = u S, * ∞ φ S * P. It then follows from the bottom triangle in (47) that is a surjection in degree 2 (this is clear as the target is zero) and an isomorphism in degrees ≥ 3. Since Rf * Z(1) is concentrated in degrees ≤ 2 this is equivalent to being an isomorphism. But the map in (69) is isomorphic to the map in (60) which we have shown to be an isomorphism in degrees ≥ 3 in the proof of Lemma 9. To see that the two maps are isomorphic in degrees ≥ 3 use the exact triangles and the isomorphism α S, * Rf * Q/Z(1) ∼ = Rf C, * (2πi)Q/Z arising from proper base change and (a G R -equivariant version of) Artin's comparison theorem between étale and analytic cohomology, where α S is the composite morphism of topoi To compute # im(φ 1 ) consider the continuation of the long exact sequences where the vertical column is also exact.

Lemma 12.
For v ∈ Σ one has Proof. Consider the commutative diagram with exact rows and columns induced by (65) The vanishing of H 1 (S v , P ) is proven in [12][(4.15)] for v ∈ Σ f and follows from For v ∈ Σ f this is because H 0 (S v , P ) = Pic(X Sv ) → Pic(X Fv ) is surjective and for v ∈ S ∞ both subgroups coincide with the kernel of the map Hence the degree map on H 0 (S v , P ) has image δ v Z which gives (72). By the snake Lemma on finds coker(ψ) Lemma 13. One has # ker(φ 6 ) = α.

Proof.
For the open subscheme U := S \ Σ f of S and any prime p the proof of Prop. 3.1 generalizes to prove a duality isomorphism AV(X U , 1) fitting into a commutative diagram of isomorphisms One then obtains diagrams (52) with S, resp. X , replaced by U , resp. X U . The triangle (54) and the left hand column in (50) then induce an isomorphism AV(U, P 0 ) fitting into a commutative diagram

One has isomorphisms
the first isomorphism holds since Tate cohomology agrees with ordinary cohomology in degrees ≥ 1, the second sinceĤ i c (U, P 0 ) is torsion for i = 1, 2 and the third by taking the limit of AV(U, P 0 ) * over all p and all ν. One finds that φ 6 is dual to the natural restriction map where the last isomorphism holds since f is smooth over U , hence P 0 coincides with the (sheaf represented by the) Neron model of J F . From the definition (63) of α we conclude that α = # coker(φ * 6 ) = # ker(φ 6 ).
where {b i } ⊆ N is a maximal linearly independent subset. One easily verifies that ∆(N ) only depends on < −, − > but not on the choice of {b i }.
Lemma 15. Let · · · → N i di − → N i+1 → · · · be an exact sequence of finitely generated abelian groups of finite length. Assume each N i is equipped with an inner product τ i so that Proof. If suffices to prove the statement for the short exact sequences A maximal linearly independent subset {b j } 1≤j≤l ⊂ ker(d i ) can be extended to a maximal linearly independent subset {b j } 1≤j≤l+k ⊂ N i and {b j } l+1≤j≤l+k ⊂ im(d i ) is then also a maximal linearly independent subset. By the snake lemma we have Zb j ] and we also have since the {b j } l+1≤j≤l+k can be modified into elements of ker(d i ) ⊥ R (by adding elements of ker(d i ) R ) without changing det(<b j ,b j ′ >) l+1≤j,j ′ ≤l+k or det(< b j , b j ′ > ) 1≤j,j ′ ≤l+k .
Recall that for each v ∈ Σ f we denote by C v the set of irreducible components of the fibre X κ(v) and by r v,i = [κ(v) i : κ(v)] the degree of the constant field of the component corresponding to i ∈ C v . Lemma 16. Let L(J F , s) be the Hasse-Weil L-function of the Jacobian of X F . Then and an exact sequence Comparing with (4) we find (see [5][Rem. 7.5] for connections with the perverse t-structure) From this (74) and (75) are immediate.
Lemma 17. For each v ∈ Σ f we have where Φ v is the component group of the Neron model of J F at v. Here ∆ is formed with respect to the Arakelov intersection pairing.
Combining (76) Proof. We have an isomorphism of G R -modules has kernel and cokernel of the same length. From this the statement is immediate.
We now have all the ingredients to compare (5) and (6) with the Birch and Swinnerton-Dyer conjecture. Taking cohomology over S of the top row in (65) gives an exact sequences where A is a finite abelian group of cardinality α defined in (63). Since the Z-rank of E (5) ) · ∆(J F (F )) · #X(J F ) · Ω(X ) · Remark 2. Let f : X → S be a flat morphism from a smooth projective surface X to a smooth projective connected curve S π − → Spec(k) over a finite field k and assume f has geometrically connected fibres. By [1][Rem. 5.4.9] K := Rf * Q l