First and second K -groups of an elliptic curve over a global field of positive characteristic
aa r X i v : . [ m a t h . K T ] N ov Ann. Inst. Fourier, Grenoble
Working version – October 8, 2018
FIRST AND SECOND K -GROUPS OF AN ELLIPTICCURVE OVER A GLOBAL FIELD OF POSITIVECHARACTERISTIC by Satoshi KONDO and Seidai YASUDA (*) Abstract.
In this paper, we show that the maximal divisible sub-group of groups K and K of an elliptic curve E over a function fieldis uniquely divisible. Further those K -groups modulo this uniquelydivisible subgroup are explicitly computed. We also calculate the mo-tivic cohomology groups of the minimal regular model of E , whichis an elliptic surface over a finite field. Sur les premier et second K -groupes d’une courbe elliptique sur uncorps global de caract´eristique positive R´esum´e.
On d´emontre que les plus grands sous-groupes divisiblesdes groupes K et K d’une courbe elliptique E sur un corps globalde caract´eristique positive sont uniquement divisibles et on d´ecritexplicitement les K -groupes modulo leurs plus grands sous-groupesdivisibles. On calcule ´egalement la cohomologie motivique du mod`eleminimal de E qui est une surface elliptique sur un corps fini. Keywords:
K-theory, function field, elliptic curve, motivic cohomology.
Math. classification:
SATOSHI KONDO AND SEIDAI YASUDA
1. Introduction
In this paper, we study the kernel and cokernel of boundary maps in thelocalization sequence of G -theory of the triple: an elliptic curve E over aglobal function field k of positive characteristic, the regular minimal model E of E that is proper flat over the curve C associated with k , and the fibersof E → C . The aim of this initial section is to provide our motivation,describe some necessary background, and present known results more gen-erally. Precise mathematical statements regarding our results are presentedin Section 1.5. Among these statements, we suggest that our most interest-ing point is represented in assertion (2) of Theorem 1.2, which relates thekernel of a boundary map to the special value L ( E,
0) of the L -function of E . From Grothendieck’s theory of motives, Beilinson envisioned the abeliancategory of mixed motives over a base (see [36]). The existence of such acategory remains a conjecture, but we now have various constructions ofthe triangulated category of (mixed) motives, which serves as the derivedcategory of the sought-after category. Here, motivic cohomology groupsare extension groups in the category of mixed motives, or, unconditionally,homomorphism groups in the triangulated category of motives. Algebraic K -theory also enters the picture here via the Atiyah-Hirzebruch-type spec-tral sequence (AHSS) of Grayson-Suslin or Levine [20], [37], [61] (see also[14, (1.8), p.211], [19]), i.e., E s,t = H s − t M ( X, Z ( − t )) ⇒ K ′− s − t ( X )or via formula [7, 9.1] (see also [19, p.59]) K ′ n ( X ) ⊗ Z Q ∼ = M i H i − n M ( X, Q ( i ))of Bloch. Also applicable here are the Riemann-Roch theorem for higher K -theory and the motivic cohomology theory by Gillet [18], Levine [34],Riou [57], and Kondo-Yasuda [32].Arithmetic geometers are individuals who study motives (or varieties)over number fields, over the function fields of curves C over finite fields (de-noted function fields for short), or over finite fields. For such a variety, the L -function, also known as the zeta function, one of the fundamental invari-ants in arithmetic geometry, is defined. The motivation for our work stems ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE L -functions. Conjectures over Q were formulated byLichtenbaum, Beilinson, and Bloch-Kato (see [27] for a summary of theseconjectures). Using the analogy between number fields and function fields,parts of these conjectures can be translated into conjectures over functionfields.The case over function fields and over finite fields are related as follows.Given a variety over a function field, take a model, i.e., a variety over thebase finite field fibered over C whose generic fiber is the given variety. Then,the motivic cohomology groups and the L -functions (over finite fields, i.e.,congruence zeta functions) are related, thus, we obtain some conjecturalstatements regarding motivic cohomology groups of the model and its con-gruence zeta function from these conjectures over function fields concerning L -functions and vice versa. Note that Geisser [15] presents a descriptionof K -theory and motivic cohomology groups of varieties over finite fieldsassuming both Parshin’s conjecture and Tate’s conjecture.Finally, let us mention the finite generation conjecture of Bass [27, p.386,Conjecture 36], which states that the K -groups of a regular scheme of finitetype over Spec Z are finitely generated. Unconditionally verifying the conjectures above is difficult, as is verify-ing the consequences of the conjectures. The key theorem here is the Rost-Voevodsky theorem (i.e., the Bloch-Kato conjecture) and, as a consequence,the Geisser-Levine theorem (i.e., a part of the Beilinson-Lichtenbaum con-jecture) [17, Theorem 1.1, Corollary 1.2]. These theorems state that themotivic cohomology with torsion coefficients and ´etale cohomology are iso-morphic in certain range of bidegrees.These theorems and the aforementioned AHSS can be used as the pri-mary ingredients for computing the K -groups and motivic cohomologygroups of the ring of integers of global fields. In these “one-dimensional”cases, the Bass conjecture on finite generation is known to hold basedon work by Quillen [56] and Grayson-Quillen [19]. In positive character-istic, these ingredients are sufficient for computing the motivic cohomologygroups, and a substantial amount of computation can also be performed inthe number field case. We refer to [67] for details on these unconditionalresults. SUBMITTED ARTICLE : 2011K1K22ND41.TEX
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Unfortunately, much less is known regarding higher dimensions. Our re-sults may be regarded as the next step in determining this in that we treatelliptic curves over a function field and elliptic surfaces over a finite field.The theorems of Rost-Voevodsky and Geisser-Levine do indeed cover manyof the bidegrees, but there remain lower bidegrees that require additionalwork.
To compute motivic cohomology with Z -coefficient, one first divides theproblem into Q -coefficient, torsion coefficient, and the divisible part in thecohomology with Z -coefficient. Then, for the prime-to-the-characteristiccoefficient, one uses the ´etale realization, i.e., a map to the ´etale cohomol-ogy. For the p -part, one uses the map to the cohomology of de Rham-Wittcomplexes. Here, the target groups are presumably easier to compute thanthe motivic cohomology groups, and these realization maps are now knownto be isomorphisms in many cases, due to Rost-Voevodsky and Geisser-Levine; however, no general method for computing the Q -coefficient anddivisible parts is known. In one-dimensional cases, we have finite gener-ation theorems, which then imply that the divisible part is zero and the Q -coefficient part is finite dimensional, and the Q -coefficient part is zeroexcept for bidegrees (0 , , , X over a finitefield. In Section 2, we prove that the divisible part of motivic cohomologygroups, which may still contain torsion, is actually uniquely divisible exceptfor bidegree (3 , X is a model of an elliptic curve over afunction field.When the coefficient in the motivic cohomology groups is the second Tatetwist, especially when the bidegree is (3 , p part. The p -partwas treated by Gros-Suwa [21], though we need a supplementary statement ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE p -part, which we provide in the Appendix. We follow their outlineclosely in Section 2 below.We note here that the series of results above are written in terms of K -cohomology groups and the associated realization maps. The K -cohomologygroups were used as a substitute for motivic cohomology groups and arenow known to be isomorphic to motivic cohomology groups in particu-lar bidegrees, as shown by Bloch, Jannsen, Landsburg, M¨uller-Stach, andRost (see [48, p.297, Bloch’s formula, Corollary 5.3, Theorem 5.4]). Someearlier results do imply some of the results in our paper via this com-parison isomorphism; however we present our exposition independent ofearlier results and ensure that our work is self-contained for two reasons.First, if we used earlier results, in possible applications of our result, somecompatibility checks would have become necessary. For example, in earlierpapers, a certain map from K -cohomology to ´etale cohomology groups wasused. Conversely, we use a cycle map from higher Chow groups given byGeisser-Levine [17]. It is nontrivial to check if these two maps are com-patible under the comparison isomorphism. Therefore, while we do referto earlier papers, we use only the non-motivic statements, reproducing themotivic statements in our paper where applicable. Second, some earlier re-sults (e.g., those of [21]) are usually stated for projective schemes. We alsoneed similar results for not necessarily projective schemes, since we treata curve over a function field as the limit of surfaces fibered over (affine)curves over a finite field. When we remove the condition that the schemeconsidered is projective, it becomes much more difficult to show that thedivisible part of motivic cohomology is uniquely divisible. K -theory In our paper, we use Chern classes to relate K -theory and motivic co-homology. We use the Riemann-Roch theorem without denominators, andwe perform some direct computations for singular curves over finite fields.We further explain these two independent issues below. Finally, in the lastparagraph of this subsection, we describe the application.The first issue is the comparison isomorphism between Levine’s mo-tivic cohomology and Bloch’s higher Chow groups, an issue we detail inSection 3. The general problem is summarized as follows. We have vari-ous constructions of motivic cohomology theory (or groups), i.e., Suslin-Voevodsky-Friedlander [66], Hanamura [23], Levine [34], Bloch [7] (higher SUBMITTED ARTICLE : 2011K1K22ND41.TEX
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Chow groups), Morel-Voevodsky [46], and the graded pieces of rational K -groups. The motivic cohomology groups defined in various ways are gener-ally known to be isomorphic, however, it is nontrivial to verify whether thecomparison isomorphisms respect other structures, such as functoriality,Chern classes (characters), localization sequences, and product structures,etc. The Riemann-Roch theorem without denominators is known to holdin the following three cases: by Gillet for cohomology theories satisfyinghis axioms [18]; by Levine for his motivic cohomology theory [34]; and byKondo-Yasuda in the context of motivic homotopy theory of Morel and Vo-evodsky [32]. None of these directly apply to the motivic cohomology groupsin our paper, namely the higher Chow groups of Bloch, hence we use thecomparison isomorphism between higher Chow groups and Levine’s motiviccohomology groups, importing the Riemann-Roch theorem of Levine intoour setting by checking various compatibilities. The compatibility proof forGysin maps and localization sequences under the comparison isomorphismare thus the focus of Section 3.The second issue is the integral construction of Chern characters forsingular curves over finite fields, which we cover more fully in Section 4.The Chern characters from K -theory (or G -theory) to motivic cohomologyare usually defined for nonsingular varieties and come with denominators inhigher degrees. In our paper, we focus on elliptic fibrations; singular curvesover a finite field appear naturally as fibers, and we must study their G -theory. We do so by constructing a generalization of Chern characters inan ad hoc manner, generalizing those in the smooth case. To be able to usesuch an approach, we make substantial use of the fact that we consider onlycurves (however singular) over finite fields. We use the known computationof motivic cohomology and K -groups of nonsingular curves over finite fields,and, as one additional ingredient, we use a result from our previous work(Lemma 8). The principal result of Section 4 is Proposition 4.1 which statesthat the 0-th and 1-st G -groups are integrally isomorphic via these Cherncharacters to the direct sum of relevant motivic cohomology groups with a Z -coefficient.Finally, in Section 5, we apply results from Sections 2, 3, and 4, to com-pute the lower K -groups of a curve over a function field in terms of motiviccohomology groups. An additional ingredient here is the computation of K of fields given by Nesterenko-Suslin [50] and Totaro [63]. ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE K -groups The aim of our paper is to explicitly compute the K and K groups,the motivic cohomology groups of an elliptic curve over a function field,and the motivic cohomology groups of an elliptic surface over a finite field.In this subsection, we provide the precise statements of our results (i.e.,Theorems 1.1 and 1.2) concerning K and K groups of an elliptic curve.We refer to Theorems 6.2, 6.3, 7.1, and 7.2 below for more detailed resultsconcerning motivic cohomology groups.Let k be a global field of positive characteristic p , and let E be an ellipticcurve over Spec k . Let C be the proper smooth irreducible curve over a finitefield whose function field is k . We regard a place ℘ of k as a closed point of C , and vice versa. Let κ ( ℘ ) denote the residue field at ℘ of C . Let f : E → C denote the minimal regular model of the elliptic curve E → Spec k . This f is a proper, flat, generically smooth morphism such that for almost allclosed points ℘ of C , the fiber E ℘ = E × C Spec κ ( ℘ ) at ℘ is a genus onecurve, and such that the generic fiber is the elliptic curve E → Spec k .Let us identify the K -theory and G -theory of regular Noetherian schemes.First, there is a localization sequence of G -theory, i.e., K i ( E ) → K i ( E ) ⊕ ∂ i℘ −−−→ M ℘ G i − ( E ℘ ) → K i − ( E ) , in which ℘ runs over all primes of k .For a scheme X , we let X be the set of closed points of X . Let us use ∂ to denote the boundary map ⊕ ∂ ℘ , ∂ : K ( E ) red → L ℘ ∈ C G ( E ℘ ) to denotethe boundary map induced by ∂ , and ∂ : K ( E ) red → L ℘ ∈ C G ( E ℘ ) todenote the boundary map induced by ⊕ ∂ ℘ . Here, for an abelian group M ,we let M div denote the maximal divisible subgroup of M , and set M red = M/M div .In Theorems 1.1 and 1.2 below, we use the following notation. We usesubscript − Q to mean − ⊗ Z Q . For a scheme X , let Irr( X ) be the set ofirreducible components of X . Let F q be the field of constants of C . For ascheme X of finite type over Spec F q and for i ∈ Z , choose a prime number ℓ = p and set L ( h i ( X ) , s ) = det(1 − Frob · q − s ; H i et ( X × Spec F q Spec F q , Q ℓ )) , where F q is an algebraic closure of F q and Frob ∈ Gal( F q / F q ) is the geo-metric Frobenius element. In all cases considered in Theorems 1.1 and 1.2,the function L ( h i ( X ) , s ) does not depend on the choice of ℓ . Let T ′ (1) denote SUBMITTED ARTICLE : 2011K1K22ND41.TEX
SATOSHI KONDO AND SEIDAI YASUDA what we call the twisted Mordell-Weil group T ′ (1) = M ℓ = p ( E ( k ⊗ F q F q ) tors ⊗ Z Z ℓ (1)) Gal( F q / F q ) . We write S (resp. S ) for the set of primes of k at which E has splitmultiplicative (resp. bad) reduction; we also regard it as a closed subschemeof C with the reduced structure; note that the set of primes S will beintroduced later. Further, for a set M , we denote its cardinality by | M | .Finally, we let r = | S | . Theorem 1.1 (see Theorem 6.1(1)(2) and Theorem 7.2(2)) . —
Supposethat S is non-empty, or equivalently that f is not smooth. (1) The dimension of the Q -vector space ( K ( E ) red ) Q is r . (2) The cokernel of the boundary map ∂ : K ( E ) → L ℘ ∈ C G ( E ℘ ) isa finite group of order ( q − | L ( h (Irr( E S )) , − || T ′ (1) | · | L ( h ( S ) , − | . (3) The group K ( E ) div is uniquely divisible, and the kernel of theboundary map ∂ : K ( E ) red → L ℘ ∈ C G ( E ℘ ) is a finite group oforder | L ( h ( C ) , − | Let L ( E, s ) denote the L -function of E (see Section 6.3). We write Jac( C )for the Jacobian of C . Theorem 1.2 (see Theorem 6.1(3)(4)) . —
Suppose that S is non-empty, or equivalently that f is not smooth. (1) The group K ( E ) div is uniquely divisible. (2) The kernel of the boundary map ∂ : K ( E ) red → L ℘ ∈ C G ( E ℘ ) is a finite group of order ( q − | T ′ (1) | · | L ( E, | . The cokernel of ∂ is a finitely generated abelian group of rank | Irr( E S ) | − | S | whose torsion subgroup is isomorphic to Jac( C )( F q ) ⊕ . In this subsection, we describe similarities between our statements andthe Birch-Tate conjecture [62, pp.206–207]. Although there is no direct con-nection between them, we hope this subsection provides some justificationfor the way our statements are formulated. The similarities were pointedout by Takao Yamazaki.
ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE K -theory are stated in terms of the boundary map inthe localization sequence. For example, in Theorem 1.2(2), in which wedescribe the K group of an elliptic curve over a function field, we considerthe boundary map to the G -groups of the fibers. Since the G -groups ofcurves over finite fields (i.e., the fibers) are known, the statement gives adescription of the K group. This type of description is not directly relatedto the conjectures of Lichtenbaum and Beilinson.The Birch-Tate conjecture focuses on the K group of a global field inany characteristic. They study the boundary map, which they denote λ ,from the K group to the direct sum of the K group of the residue fields.A part of their conjecture is that the order of the kernel of the boundarymap λ is expressed using the special value ζ F ( −
1) and the invariant w F ,which is expressed in terms of the number of roots of unity.In our theorem, we consider the K group instead of the K group,an elliptic curve over a function field instead of the function field (i.e., aglobal field with positive characteristic), the Hasse-Weil L -function L ( E, s )instead of the zeta function of the global field. Note that the value | T ′ (1) | inour setting plays the role of w F .Next, we note that there is no counterpart for the factor ( q − in theirconjecture, because there is more than one degree for which the cohomologygroups are conjecturally nonzero in our setting, in turn because the varietyis one-dimensional as opposed to zero-dimensional; the global field itself isregarded as a zero-dimensional variety over the global field.Finally, while it may be interesting to degenerate the elliptic curve E tocompare our results with the conjecture, we have not pursued this point. In this subsection, we speculate on higher genus cases, which in turnpoints to the reason why we restrict our efforts to elliptic curves and towhere the difficulty lies in our results. More specifically, the results of Sec-tions 2, 3, 4, and 5 are valid and proved for curves of arbitrary genus.Only in Sections 6 and 7 do we use the fact that E is of genus 1, whichwe use in two ways, one being through Theorem 1.3, the other throughLemma 18. The former is more motivic, whereas the second is primarilyused in the computation of the ´etale cohomology of elliptic surfaces and isnot so motivic in nature.Note that we separately treat cases j > j SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA use Theorem 1.3. We use some consequences of Lemma 18 that expressthe cohomology of an elliptic surface in terms of the base curve, hence ourresults appear as if there are little contributions from the cohomology of E .The second author suspects that a result analogous to Lemma 18 shouldalso hold true in higher genus cases, thus we may formulate statementsthat look similar to those given in our paper.The results of Section 6, i.e., for the j Theorem 1.3 ([31, Theorem 1.1]) . —
Let the notations be as in Section 1.5.For an arbitrary set S of closed points of C , the homomorphism inducedby the boundary map ∂ , i.e., K ( E ) Q ⊕ ∂ ℘ Q −−−→ M ℘ ∈ S G ( E ℘ ) Q is surjective. Theorem 1.3 is a consequence of Parshin’s conjecture, hence a similar state-ment is expected to hold even if we replace E by a curve of higher genus,but the proof is not known. In [8], Chida et al. constructed curves otherthan elliptic curves for which surjectivity similar to that of Theorem 1.3holds true. We might be able to draw some consequences similar to thetheorems in our present paper for those curves. Finally, in this subsection, we provide a short explanation of the contentof each section. As seen above, the motivation and brief explanation re-garding the results of Section 2 and Sections 3-5 are given in Sections 1.3and 1.4, respectively. Each of these sections is fairly independent.In Section 2, we compute the motivic cohomology groups of an arbi-trary smooth surface X over finite fields. The difficult case is that of H M ( X, Z (2)), which is therefore the focus of Section 2.2. In Section 3,we prove the compatibility of Chern characters and the localization se-quence of motivic cohomology. In Section 4, we define Chern characters forsingular curves over finite fields, though the treatment is quite ad hoc. InSection 5, we then give the relation via the Chern class map between the K and K groups of curves over function fields and motivic cohomologygroups. We present our main result in Section 6. Then, applying results ofSections 2, 4, and 5, and using special features of elliptic surfaces, including ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE p -part separately in Appendix A; see its introduction for moretechnical details. In Section 7, we use the Bloch-Kato conjecture, as provedby Rost and Voevodsky (see Theorem 2.1), and generalize our results inSection 6.
2. Motivic cohomology groups of smooth surfaces
We begin this section by referring to Section 1.3, which provides a generaloverview of the contents of this section.Aside from the uniquely divisible part, we understand the motivic co-homology groups of smooth surfaces over finite fields fairly well. It followsfrom the Bass conjecture that the divisible part is zero [27, Conjecture 37,p.387].The main goal of this section is to prove Theorem 2.2, therefore let usprovide a brief description of the statement. Let X be a smooth surfaceover a finite field. We find that H i M ( X, Z ( j )) is an extension of a finitelygenerated abelian group by a uniquely divisible group except when ( i, j ) =(3 , i, j ) = (0 , , (1 , , (3 , , X . We refer to Theorem 2.2, thefollowing table, and the following paragraph for further details. Note thatfor a prime number ℓ , we let | | ℓ : Q ℓ → Q denote the ℓ -adic absolute valuenormalized such that | ℓ | ℓ = ℓ − . Let F q be a field of cardinal q of characteristic p . For a separated scheme X which is essentially of finite type over Spec F q , we define the motiviccohomology group H i M ( X, Z ( j )) as the homology group H i M ( X, Z ( j )) = H j − i ( z j ( X, • )) of Bloch’s cycle complex z j ( X, • ) [7, Introduction, p. 267](see also [17, 2.5, p. 60] to remove the condition that X is quasi-projective).When X is quasi-projective, we have H i M ( X, Z ( j )) = CH j ( X, j − i ),where the right-hand side is the higher Chow group of Bloch [7]. Wesay that a scheme X is essentially smooth over a field k if X is a lo-calization of a smooth scheme of finite type over k . When X is essentiallysmooth over Spec F q , it coincides with the motivic cohomology group de-fined in [34, Part I, Chapter I, 2.2.7, p. 21] or [66] (cf. [35, Theorem 1.2, SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA p. 300], [64, Corollary 2, p. 351]). For a discrete abelian group M , we set H i M ( X, M ( j )) = H j − i ( z j ( X, • ) ⊗ Z M ).We note here that this notation is inappropriate if X is not essentiallysmooth, for in that case it would be a Borel-Moore homology group. A rea-son for using this notation is that in Section 4.2, we define Chern classes forhigher Chow groups of low degrees as if higher Chow groups were forminga cohomology theory.First, let us recall that the groups H i M ( X, Z ( j )) have been known for j
1. By definition, H i M ( X, Z ( j )) = 0 for j i, j ) = (0 , H M ( X, Z (0)) = H ( X, Z ). We have H i M ( X, Z (1)) = 0 for i = 1 , H M ( X, Z (1)) = H ( X, G m ), and H M ( X, Z (1)) = Pic( X ).Below is a conjecture of Bloch-Kato [28, §
1, Conjecture 1, p. 608], whichhas been proved by Rost, Voevodsky, Haesemeyer, and Weibel.
Theorem 2.1 (Rost-Voevodsky; the Bloch-Kato conjecture) . —
Let j > be an integer. Then, for any finitely generated field K over F q and anypositive integer ℓ = p , the symbol map K Mj ( K ) → H j et (Spec K, Z /ℓ ( j )) issurjective. Definition 1. —
Let M be an abelian group. We say that M is finitemodulo a uniquely divisible subgroup (resp. finitely generated modulo auniquely divisible subgroup ) if M div is uniquely divisible and M red is finite(resp. M div is uniquely divisible and M red is finitely generated). We note that if M is finite modulo a uniquely divisible subgroup, then M tors is a finite group and M = M div ⊕ M tors .Recall that for a scheme X , we let Irr( X ) denote the set of irreduciblecomponents of X . The aim of Section 2.1 is to prove the below theorem. Theorem 2.2. —
Let X be a smooth surface over F q . Let R ⊂ Irr( X ) denote the subset of irreducible components of X that are projective over Spec F q . For X ′ ∈ Irr( X ) , let q X ′ denote the cardinality of the field ofconstants of X ′ . (1) H i M ( X, Z (2)) is finitely generated modulo a uniquely divisible sub-group if i = 3 or if X is projective. More precisely, (a) H i M ( X, Z (2)) is zero for i > . (b) H M ( X, Z (2)) is a finitely generated abelian group of rank | R | . (c) If i or if X is projective and i , the group H i M ( X, Z (2)) is finite modulo a uniquely divisible subgroup. (d) H M ( X, Z (2)) is finitely generated modulo a uniquely divisiblesubgroup. ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE
For i , the group H i M ( X, Z (2)) tors is canonically isomor-phic to the direct sum L ℓ = p H i − ( X, Q ℓ / Z ℓ (2)) . In particular, H i M ( X, Z (2)) is uniquely divisible for i . (f) If X is projective, then H M ( X, Z (2)) tors is isomorphic to thedirect sum of the group L ℓ = p H ( X, Q ℓ / Z ℓ (2)) and a finite p -group of order | Hom(Pic oX/ F q , G m ) | · | L ( h ( X ) , | − p . Here, welet Hom(Pic oX/ F q , G m ) denote the set of morphisms Pic oX/ F q → G m of group schemes over Spec F q . (2) Let j > be an integer. Then, for any integer i , the group H i M ( X, Z ( j )) is finite modulo a uniquely divisible subgroup. More precisely, (a) H i M ( X, Z ( j )) is zero for i > max(6 , j + 1) , is isomorphic to L X ′ ∈ R Z / ( q j − X ′ − for ( i, j ) = (5 , , (5 , , and is finite for ( i, j ) = (4 , . (b) H i M ( X, Z ( j )) tors is canonically isomorphic to the direct sum L ℓ = p H i − ( X, Q ℓ / Z ℓ ( j )) . In particular, H i M ( X, Z ( j )) is uniquelydivisible for i or i j , and H M ( X, Z ( j )) tors is iso-morphic to the direct sum L X ′ ∈ Irr( X ) Z / ( q jX ′ − . In the table below, we summarize the description of the groups H i M ( X, Z ( j ))stated in Theorem 2.2. Here, we write u.d, f./u.d., f.g./u.d., f., and f.g. foruniquely divisible, finite modulo a uniquely divisible subgroup, finite gen-erated modulo a uniquely divisible subgroup, finite, and finitely generated,respectively. j \ i < < i < j j ( = 0) j + 1 j + 2 > j + 30 0 H ( Z ) - 01 0 - H ( G m ) Pic( X ) 02 u. d. f./u. d. f. g./u. d. ? f. g. 0f./u. d. if projective3 u. d. f./u. d. f. 04 u. d. f./u. d. f. 0 > i j Lemma 1. —
Let X be a separated scheme essentially of finite type over Spec F q . Let i and j be integers. If both H i − M ( X, Q / Z ( j )) and lim ←− m H i M ( X, Z /m ( j )) are finite, then H i M ( X, Z ( j )) is finite modulo a uniquely divisible subgroupand its torsion subgroup is isomorphic to H i − M ( X, Q / Z ( j )) . SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA
Proof. —
Let us consider the exact sequence(2.1) 0 → H i − M ( X, Z ( j )) ⊗ Z Q / Z → H i − M ( X, Q / Z ( j )) → H i M ( X, Z ( j )) tors → . Since H i − M ( X, Q / Z ( j )) is a finite group, all groups in the above exactsequence are finite groups. Then, H i − M ( X, Z ( j )) ⊗ Z Q / Z must be zero,since it is finite and divisible, hence we have a canonical isomorphism H i − M ( X, Q / Z ( j )) → H i M ( X, Z ( j )) tors . The finiteness of H i M ( X, Z ( j )) tors implies that the divisible group H i M ( X, Z ( j )) div is uniquely divisible andthe canonical homomorphism H i M ( X, Z ( j )) red → lim ←− m H i M ( X, Z ( j )) /m is injective. The latter group lim ←− m H i M ( X, Z ( j )) /m is canonically embed-ded in the finite group lim ←− m H i M ( X, Z /m ( j )), hence we conclude that H i M ( X, Z ( j )) red is finite. This proves the claim. (cid:3) Lemma 2. —
Let X be a smooth projective surface over F q . Let j bean integer. Then, H i M ( X, Q / Z ( j )) and lim ←− m H i M ( X, Z /m ( j )) are finite if i = 2 j or j > .Proof. — The claim for j j = 2. Then,the claim for i > p ∤ m , from the theorem of Geisser and Levine [17, Corollary1.2, p. 56] (see also [Corollary 1.4]) and the theorem of Merkurjev andSuslin [40, (11.5), Theorem, p. 328], it follows that the cycle class map H i M ( X, Z /m (2)) → H i et ( X, Z /m (2)) is an isomorphism for i i = 3. By [10, Th´eor`eme 2, p. 780] and the exact sequence[10, 2.1 (29) p. 781], for i
3, the group lim −→ m, p ∤ m H i et ( X, Z /m (2)) and thegroup lim ←− m, p ∤ m H i et ( X, Z /m (2)) are finite.Let W n Ω • X, log denote the logarithmic de Rham-Witt sheaf (cf. [26, I, 5.7,p. 596]), which was introduced by Milne in [42]. There is an isomorphism H i M ( X, Z /p n (2)) ∼ = H i − ( X, W n Ω X, log ) (cf. [16, Theorem 8.4, p. 491]). Inparticular, we have H i M ( X, Q p / Z p (2)) = 0 for i §
2, Th´eor`eme 3, p. 782], lim −→ n H i et ( X, W n Ω X, log ) is a finite groupfor i = 0 ,
1. By [10, Pas n o
1, p. 783], the projective system { H i et ( X, W n Ω X, log ) } n satisfies the Mittag-Leffler condition. Using the same argument used in [10,Pas n o
4, p. 784], we obtain an exact sequence0 → lim ←− n H i et ( X, W n Ω X, log ) ⊗ Z Q p / Z p → lim −→ n H i et ( X, W n Ω X, log ) → lim ←− n H i +1et ( X, W n Ω X, log ) tors → . ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE ←− n H i et ( X, W n Ω X, log ) is also finite for i = 0 , −→ n H i − ( X, W n Ω X, log ). Since the homomorphism H i Zar ( X, W n Ω X, log ) → H i et ( X, W n Ω X, log )induced by the change of topology ε : X et → X Zar , is an isomorphism for i = 0 and is injective for i = 1, we observe that lim ←− n H M ( X, Z /p n (2)) iszero and that both H M ( X, Q p / Z p (2)) and lim ←− n H M ( X, Z /p n (2)) are finitegroups. This proves the claim for j = 2.Suppose j >
3. When i > j , since X is a surface, the cycle com-plex defining the higher Chow groups are zero in the negative degrees,hence the claim holds true. Since j >
3, we have H i M ( X, Z /p n ( j )) ∼ = H i − j Zar ( X, W n Ω jX, log ) = 0. Using the theorem of Rost and Voevodsky (The-orem 2.1), by the theorem of Geisser-Levine [17, Theorem 1.1, p. 56], thegroup H i M ( X, Z /m ( j )) is isomorphic to H i Zar ( X, τ j Rε ∗ Z /m ( j )) if p ∤ m .Since any affine surface over F q has ℓ -cohomological dimension three forany ℓ = p , it follows that H i Zar ( X, τ j Rε ∗ Z /m ( j )) ∼ = H i et ( X, Z /m ( j )) forall i . Hence, by [10, Th´eor`eme 2, p. 780] and the exact sequence [10, 2.1 (29)p. 781], for i j −
1, the groups H i M ( X, Q / Z ( j )) and lim ←− m H i M ( X, Z /m ( j ))are finite. This proves the claim for j > (cid:3) Lemma 3. —
Let Y be a scheme of dimension d of finite typeover Spec F q . Then H i M ( Y, Z ( j )) is a torsion group unless j d and j i j .Proof. — By taking a smooth affine open subscheme of Y red whose com-plement is of dimension zero, and using the localization sequence of motiviccohomology, we are reduced to the case in which Y is connected, affine,and smooth over Spec F q . When d = 0 (resp. d = 1), the claim followsfrom the results of Quillen [55, Theorem 8(i), p. 583] (resp. Harder [24,Korollar 3.2.3, p. 175] (see [19, Theorem 0.5, p. 70] for the correct inter-pretation of his results)) on the structure of the K -groups of Y , combinedwith the Riemann-Roch theorem for higher Chow groups [7, Theorem 9.1,p. 296]. (cid:3) Lemma 4. —
Let ϕ : M → M ′ be a homomorphism of abelian groupssuch that Ker ϕ is finite and (Coker ϕ ) div = 0 . If M div or M ′ div is uniquelydivisible, then ϕ induces an isomorphism ϕ div : M div ∼ = −→ M ′ div .Proof. — First, we show that ϕ div is surjective. Since (Coker ϕ ) div = 0,for any a ∈ M ′ div , we have ϕ ( a ) − = ∅ . Let N be a positive integer such that N (Ker ϕ ) = 0. Set M = ϕ − ( M ′ div ) and M = N M . Since ϕ induces asurjection M → M ′ div , it suffices to show that M is divisible. Let x ∈ M SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA and n ∈ Z > . Take y ∈ M such that x = N y ; z ∈ M ′ div such that nz = ϕ ( y ); y ′ ∈ M such that ϕ ( y ′ ) = z . Set x ′ = N y ′ ∈ M . Then ϕ ( y − ny ′ ) = 0 implies x − nx ′ = N y − nN y ′ = 0. This shows that x isdivisible, hence proving the surjectivity.Next, let us prove the injectivity. Suppose M div is uniquely divisible.Then, since Ker ϕ is torsion, Ker ϕ ∩ M div = 0, and ϕ div is injective. Suppose M ′ div is uniquely divisible. Then, the torsion subgroup of M div is containedin Ker ϕ . Since a nonzero torsion divisible group is infinite, M div mustbe uniquely divisible, hence we are reduced to the case above and ϕ div isinjective. (cid:3) Proof of Theorem 2.2. —
Without loss of generality, we may assumethat X is connected. We first prove the claims assuming X is projective.It is clear that the group H i M ( X, Z ( j )) is zero for i > min( j + 3 , j + 1). Itfollows from [10, p. 787, Proposition 4] that the degree map H M ( X, Z (2)) =CH ( X ) → Z has finite kernel and cokernel, proving the claim for i > min( j + 3 , j ). Next, we fix j >
2. For i j −
1, the group H i M ( X, Z ( j ))is finite modulo a uniquely divisible subgroup by Lemmas 1 and 2. Theclaim on the identification of H i M ( X, Z ( j )) tors with the ´etale cohomologyfollows immediately from the argument in the proof of Lemma 2 except forthe p -primary part of H M ( X, Z (2)), which follows from Proposition A.1.To finish the proof, it remains to prove that H i M ( X, Z ( j )) div is zero for j > i = j + 1 , j + 2. It suffices to prove that H i M ( X, Z ( j )) is a torsiongroup for j > i > j + 1. Consider the limitlim −→ Y H i − M ( Y, Z ( j − → H i M ( X, Z ( j )) → lim −→ Y H i M ( X \ Y, Z ( j ))of the localization sequence in which Y runs over the reduced closed sub-schemes of X of pure codimension one. H i − M ( Y, Z ( j − −→ Y H i − M ( X \ Y, Z ( j − X is projective.For a general connected surface X , take an embedding X ֒ → X ′ of X into a smooth projective surface X ′ over F q such that Y = X ′ \ X is of purecodimension one in X ′ . We can show that such an X ′ exists via [49] and aresolution of singularities [3, p. 111], [38, p. 151]. Then, the claims, exceptfor that of the identification of H i M ( X, Z ( j )) tors with the ´etale cohomology,easily follow from Lemma 4 and the localization sequence · · · → H i − M ( Y, Z ( j − → H i M ( X ′ , Z ( j )) → H i M ( X, Z ( j )) → · · · . ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE H i M ( X, Z ( j )) tors with the ´etale cohomol-ogy can be obtained using a similar approach to that used in the proof ofLemma 2. This completes the proof. (cid:3) H M ( X, Z (2)) tors Proposition 2.3. —
Let X be a smooth surface over F q . Let X ֒ → X ′ be an open immersion such that X ′ is smooth projective over F q and Y = X ′ \ X is of pure codimension one in X ′ . Then, the following conditionsare equivalent. (1) H M ( X, Z (2)) is finitely generated modulo a uniquely divisible sub-group. (2) H M ( X, Z (2)) tors is finite. (3) The pull-back map H M ( X ′ , Z (2)) → H M ( X, Z (2)) induces an iso-morphism H M ( X ′ , Z (2)) div ∼ = −→ H M ( X, Z (2)) div . (4) The kernel of the pull-back map H M ( X ′ , Z (2)) → H M ( X, Z (2)) isfinite. (5) The cokernel of the boundary map ∂ : H M ( X, Z (2)) → H M ( Y, Z (1)) is finite.Further, if the above equivalent conditions are satisfied, then the torsiongroup H M ( X, Z (2)) tors is isomorphic to the direct sum of a finite group of p -power order and the group L ℓ = p H ( X, Q ℓ / Z ℓ (2)) red , and the localizationsequence induces a long exact sequence (2.2) · · · → H i − M ( Y, Z (1)) → H i M ( X ′ , Z (2)) red → H i M ( X, Z (2)) red → · · · of finitely generated abelian groups.Proof. — Condition (1) clearly implies condition (2). The localization se-quence shows that conditions (4) and (5) are equivalent and that condition(3) implies condition (1). From the localization sequence and Lemma 4,condition (4) implies condition (3).We claim that condition (2) implies condition (4). Assume condition (2)and suppose that condition (4) is not satisfied. We set M = Ker[ H M ( X ′ , Z (2)) → H M ( X, Z (2))]. The localization sequence shows that M is finitely gener-ated. By assumption, M is not torsion. Since H M ( X ′ , Z (2)) is finite moduloa uniquely divisible subgroup, the intersection H M ( X ′ , Z (2)) div ∩ M is anontrivial free abelian group of finite rank, hence H M ( X, Z (2)) contains agroup isomorphic to H M ( X ′ , Z (2)) div / ( H M ( X ′ , Z (2)) div ∩ M ), which con-tradicts condition (2), hence condition (2) implies condition (4). This com-pletes the proof of the equivalence of conditions (1)-(5). SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA
Suppose that conditions (1)-(5) are satisfied. The localization sequenceshows that the kernel (resp. the cokernel) of the pull-back H i M ( X ′ , Z (2)) → H i M ( X, Z (2)) is a torsion group (resp. has no nontrivial divisible subgroup)for any i ∈ Z , hence by Lemma 4, H i M ( X, Z (2)) div is uniquely divisible andsequence (2.2) is exact. Condition (2) and exact sequence (2.1) for ( i, j ) =(3 ,
2) yield an isomorphism H M ( X, Z (2)) tors ∼ = H M ( X, Q / Z (2)) red . Then,the claim on the structure of H M ( X, Z (2)) tors follows from the theorem ofGeisser and Levine [17, Corollary 1.2, p. 56. See also Corollary 1.4] andthe theorem of Merkurjev and Suslin [40, (11.5), Theorem, p. 328]. Thiscompletes the proof. (cid:3) Let X be a smooth projective surface over F q . Suppose that X admits aflat, surjective, and generically smooth morphism f : X → C to a connectedsmooth projective curve C over F q . For each point ℘ ∈ C , let X ℘ = X × C ℘ denote the fiber of f at ℘ . Corollary 1. —
Let the notations be as above. Let η ∈ C denotethe generic point. Suppose that the cokernel of the homomorphism ∂ : H M ( X η , Z (2)) → L ℘ ∈ C H M ( X ℘ , Z (1)) , which is the inductive limit of theboundary maps of the localization sequences, is a torsion group. Then, thegroup H i M ( X η , Z (2)) div is uniquely divisible for all i ∈ Z and the inductivelimit of localization sequences induces a long exact sequence · · · → M ℘ ∈ C H i − M ( X ℘ , Z (1)) → H i M ( X, Z (2)) red → H i M ( X η , Z (2)) red → · · · . Proof. —
Since L ℘ ∈ C H i M ( X ℘ , Z (1)) has no nontrivial divisible sub-group for all i ∈ Z and is torsion for i = 1 by Lemma 3, the claim followsfrom Lemma 4. (cid:3)
3. The compatibility of Chern characters and thelocalization sequence
We first refer the reader to the first half of Section 1.4 for a generaloverview of the contents of this section.The aim of this section is to prove Lemma 5. Only Lemma 5 and Re-mark 1 will be used beyond this section.In this paper, for the Chern class, we use the Chern class for motivic coho-mology of Levine (see below) combined with the comparison isomorphismbetween Levine’s motivic cohomology groups and higher Chow groups. Wealso use the Riemann-Roch theorem for such Chern classes by checking thecompatibility of the localization sequences with the comparison isomor-phisms (i.e., Lemmas 6 and 7).
ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE Given an essentially smooth scheme X (see Section 2.1 for the def-inition of “essentially smooth”) over Spec F q and integers i, j >
0, let c i,j : K i ( X ) → H j − i M ( X, Z ( j )) be the Chern class map. Several approachesto constructing the map c i,j have been proposed, including [7, p. 293],[34, Part I, Chapter III, 1.4.8. Examples. (i), p. 123], [54, Definition 5,p. 315]; all of these approaches are based on Gillet’s work [18, p. 228–229, Definition 2.22]. In this paper, we adopt the definition of Levine [34,Part I, Chapter III, 1.4.8. Examples. (i), p. 123] in which c i,j is denotedby c j, j − iX , given in [34, Part I, Chapter I, 2.2.7, p. 21]. The definition ofthe target H j − i ( X, Z ( j )) = H j − iX ( X, Z ( j )) of c j, j − iX , which we denote by H j − i L ( X, Z ( j )), is different from the definition of the group H j − i M ( X, Z ( j ));however, by combining (ii) and (iii) of [34, Part I, Chapter II, 3.6.6. Theo-rem, p. 105], we obtain a canonical isomorphism(3.1) β ij : H i M ( X, Z ( j )) ∼ = −→ H i L ( X, Z ( j )) , which is compatible with the product structures. The precise definition ofour Chern class map c i,j is the composition c i,j = ( β ij ) − ◦ c j, j − iX .The map c i,j is a group homomorphism if i > i, j ) = (0 , , : K ( X ) → H M ( X, Z (0)) ∼ = H ( X, Z ) denote the homomorphismthat sends the class of locally free O X -module F to the rank of F . For i > a ∈ K i ( X ), we put formally ch i, ( a ) = 0. Lemma 5. —
Let X be a scheme which is a localization of a smoothquasi-projective scheme over Spec F q . Let Y ⊂ X be a closed subschemeof pure codimension d , which is essentially smooth over Spec F q . Then for i, j > or ( i, j ) = (0 , , the diagram (3.2) K i ( Y ) α i,j −−−−→ H j − i − d M ( Y, Z ( j − d )) y y K i ( X ) c i,j −−−−→ H j − i M ( X, Z ( j )) y y K i ( X \ Y ) c i,j −−−−→ H j − i M ( X \ Y, Z ( j )) y y K i − ( Y ) α i − ,j −−−−→ H j − i − d +1 M ( Y, Z ( j − d )) SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA is commutative. Here, the homomorphism α i,j is defined as follows: for a ∈ K i ( Y ) , the element α i,j ( a ) equals G d,j − d (ch i, ( a ) , c i, ( a ) , . . . , c i,j − d ( a ); c , ( N ) , . . . , c ,j − d ( N )) , where G d,j − d is the universal polynomial in [2, Expos´e 0, Appendice, Propo-sition 1.5, p. 37] , N is the conormal sheaf of Y in X , and the left (resp.the right) vertical sequence is the localization sequence of K -theory (resp.of higher Chow groups established in [6, Corollary (0.2), p. 357] ).Proof. — We may assume that X is quasi-projective and smooth overSpec F q . It follows from [34, Part I, Chapter III, 1.5.2, p. 130] and theRiemann-Roch theorem without denominators [34, Part I, Chapter III,3.4.7. Theorem, p. 174] that diagram (3.2) is commutative if we replace theright vertical sequence by Gysin sequence(3.3) H j − i − d L ( Y, Z ( j − d )) → H j − i L ( X, Z ( j )) → H j − i L ( X \ Y, Z ( j )) → H j − i − d +1 L ( Y, Z ( j − d ))in [34, Part I, Chapter III, 2.1, p. 132]. It suffices to show that Gysin se-quence (3.3) is identified with the localization sequence of higher Chowgroups. Here, we use the notations from [34, Part I, Chapter I, II]. Let S = Spec F q and V denote the category of schemes that is essentiallysmooth over Spec F q . Let A mot ( V ) be the DG category defined in [34, PartI, Chapter I, 1.4.10 Definition, p. 15]. For an object Z in V and a morphism f : Z ′ → Z in V that admits a smooth section, and for j ∈ Z , we have theobject Z Z ( j ) f in A mot ( V ). When f = id Z is the identity, we abbreviate Z Z ( j ) id Z by Z Z ( j ). For a closed subset W ⊂ Z , let Z Z,W ( j ) be the objectintroduced in [34, Part I, Chapter I, (2.1.3.1), p. 17]; this is an object inthe DG category C b mot ( V ) of bounded complexes in A mot ( V ). The object Z Z ( j ) f belongs to the full subcategory A mot ( V ) ∗ of A mot ( V ) introduced in[34, Part I, Chapter I, 3.1.5, p. 38], and the object Z Z,W ( j ) belongs to theDG category C b mot ( V ) ∗ of bounded complexes in A mot ( V ) ∗ . For i ∈ Z , weset H i L ,W ( Z, Z ( j )) = Hom D b mot ( V ) (1 , Z Z,W ( j )[ i ]) where 1 denotes the ob-ject Z Spec F q (0) and D b mot ( V ) denotes the category introduced in [34, PartI, Chapter I, 2.1.4 Definition, p. 17–18].Let X , Y be as in the statement of Lemma 5. Let K b mot ( V ) be the ho-motopy category of C b mot ( V ). Then, we have a distinguished triangle Z X,Y ( j ) → Z X ( j ) → Z X \ Y ( j ) +1 −−→ ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE K b mot ( V ). This distinguished triangle yields a long exact sequence(3.4) · · · → H i L ,Y ( X, Z ( j )) → H i L ( X, Z ( j )) → H i L ( X \ Y, Z ( j )) → H i +1 L ,Y ( X, Z ( j )) → · · · . In [34, Part I, Chapter III, (2.1.2.2), p. 132], Levine constructs an iso-morphism ι ∗ : Z Y ( j − d )[ − d ] → Z X,Y ( j ) in D b mot ( V ). This isomorphisminduces an isomorphism ι ∗ : H i − d L ( Y, Z ( j − d )) ∼ = −→ H i L ,Y ( X, Z ( j )). This lat-ter isomorphism, together with long exact sequence (3.4) yields the Gysinsequence (3.3).We set z jY ( X, −• ) = Cone( z j ( X, −• ) → z j ( X \ Y, −• ))[ −
1] and definecohomology with support H i M ,Y ( X, Z ( j )) = H i − j ( z jY ( X, −• )). The dis-tinguished triangle z jY ( X, −• ) → z j ( X, −• ) → z j ( X \ Y, −• ) +1 −−→ in the derived category of abelian groups induces a long exact sequence(3.5) · · · → H i M ,Y ( X, Z ( j )) → H i M ( X, Z ( j )) → H i M ( X \ Y, Z ( j )) → H i +1 M ,Y ( X, Z ( j )) → · · · . The push-forward map z j − d ( Y, −• ) → z j ( X, −• ) of cycles gives a homo-morphism z j − d ( Y, −• ) → z jY ( X, −• ) of complexes of abelian groups, whichis known to be a quasi-isomorphism by [6, Theorem (0.1), p. 537], henceit induces an isomorphism ι ∗ : H j − i − d M ( Y, Z ( j − d )) ∼ = −→ H j − i M ,Y ( X, Z ( j )).Then, the claim follows from Lemmas 6 and 7 below. (cid:3) Lemma 6. —
Let X and Y be as in Lemma 5. For each i, j ∈ Z , thereexists a canonical isomorphism β iY,j : H i M ,Y ( X, Z ( j )) ∼ = −→ H i L ,Y ( X, Z ( j )) such that the long exact sequence (3.4) is identified with the long exactsequence (3.5) via this isomorphism and the isomorphism (3.1).Proof. — First, let us recall the relation between the group H i M ( X, Z ( j )) ∼ = −→ β ij H i L ( X, Z ( j ))and the naive higher Chow group introduced by Levine [34, 2.3.1. Defi-nition, p.70]. For an object Γ of C b mot ( V ), the naive higher Chow groupCH naif (Γ , p ) is by definition the cohomology group H − p ( Z mot (Γ , ∗ )). Here, SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA Z mot ( , ∗ ) is as in [34, Part I, Chapter II, 2.2.4. Definition, p. 68], which isa DG functor from the category C b mot ( V ) ∗ to the category of complexes ofabelian groups bounded from below. Since X is a localization of a smoothquasi-projective scheme over k , it follows from [34, 2.4.1., p.71] that wehave a natural isomorphism CH naif ( Z X ( j )[2 j ] , j − i ) ∼ = H i L ( X, Z ( j )).Observe that the functor Z mot [34, Part I, Chapter I, (3.3.1.2), p. 40]from the category C b mot ( V ) to the category of bounded complexes of abeliangroups is compatible with taking cones, hence the DG functor Z mot ( , ∗ )is also compatible with taking cones. Since Z mot ( Z X ( j ) id X , ∗ ) is canonicallyisomorphic to the cycle complex z j ( X, −• ), the complex Z mot ( Z X,Y ( j ) id X , ∗ )is canonically isomorphic to z jY ( X, −• ). For an object Γ in C b mot ( V ) ∗ , let CH (Γ , p ) be the higher Chow group defined in [34, Part I, Chapter II,2.5.2. Definition, p. 76]. From the definition of CH (Γ , p ), we obtain canoni-cal homomorphisms H j − i M ( X, Z ( j )) → CH ( Z X ( j ) , i ), H j − i M ( X \ Y, Z ( j )) →CH ( Z X \ Y ( j ) , i ), and H j − i M ,Y ( X, Z ( j )) → CH ( Z X,Y ( j ) , i ) such that the dia-gram(3.6) H j − i M ,Y ( X, Z ( j )) −−−−→ CH ( Z X,Y ( j ) , i ) y y H j − i M ( X, Z ( j )) −−−−→ CH ( Z X ( j ) , i ) y y H j − i M ( X \ Y, Z ( j )) −−−−→ CH ( Z X \ Y ( j ) , i ) y y H j − i +1 M ,Y ( X, Z ( j )) −−−−→ CH ( Z X,Y ( j ) , i − CH (Γ) = CH (Γ , → Hom D b mot ( V ) (1 , Γ) [34, p. 76] for an object Γ ∈ C b mot ( V ) ∗ . Also recall that CH (Γ) = lim −→ Γ → Γ e U H ( Z mot (Tot Γ e U , ∗ )) where Γ → Γ e U runs over the hyper-resolutions of Γ [34, Part I, Chapter II, 1.4.1. Definition, p. 59] and Tot : C b ( C b mot ( V ) ∗ ) → C b mot ( V ) ∗ denotes the total complex functor in [34, PartI, Chapter II, 1.3.2, p. 58]. The homomorphism cl(Γ) is defined as theinductive limit of the compositioncl naif (Tot Γ e U ) : H ( Z mot (Tot Γ e U , ∗ )) → Hom D b mot ( V ) (1 , Tot Γ e U ) ∼ = ←− Hom D b mot ( V ) (1 , Γ) . ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE C b mot ( V ) ∗ , the homomorphism cl naif (Γ) is, by definition[34, Part I, Chapter II, (2.3.6.1), p. 71], equal to the composition H ( Z mot (Γ , ∗ )) ∼ = ←− H ( Z mot (Σ N (Γ)[ N ])) ∼ = ←− Hom K b mot ( V ) ( e ⊗ a ⊗ , Σ N (Γ)[ N ]) → Hom D b mot ( V ) ( e ⊗ a ⊗ , Σ N (Γ)[ N ]) ∼ = ←− Hom D b mot ( V ) (1 , Γ)for sufficiently large integers
N, a >
0. Here, Σ N is the suspension functorin [34, Part I, Chapter II, 2.2.2. Definition, p. 68], and e is the object in[34, Part I, Chapter I, 1.4.5, p. 13] that we regard as an object in A mot ( V ).Let Γ → Γ ′ be a morphism in C b mot ( V ) and set Γ ′′ = Cone(Γ → Γ ′ )[ − N is compatible with taking cones, the diagram(3.7) CH (Γ ′′ , i ) cl(Γ ′′ [ − i ]) −−−−−−→ Hom D b mot ( V ) (1 , Γ ′′ [ − i ]) y y CH (Γ , i ) cl(Γ[ − i ]) −−−−−→ Hom D b mot ( V ) (1 , Γ[ − i ]) y y CH (Γ ′ , i ) cl(Γ ′ [ − i ]) −−−−−−→ Hom D b mot ( V ) (1 , Γ ′ [ − i ]) y y CH (Γ ′′ , i − cl(Γ ′′ [ − i +1]) −−−−−−−−→ Hom D b mot ( V ) (1 , Γ ′′ [ − i + 1])is commutative.The homomorphism β ij : H i M ( X, Z ( j )) → H j − i L ( X, Z ( j )) is, by defini-tion, equal to the composition H i M ( X, Z ( j )) → CH ( Z X ( j ) , j − i ) cl( Z X ( j )[ i − j ]) −−−−−−−−−−→ Hom D b mot ( V ) (1 , Z X ( j )[ i ]) = H i L ( X, Z ( j )) . We define β iY,j : H i M ,Y ( X, Z ( j )) → H j − i L ,Y ( X, Z ( j )) to be the composition H i M ,Y ( X, Z ( j )) → CH ( Z X,Y ( j ) , j − i ) cl( Z X,Y ( j )[ i − j ]) −−−−−−−−−−−→ Hom D b mot ( V ) (1 , Z X,Y ( j )[ i ]) = H i L ,Y ( X, Z ( j )) . SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA
By (3.6) and (3.7), we have a commutative diagram H j − i M ,Y ( X, Z ( j )) β j − iY,j −−−−→ H j − i L ,Y ( X, Z ( j )) y y H j − i M ( X, Z ( j )) β j − ij −−−−→ ∼ = H j − i L ( X, Z ( j )) y y H j − i M ( X \ Y, Z ( j )) β j − ij −−−−→ ∼ = H j − i L ( X \ Y, Z ( j )) y y H j − i +1 M ,Y ( X, Z ( j )) β j − i +1 j −−−−−→ H j − i +1 L ,Y ( X, Z ( j )) , where the right vertical arrow is the long exact sequence (3.4), hence β j − iY,j is an isomorphism and the claim follows. (cid:3) Lemma 7. —
The diagram H j − i − d M ( Y, Z ( j − d )) ι ∗ −−−−→ ∼ = H j − i M .Y ( X, Z ( j )) β j − i − dj − d y ∼ = β j − iY,j y ∼ = H j − i − d L ( Y, Z ( j − d )) ι ∗ −−−−→ ∼ = H j − i L ,Y ( X, Z ( j )) is commutative.Proof. — Recall the construction of the upper horizontal isomorphism ι ∗ in [34, Part I, Chapter III, (2.1.2.2), p. 132]. Let Z be the blow-upof X × Spec F q A F q along Y × Spec F q { } . Let W be the proper transform of Y × Spec F q A F q to Z . Then, W is canonically isomorphic to Y × Spec F q A F q . Let P be the inverse image of Y × Spec F q { } under the map Z → X × Spec F q A F q and let Q = P × Z W . We set Z ′ = Z ∐ ( X × Spec F q { } ) ∐ P and let f : Z ′ → Z denote the canonical morphism. We then have canonical morphisms Z P,Q ( j ) ← Z Z,W ( j ) f → Z X × Spec F q { } ,Y × Spec F q { } ( j ) = Z X,Y ( j )in C b mot ( V ) ∗ , which become isomorphisms in the category D b mot ( V ).Let g : P → Y × Spec F q { } ∼ = Y be the canonical morphism. The restric-tion of g to Q ⊂ P is an isomorphism, hence giving a section s : Y → P to g . ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE dP,Q ( Q ) ∈ H dQ ( P, Z ( d )) in [34, Part I, Chapter I, (3.5.2.7),p. 48] comes from the map [ Q ] Q : e ⊗ → Z P,Q ( d )[2 d ] in C b mot ( V ), asdefined in [34, Part I, Chapter I, (2.1.3.3), p. 17]. We then have morphisms(3.8) e ⊗ Z P ( j − d )[ − d ] → Z P,Q ( d ) ⊗ Z P ( j − d ) γ −→ Z P × Spec F q P,Q × Spec F q P ( j ) ← Z P × Spec F q P,Q × Spec F q P ( j ) f ′ ∆ ∗ P −−→ Z P,Q ( j )in C b mot ( V ). Here, γ is the map induced from the external products, i.e., ⊠ P,P : Z P ( d ) ⊗ Z P ( j − d ) → Z P × Spec F q P ( j ) and ⊠ Q,P : Z Q ( d ) ⊗ Z P ( j − d ) → Z Q × Spec F q P ( j ); further, ∆ P : P → P × Spec F q P denotes the diagonalembedding and f ′ is the morphism f ′ = id P × Spec F q P ∐ ∆ P : P × Spec F q P ∐ P → P × Spec F q P. The morphisms in (3.8) above induce morphism δ : Z P ( j − d )[ − d ] → Z P,Q ( j ) in D b mot ( V ). The composite morphism Z Y ( j − d )[ − d ] q ∗ −→ Z P ( j − d )[ − d ] δ −→ Z P,Q ( j ) induces a homomorphism δ ∗ : H i − d L ( Y, Z ( j − d )) → H i L ,Q ( P, Z ( j )) for each i ∈ Z .From the construction of δ ∗ , we observe that the diagram H i − d M ( Y, Z ( j − d )) s ∗ −−−−→ H i M ,Q ( P, Z ( j )) β i − dj − d y β iQ,j y H i − d L ( Y, Z ( j − d )) δ ∗ −−−−→ H i L ,Q ( P, Z ( j ))is commutative. Here, the upper horizontal arrow s ∗ is the homomor-phism that sends the class of a cycle V ∈ z j − d ( Y, j − i ) to the class in H i M ,Q ( P, Z ( j )) of the cycle s ( V ), which belongs to the kernel of z j ( P, j − i ) → z j ( P \ Q, j − i ). Next, the isomorphism ι ∗ : H j − i − d L ( Y, Z ( j − d )) ∼ = −→ H j − i L ,Y ( X, Z ( j )) equals the composition H j − i − d L ( Y, Z ( j − d )) δ ∗ −→ H j − i L ,Q ( P, Z ( j )) ← H j − i L ,W ( Z, Z ( j )) → H j − i L ,Y × Spec F q { } ( X × Spec F q { } , Z ( j )) = H j − i L ,Y ( X, Z ( j )) . We can easily verify that the isomorphism ι ∗ : H j − i − d M ( Y, Z ( j − d )) ∼ = −→ H j − i M ,Y ( X, Z ( j )) equals the composition H j − i − d M ( Y, Z ( j − d )) s ∗ −→ H j − i M ,Q ( P, Z ( j )) ← H j − i M ,W ( Z, Z ( j )) → H j − i M ,Y × Spec F q { } ( X × Spec F q { } , Z ( j )) = H j − i M ,Y ( X, Z ( j )) . Given this, the claim follows. (cid:3)
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Remark 1. —
For j = d , we have α i,d = ( − d − ( d − · ch i, . For i > and j = d + 1 , we have α i,d +1 = ( − d d ! · c i, .Suppose that d = 1 and N ∼ = O Y . Then, we have α i, = ch i, and α i,j ( a ) = ( − j − Q j − ( c i, ( a ) , . . . , c i,j − ( a )) for i > , j > , where Q j − denotes the ( j − -st Newton polynomial, which expresses the ( j − -stpower sum polynomial in terms of the elementary symmetric polynomials.In particular, we have α i, = − c i, for i > and α i,j = − ( j − c i,j − for i > , j > .
4. Motivic Chern characters for singular curves over finitefields
Before embarking on this section, we refer to the latter half of Section 1.4for a general overview. Note that the output of this section consists ofLemma 8 and Proposition 4.1.In this section, we construct Chern characters of low degrees for singu-lar curves over finite fields with values in the higher Chow groups in anad hoc manner. Bloch defines Chern characters with values in the higherChow groups tensored with Q in [7, (7.4), p. 294]. We restrict ourselves toone-dimensional varieties over finite fields, but the target group lies withcoefficients in Z . Below, we first state a lemma to be used in this section and later inLemma 13. For a scheme X , we let O ( X ) = H ( X, O X ) denote the coor-dinate ring of X . Lemma 8. —
Let X be a connected scheme of pure dimension one, sep-arated and of finite type over Spec F q . Then, the push-forward map α X : H M ( X, Z (2)) → H M (Spec O ( X ) , Z (1)) is an isomorphism if X is proper, and H M ( X, Z (2)) is zero if X is notproper.Proof. — This follows from Theorem 1.1 of [33]. (cid:3)
ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE Let Z be a scheme over Spec F q of pure dimension one, separated and offinite type over Spec F q . We construct a canonical homomorphism ch ′ i,j : G i ( Z ) → H j − i M ( Z, Z ( j )) for ( i, j ) = (0 , , , , ′ i,i , ch ′ i,i +1 ) : G i ( Z ) → H i M ( Z, Z ( i )) ⊕ H i +2 M ( Z, Z ( i + 1))is an isomorphism for i = 0 ,
1. Since the G -theory of Z and the G -theoryof Z red are isomorphic, and the same holds for the motivic cohomology, itsuffices to treat the case where Z is reduced.Consider a dense affine open smooth subscheme Z (0) ⊂ Z , and let Z (1) = Z \ Z (0) be the complement of Z (0) with the reduced scheme structure. Wedefine ch ′ , to be the composition G ( Z ) → K ( Z (0) ) ch , −−−→ H M ( Z (0) , Z (0)) ∼ = H M ( Z, Z (0)) . We then use the following lemma.
Lemma 9. —
For i = 0 (resp. i = 1 ), the diagram K i +1 ( Z (0) ) −−−−→ K i ( Z (1) ) c i +1 ,i +1 y y ch , ( resp. c , ) H i +1 M ( Z (0) , Z ( i + 1)) −−−−→ H i M ( Z (1) , Z ( i )) , where each horizontal arrow is a part of the localization sequence, is com-mutative.Proof. — Let e Z denote the normalization of Z . We write e Z (0) = Z (0) × Z e Z ( ∼ = Z (0) ) and e Z (1) = ( Z (1) × Z e Z ) red . Comparing diagrams K i +1 ( e Z (0) ) → K i ( e Z (1) ) ↓ ↓ K i +1 ( Z (0) ) → K i ( Z (1) ) and H i +1 M ( e Z (0) , Z ( i + 1)) → H i M ( e Z (1) , Z ( i )) ↓ ↓ H i +1 M ( Z (0) , Z ( i + 1)) → H i M ( Z (1) , Z ( i ))reduces us to proving the same claim for e Z (0) and e Z (1) . This then followsfrom Lemma 5. (cid:3) We define ch ′ , to be the opposite of the composition G ( Z ) → Ker[ K ( Z (0) ) → K ( Z (1) )] c , −−→ Ker[ H M ( Z (0) , Z (1)) → H M ( Z (1) , Z (0))] ∼ = H M ( Z, Z (1)) . SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA
Next, we define ch ′ , when Z is connected. If Z is not proper, then H M ( Z, Z (2)) is zero by Lemma 8. We set ch ′ , = 0 in this case. If Z isproper, then the push-forward map H M ( Z, Z (2)) → H M (Spec H ( Z, O Z ) , Z (1)) ∼ = K (Spec H ( Z, O Z ))is an isomorphism by Lemma 8. We define ch ′ , to be ( − G ( Z ) → K (Spec H ( Z, O Z )) ∼ = H M ( Z, Z (2)) . Next, we define ch ′ , for non-connected Z to be the direct sum of ch ′ , foreach connected component of Z .Observe that the group G ( Z ) is generated by the two subgroups M = Im[ K ( Z (1) ) → G ( Z )] and M = Im[ K ( e Z ) → G ( Z )] . Using Lemma 9 and the localization sequences, the isomorphism ch , : K ( Z (1) ) ∼ = −→ H M ( Z (1) , Z (0)) induces a homomorphism ch ′ , : M → H M ( Z, Z (1)).The kernel of K ( e Z ) → G ( Z ) is contained in the image of K ( e Z (1) ) → K ( e Z ). It can then be easily verified that the composition K ( e Z (1) ) → K ( e Z ) c , −−→ H M ( e Z, Z (1)) → H M ( Z, Z (1))equals the composition K ( e Z (1) ) → K ( Z (1) ) ։ M ′ , −−−→ H M ( Z, Z (1)) . Given the above, the homomorphism c , : K ( e Z ) → H M ( e Z, Z (1)) inducesa homomorphism ch ′ , : M → H M ( Z, Z (1)) such that the two homomor-phisms ch ′ , : M i → H M ( Z, Z (1)), i = 1 ,
2, coincide on M ∩ M . Thus,we obtain a homomorphism ch ′ , : G ( Z ) → H M ( Z, Z (1)).Next, we observe that the four homomorphisms ch ′ , , ch ′ , , ch ′ , , andch ′ , do not depend on the choice of Z (0) . Proposition 4.1. —
The homomorphism (4.1) for i = 0 , is an iso-morphism.Proof. — It follows from [4, Corollary 4.3, p. 95] that the Chern classmap c , : K ( Z (0) ) → H M ( Z (0) , Z (1)) is an isomorphism, hence by con-struction, ch ′ , is surjective and its kernel equals the image of K ( Z (1) ) → G ( Z ). It follows from the vanishing of K groups of finite fields that thehomomorphism c , : K ( Z (0) ) → H M ( Z (0) , Z (2)) is an isomorphism. Wethen have isomorphismsIm[ K ( Z (1) ) → G ( Z )] ∼ = Im[ H M ( Z (1) , Z (1)) → H M ( Z, Z (2))] ∼ = H M ( Z, Z (2)) , ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE ′ , ֒ → G ( Z ) ch ′ , −−−→ H M ( Z, Z (2)) is anisomorphism. This proves the claim for G ( Z ).By the construction of ch ′ , , the image of ch ′ , contains the image of H M ( Z (1) , Z (0)) → H M ( Z, Z (1)), and the composition K ( e Z ) → G ( Z ) ch ′ , −−−→ H M ( Z, Z (1)) → H M ( Z (0) , Z (1)) equals the composition K ( e Z ) → K ( e Z (0) ) ∼ = K ( Z (0) ) c , −−→ H M ( Z (0) , Z (1)) . The above implies that ch ′ , is surjective and the homomorphismKer ch ′ , → Ker[ K ( Z (0) ) c , −−→ H M ( Z (0) , Z (1))]is an isomorphism. This proves the claim for G ( Z ). (cid:3) K -groups and motivic cohomology of curves over afunction field Please note that the last paragraph of Section 1.4 presents an overviewof the contents of this section.In this section, we focus on the following setup. Let C be a smooth pro-jective geometrically connected curve over a finite field F q . Let k denote thefunction field of C . Let X be a smooth projective geometrically connectedcurve over k . Let X be a regular model of X , which is proper and flat over C .From the computations of the motivic cohomology of a surface with afibration (e.g., X or X with some fibers removed), we deduce some resultsconcerning the K -groups of low degrees of the generic fiber X . We relatethe two using the Chern class maps and by taking the limit. Lemma 10. —
The map K ( X ) ( c , ,c , ) −−−−−−→ k × ⊕ H M ( X, Z (2)) is an isomorphism. The group H M ( X, Z (3)) is a torsion group and thereexists a canonical short exact sequence → H M ( X, Z (3)) β −→ K ( X ) c , −−→ H M ( X, Z (2)) → , such that the composition c , ◦ β equals the multiplication-by- map. SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA
Proof. —
Let X denote the set of closed points of X . Let κ ( x ) denote theresidue field at x ∈ X . We construct a commutative diagram by connectingthe localization sequence L x ∈ X K ( κ ( x )) → K ( X ) → K ( k ( X )) → L x ∈ X K ( κ ( x )) → K ( X ) → K ( k ( X )) → L x ∈ X K ( x )with the localization sequence L x ∈ X H M (Spec κ ( x ) , Z (1)) → H M ( X, Z (2)) → H M (Spec k ( X ) , Z (2)) → L x ∈ X H M (Spec κ ( x ) , Z (1)) → H M ( X, Z (2)) → H M (Spec k ( X ) , Z (2)) , using the Chern class maps. Since H M (Spec κ ( x ) , Z (1)) = 0 and the K -groups and motivic cohomology groups of fields agree in low degrees, theclaim for K ( X ) follows from diagram chasing.It also follows from diagram chasing that K ( k ( X )) → M x ∈ X K ( κ ( x )) → K ( X ) c , −−→ H M ( X, Z (2)) → H M (Spec k ( X ) , Z (3)) and H M (Spec κ ( x ) , Z (2)) for each x ∈ X are isomorphic to the Milnor K -groups K M ( k ( X )) and K M ( κ ( x )), respec-tively. From the definition of these isomorphisms in [63], the boundarymap H M ( k ( X ) , Z (3)) → H M (Spec κ ( x ) , Z (2)) is identified under these iso-morphisms with the boundary map K M ( k ( X )) → K M ( κ ( x )). Given theabove, by [41, Proposition 11.11, p. 562], we obtain the first of the twoisomorphismsCoker[ K ( k ( X )) → L x ∈ X K ( κ ( x ))] ∼ = −→ Coker[ H M ( k ( X ) , Z (3)) → L x ∈ X H M (Spec κ ( x ) , Z (2))] ∼ = −→ H M ( X, Z (3)) . This gives us the desired short exact sequence. The identity c , ◦ β = 2follows from Remark 1. Since H M (Spec κ ( x ) , Z (2)) is a torsion group foreach x ∈ X , the group H M ( X, Z (3)) is also a torsion group. This completesthe proof. (cid:3) Lemma 11. —
Let U ⊂ C be a non-empty open subscheme. We use X U to denote the complement X \ X × C U with the reduced scheme structure.Then, for ( i, j ) = (0 , , (0 , or (1 , , the diagram (5.1) K i +1 ( X ) −−−−→ G i ( X U ) c i +1 ,j +1 y ( − j ch ′ i,j y H j − i +1 M ( X, Z ( j + 1)) −−−−→ H j − i M ( X U , Z ( j )) , ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE where each horizontal arrow is a part of the localization sequence, is com-mutative.Proof. — Let X U sm ⊂ X U denote the smooth locus. The commutativityof diagram (5.1) for ( i, j ) = (1 ,
1) (resp. for ( i, j ) = (0 , K i +1 ( X ) −−−−→ K i ( X U sm ) c i +1 ,j +1 y y − c , , H j − i +1 M ( X, Z ( j + 1)) −−−−→ H j − i M ( X U sm , Z ( j ))and the injectivity of H j − i M ( X U , Z ( j )) → H j − i M ( X U sm , Z ( j )).By Lemma 10, the group K ( X ) is generated by the image of the push-forward L x ∈ X K ( κ ( x )) → K ( X ) and the image of the pull-back K ( k ) → K ( X ). Then, the commutativity of diagram (5.1) for ( i, j ) = (0 ,
1) followsfrom the commutativity of the diagram K ( Y ) −−−−→ G ( X U ) ch , y ch ′ , y H M ( Y, Z (0)) −−−−→ H M ( X U , Z (1))for any reduced closed subscheme Y ⊂ X U of dimension zero, where thehorizontal arrows are the push-forward maps by closed immersion, and thefact that the composition K ( C \ U ) f U ∗ −−→ G ( X U ) ch ′ , −−−→ H M ( X U , Z (1))is zero. Here, f U : X U → C \ U denotes the morphism induced by themorphism X → C . (cid:3) Lemma 12. —
The diagram (5.2) → H M ( X, Z (3)) → K ( X ) c , −−→ H M ( X, Z (2)) → ↓ ↓ ↓ → M ℘ ∈ C H M ( X ℘ , Z (2)) → M ℘ ∈ C G ( X ℘ ) − ch ′ , −−−−→ M ℘ ∈ C H M ( X ℘ , Z (1)) → , where the first row is as in Lemma 10, the second row is obtained fromProposition 4.1, and the vertical maps are the boundary maps in the local-ization sequences, is commutative.Proof. — It follows from Lemma 11 that the right square is commutative.For each closed point x ∈ X , let D x denote the closure of x in X and write D x,℘ = D x × C Spec κ ( ℘ ) for ℘ ∈ C . Then, the commutativity of the left SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA square in (5.2) follows from the commutativity of the diagram H M ( X, Z (3)) ← H M (Spec κ ( x ) , Z (2)) ∼ = K ( κ ( x )) → K ( X ) ↓ ↓ ↓ ↓ M ℘ ∈ C H M ( X ℘ , Z (2)) ← M ℘ ∈ C H M ( D x,℘ , Z (1)) ∼ = M ℘ ∈ C G ( D x,℘ ) → M ℘ ∈ C G ( X ℘ ) . Here, each vertical map is a boundary map, the middle horizontal arrowsare Chern classes, and the left and right horizontal arrows are the push-forward maps by the closed immersion. (cid:3)
Lemma 13. —
Let U be an open subscheme of C such that U = C . Let ∂ : H M ( X × C U, Z (3)) → H M ( X U , Z (2)) denote the boundary map of thelocalization sequence. Then, the following composition is an isomorphism: α : Coker ∂ ֒ → H M ( X , Z (3)) → H M (Spec F q , Z (1)) ∼ = F × q . Here the first map is induced by the push-forward map by the closed im-mersion, and the second map is the push-forward map by the structuremorphism.Proof. —
For each closed point x ∈ X , let D x denote the closure of x in X . We set D x,U = D x × C U . Let D Ux denote the complement D x \ D x,U with the reduced scheme structure. Let ι x : D x ֒ → X , ι x,U : D x,U ֒ → X U , ι Ux : D Ux ֒ → X U denote the canonical inclusions. Let us consider thecommutative diagram H M ( D x,U , Z (2)) −−−−→ H M ( D Ux , Z (1)) β −−−−→ H M ( D x , Z (2)) ι x,U ∗ y ι Ux ∗ y H M ( X × C U, Z (3)) ∂ −−−−→ H M ( X U , Z (2)) −−−−→ Coker ∂ → , where the first row is the localization sequence. Since X is geometricallyconnected, it follows from [22, Corollaire (4.3.12), p. 134] that each fiberof X → C is connected. In particular, D x intersects every connected com-ponent of X U , which implies that the homomorphism ι Ux ∗ in the abovediagram is surjective, hence we have a surjective homomorphism Im β ։ Coker ∂ . Let F ( x ) denote the finite field H ( D x , O D x ). Then, the isomor-phism H M ( D x , Z (2)) → H M (Spec F ( x ) , Z (1)) ∼ = F ( x ) × (see Lemma 8)gives an isomorphism Im β ∼ = F ( x ) × . Hence | Coker ∂ | divides gcd x ∈ X ( | F ( x ) × | ) = q −
1, where the equality follows from [60, 1.5.3 Lemme 1, p. 325]. We caneasily verify that the composition F ( x ) × ∼ = Im β ։ Coker ∂ α −→ F × q ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE F ( x ) × → F × q – which implies | Coker ∂ | > q −
1, hence | Coker ∂ | = q − α is an isomorphism. The claimis proved. (cid:3)
6. Main results for j The reader is referred to Sections 1.6 and 1.7 for some remarks concerningthe contents of this section.The objective is to prove Theorems 6.1, 6.2, and 6.3. The statements givesome information on the structures of K -groups and motivic cohomologygroups of elliptic curves over global fields and of the (open) complements ofsome fibers of an elliptic surface over finite fields. We compute the ordersof some torsion groups, in terms of the special values of L -functions, thetorsion subgroup of (twisted) Mordell-Weil groups, and some invariants ofthe base curve. We remark that Milne [43] expresses the special values ofzeta functions in terms of the order of arithmetic ´etale cohomology groups.Our result is similar in spirit.Let us give the ingredients of the proof. Using Theorem 1.3, we deducethat the torsion subgroups we are interested in are actually finite. Then thetheorem of Geisser and Levine and the theorem of Merkurjev and Suslinrelate the motivic cohomology groups modulo their uniquely divisible partsto the ´etale cohomology and cohomology of de Rham-Witt complexes. Weuse the arguments that appear in [42], [10], and [21] to compute such coho-mology groups (see Section 1.3 in which we explain our general strategy).The computation of the exact orders of torsion may be new.One geometric property of an elliptic surface that makes this explicitcalculation possible is that the abelian fundamental group is isomorphic tothat of the base curve. This follows from a theorem in [58] for the prime-to- p part.We also use class field theory of Kato and Saito for surfaces over finitefields [29]. We combine their results and Lemma 18 to show that the groupsof zero-cycles on the elliptic surface and on the base curve are isomorphic.Let us give a brief outline. Recall in Section 6.3 that the main part of thezeta function of E is the L -function of E . Note that there are contributionsfrom the L -function of the base curve C and also of the singular fibersof E (Lemma 16, Corollary 6). We relate the G -groups of these singularfibers to the motivic cohomology groups, then they are in turn related tothe values of the L -function (Lemmas 21 and 22). The technical input forthese lemmas originate from our results in [33] and the classification of thesingular fibers of an elliptic fibration. SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA
Let k , E , S , S , r , C , and E be as defined in Section 1. We also let S denote the set of primes of k at which E has multiplicative reduction,thus we have S ⊂ S ⊂ S . Let p denote the characteristic of k . Theclosure of the origin of E in E gives a section to E → C , which we denoteby ι : C → E . Throughout this section, we assume that the structuremorphism f : E → C is not smooth, which in particular implies that it isnot isotrivial. For any scheme X over C , let E X denote the base change E × C X . For any non-empty open subscheme U ⊂ C , we use E U to denotethe complement E \ E U with the reduced scheme structure.Let F q denote the field of constants of C . We take an algebraic closure F q of F q . Let Frob ∈ G F q = Gal( F q / F q ) denote the geometric Frobenius.For a scheme X over Spec F q , we use X to denote its base change X = X × Spec F q Spec F q to F q . We often regard the set Irr( X ) of irreduciblecomponents of X as a finite ´etale scheme over Spec F q corresponding tothe G F q -set Irr( X ). We set T = E ( k ⊗ F q F q ) tors . For each integer j ∈ Z , we let T ′ ( j ) = L ℓ = p ( T ⊗ Z Z ℓ ( j )) G F q . Theorem 6.1. —
Let the notations and assumptions be as above. Let L ( E, s ) denote the L -function of the elliptic curve E over global field k (seeSection 6.3). (1) The Q -vector space ( K ( E ) red ) Q is of dimension r . (2) The cokernel of the boundary map ∂ : K ( E ) → L ℘ ∈ C G ( E ℘ ) isa finite group of order ( q − | L ( h (Irr( E S )) , − || T ′ (1) | · | L ( h ( S ) , − | . (3) The group K ( E ) div is uniquely divisible. (4) The kernel of the boundary map ∂ : K ( E ) red → L ℘ ∈ C G ( E ℘ ) is a finite group of order ( q − | T ′ (1) | · | L ( E, | . The cokernel of ∂ is a finitely generated abelian group of rank | Irr( E S ) | −| S | whose torsion subgroup is isomorphic to Jac( C )( F q ) ⊕ , where Jac( C ) denotes the Jacobian of C (when the genus of C is , weunderstand it to be a point). ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE X be a scheme of finite type over Spec F q . For an integer i ∈ Z and aprime number ℓ = p , we set L ( h i ( X ) , s ) = det(1 − Frob · q − s ; H i et ( X, Q ℓ )).In all cases considered in our paper, L ( h i ( X ) , s ) does not depend on thechoice of ℓ .For each non-empty open subscheme U ⊂ C , let T U denote the torsionsubgroup of the group Div( E U ) / ∼ alg of divisors on E U modulo algebraicequivalence. For each integer j ∈ Z , we set T ′ U, ( j ) = L ℓ = p ( T U ⊗ Z Z ℓ ( j )) G F q .We deduce from [58, Theorem 1.3, p. 214] that the canonical homomor-phism lim −→ U ′ T U ′ → T , where the limit is taken over the open subschemes of C , is an isomorphism. We easily verify that the canonical homomorphism T U → lim −→ U ′ T U ′ is injective and is an isomorphism if E U → U is smooth.In particular, we have an injection T ′ U, ( j ) ֒ → T ′ ( j ) , which is an isomorphismif E U → U is smooth.In Section 6.7, we deduce Theorem 6.1 from the following two theorems. Theorem 6.2. —
Let the notations and assumptions be as above. Let ∂ i M ,j : H i M ( E, Z ( j )) red → L ℘ ∈ C H i − M ( E ℘ , Z ( j − denote the homomor-phism induced by the boundary map of the localization sequence estab-lished in [6, Corollary (0.2), p. 537] . (1) For any i ∈ Z , the group H i M ( E, Z (2)) div is uniquely divisible. (2) For i , the cohomology group H i M ( E, Z (2)) is uniquely divisible. H M ( E, Z (2)) is finite modulo a uniquely divisible subgroup and H M ( E, Z (2)) tors is cyclic of order q − . (3) The kernel (resp. cokernel) of the homomorphism ∂ M , is a finitegroup of order | L ( h ( C ) , − | (resp. of order ( q − | L ( h (Irr( E S )) , − || T ′ (1) | · | L ( h ( S ) , − | ) . (4) The kernel (resp. cokernel) of the homomorphism ∂ M , is a finitegroup of order ( q − | T ′ (1) |·| L ( E, | (resp. is isomorphic to Pic( C ) ). (5) For i > , the group H i M ( E, Z (2)) is zero. (6) H M ( E, Z (3)) is a torsion group, and the cokernel of the homomor-phism ∂ M , is a finite cyclic group of order q − . Theorem 6.3. —
Let U ⊂ C be a non-empty open subscheme. Then,we have the following. (1) For any i ∈ Z , the group H i M ( E U , Z (2)) is finitely generated moduloa uniquely divisible subgroup. (2) For i , the cohomology group H i M ( E U , Z (2)) is uniquely divis-ible, H M ( E U , Z (2)) is finite modulo a uniquely divisible subgroupand H M ( E U , Z (2)) tors is cyclic of order q − . SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA (3)
The rank of H M ( E U , Z (2)) red is | S \ U | . If U = C (resp. U = C ),the torsion subgroup of H M ( E U , Z (2)) red is of order | L ( h ( C ) , − | (resp. of order | T ′ U, (1) | · | L ( h ( C ) , − L ( h ( C \ U ) , − | / ( q − ). (4) If U = C (resp. U = C ), the cokernel of the boundary homomor-phism H M ( E U , Z (2)) → H M ( E U , Z (1)) is zero (resp. is finite oforder ( q − | L ( h (Irr( E U )) , − || T ′ U, (1) | · | L ( h ( C \ U ) , − | ) . (5) The rank of H M ( E U , Z (2)) red is max( | C \ U | − , . If U = C (resp. U = C ), the torsion subgroup H M ( E U , Z (2)) tors is finite of order | L ( h ( E ) , | (resp. of order | T ′ U, (1) | · | L ( h ( E ) , L ∗ ( h ( E U ) , L ( h ( C \ U ) , − | ( q − | L ( h (Irr( E U )) , − | ) . Here, L ∗ ( h ( E U ) ,
0) = lim s → ( s log q ) −| S \ U | L ( h ( E U ) , s ) is the leading coefficient of L ( h ( E U ) , s ) at s = 0 . (6) H M ( E U , Z (2)) is canonically isomorphic to Pic( U ) . For i > , thegroup H i M ( E U , Z (2)) is zero. L ( E, s ) and the congruence zeta functionof E . Let ℓ = p be a prime number. Take an open subset j : U → C such thatthe restriction f U : E U → U is (proper) smooth.By the Grothendieck-Lefschetz trace formula, we have L ( E, s ) = Y i =0 det(1 − Frob · q − s ; H i et ( C, j ∗ R f U ∗ Q ℓ )) ( − i − . Note that by the adjunction id → j ∗ j ∗ , we have a canonical morphism R f ∗ Q ℓ → j ∗ j ∗ R f ∗ Q ℓ ∼ = j ∗ R f U ∗ Q ℓ . We will prove the following lemma:
Lemma 6.4. —
The map above is an isomorphism.Proof. —
Let ℘ be a closed point of C , and let S be strict henselizationof C at ℘ . Let s be the closed point of S and let η denote the spectrum ofa separable closure of the function field of S . ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE H ( E s , Q ℓ ) → H ( E η , Q ℓ ). This is identi-fied with H ( π ( E ¯ η ) , Q ℓ ) → H ( π ( E ¯ s ) , Q ℓ ), and is injective, since π ( E ¯ s ) → π ( E ¯ η ) is surjective ([1, p.270, Exp X, Corollaire 2.4]).Set V = H ( E η , Q ℓ ). For surjectivity, we prove that the image of thespecialization map is equal to the invariant part V I s under the action ofthe inertia group I s at s . It suffices to show that H ( E s , Q ℓ ) and V I s havethe same dimension.We use the Kodaira-N´eron-Tate classification of bad fibers (cf. [39, 10.2.1,p. 484–489]) to compute that H ( E s , Q ℓ ) is of dimension 2, 1, 0 if E hasgood reduction, semistable bad reduction, and additive reduction at ℘ ,respectively. It is well known that V I s is of dimension 2 if and only if E has good reduction at p . If E has semistable bad reduction, then the actionof I s on V is nontrivial and unipotent. Hence V I s is of dimension 1 in thiscase. Suppose that E has additive reduction and V I s = 0. Then V I s is ofdimension one. Poincare duality implies that I s acts trivially on V /V I s ,hence the action of I s on V is unipotent. Therefore, E must have eithergood or semistable reduction, which leads to a contradiction. (cid:3) Corollary 2. —
We have L ( E, s ) = Y i =0 det(1 − Frob · q − s ; H i et ( C, R f ∗ Q ℓ )) ( − i − . Below, we will work with this expression.
Lemma 14. —
Let D be a scheme of dimension that is proper over Spec F q . Let ℓ = p be an integer. Then, the group H i et ( D, Z ℓ ) is torsionfree for any i ∈ Z and is zero for i = 0 , , . The group H i et ( D, Q ℓ ) is pureof weight i for i = 1 and is mixed of weight { , } for i = 1 . The group H ( D, Q ℓ ) is pure of weight one (resp. pure of weight zero) if D is smooth(resp. every irreducible component of D is rational).Proof. — We may assume that D is reduced. Let D ′ be the normalizationof D . Let π : D ′ → D denote the canonical morphism. Let F n denote thecokernel of the homomorphism Z /ℓ n → π ∗ ( Z /ℓ n ) of ´etale sheaves. Thesheaf F n is supported on the singular locus D sing of D and is isomorphicto i ∗ (Coker[ Z /ℓ n → π sing ∗ ( Z /ℓ n )]), where i : D sing ֒ → D is the canonicalinclusion and π sing : D ′ × D D sing → D sing is the base change of π . Then,the claim follows from the long exact sequence · · · → H i et ( D, Z /ℓ n ) → H i et ( D ′ , Z /ℓ n ) → H i et ( D, F n ) → · · · . (cid:3) SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA
Lemma 15. — If i = 1 , then H i et ( C, R f ∗ Q ℓ ) = 0 .Proof. — For any point x ∈ C ( F q ) lying over a closed point ℘ ∈ C , thecanonical homomorphism H ( C, R f ∗ Q ℓ ) → H ( E x , Q ℓ ) is injective since H ,c ( C \ { x } , R f ∗ Q ℓ ) = 0. By Lemma 14, the module H ( E x , Q ℓ ) is pureof weight one (resp. of weight zero) if E x is smooth (resp. is not smooth).Since we have assumed that f : E → C is not smooth, H ( C, R f ∗ Q ℓ ) = 0.Consider a non-empty open subscheme U ⊂ C such that f U : E U → U is smooth. The group H ( C, R f ∗ Q ℓ ) ∼ = H ,c ( U , R f U ∗ Q ℓ ) is the dual of H ( U , R f U ∗ Q ℓ (1)) by Poincar´e duality. Next, assume H ( U , R f U ∗ Q ℓ (1)) =0. Let T ℓ ( E ) denote the ℓ -adic Tate module of E . The ´etale fundamentalgroup π ( U ) acts on T ℓ ( E ). By the given assumption, the π ( U )-invariantpart V = ( T ℓ ( E ) ⊗ Q ℓ ) π ( U ) is nonzero. Since f is not smooth, V is one-dimensional, hence we have a nonzero homomorphism π ( U ) ab → Hom( T ℓ ( E ) ⊗ Q ℓ /V, V ) of G F q -modules. By the weight argument, we observe that this isimpossible, hence H ( C, R f ∗ Q ℓ (1)) = 0. (cid:3) As an immediate consequence, we obtain the following corollary.
Corollary 3. —
The spectral sequence E i,j = H i et ( C, R j f ∗ Q ℓ ) ⇒ H i + j et ( E , Q ℓ ) is E -degenerate. (cid:3) Lemma 16. —
Let U ⊂ C be a non-empty open subscheme such that f U : E U → U is smooth. Let Irr ( E U ) ⊂ Irr( E U ) denote the subset ofirreducible components of E U that do not intersect ι ( C ) . We regard Irr ( E U ) as a closed subscheme of Irr( E U ) (recall the convention in Section 6.1 on Irr( E U ) ). Then, L ( h i ( E ) , s ) = (1 − q − s ) , if i = 0 ,L ( h ( C ) , s ) , if i = 1 , (1 − q − s ) L ( E, s ) L ( h (Irr ( E U )) , s − , if i = 2 ,L ( h ( C ) , s − , if i = 3 , (1 − q − s ) , if i = 4 . Proof. —
We prove the lemma for i = 2; the other cases are straightfor-ward. Since R f U ∗ Q ℓ ∼ = Q ℓ ( − → H ( C, R f ∗ Q ℓ ) → H ( E U , Q ℓ ) → H ,c ( U , Q ℓ ( − . The map H ( E U , Q ℓ ) → H ,c ( U , Q ℓ ( − H ( E U , Q ℓ ) → H ( C \ U , Q ℓ ( − → H ,c ( U , Q ℓ ( − . ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE H ( C, R f ∗ Q ℓ ) is isomorphic to the inverse image of theimage of the homomorphism H ( C, Q ℓ ( − → H ( C \ U , Q ℓ ( − H ( E U , Q ℓ ) → H ( C \ U , Q ℓ ( − (cid:3) E Lemma 17. —
For i = 0 , , the pull-back H i ( C, O C ) → H i ( E , O E ) is anisomorphism.Proof. — The claim for i = 0 is clear, thus we prove the claim for i = 1.Let us write L = R f ∗ O E . It suffices to prove H ( C, L ) = 0. We note that L is an invertible O C -module since the morphism E → C has no multiplefiber. The Leray spectral sequence E i,j = H i ( C, R j f ∗ O E ) ⇒ H i + j ( E , O E )shows that the Euler-Poincar´e characteristic χ ( O E ) equals χ ( O C ) − χ ( L ) = − deg L . By the well-known inequality χ ( O E ) > L <
0, which proves H ( C, L ) = 0. (cid:3) Lemma 18. — (1)
The canonical homomorphism π ab1 ( E ) → π ab1 ( C ) between the abelian (´etale) fundamental groups is an isomorphism. (2) The canonical morphism
Pic oC/ F q → Pic o E / F q between the identitycomponents of the Picard schemes is an isomorphism.Proof. — The homomorphism Pic oC/ F q → Pic o E / F q , red is an isomorphismby [58, Theorem 4.1, p. 219]. This, combined with the cohomology long ex-act sequence of the Kummer sequence, implies that if p ∤ m , then H ( C, Z /m ) → H ( E , Z /m ) is an isomorphism, hence to prove (1), we are reduced to show-ing that H ( C, Z /p n ) → H ( E , Z /p n ) is an isomorphism for all n >
1. Forany scheme X that is proper over Spec F q , there exists an exact sequence0 → Z /p n Z → W n O X − σ −−−→ W n O X → W n O X is the sheaf of Witt vectors and σ : W n O X → W n O X is the Frobenius endomorphism. This gives rise to the followingcommutative diagram with exact rows · · · − σ −−−→ H ( C, W n O C ) → H ( C, Z /p n ) → H ( C, W n O C ) − σ −−−→ · · ·↓ ↓ ↓· · · − σ −−−→ H ( E , W n O E ) → H ( E , Z /p n ) → H ( E , W n O E ) − σ −−−→ · · · . By Lemma 17 and induction on n , we observe that the homomorphism H i ( C, W n O C ) → H i ( E , W n O E ) is an isomorphism for i = 0 ,
1, thus the
SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA map H ( C, Z /p n ) → H ( E , Z /p n ) is an isomorphism. This proves claim(1).For (2), it suffices to prove that homomorphism Lie Pic C/ F q → Lie Pic E / F q between the tangent spaces is an isomorphism. Since this homomorphism isidentified with homomorphism H ( C, O C ) → H ( E , O E ), claim (2) followsfrom Lemma 17. (cid:3) Remark 2. —
Using Lemma 18 (1), we can prove that the homomor-phism π ( E ) → π ( C ) is an isomorphism. Since it is not used in this paper,we only sketch the proof below.Let x → C be a geometric point. Since the morphism f : E → C hasa section, fiber E x of f at x has a reduced irreducible component. This,together with the regularity of E and C , shows that the canonical ringhomomorphism H ( x, O x ) → H ( Y, O Y ) is an isomorphism for any con-nected finite ´etale covering Y of E x , hence by the same argument as in theproof of [1, X, Proposition 1.2, Th´eor`eme 1.3, p. 262] , we have an exactsequence π ( E x ) → π ( E ) → π ( C ) → . In particular here, the kernel of π ( E ) → π ( C ) is abelian. Applying Lemma 18(1)to E × C C ′ → C ′ for each finite connected ´etale cover C ′ → C , we obtainthe bijectivity of π ( E ) → π ( C ) .More generally, the statements in Lemma 18 and the statement abovethat the fundamental groups are isomorphic are also valid if E is a regularminimal elliptic fibration that is proper flat non-smooth over C with asection, where C is a proper smooth curve over an arbitrary perfect basefield. Corollary 4. —
For any prime number ℓ = p and for any i ∈ Z , thegroup H i et ( E , Q ℓ / Z ℓ ) is divisible.Proof. — The claim for i = 1 , H ( E , Q ℓ / Z ℓ ) ∼ = H ( C, Q ℓ / Z ℓ ), hence H ( E , Q ℓ / Z ℓ ) is divisible. The group H ( E , Q ℓ / Z ℓ ) is divisible since H ( E , Q ℓ / Z ℓ ) red is isomorphic to the Pon-tryagin dual of H ( E , Q ℓ / Z ℓ (2)) red . (cid:3) Corollary 5. —
For i ∈ Z , we set M ij = L ℓ = p H i et ( E , Q ℓ / Z ℓ ( j )) . Fora rational number a , we write | a | ( p ′ ) = | a | · | a | p . (1) For i − or i > , the group M ij is zero. (2) For j = 2 (resp. j = 2 ), the group M j is zero (resp. is isomorphicto L ℓ = p Q ℓ / Z ℓ ). ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE
For j = 0 , the group M j is cyclic of order q | j | − . The group M is isomorphic to L ℓ = p Q ℓ / Z ℓ . (4) For j = 0 , the group M j is finite of order | L ( h ( C ) , − j ) | ( p ′ ) . (5) For j = 1 , the group M j is finite of order | L ( h ( E ) , − j ) | ( p ′ ) . (6) For j = 1 , group M j is finite of order | L ( h ( C ) , − j ) | ( p ′ ) . (7) For j = 2 , the group M j is cyclic of order q | − j | − . The group M is isomorphic to L ℓ = p Q ℓ / Z ℓ .Proof. — By Corollary 4, if i = 2 j +1 and ℓ = p , the group H i et ( E , Q ℓ / Z ℓ ( j ))is isomorphic to H i et ( E , Q ℓ / Z ℓ ( j )) G F q . Then, we have | H i et ( E , Q ℓ / Z ℓ ( j )) | = | L ( h − i ( E ) , − j ) | − ℓ by Poincar´e duality for i = 2 j, j +1, hence the claim follows from Lemma 16. (cid:3) We first fix a non-empty open subscheme U ⊂ C . Lemma 19. —
Let ℓ = p be a prime number. For i ∈ Z , let γ i denotethe pull-back γ i : H i et ( E , Z ℓ ) → H i et ( E U , Z ℓ ) . (1) For i = 0 , , the homomorphism γ i is zero. (2) The cokernel (Coker γ ) Q ℓ is isomorphic to the kernel of H ( C \ U , Q ℓ ( − → H (Spec F q , Q ℓ ( − . (3) There exists a canonical isomorphism
Hom Z ( T U , Q ℓ / Z ℓ ( − ∼ = (Coker γ ) tors . Proof. —
By Lemma 18, the pull-back H ( C, Z ℓ ) → H ( E , Z ℓ ) is anisomorphism, hence the homomorphism H ( E , Z ℓ ) → H ( E U , Z ℓ ) is zero.Claim (1) follows.Let NS( E ) = Pic E / F q ( F q ) / Pic o E / F q ( F q ) denote the N´eron-Severi group of E . For a prime number ℓ , we set T ℓ M = Hom( Q ℓ / Z ℓ , M ).We have the following exact sequence from Kummer theory:0 → NS( E ) ⊗ Z Z ℓ cl ℓ −−→ H ( E , Z ℓ (1)) → T ℓ H ( E , G m ) → . We note that T ℓ H ( E , G m ) is torsion free. For D ∈ Irr( E U ), let [ D ] ∈ NS( E ) denote the class of the Weil divisor D red on E . By [11, Cycle,Definition 2.3.2, p. 145], the D -component of the homomorphism γ : SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA H ( E , Z ℓ ) → H ( E U , Z ℓ ) ∼ = Map(Irr( E U ) , Z ℓ ( − H ( E , Z ℓ ) ∪ cl ℓ ([ D ]) −−−−−−→ H ( E , Z ℓ (1)) ∼ = Z ℓ ( − . Let M ⊂ NS( E ) denote the subgroup generated by { [ D ] | D ∈ Irr( E U ) } .By Corollary 4, the cup-product H ( E , Z ℓ (1)) × H ( E , Z ℓ (1)) → H ( E , Z ℓ (2)) ∼ = Z ℓ is a perfect pairing, hence the image of γ is identified with the image ofthe compositionHom Z ℓ ( H ( E , Z ℓ (1)) , Z ℓ ( − α ∗ −−→ Hom Z ( M, Z ℓ ( − β ∗ −→ Map(Irr( E U ) , Z ℓ ( − ∼ = H ( E U , Z ℓ ) , where α ∗ is the homomorphism induced by the restriction α : M ⊗ Z Z ℓ ֒ → H ( E , Z ℓ (1)) of the cycle class map cl ℓ to M , and the homomorphism β ∗ is induced by the canonical surjection β : L D ∈ Irr( E U ) Z ։ M . Since β issurjective, the homomorphism β ∗ is injective and we have an exact sequence0 → Coker α ∗ → Coker γ → Coker β ∗ → . Since α is a homomorphism of finitely generated Z ℓ -modules that is in-jective, the cokernel Coker α ∗ is a finite group. Further, the group M isa free abelian group with basis Irr ( E U ) ∪ { D ′ } , where D ′ is an arbitraryelement in Irr( E U ) \ Irr ( E U ), hence Coker β ∗ is isomorphic to the groupHom Z (Ker β, Z ℓ ( − γ is identified with the group Coker α ∗ . Thehomomorphism α ∗ is the composition of the two homomorphismsHom Z ℓ ( H ( E , Z ℓ (1)) , Z ℓ ( − cl ∗ ℓ −−→ Hom Z (NS( E ) , Z ℓ ( − ι ∗ −→ Hom Z ( M, Z ℓ ( − , where the first (resp. second) homomorphism is that induced by the cycleclass map cl ℓ : NS( E ) ⊗ Z Z ℓ ֒ → H ( E , Z ℓ (1)) (resp. the inclusion M ⊗ Z Z ℓ ֒ → NS( E )). Since Coker cl ℓ is torsion free, as noted above, the homomorphismcl ∗ ℓ is surjective, hence we have isomorphismsCoker α ∗ ∼ = Coker ι ∗ ∼ = Ext Z (NS( E ) /M, Z ℓ ( − ∼ = Hom Z (NS( E ) /M, Q ℓ / Z ℓ ( − Z (Div( E U ) / ∼ alg , Q ℓ / Z ℓ ( − . Given the above, we have claim (3). (cid:3)
ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE Corollary 6. —
For i = 3 , the group H ic, et ( E U , Z ℓ ) is torsion freeand the group H c, et ( E U , Z ℓ ) tors is canonically isomorphic to the group Hom Z ( T U , Q ℓ / Z ℓ ( − . We set L ( h ic,ℓ ( E U ) , s ) = det(1 − Frob · q − s ; H ic, et ( E U , Q ℓ )) . Then if U = C , we have L ( h ic,ℓ ( E U ) , s ) = , if i or i > , L ( h ( C ) ,s ) L ( h ( C \ U ) ,s )1 − q − s if i = 1 , L ( h ( E ) ,s ) L ( h ( E U ) ,s ) L ( h ( C \ U ) ,s − − q − s ) L ( h ( E U ) ,s ) if i = 2 , L ( h ( C ) ,s − L ( h ( C \ U ) ,s − − q − s if i = 3 , − q − s , if i = 4 . Proof. —
The above follows from Lemmas 16 and 19, as well as the longexact sequence · · · → H i et ,c ( E U , Z ℓ ) → H i et ( E , Z ℓ ) → H i et ( E U , Z ℓ ) → · · · . (cid:3) Remark 3. —
Corollary 6 in particular shows that the function L ( h ic,ℓ ( E U ) , s ) is independent of ℓ = p . We can show the ℓ -independence of L ( h ic,ℓ ( X ) , s ) =det(1 − Frob · q − s ; H ic, et ( X, Q ℓ )) for any normal surface X over F q , whichis not necessarily proper. Since we will not need this further, we only showa sketch here. There is a proper smooth surface X ′ and a closed subset D ⊂ X ′ of pure codimension one such that X = X ′ \ D . We can expressthe cokernel and kernel of the restriction map H ( X ′ , Q l ) → H ( D, Q l ) interms of Pic X ′ / F q and the Jacobian of the normalization of each irreduciblecomponent of D . Then, we apply the same method as above to obtain theresult. Corollary 7. —
Suppose that U = C . Then, we have the following. (1) H i et ( E U , Q ℓ / Z ℓ ( j )) is zero for i − or i > . (2) For j = 0 , the group H ( E U , Q ℓ / Z ℓ ( j )) is isomorphic to Z ℓ / ( q j − ,and H ( E U , Q ℓ / Z ℓ (0)) = Q ℓ / Z ℓ . (3) For j = 0 , , the group H ( E U , Q ℓ / Z ℓ ( j )) is finite of order | T ′ U, ( j − | − ℓ · | L ( h ( C ) , − j ) L ( h ( C \ U ) , − j ) | − ℓ | q j − − | − ℓ . The group H ( E U , Q ℓ / Z ℓ (0)) is isomorphic to the direct sum of Q ℓ / Z ℓ and a finite group of order | T ′ U, ( − | − ℓ · | L ( h ( C ) , L ( h ( C \ U ) , | − ℓ | q − | − ℓ . SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA (4)
For j = 1 , , the cohomology group H ( E U , Q ℓ / Z ℓ ( j )) is finite oforder | T ′ U, ( j − | − ℓ · | L ( h ( E ) , − j ) L ( h ( E U ) , − j ) L ( h ( C \ U ) , − j ) | − ℓ | ( q j − − L ( h ( E U ) , − j ) | − ℓ . (5) For j = 1 , , the group H ( E U , Q ℓ / Z ℓ ( j )) is finite of order | L ( h ( C ) , − j ) L ( h ( C \ U ) , − j ) | − ℓ | q j − − | − ℓ . The cohomology group H ( E U , Q ℓ / Z ℓ (1)) is isomorphic to the di-rect sum of ( Q ℓ / Z ℓ ) ⊕| C \ U |− and a finite group of order | L ( h ( C ) , L ( h ( C \ U ) , | − ℓ | q − | − ℓ . (6) For j = 2 (resp. j = 2 ), the group H ( E U , Q ℓ / Z ℓ ( j )) is zero (resp.is isomorphic to ( Q ℓ / Z ℓ ) ⊕| C \ U |− ).Proof. — The cohomology group H i et ( E U , Q ℓ / Z ℓ ( j )) is the Pontryagindual of the group H − i et ,c ( E U , Z ℓ (2 − j )). The claims follow from Corollary 6and the short exact sequence0 → H − ic, et ( E U , Z ℓ (2 − j )) G F q → H − ic, et ( E U , Z ℓ (2 − j )) → H − ic, et ( E U , Z ℓ (2 − j )) G F q → . (cid:3) Lemma 20. —
Suppose that U = C . Then, H ( E U , Q ℓ / Z ℓ (2)) red is fi-nite of order | T ′ U, (1) | − ℓ · | L ( h ( E ) , L ∗ ( h ( E U ) , L ( h ( C \ U ) , − | − ℓ | ( q − L ( h (Irr( E U )) , − | − ℓ . Proof. —
We note that the group H ( E U , Q ℓ / Z ℓ (2)) red is canonicallyisomorphic to the group H ( E U , Z ℓ (2)) tors . Let us consider the long exactsequence · · · → H i E U , et ( E , Z ℓ (2)) µ i −→ H i et ( E , Z ℓ (2)) → H i et ( E U , Z ℓ (2)) → · · · . The group Ker µ is isomorphic to the Pontryagin dual of the cokernel ofthe homomorphism H ( E , Q ℓ / Z ℓ ) → H ( E U , Q ℓ / Z ℓ ). By Lemma 18, thishomomorphism factors through H ( C \ U, Q ℓ / Z ℓ ) → H ( E U , Q ℓ / Z ℓ ). Inparticular, the group (Ker µ ) tors is isomorphic to the Pontryagin dual of( H ( E U , Q ℓ / Z ℓ ) G F q ) red . By the weight argument, we observe that Coker µ is a finite group. It then follows that | H ( E U , Z ℓ (2)) tors | = | L ∗ ( h ( E U ) , | · | Coker µ | . ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE µ ′ denote the homomorphism H E U , et ( E , Z ℓ (2)) → H ( E , Z ℓ (2)). Wehave an exact sequence(6.1) Ker µ → H E U , et ( E , Z ℓ (2)) G F q → (Coker µ ′ ) G F q → Coker µ → . Since Ker µ ∼ = Coker[ H ( E , Z ℓ (2)) → H ( E U , Z ℓ (2))], the cokernel ofKer µ → H E U , et ( E , Z ℓ (2)) G F q is isomorphic to the cokernel of the homo-morphism ν ′ : H ( E U , Z ℓ (2)) G F q → H E U , et ( E , Z ℓ (2)) G F q . Consider the following diagram with exact rows0 −−−−→
Coker µ ′ −−−−→ H ( E U , Z ℓ (2)) ν −−−−→ H E U , et ( E , Z ℓ (2)) − Frob y − Frob y − Frob y −−−−→ Coker µ ′ −−−−→ H ( E U , Z ℓ (2)) ν −−−−→ H E U , et ( E , Z ℓ (2)) . Since (Coker ν ) G F q ⊂ H ( E , Z ℓ (2)) G F q = 0, Coker ν ′ is isomorphic to thekernel of (Coker µ ′ ) G F q → H ( E U , Z ℓ (2)) G F q , hence by (6.1), | Coker µ | equals the order of M ′′ = Im[(Coker µ ′ ) G F q → H ( E U , Z ℓ (2)) G F q ]= Im[ H ( E , Z ℓ (2)) G F q → H ( E U , Z ℓ (2)) G F q ] . Next, we set M ′ = Im[ H ( E , Z ℓ (2)) → H ( E U , Z ℓ (2))]. From the commu-tative diagram with exact rows(6.2) → NS( E ) ⊗ Z Z ℓ → H ( E , Z ℓ (1)) → T ℓ H ( E , G m ) → ↓ ↓ ↓ → (Div( E U ) / ∼ alg ) ⊗ Z Z ℓ → H ( E U , Z ℓ (1)) → T ℓ H ( E U , G m ) → and the exact sequence0 → H ( E , G m ) → H ( E U , G m ) → H ( E U , Q / Z ) , we obtain an exact sequence0 → M ′ → H ( E U , Z ℓ (2)) → T ℓ H ( E U , Q ℓ / Z ℓ (1)) . By the weight argument, we obtain ( T ℓ H ( E U , Q ℓ / Z ℓ (1))) G F q = 0, hencethe canonical surjection M ′ G F q → M ′′ is an isomorphism. From (6.2), wehave an exact sequence0 → (Div( E U ) / ∼ alg ) ⊗ Z Z ℓ (1) → M ′ → T ℓ H ( E , G m )(1) → . By the weight argument, we yield ( T ℓ H ( E , G m )(1)) G F q = 0, hence0 → ((Div( E U ) / ∼ alg ) ⊗ Z Z ℓ (1)) G F q → M ′ G F q → ( T ℓ H ( E , G m )(1)) G F q → SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA is exact. Therefore, | Coker µ | = | M ′ G F q | equals | ( T U ⊗ Z Z ℓ (1)) G F q | · | det(1 − Frob; H ( E , Q ℓ (2))) | − ℓ | det(1 − Frob; Ker[NS( E ) → Div( E U ) / ∼ alg ] ⊗ Z Q ℓ (1)) | − ℓ . This proves the claim. (cid:3)
Fix a non-empty open subscheme U ⊂ C . Let f U : E U → C \ U denotethe structure morphism and let ι U : C \ U → E U denote the morphisminduced from ι : C → E . Lemma 21. —
The homomorphism (ch ′ , , f U ∗ ) : G ( E U ) → H M ( E U , Z (1)) ⊕ K ( C \ U ) is an isomorphism.Proof. — The morphism f U : E U → C \ U has connected fibers, hencethe claim follows from Proposition 4.1 and the construction of ch ′ , . (cid:3) Lemma 22. —
The group H M ( E U , Z (1)) is finitely generated of rank | C \ U | . Moreover, H M ( E U , Z (1)) tors is of order | L ∗ ( h ( E U ) , | .Proof. — It suffices to prove the following claim: if E has good reduction(resp. non-split multiplicative reduction, resp. split multiplicative or addi-tive reduction) at ℘ ∈ C , then H M ( E ℘ , Z (1)) is a finitely generated abeliangroup of rank one, and | H M ( E ℘ , Z (1)) tors | equals |E ℘ ( κ ( ℘ )) | (resp. rank two,resp. rank one). We set E ℘, (0) = ( E ℘, red ) sm \ ι ( ℘ ) and E ℘, (1) = E U \ E ℘, (0) .We then have an exact sequence H M ( E ℘, (0) , Z (1)) → H M ( E ℘, (1) , Z (0)) → H M ( E ℘ , Z (1)) → Pic( E ℘, (0) ) → . First, suppose that E does not have non-split multiplicative reduction at ℘ or that E has non-split multiplicative reduction at ℘ and E ℘ ⊗ κ ( ℘ ) F q hasan even number of irreducible components. Then, using the classificationof Kodaira, N´eron, and Tate (cf. [39, 10.2.1, p. 484–489]) of singular fibersof E → C , we can verify the equalityIm[ H M ( E ℘, (1) , Z (0)) → H M ( E ℘ , Z (1))]= Im[ ι ∗ : H M (Spec κ ( ℘ ) , Z (0)) → H M ( E ℘ , Z (1))] . This shows that the group H M ( E ℘ , Z (1)) is isomorphic to the direct sumof Picard group Pic( E ℘, (0) ) and H M (Spec κ ( ℘ ) , Z (0)) ∼ = Z . In particular, ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE H M ( E ℘ , Z (1)) tors ∼ = Pic( E ℘, (0) ), from which we easily deduce theclaim.Next, suppose that E has non-split multiplicative reduction at ℘ and E ℘ ⊗ κ ( ℘ ) F q has an odd number of irreducible components. In this case, wecan directly verify that the image of H M ( E ℘, (1) , Z (0)) → H M ( E ℘ , Z (1)) isisomorphic to Z ⊕ Z / E ℘, (0) ) = 0. The claim in this case follows. (cid:3) Lemma 23. —
The diagram K ( E ) −−−−→ G ( E U ) ι ∗ y ι U ∗ y K ( k ) −−−−→ K ( C \ U ) is commutative.Proof. — The group K ( E ) is generated by the image of f ∗ : K ( k ) → K ( E ) and the image of L x ∈ E K ( κ ( x )) → K ( E ). The claim followsfrom the fact that the localization sequence in G -theory commutes withflat pull-backs and finite push-forwards. (cid:3) Lemma 24. —
For any non-empty open subscheme U ⊂ C , the cokernelof the boundary map ∂ U : H M ( E U , Z (2)) → H M ( E U , Z (1)) is finite.Proof. — If suffices to prove the claim for sufficiently small U , hence weassume that E U → U is smooth. Since K ( E U ) Q → K ( E ) Q is an isomor-phism in this case, the claim follows from Theorem 1.3 and Lemma 11. (cid:3) Proof of Theorem 6.3. —
Claims (1) and (2) follow from Theorem 2.2,Proposition 2.3, and Lemma 24. Proposition 2.3 gives an exact sequence0 → H M ( E , Z (2)) tors → H M ( E U , Z (2)) red ∂ U −−→ H M ( E U , Z (1)) → H M ( E , Z (2)) tors → H M ( E U , Z (2)) red ∂ U −−→ H M ( E U , Z (1)) → CH ( E ) → CH ( E U ) → . From Lemma 24, it follows that Coker ∂ U is a finite group, which im-plies that the group H M ( E U , Z (2)) red is of rank | S \ U | . By Theorem 2.2, | H M ( E U , Z (2)) tors | equals Y ℓ = p | H ( E U , Q ℓ / Z ℓ (2)) | . SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA
By Corollaries 5 and 7, this equals | T ′ U, (1) | · | L ( h ( C ) , − L ( h ( C \ U, − / ( q − | . The above proves claim (3).As noted in the proof of Theorem 2.2 (1), the group CH ( E ) is a finitelygenerated abelian group of rank one and CH ( E U ) is finite if U = C . ByLemma 22, H M ( E U , Z (1)) is a finitely generated abelian group of rank | C \ U | , hence the rank of H M ( E U , Z (2)) red equals max( | C \ U | − , ( E ) → Pic( C ) is an isomorphism, hence the homo-morphism H M ( E U , Z (1)) → CH ( E ) ∼ = Pic( C ) factors through the push-forward map f U ∗ : H M ( E U , Z (1)) → H M ( C \ U, Z (0)). By the surjectivityof f U ∗ , we have isomorphismsCH ( E U ) ∼ = Coker[ H M ( C \ U, Z (0)) → Pic( C )] ∼ = Pic( U ) , which proves claim (6). Since the group H M ( C \ U, Z (0)) is torsion free,the image of H M ( E U , Z (1)) tors in CH ( E ) is zero, thus we have an exactsequence 0 → Coker ∂ U → H M ( E , Z (2)) tors → H M ( E U , Z (2)) tors → H M ( E U , Z (1)) tors → . By Proposition 2.3 and Lemma 20, the group H M ( E U , Z (2)) tors is finite oforder p m | T ′ U, (1) | · | L ( h ( E ) , L ∗ ( h ( E U ) , L ( h ( C \ U ) , − | ( q − | L ( h (Irr( E U )) , − | for some m ∈ Z . By Lemma 22, the group H M ( E U , Z (1)) tors is finite of or-der | L ∗ ( h ( E U ) , | . By Lemma 18, the Picard scheme Pic o E / F q is an abelianvariety and, in particular, Hom(Pic o E / F q , G m ) = 0, hence by Theorem 2.2and Corollary 5, the group H M ( E , Z (2)) tors is of order | L ( h ( E ) , | . There-fore, | Coker ∂ U | = | H M ( E , Z (2)) tors |·| H M ( E U , Z (1)) tors || H M ( E U , Z (2)) tors | = p − m ( q − | L ( h (Irr( E U )) , − || T ′ U, (1) |·| L ( h ( C \ U ) , − | . Since | Coker ∂ U | is prime to p , we have m = 0. This proves claims (4) and(5) and completes the proof. (cid:3) Proof of Theorem 6.2. —
Claim (5) is clear. Claim (1) follows fromCorollary 1 and Theorem 1.3. We easily verify that H i M ( E U , Z (1)) is zero for i
0. By the localization sequence of higher Chow groups (cf. [6, Corollary
ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE H i M ( E , Z (2)) ∼ = H i M ( E U , Z (2)) for i
1. Taking theinductive limit with respect to U , we obtain claim (2).By Corollary 1, we have an exact sequence(6.3) 0 → H M ( E , Z (2)) tors α −→ H M ( E, Z (2)) red ∂ M , −−−→ M ℘ ∈ C H M ( E ℘ , Z (1)) → H M ( E , Z (2)) tors → H M ( E, Z (2)) red ∂ M , −−−→ M ℘ ∈ C H M ( E ℘ , Z (1)) → Pic( C ) → . Hence by Theorem 2.2 and Corollary 5, the group Ker ∂ M , is finite oforder | L ( h ( C ) , − | . For a non-empty open subscheme U ⊂ C , considerthe group Coker ∂ U in the proof of Theorem 6.3. For two non-empty opensubschemes U ′ , U ⊂ C with U ′ ⊂ U , the homomorphism Coker ∂ U → Coker ∂ U ′ is injective since both Coker ∂ U and Coker ∂ U ′ canonically injectinto H M ( E , Z (2)) tors . Claim (3) follows from claim (4) of Theorem 6.3 bytaking the inductive limit. Claim (4) follows from exact sequence (6.3) andLemma 16.From the localization sequence, it follows that the push-forward ho-momorphism L x ∈ E H M (Spec κ ( x ) , Z (2)) → H M ( E, Z (3)) is surjective,hence H M ( E, Z (3)) is a torsion group and claim (6) follows from Lemma 13.This completes the proof. (cid:3) Proof of Theorem 6.1. —
Consider the restriction γ : Ker c , → H M ( E, Z (2))of c , to Ker c , . By Lemma 10, both Ker γ and Coker γ are annihi-lated by the multiplication-by-2 map, which implies that the image of γ contains H M ( E, Z (2)) div and that Ext Z ( H M ( E, Z (2)) div , Ker γ ) is zero.From this, it follows that the homomorphism γ induces an isomorphism(Ker c , ) div ∼ = −→ H M ( E, Z (2)) div , which shows that the homomorphism K ( E ) red → H M ( E, Z (2)) red induced by c , is surjective with torsion ker-nel, thus claim (1) follows from Theorem 6.2 (3).Claim (3) follows from Theorem 6.2 (1) and Lemma 10.For ℘ ∈ C , let ι ℘ : Spec κ ( ℘ ) → E ℘ denote the reduction at ℘ of themorphism ι : C → E . Diagram (5.2) gives an exact sequenceCoker ∂ M , → Coker ∂ → Coker ∂ M , → . By Lemma 13, we have an isomorphism Coker ∂ M , ∼ = F × q . By the con-struction of this isomorphism, we see that the composition F × q ∼ = Coker ∂ M , → Coker ∂ ֒ → K ( E ) → K (Spec F q ) ∼ = F × q SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA equals the identity, hence the map Coker ∂ M , → Coker ∂ is injective.Then, claim (2) follows from Theorem 6.2 (3).From Proposition 4.1 and Lemmas 10, 11, and 23, it follows that the ho-momorphism ∂ : K ( E ) red → L ℘ ∈ C G ( E ℘ ) is identified with the directsum of the map ∂ ′ : k × → L ℘ ∈ C H M (Spec κ ( ℘ ) , Z (0)) → L ℘ H M ( E ℘ , Z (0))and the map ∂ M , : H M ( E, Z (2)) red → L ℘ H M ( E ℘ , Z (1)). We then haveisomorphismsKer ∂ ′ ∼ = F × q , Coker ∂ ′ ∼ = Pic( C ) ⊕ L ℘ Z | Irr( E ℘ ) |− , Ker ∂ M , ∼ = H M ( E , Z (2)) tors / Coker ∂ M , , Coker ∂ M , ∼ = Pic( C ) . Claim (4) follows, which completes the proof of Theorem 6.1. (cid:3)
7. Results for j > In this section, we obtain results for j >
3, generalizing the theorems ofSection 6. The proofs here are simpler than those of Section 6 in that we donot use tools such as the class field theory of Kato-Saito [29] or Theorem 1.3.We also refer the reader to Section 1.7 for remarks concerning the contentsof this section. Finally, we note that the notation we use is as in Section 6.
For integers i, j , consider the boundary map ∂ i M ,j : H i M ( E, Z ( j )) red → M ℘ ∈ C H i − M ( E ℘ , Z ( j − . Theorem 7.1. —
Let j > be an integer. (1) For any i ∈ Z , both Ker ∂ i M ,j and Coker ∂ i M ,j are finite groups. (2) We have | Ker ∂ i M ,j | = , if i or i > ,q j − , if i = 1 , | L ( h ( C ) , − j ) | , if i = 2 , | T ′ ( j − |·| L ( h ( E ) , − j ) | q j − − , if i = 3 , | L ( h ( C ) , − j ) | , if i = 4 . Further, the group
Ker ∂ M ,j is cyclic of order q j − . ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE
We have | Coker ∂ i M ,j | = , if i , i = 3 , or i > , q j − − | T ′ ( j − | , if i = 2 ,q j − − if i = 4 . (4) Let U ⊂ C be a non-empty open subscheme. Then, the group H i M ( E U , Z ( j )) is finite modulo a uniquely divisible subgroup forany i ∈ Z . The group H i M ( E U , Z ( j )) is zero if i > max(6 , j ) and isfinite for ( i, j ) = (4 , , (5 , , (4 , , (5 , , or (5 , . (5) H i M ( E U , Z ( j )) is uniquely divisible for i or i j , and H M ( E U , Z ( j )) tors is cyclic of order q j − . (6) Suppose that U = C (resp. U = C ). The group H M ( E U , Z ( j )) tors is of order | L ( h ( C ) , − j ) | (resp. of order | T ′ U, ( j − | · | L ( h ( C ) , − j ) L ( h ( C \ U ) , − j ) | q j − − . The group H M ( E U , Z ( j )) tors is of order | L ( h ( E ) , − j ) | (resp. oforder | T ′ U, ( j − | · | L ( h ( E ) , − j ) L ( h ( E U ) , − j ) L ( h ( C \ U ) , − j ) | ( q j − − | L ( h (Irr( E U )) , − j ) | ) . The group H M ( E U , Z ( j )) tors is of order | L ( h ( C ) , − j ) | (resp. oforder | L ( h ( C ) , − j ) L ( h ( C \ U ) , − j ) | q j − − . The group H M ( E U , Z ( j )) tors is cyclic of order q j − − (resp. iszero). Theorem 7.2. —
The following statements hold. (1)
The group K ( E ) div is uniquely divisible and c , induces an iso-morphism K ( E ) div ∼ = H M ( E, Z (2)) div . (2) The kernel of the boundary map ∂ : K ( E ) red → L ℘ ∈ C G ( E ℘ ) isa finite group of order | L ( h ( C ) , − | . Lemma 25. —
Let X be a smooth projective geometrically connectedcurve over a global field k ′ . Let k ′ ( X ) denote the function field of X . Then,the Milnor K -group K Mn ( k ′ ( X )) is torsion for n > X ) , and isof exponent two (resp. zero) for n > X ) if char( k ′ ) = 0 (resp. SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA char( k ′ ) > ). Here, gon( X ) denotes the gonality of X , i.e., the minimaldegree of morphisms from X to P k ′ .Proof. — The field k ′ ( X ) is an extension of degree gon( X ) of a subfield K of the form K = k ′ ( t ). From the split exact sequence0 → K Mn ( k ′ ) → K Mn ( K ) → M P K Mn − ( k ′ [ t ] /P ) → P runs over the irreducible monic poly-nomials in k ′ [ t ], and using [5, Chapter II, (2.1), p. 396], we observe that K Mn ( K ) is torsion for n > n > k ′ ) = 0 (resp. char( k ′ ) > K = V ⊂ V ⊂ · · · ⊂ V gon( X ) = k ′ ( X ) of K -subspaces of k ′ ( X ) with dim K V i = i . For each i , we set V ∗ i = V i \ { } .Suppose i > α, β ∈ V i \ V i − . Then, thereexist a, b ∈ K × such that γ = aα + bβ ∈ V i − . If γ = 0 (resp. γ = 0),then { aα, bβ } = 0 (resp. { aα/γ, bβ/γ } = 0) in K M ( k ′ ( X )). Expandingthis equality, we obtain an expression for { α, β } . We observe that { β, γ } belongs to the subgroup of K M ( k ′ ( X )) generated by { V ∗ i , V ∗ i − } , hencefor n > gon( X ) −
1, the group K Mn ( k ′ ( X )) is generated by the image of { V ∗ gon( X ) , . . . , V ∗ } × K Mn − gon( X )+1 ( K ). This proves the claim. (cid:3) Lemma 26. —
The push-forward homomorphism H M ( E U , Z ( j − → H M ( E , Z ( j )) is zero.Proof. — Consider the composition H M ( E U , Z ( j − → H M ( E , Z ( j )) f ∗ −→ H M ( C, Z ( j − H M ( C \ U, Z ( j − H M ( E U , Z ( j − f ∗ , tors : H M ( E , Z ( j )) tors → H M ( C, Z ( j − tors induced by f ∗ is an isomorphism.Next, consider the commutative diagram H M ( E , Z ( j )) tors f ∗ , tors −−−−→ H M ( C, Z ( j − tors ∼ = x ∼ = x H M ( E , Q / Z ( j )) −−−−→ H M ( C, Q / Z ( j − ∼ = y ∼ = yM ℓ = p H ( E , Q ℓ / Z ℓ ( j )) G F q −−−−→ M ℓ = p H ( C, Q ℓ / Z ℓ ( j − G F q . ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE → Z → Q → Q / Z →
0, and the lower vertical arrows are those obtained from Theo-rem 2.2(2)(b) (the same argument also applies to curves) further using theweight argument. The homomorphism at the bottom is an isomorphism byLemma 18, hence f ∗ , tors is an isomorphism, as desired. (cid:3) Proof of Theorem 7.1. —
From Theorem 2.2(2) and Lemma 25, claims(4) and (5) follow. Claim (6) follows from Theorem 2.2(2) and Corollary 7.Using an approach similar to that of the proof of Corollary 1, we are ableto show that the pull-back map induces an isomorphism H i M ( E , Z ( j )) div ∼ = H i M ( E, Z ( j )) div for all i ∈ Z , and that the localization sequence induces along exact sequence(7.1) · · · → M ℘ ∈ C H i − M ( E ℘ , Z ( j − → H i M ( E , Z ( j )) tors → H i M ( E, Z ( j )) tors → · · · . Using the theorem introduced by Rost and Voevodsky (i.e., Theorem 2.1above), we observe that for any ℘ ∈ C , even if E ℘ is singular, the group H i M ( E ℘ , Z ( j − i , is zero for i i >
4, and is cyclic oforder q j − ℘ −
1, where q ℘ = | κ ( ℘ ) | is the cardinality of the residue field at ℘ , for i = 1. From the exact sequence (7.1) and Lemma 26, we can deduceclaims (1), (2), and (3) from claims (4), (5), and (6), thus completing theproof. (cid:3) Proof of Theorem 7.2. —
Let U = C . Then, by Lemmas 13 and 26, thefollowing sequence is exact:0 → H M ( E , Z (3)) → H M ( E U , Z (3)) ∂ −→ H M ( E U , Z (2)) α −→ F × q → . Here, ∂ and α are as in Lemma 13, and the second map is the pull-back.By taking the inductive limit, we obtain an exact sequence(7.2) 0 → H M ( E , Z (3)) → H M ( E, Z (3)) ∂ M , −−−→ L ℘ ∈ C H M ( E ℘ , Z (2)) → F × q → . By Theorem 2.2 and Corollary 1, the group H M ( E, Z (3)) div is zero,hence using Lemma 10, we obtain K ( E ) div ⊂ Ker c , . From the proof ofTheorem 6.1, we saw that map c , induces an isomorphism (Ker c , ) div ∼ = −→ H M ( E, Z (2)) div , hence c , induces an isomorphism K ( E ) div ∼ = H M ( E, Z (2)) div , SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA which proves claim (1). Claim (2) follows from Theorems 6.1 and 6.2, thecommutative diagram (5.2), and exact sequence (7.2). This completes theproof. (cid:3)
Appendix A. A proposition on the p -part The aim of this Appendix is to provide a proof of Proposition A.1 below,which we used in the proof of Theorem 2.2. Nothing in this Appendix isnew except for the definition of the Frobenius map on the inductive limit(not on the inverse limit) given in Section A.3. A similar presentation hasalready been provided in the work of Milne [42] and Nygaard [51].
Proposition A.1. —
Let X be a smooth projective geometrically con-nected surface over a finite field F q of cardinal q of characteristic p . Let W n Ω iX, log denote the logarithmic de Rham-Witt sheaf (cf. [26, I, 5.7, p. 596] ).Then, the inductive limit lim −→ n H ( X, W n Ω X, log ) with respect to multiplication-by- p is finite of order | Hom(Pic oX/ F q , G m ) | − p ·| L ( h ( X ) , | − p . Here, Hom(Pic oX/ F q , G m ) denotes the set of homomorphisms Pic oX/ F q → G m of F q -group schemes, and L ( h ( X ) , s ) is the (Hasse-Weil) L -function of h ( X ) . A.1. The de Rham-Witt complex
In this Appendix, let k be a perfect field of characteristic p . Let X be ascheme of dimension δ that is proper over Spec k . For i, n ∈ Z , let W n Ω • X denote the de Rham-Witt complex (cf. [26, I, 1.12, p. 548]) of the ringedtopos of schemes over X with Zariski topology. We let R : W n Ω iX → W n − Ω iX , F : W n Ω iX → W n − Ω iX , and V : W n Ω iX → W n +1 Ω iX denotethe restriction, the Frobenius, and the Verschiebung, respectively. For each i ∈ Z , the sheaf W n Ω iX has a canonical structure of coherent W n O X -module, which enables us to regard W n Ω iX as an ´etale sheaf. Therefore, inthis Appendix, we focus on the category of ´etale sheaves on schemes over X . A.2. Logarithmic de Rham-Witt sheaves
For n ∈ Z , let W n Ω iX, log ⊂ W n Ω iX denote the logarithmic de Rham-Wittsheaf (cf. [26, I, 5.7, p .596]). ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE Lemma 27. —
The homomorphism V : W n Ω iX → W n +1 Ω iX sends W n Ω iX, log into W n +1 Ω iX, log .Proof. — Let x ∈ W n Ω iX, log be an ´etale local section. By the definitionof W n Ω iX, log , there exists an ´etale local section y ∈ W n +1 Ω iX, log such that x = Ry . We observe that Ry = F y , hence
V x = V Ry = V F y = py ∈ W n +1 Ω iX, log . (cid:3) Let CW Ω iX denote the inductive limit CW Ω iX = lim −→ n, V W n Ω iX withrespect to V . The above lemma enables us to define the inductive limit CW Ω iX, log = lim −→ n, V W n Ω iX, log . A.3. Modified Frobenius operator
In this subsection we define an operator F ′ : CW Ω iX → CW Ω iX suchthat the sequence(A.1) 0 → CW Ω iX, log → CW Ω iX − F ′ −−−→ CW Ω iX → n >
0, let f W n Ω iX denote the cokernel of the homomorphism V n :Ω iX = W Ω iX → W n +1 Ω iX . The homomorphisms R , F and V on W n +1 Ω iX induce homomorphisms on f W n Ω iX , which we denote using the same no-tation. If n >
1, the homomorphisms
R, F : W n +1 Ω iX → W n Ω iX fac-tor through the canonical surjection W n +1 Ω iX → f W n Ω iX . We let e R, e F : f W n Ω iX → W n Ω iX denote the induced homomorphisms. Then, both e R and e F commute with R , F and V . For n >
0, we let f W n Ω iX, log denote the imageof W n +1 Ω iX, log by the canonical surjection W n +1 Ω iX → f W n Ω iX . The restric-tion of e R : f W n Ω iX → W n Ω iX to f W n Ω iX, log gives a surjective homomorphism e R log : f W n Ω iX, log → W n Ω iX, log . Lemma 28. —
The homomorphisms e R , e R log induce isomorphisms lim −→ n > , V f W n Ω iX ∼ = CW Ω iX , lim −→ n > , V f W n Ω iX, log ∼ = CW Ω iX, log . Proof. —
Surjectivity here is clear. From [26, I, Proposition 3.2, p. 568], itfollows that the kernel of e R equals the image of the composition W Ω iX dV n −−−→ W n +1 Ω iX ։ f W n Ω iX . Since V d = pdV , we have V (Ker e R ) = 0. This provesthe injectivity. (cid:3) SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA
We observe that f W n Ω iX, log is contained in the kernel of e R − e F : f W n Ω iX → W n Ω iX , hence(A.2) 0 → f W n Ω iX, log → f W n Ω iX e R − e F −−−→ W n Ω iX → Lemma 29. —
The inductive limit → lim −→ n > , V f W n Ω iX, log → lim −→ n > , V f W n Ω iX → CW Ω iX → of (A.2) with respect to V is exact.Proof. — The argument in the proof of [26, I, Th´eor`eme 5.7.2, p. 597]shows that the kernel of R − F : W n +1 Ω iX → W n Ω iX is contained in W n +1 Ω iX, log + Ker R , hence the claim follows from Lemma 28. (cid:3) The inductive limit of e F : f W n Ω iX → f W n +1 Ω iX gives the endomorphism F ′ : CW Ω iX ∼ = lim ←− n > , V f W n Ω iX → CW Ω iX . By Lemmas 28 and 29, wehave a canonical exact sequence (A.1). A.4. The duality
Let H ∗ ( X, W n Ω iX ) denote the cohomology groups of W n Ω iX with respectto the Zariski topology.The trace map Tr : H δ ( X, W n Ω δX ) ∼ = W n ( F q ) is defined in [25, 4.1.3,p. 49]. This commutes with homomorphisms R , F and V . For 0 i, j δ ,the product m : W n Ω iX × W n Ω δ − iX → W n Ω δX gives a W n ( k )-bilinear paring( , ) : H j ( X, W n Ω iX ) × H δ − j ( X, W n Ω δ − iX ) → H δ ( X, W n Ω δX ) Tr −→ W n ( k ) . By [25, Corollary 4.2.2, p. 51], this pairing is perfect.Since m ◦ (id ⊗ V ) = V ◦ m ◦ ( F ⊗ id), the diagram W n +1 Ω iX × W n +1 Ω δ − iX −−−−→ W n +1 ( k ) F y V x V x W n Ω iX × W n Ω δ − iX −−−−→ W n ( k )is commutative, hence this induces an isomorphism(A.3) H δ − j ( X, CW Ω δ − iX ) ∼ = lim −→ n Hom W n ( k ) ( H j ( X, W n Ω iX ) , W n ( k )) , where the transition map in the inductive limit of the right hand side isgiven by f V ◦ f ◦ F . We endow each H j ( X, W n Ω iX ) with the discrete ANNALES DE L’INSTITUT FOURIER AND K OF AN ELLIPTIC CURVE H j ( X, W ′ Ω iX ) = lim ←− n, F H j ( X, W n Ω iX ) and endow it withthe induced topology. Next, we turn H j ( X, W ′ Ω iX ) into a W ( k )-module byletting a · ( b n ) = ( σ − n ( a ) b n ) for a ∈ W ( k ), b n ∈ H j ( X, W n Ω iX ). We set D = lim −→ n, V W n ( k ) and endow it with the discrete topology. We turn D intoa W ( k )-module by letting a · c n = σ − n ( a ) c n for a ∈ W ( k ), c n ∈ W n ( k ).Then, the right hand side of (A.3) equals Hom W ( k ) , cont ( H j ( X, W ′ Ω iX ) , D ).The homomorphism R : H j ( X, W n Ω iX ) → H j ( X, W n − Ω iX ) induces anendomorphism R ′ : H j ( X, W ′ Ω iX ) → H j ( X, W ′ Ω iX ). The Frobenius endo-morphism σ : W n ( k ) → W n ( k ) induces an endomorphism σ : D → D . Lemma 30. —
Under the isomorphism (A.3), the endomorphism F ′ : H δ − j ( X, CW Ω δ − iX ) → H δ − j ( X, CW Ω δ − iX ) is identified with the endomorphism of Hom W ( k ) , cont ( H j ( X, W ′ Ω iX ) , D ) ,that sends a homomorphism f : H j ( X, W ′ Ω iX ) → D to the homomorphism σ ◦ f ◦ R ′ .Proof. — The proof here is immediate from the definition of the isomor-phism (A.3) and the module D . (cid:3) A.5. The degree zero case
We are primarily concerned with the case in which i = 0. We denote H j ( X, W ′ Ω X ) by H j ( X, W ′ O X ). Recall that F : W n Ω X → W n − Ω X equals the composition W n O X σ −→ W n O X R −→ W n − O X . From [26, II,Proposition 2.1, p. 607], it follows that H j ( X, W Ω iX ) → lim ←− n, R H j ( X, W n Ω iX )is an isomorphism, hence H j ( X, W ′ O X ) is isomorphic to the projectivelimit e H j ( X, W O X ) = lim ←− [ · · · σ −→ H j ( X, W O X ) σ −→ H j ( X, W O X )] . The endomorphism σ : H j ( X, W O X ) → H j ( X, W O X ) induces an auto-morphism σ : e H j ( X, W O X ) ∼ = −→ e H j ( X, W O X ). We observe here that theendomorphism R ′ on H j ( X, W ′ O X ) corresponds to the endomorphism σ − on e H j ( X, W O X ).Let K = Frac W ( k ) denote the field of fractions of W ( k ). The homo-morphism σ n /p n : W n ( k ) → K/W ( k ) for each n > D ∼ = K/W ( k ) of W ( k )-modules that commutes with the ac-tion of σ . SUBMITTED ARTICLE : 2011K1K22ND41.TEX SATOSHI KONDO AND SEIDAI YASUDA
A.6. Proof of Proposition A.1
Suppose that k = F q . Then by Lemma 30, H ( X, CW Ω dX ) is isomorphicto the Pontryagin dual of e H δ ( X, W O X ), hence the group H ( X, CW Ω δX, log ) ∼ = Ker[ H ( X, CW Ω δX ) − F ′ −−−→ H ( X, CW Ω δX )]is isomorphic to the Pontryagin dual of the cokernel of 1 − σ − on e H δ ( X, W O X ). Proposition A.2. —
Let k = F q be a finite field and X be a scheme ofdimension δ that is smooth and projective over Spec k . Suppose that the V -torsion part T of H δ ( X, W O X ) is finite. Then, H ( X, CW Ω δX ) is a finitegroup of order | T σ | · | L ( h δ ( X ) , | − p . Here, T σ denotes the σ -invariant partof T .Proof. — By the argument above, the order of H ( X, CW Ω δX ) equalsthe order of the cokernel of 1 − σ on e H δ ( X, W O X ) if it is finite. Thetorsion subgroup of e H δ ( X, W O X ) is finite since it injects into T . By [26,II, Corollaire 3.5, p. 616], e H δ ( X, W O X ) ⊗ Z p Q p is isomorphic to the slope-zero part of H δ crys ( X/W ( k )) ⊗ Z p Q p , hence the claim follows. (cid:3) Proof of Proposition A.1. —
Let the notation be as above and supposethat δ = 2. Then, by [26, II, Remarque 6.4, p. 641], the module T in theabove proposition is canonically isomorphic to the groupHom W ( F q ) ( M (Pic oX/ F q / Pic oX/ F q , red ) , K/W ( F q )) , where M ( ) denotes the contravariant Dieudonn´e module functor. In par-ticular, T is a finite group. Let T σ denote the σ -coinvariants of T . Then,by Dieudonn´e theory (cf. [12]), Hom W ( F q ) ( T σ , K/W ( F q )) is canonically iso-morphic to Hom(Pic oX/ F q , G m ), hence the claim follows from Proposition A.2. (cid:3) BIBLIOGRAPHY [1]
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