Conjectures on L-functions for flag bundles on Dedekind domains
aa r X i v : . [ m a t h . AG ] A ug CONJECTURES ON L-FUNCTIONS FOR PARTIAL FLAGBUNDLES ON DEDEKIND DOMAINS
HELGE ¨OYSTEIN MAAKESTAD
Abstract.
The aim of this paper is to give evidence for the Beilinson-Soul´evanishing conjecture and Soul´e conjecture on L-functions for partial flag bun-dles over Dedekind domains. Let O K be the ring of integers in an algebraicnumber field K with S := Spec( O K ). Let T , . . . , T n be regular schemes offinite type over S , and let X be a scheme of finite type over T n with a strati-fication (a generalized cellular decomposition) of closed subschemes ∅ = X − ⊆ X ⊆ · · · ⊆ X n − ⊆ X n := X with X i − X i − = E i , where E i is a vector bundle of rank d i on T i . We provethat if the Beilinson-Soul´e vanishing conjecture and the Soul´e conjecture on L-functions holds for T i , it follows the same conjectures hold for X . As Corollarywe prove the Beilinson-Soul´e vanishing conjecture and the Soul´e conjecture onL-functions for any partial flag bundle F ( N, E ) where E is a locally trivial O S -module of rank n . We also reduce the study of the Beilinson-Soul´e vanishingconjecture and the Soul´e conjecture on L-functions to the study of affine regularschemes of finite type over Z . Hence we get an approach to the Birch andSwinnerton-Dyer conjecture for abelian schemes using affine regular schemesof finite type over Z . For a partial flag bundle F ( N, E ) over O K we give anexplicit formula for the L-funtion in terms of the L-function of O K . Hence theBloch-Kato conjecture on the Tamagawa number of F ( N, E ) is reduced to thestudy of the same conjecture for O K . Contents
1. Introduction 22. Algebraic K-theory and Adams operations 33. Reduction of the Beilinson-Soul´e vanishing conjecture and Soul´econjecture on L-functions to the affine regular case 54. Conjectures on L-functions for projective bundles and flag bundles onDedekind domains. 125. Appendix A: The weight space decomposition for algebraic K-theory ofprojective bundles 226. Appendix B: Some general properties of formal power series 26References 27
Date : June 2020.1991
Mathematics Subject Classification.
Key words and phrases. algebraic K-theory, vanishing, L-function, special values, Beilinson-Soul´e vanishing conjecture, Tate, Lichtenbaum, Deligne, Bloch, Beilinson conjecture. Introduction If O K is the ring of integers in an algebraic number field K , it follows the rankof the m’th K-group K ′ m ( O K ) and the rank of the weight space K ′ m ( O K ) ( i ) is wellknown for all integers m ≥ , i ≥ O K , s ) of O K and the K-theoretic j’th Euler characteristic χ ( O K , j ) := X m ≥ ( − m +1 dim Q (K ′ m ( O K ) ( j ) ) . (1.0.1)The function L( O K , s ) is the well known Dedekind L-function of the number field K . Borel proved in [3] that χ ( O K , j ) is an integer for any number field K and any integer j .(1.0.2)He also proved the relationship χ ( O K , j ) = ord s = j (L( O K , s ))(1.0.3)between the Euler characteristic and L-function of O K . In the litterature Conjec-ture 1.0.2 is referred to as the Beilinson-Soul´e vanishing conjecture, and Conjecture1.0.3 is referred to as the Soul´e conjecture. In the paper [19] Conjectures 1.0.2 and1.0.3 are formulated for any quasi projective scheme of finite type over Z . The aimof this paper is to prove the Beilinson-Soul´e vanishing conjecture and the Soul´econjecture for a class of schemes of finite type over the ring O K called partial flagbundles.Let k be a field and W an n -dimensional vector space over k . Let N := { n , .., n l } be a sequence of positive integers with P i n i = n and let F ( N, W ) be the flag varietyof flags of type N in W . It follows the set of k -rational points of F ( N, W ) are inone-to-one correspondence with the set of flags { W i } of type N in W . A flag oftype N in W is a sequence of k -vector spaces W ⊆ W ⊆ · · · ⊆ W l − ⊆ W with dim k ( W i ) = n + · · · + n i . If l = 2 and n < n , it follows the flag variety F ( N, W ) is the grassmannian variety G ( n , W ) of n -dimensional sub spaces of W .A partial flag bundle is a relative version of F ( N, W ). Let S be a scheme and let E be a locally trivial O S -module of rank n . The flag bundle F ( N, E ) is a schemeequipped with a surjective morphism of schemes π : F ( N, E ) → S such that the fiber π − ( s ) at any point s ∈ S is isomorphic to the flag variety F ( N, E ( s )) of flags of κ ( s )-vector spaces of type N in the fiber E ( s ) of E at s .Let T be a regular scheme of finite type over O K and X be a scheme of finitetype over T with a cellular decomposition X i ⊆ X of closed subschemes, such that X i − X i − is a finite disjoint union of affine space A iT over T . In Theorem 4.15 weprove that if the Beilinson-Soul´e vanishing conjecture and the Soul´e conjecture onL-functions holds for T , it follows the same conjectures hold for X . In particularit follows there is an equality of integers χ ( X, j ) = ord s = j (L( X, s ))(see Theorem 4.15). We prove a similar result for any scheme X equipped witha generalized cellular decomposition in Lemma 4.20. As a Corollary we provethe Beilinson-Soul´e vanishing conjecture and the Soul´e conjecture on L-functionsfor any partial flag bundle F ( N, E ) on Spec( O K ) (see Corollary 4.16). This gives ONJECTURES ON L-FUNCTIONS FOR PARTIAL FLAG BUNDLES ON DEDEKIND DOMAINS3 an infinite number of non-trivial examples of partial flag bundles F ( N, E ) whereConjecture 3.3.1 and 3.3.3 hold (see Example 4.17). If A is an abelian scheme over O K it follows the Soul´e Conjecture 3.3.3 for A is one way to formulate a version ofthe Birch and Swinnerton-Dyer conjecture for A using algebraic K-theory.We reduce the study of the Beilinson-Soul´e vanishing conjecture and the Soul´econjecture on L-functions to the study of affine regular schemes of finite type over Z . Hence we get an approach to the Birch and Swinnerton-Dyer conjecture forabelian schemes using affine regular schemes of finite type over Z .We use the projective bundle formula for algebraic K-theory and an elementaryconstruction of eigenvectors for the Adams operator to calculate the weight spaceK ′ m ( P ( E ∗ )) ( i ) for any pair of integers m ≥ , i ≥
1, any finite rank projective O K -module E forany algebraic number field K (see Theorem 5.4). This illustrates the possiblility todo explicit computations for the K-theory of projective bundles and more generalflag bundles. 2. Algebraic K-theory and Adams operations
Let O K be the ring of integers in an algebraic number field K and let S =Spec( O K ). Let X be a scheme of finite type over S .In this section we introduce some notation from Soul´e ’s original paper [19]: Let M ( X ) denote the category of coherent O X -modules and let BQM ( X ) denote thesimplicial classifying set of M ( X ). Let BQP ( X ) denote the simplicial classifyingset of P ( X ), where P ( X ) is the category of locally trivial finite rank O X -modules.By definition K m ( X ) := π m +1 ( BQM ( X ))(2.0.1) K m ( X ) := π m +1 ( BQP ( X ))(2.0.2)where m is an arbitrary integer. If X is a regular scheme it follows K m ( X ) =K m ( X ) and K m ( X ) = 0 for m <
0. Assume X is a scheme of finite type over Z andassume u : X → M is a closed immersion into a scheme M where M is a regularscheme of finite type over Z of dimension D . Define K Xm ( M ) as the homotopy groupof the fiber of the canonical map BQP ( M ) → BQP ( M − X ) Definition 2.1. If Y is a regular scheme of finite type over Z , there is for everypositive integer k ≥ ψ k : K m ( Y ) → K m ( Y )with the following properties: If L is the class of a line bundle in K ( Y ) it follows ψ k ( L ) := L k . The map ψ k is the k ’th Adams operator for K m ( Y ).The map ψ k is functorial in the sense that for any map p : Y → Y ′ of regularschemes Y, Y ′ of finite type over Z it follows ψ k ( p ∗ x ) = p ∗ ( ψ k ( x )) HELGE ¨OYSTEIN MAAKESTAD for any element x ∈ K m ( Y ′ ). The abelian group K ∗ ( Y ) := ⊕ m ≥ K m ( Y ) is a gradedcommutative ring and the endomorphism ψ k : K ∗ ( Y ) → K ∗ ( Y )is a ring homomorphism: ψ k ( xy ) = ψ k ( x ) ψ k ( y ) for any x ∈ K m ( Y ) , y ∈ K n ( Y ).The operation ψ k induce canonically a ring homomorphism ψ k : K ∗ ( Y ) ⊗ Q → K ∗ ( Y ) ⊗ Q (let K m ( Y ) Q := K m ( Y ) ⊗ Q ) and we defineK m ( Y ) ( i ) Q := { x ∈ K m ( Y ) Q : such that ψ k ( x ) = k i x . } There is a direct sum decompositionK m ( Y ) Q ∼ = ⊕ i ∈ Z K m ( Y ) ( i ) Q and the space K m ( Y ) ( i ) Q is independent of choice of positive integer k . By definitionwe let K m ( X ) ( i ) := K Xm ( M ) ( D − i ) Q . (2.1.1)When X is regular we may choose M = X . It followsK m ( X ) ( i ) = K m ( X ) ( D − i ) Q where D = dim ( X ). Hence when X is a regular scheme of finite type over Z wemay use the K-theory of the category P ( X ) of finite rank algebraic vector bundleson X and the Adams operations on K m ( X ) Q to calculate the group K m ( X ) ( i ) introduced in Soul´e s paper. Definition 2.2.
Let X be a scheme of finite type over Z and let i : X → M be aclosed embedding into a regular scheme M of finite type over Z with D := dim ( M ).Define K ′∗ ( X ) := ⊕ m ∈ Z K Xm ( M )and K ′∗ ( X ) Q := K ′∗ ( X ) ⊗ Q . Define K ′ m ( X ) ( j ) := K Xm ( M ) ( D − j ) Q . The Q -vector spaceK ′ m ( X ) ( j ) is the weight space of weight j .The following result calculates K m ( O K ) Q and K m ( O K ) ( i ) Q for all m, i : Theorem 2.3.
Let K be a number field with ring of integers O K and real andcomplex places r , r . The following holds: K m ( O K ) Q = 0 for all m < ( O K ) Q = Q (2.3.2) K m ( O K ) Q = 0 for m = 2 i, i = 0(2.3.3) K m ( O K ) Q = Q r + r for m ≡ mod m ( O K ) Q = Q r for m ≡ mod . (2.3.5) Moreover K i − ( O K ) ( i ) Q = Q r + r for i ≡ mod i − ( O K ) ( i ) Q = Q r for i ≡ mod . (2.3.7) ONJECTURES ON L-FUNCTIONS FOR PARTIAL FLAG BUNDLES ON DEDEKIND DOMAINS5
The history of the calculation of the groups K ′ m ( O K ) and K ′ m ( O K ) Q is long andcomplicated, and the reader should consult to [15] and [18]. The calculation ofK ′ m ( O K ) ⊗ Q for m = 0 follows from the fact the ideal class group of O K is finite,a result going back to Minkowski. The case m = 1 is Dirichlet’s unit theorem. For m ≥ Reduction of the Beilinson-Soul´e vanishing conjecture and Soul´econjecture on L-functions to the affine regular case
In this section we reduce the study of the Beilinson-Soul´e vanishing conjectureand Soul´e ’s conjecture on L-functions to the study of affine regular schemes offinite type over Z . Let K be an algebraic number field with ring of integers O K and let S := Spec( O K ). Definition 3.1.
Let X be a quasi projective scheme of finite type over S and let i ∈ Z . Let χ ( X, i ) := X m ∈ Z ( − m +1 dim Q (K m ( X ) ( i ) )be the Euler characteristic of X of type i . Definition 3.2.
Let X be a scheme of finite type over S . LetL( X, s ) := Y x ∈ X cl − N ( x ) − s be the L-function of X . Here we view s as a complex variable and the infiniteproduct is taken over the set of closed points x in X cl . By definition N ( x ) := κ ( x )where κ ( x ) is the residue field of x .Note: Since X is of finite type over Z and x is a closed point, it follows κ ( x ) isa finite field. Example 3.3.
The Dedekind L-function. If K is an algebraic number field with ring of integers O K and S := Spec( O K ),it follows L( S, s ) is the
Dedekind L-function of K . In particular L(Spec( Z ) , s ) isthe Riemann zeta function .In Soul´e s paper [19] the following conjecture is stated:
Conjecture 1. (Conjecture 2.2 in [19] ) Let X be a quasi projective scheme of finitetype over Z and let i ∈ Z be an integer.For fixed integer i the group K m ( X ) ( i ) is zero for almost all integers m . (3.3.1) dim Q (K m ( X ) ( i ) ) is finite for all m, i . (3.3.2) χ ( X, i ) = ord s = i (L( X, s )) for all i ∈ Z (3.3.3)Note: The Conjecture 3.3.3 is mentioned in Wiles’ CLAY Math description of the Birch and Swinnerton-Dyer conejcture (one of the
Millenium Problems , see [23]).In [23] Conjecture 3.3.3 is referred to as due to Tate, Lichtenbaum, Deligne, Bloch,Beilinson and others. Conjecture 3.3.1 is sometimes referred to as the
Beilinson-Soul´e vanishing conjecture . If E is a relative elliptic curve over O K , it followsConjecture 3.3.3 is a version of the Birch and Swinnerton-Dyer conjecture for E using K-theory. The version given in [23] is formulated for an elliptic curve E over Q HELGE ¨OYSTEIN MAAKESTAD and the group of rational points E ( Q ) of E . There is an embedding E ( Q ) ⊆ Pic( E )and K ( E ) = Pic( E ) ⊕ Z , hence the conjecture in [23] is similar to Conjecture3.3.3. Hence we may view the conjecture mentioned in [23] as a special case ofConjecture 3.3.3. Note morover that if X red is the reduced scheme of X it followsL( X, s ) = L( X red , s ) and K ′ m ( X ) = K ′ m ( X red ), hence Conjecture 1 holds for X ifand only if it holds for X red . Lemma 3.4.
Let U be a scheme over Z and let k be an integer. It follows χ ( U, k ) is an integer if and only if Conjecture 3.3.1 holds for i = k .Proof. The proof is immediate. (cid:3)
Example 3.5.
Conjecture 1 for Dedekind L-functions. If S := Spec( O K ) with K an algebraic number field, it follows 3.3.1, 3.3.2 and3.3.3 holds by the work of Borel [3]. Example 3.6.
Conjecture 1 for finite fields.
Let k be a finite field. It follows K ′ m ( k ) Q = 0 hence K ′ m ( k ) ( j ) = 0 for all integers m, j , and it follows 3.3.1 holds for S := Spec( k ). One also checks 3.3.3 holds for S .Note: In the case when X is a regular scheme of dimension D it follows there isan equality of groups K m ( X ) ( i ) ∼ = K m ( X ) ( D − i ) Q K m ( X ) is the K-theory of the category P ( X ) of locally trivial finite rank O X -modules.Recall the following results: Lemma 3.7.
Let X be of finite type over S with X = U ∪ V a disjoint union oftwo subschemes U, V . It follows L( X, s ) = L(
U, s ) L(
V, s ) . If U ⊆ X is an opensubscheme with Z := X − U it follows L( X, s ) = L(
U, s ) L(
Z, s ) . Assume X, Y are schemes of finite type over S such that for any closed point t ∈ S there isan isomorphism X t ∼ = Y t of fibers. It follows there is an equality of L-functions L( X, s ) = L(
Y, s ) . There is an equality L( A dX , s ) = L( X, s − d ) . More generally if E is a vector bundle of rank d on X it follows L( E, s ) = L(
X, s − d ) .Proof. Assume we may write X as a disjoint union X = U ∪ V . It follows X cl = U cl ∪ V cl . We getL( X, s ) = Y x ∈ X cl − N ( x ) − s = Y x ∈ U cl − N ( x ) − s Y x ∈ V cl − N ( x ) − s =L( U, s ) L(
V, s )and the first claim follows. We moreover getL(
X, s ) = Y x ∈ X cl − N ( x ) − s = Y t ∈ S cl Y x ∈ X clt − N ( x ) − s = Y t ∈ S cl L( X t , s ) = Y t ∈ S cl L( Y t , s ) = L( Y, s ) . By Exercise 5.3 and 5.4 in Appendix C in [13] we get the following: If T is ascheme of finite type over a finite field k with q elements and Z ( T, t ) is the Weil zeta
ONJECTURES ON L-FUNCTIONS FOR PARTIAL FLAG BUNDLES ON DEDEKIND DOMAINS7 function of T , then Z ( T × k A dk , t ) = Z ( T, q d t ). Moreover L( T, s ) = Z ( T, q − s ). Weget the following: If x ∈ X is a closed point and T := Spec( κ ( x )) and q := κ ( x ),it follows the fiber of the map p : A dX → X at T is A dT . It follows L( A dX , s ) = Y x ∈ X cl L( A dT , s ) . We getL( A dT , s ) = Z ( A dT , q − s ) = Z ( T, q d q − s ) = Z ( T, q − ( s − d ) ) = L( T, s − d ) . It follows L( A dX , s ) = Y x ∈ X cl L( A dT , s ) = Y x ∈ X cl L( T, s − d ) = L( X, s − d ) . Since A dX and E have the same fibers it follows L( E, s ) = L( A dX , s ), hence L( E, s ) =L(
X, s − d ) . The Lemma follows. (cid:3)
Example 3.8.
A cohomological description of the local L-factors of L( X, s ) . Assue π : X → S is a scheme of finite type over S where S := Spec( O K ) and K is a number field. Assume X s := π − ( s ) is smooth and projective for any closedpoint s ∈ S . We may express the L-function L( X s , s ) of the fiber X s in terms ofthe Weil zeta function ζ X s ( t ) as follows:L( X s , s ) = ζ X s ( q − s )where κ ( s ) = F q and q = p n with p > X is a smooth projective schemeof finite type over κ ( s ), it is well known the function ζ X ( t ) is a rational function ζ X ( t ) = P ( t ) · · · P n − ( t ) P ( t ) · · · P n ( t )where n := dim ( X ). There is moreover a determinantal formula P i ( t ) = det(1 − f ∗ t ; H i ( X, Q l ))(3.8.1)where f : X → X is the Frobenius morphism and H i ( X, Q l ) is l-adic etale cohomol-ogy (see [7], [13]). Hence the global L-function L( X, s ) is calculated by the l-adicetale cohomology groups H ∗ ( X s , Q l s ) for varying primes l s = char ( κ ( s )) via theformula L( X, s ) = Y s ∈ S cl ζ X s ( q − s )and Formula 3.8.1. The determinantal formula 3.8.1 may be proved using otherp-adic cohomology theories (rigid cohomology, cristalline cohomology, prismaticcohomology etc.). The rationality of the zeta function ζ X ( t ) was first proved byDwork in [7] in 1960 using p-adic methods. Corollary 3.9.
Let
E, F be locally trivial O K -modules of rank d + 1 . It follows L( P ( E ∗ ) , s ) = L( P ( F ∗ ) , s ) . Assume T is a regular scheme of finite type over O K and A dT is affine d-space over T . It follows conjecture 3.3.3 holds for T if and onlyif holds for A dT . More generally: If π : E → T is a vector bundle of rank l on T isfollows 3.3.3 holds for T if and only if it holds for E . Moreover L( P ( E ∗ ) , S ) = L( S, s ) L(
S, s − · · · L( S, s − d ) . (3.9.1) HELGE ¨OYSTEIN MAAKESTAD
Proof.
Since P ( E ∗ ) and P ( F ∗ ) have the same fibers, it follows from Lemma 3.7there is an equality L( P ( E ∗ ) , s ) = L( P ( F ∗ ) , s )Let dim ( T ) = n . We get χ ( A dT , k ) = X m ∈ Z ( − m +1 dim Q (K ′ m ( A dT ) ( k ) ) = X m ∈ Z ( − m +1 dim Q (K m ( A dT ) ( d + n − k ) Q ) = X m ∈ Z ( − m +1 dim Q (K m ( T ) ( n − ( k − d )) Q ) = X m ∈ Z ( − m +1 dim Q (K ′ m ( T ) ( k − d ) ) = χ ( T, k − d ) . Hence χ ( A dT , k ) = χ ( T, k − d ) . Assume ord s = k (L( T, s )) = χ ( T, k ). We get ord s = k (L( A dT , s )) = ord s = k (L( T, s − d )) . Let t := s − d , we get ord t = k − d (L( T, t )) = χ ( T, k − d ) = χ ( A dT , k )hence the conjecture holds for A dT . The converse is proved similarly. Since L( E, s ) =L( A lT , s ) and χ ( E, k ) = χ ( A lT , k ), it follows 3.3.3 holds for E if and only if it holdsfor A lT which is if and only if 3.3.3 holds for T . There is a stratification ∅ = X − ⊆ X ⊆ · · · ⊆ X d := P ( E ∗ )with X i − X i − = A iS . Since L( X i − X i − , s ) = L( A iS , s ) = L( S, s − i ) Formula 3.9.1follows using induction. The Corollary is proved. (cid:3) Corollary 3.10.
Let U be a scheme over Z with χ ( U, k ) an integer for all k ∈ Z .Let E be a locally trivial O U -module of rank d + 1 . It follows χ ( P ( E ∗ ) , j ) = d X i =0 χ ( U, k − i ) . Hence it follows χ ( P ( E ∗ ) , k ) is an integer for all k ∈ Z .Proof. Since P ( E ∗ ) and P dU have the same fibers it follows χ ( P ( E ∗ ) , k ) = χ ( P dU , k ).By induction it follows χ ( P dU , j ) = d X i =0 χ ( U, k − i )and the Corollary follows since χ ( U, k ) is an integer for all integers k . (cid:3) The following Lemma is by some authors referred to as the
Jouanolou trick : Lemma 3.11.
Let T := Spec( B ) be an affine scheme of finite type over Z and let X ⊆ P nT be a quasi projective scheme over T . It follows there is an affine scheme W := Spec( B ) and a surjective map π : W → X where the fibers of π is affine l -space A l .Proof. This is proved in [14], Lemma 1.5. (cid:3)
ONJECTURES ON L-FUNCTIONS FOR PARTIAL FLAG BUNDLES ON DEDEKIND DOMAINS9
The affine A l -fibration W constructed in Lemma 3.11 is an affine torsor for U .Note that if U ⊆ P n Z is a quasi projective scheme and π : W → U is an affinetorsor with fiber A l constructed in Lemma 3.11 it follows L( W, s ) = L( A lU , s ) since W and A lU have the same fibers. By construction there is an isomorphism π ∗ : K ′ m ( U ) ∼ = K ′ m ( W )of abelian groups inducing an isomorphism π ∗ ( j − l ) : K ′ m ( U ) ( j − l ) ∼ = K ′ m ( W ) ( j ) (3.11.1)for all integers j . Since W has fibers A l it follows dim ( W ) = d + l where d := dim ( U ). Hence we get the following result: Lemma 3.12.
Let U ⊆ P n Z be a quasi projective scheme and let p : W → U be thetorsor constructed in Lemma 3.11 with fiber A l . It follows L( W, s ) = L( A lU , s ) and χ ( W, j ) = χ ( U, j − l ) for all integers j .Proof. Since W and A lU have the same fibers it follows L( W, s ) = L( A lU , s ) is anequality of L-functions. By Formula 3.11.1 we get an equality χ ( U, j − l ) := X m ∈ Z ( − m +1 dim Q (K ′ m ( U ) ( j − l ) ) = X m ∈ Z ( − m +1 dim Q (K ′ m ( W ) ( j ) ) = χ ( W, j ) , hence χ ( U, j − l ) = χ ( W, j ) and the Lemma follows. (cid:3)
Lemma 3.13.
Let X be a scheme of finite type over Z and let Z ⊆ X be a closedsubscheme with open complement U := X − Z . If conjecture 3.3.1 and 3.3.3 holdsfor Z and U it follows conjecture 3.3.1 and 3.3.3 holds for X . There is for allintegers k ∈ Z an equality χ ( X, k ) = χ ( U, k ) + χ ( Z, k ) . Proof.
Assume K ′ m ( Z ) ( j ) = K ′ m ( U ) ( j ) = 0 for almost all m . There is a long exactlocalization sequence · · · → K ′ m ( Z ) ( j ) → K ′ m ( X ) ( j ) → K ′ m ( U ) ( j ) →→ K ′ m − ( Z ) ( j ) → K ′ m − ( X ) ( j ) → K ′ m − ( U ) ( j ) → · · · hence there are integers m ≤ m with the following properties: For all integers m with m ≤ m or m ≤ m it follows K ′ m ( Z ) ( j ) = K ′ m ( U ) ( j ) = 0. It follows by thelong exact localization sequence that K ′ m ( X ) ( j ) = 0 for all m ≤ m and m ≤ m ,hence Conjecture 3.3.1 holds for X . If Conjecture 3.3.3 holds for Z and U we getthe following: L( X, s ) = L(
Z, s ) L(
U, s ). We get ord s = k (L( X, s )) = ord s = k (L( Z, s )) + ord s = k (L( U, s )) = χ ( Z, k ) + χ ( U, k ) = χ ( X, k )since the Euler characteristic is additive with respect to
Z, U , hence Conjecture3.3.3 holds for X . The Lemma follows. (cid:3) We may reduce the study of Conjecture 3.3.1 and 3.3.3 to the study of affineregular schemes of finite type over Z , with a systematic use of localization, inductionon dimension and the Jouanolou trick from Lemma 3.11: Theorem 3.14.
Assume Conjecture 3.3.1 and 3.3.3 holds for any affine regularscheme of finite type over Z . It follows Conjecture 3.3.1 and 3.3.3 holds for anyquasi projective scheme U of finite type over Z .Proof. One first proves using induction, the long exact localization sequence andJouanolous trick that Conjecture 3.3.1 holds for any affine scheme S := Spec( A ) offinite type over Z . Then again using Jouanolous trick, one proves Conjecture 3.3.1holds for any quasi projective scheme U ⊆ P n Z of finite type over Z .Assume Conjecture 3.3.3 holds for all affine regular schemes S := Spec( A ) offinite type over Z . Let dim ( S ) = 1. It follows the singular subscheme S s ⊆ S is afinite set of closed points with finite residue fields and Conjecture 3.3.3 holds for S s .We use here the fact that the K-theory of a scheme X is the same as the K-theory ofthe associated reduced scheme X red . The singular scheme S s may be non-reducedbut we can pass to the reduced scheme associated to S s . Let U := S − S s . Itfollows U ⊆ P n Z is a quasi projective regular scheme and hence there is a affinetorsor p : W → U with fibers affine l -space A l . It follows since W is an A l -fibrationthat W has the same fibers as relative affine space A lU over U . Hence by Lemma3.9 it follows there is an equality of L-functionsL( W, s ) = L( A lU , s ) . Since W is an affine regular scheme of finite type over Z it follows Conjecture 3.3.3holds for W . We get by Lemma 3.12 ord s = k (L( A lU , s )) = ord s = k (L( W, s )) = χ ( W, k ) = χ ( U, k − l ) = χ ( A lU , k ) . Hence Conjecture 3.3.3 holds for A lU . By Lemma 3.9 since Conjecture 3.3.3 holdsfor A lU it holds for U . Hence Conjecture 3.3.3 holds for S s and U and hence it holdsfor S . By induction on the dimension it follows 3.3.3 holds for any affine scheme S of finite type over Z .Assume U ⊆ P n Z is a quasi projective scheme and let p : W → U be an affinetorsor with W := Spec( B ) where B is a finitely generated Z -algebra. It followsby assumption 3.3.3 holds for W . By the same argument as above it follows 3.3.3holds for A lU and again by Lemma 3.9 it follows 3.3.3 holds for U . The Theoremfollows. (cid:3) Note: A result similar to Theorem 3.14 for Conjecture 3.3.2 is mentioned inSoul´e ’s original paper [19] in Example 2.4. Theorem 3.14 is obtained using slightlydifferent techniques in [15], Lemma 43. The proof of the theorem is not difficult,but I prefer to call it a Theorem, since it is a significant reduction. The Jouanolou-Thomason trick in its most general form is a generalization of Lemma 3.11 tothe case of a quasi compact quasi separated scheme with an ample family of linebundles. Conjecture 3.3.1 and 3.3.3 is stated for quasi projective schemes of finitetype over Z . Example 3.15.
Conjecture 1 for Abelian schemes.
Let A ⊆ P nT is a projective abelian scheme of finite type over T := Spec( B ),where K is an algebraic number field and B a finitely generated and regular O K -algebra. If Conjecture 3.3.1 and 3.3.3 holds for all affine regular schemes Spec( A )of finite type over Z , it follows from Theorem 3.14 Conjecture 3.3.1 and 3.3.3 holdsfor any abelian scheme A ⊆ P nT . Hence we have reduced the study of the Birch ONJECTURES ON L-FUNCTIONS FOR PARTIAL FLAG BUNDLES ON DEDEKIND DOMAINS11 and Swinnerton-Dyer conjecture for abelian schemes to the study of affine regularschemes Spec( A ) of finite type over Z . Example 3.16.
Algebraic K-theory for an affine regular scheme of finite type over Z . Let S := Spec( A ) where A is a finitely generated and regular Z -algebra. Itfollows from [22], Section IV, 1.16.1 there is an embeddingK ∗ ( S ) ⊗ Q ⊆ H ∗ (GL( A ) , Q )(3.16.1)where GL( A ) is the infinite general linear group of A . The embedding in 3.16.1 re-alize K ∗ ( S ) ⊗ Q as the primitive elements in the Hopf algebra H ∗ (GL( A ) , Q ). Thereare Adams operators on H ∗ (GL( A ) , Q ) inducing the classical Adams operators onK ∗ ( S ) ⊗ Q , hence the weight spaces K ′ m ( S ) ( j ) may be constructed using the Hopfalgebra structure on H ∗ (GL( A ) , Q ). In the paper [3] Borel calculates the K-groupsK ∗ ( O K ) ⊗ Q for any algebraic number field K using the embedding 3.16.1. This isTheorem 2.3.If gl ( A ) is the Lie algebra of infinite matrices with coefficients in A , and A isa Q -algebra, it follows by the Loday-Quillen-Tsygan Theorem (see [21], Theorem9.10.10) there is an isomorphismPrim n (H ∗ ( gl ( A ) , Q )) ∼ = HC n − ( A ) , where HC n − ( A ) is cyclic homology of A . Hence in this case there is an explicitformula for the space of primitive elements in terms of cyclic homology. One may askfor a ”similar” explicit formula for the space of primitive elements in H ∗ (GL( A ) , Q ).The space Prim(H ∗ ( gl ( A ) , Q ))is sometimes referred to as the additive K-theory of A . Example 3.17.
Some speculations on a cohomological formulation of the Soul´econjecture.
Let S := Spec( A ) and let T := Spec( O K ). There are operators φ m,i : H ∗ (GL( A ) , Q ) → H ∗ (GL( A ) , Q )with the property that the induced morphism φ m,i : Prim m (H ∗ (GL( A ) , Q )) → Prim m (H ∗ (GL( A ) , Q ))has the following property: Let E ( m, i ) be the set of elements x with φ m,i ( x ) = x .It follows there is an equality E ( m, i ) = K ′ m ( S ) ( i ) . Hence we may define the Eulercharacteristic χ ( S, i ) using the homology H ∗ (GL( A ) , Q ) of the infinite general lineargroup: χ ( S, i ) := X m ∈ Z ( − m +1 dim Q ( E ( m, i )) . (3.17.1)In 3.17.1 we have not used algebraic K-theory K ′ m ( S ) ( i ) to define χ ( S, i ). By 3.8we may define the local L-factors L( S t , s ) for any closed point t ∈ T using a p-adiccohomology theory (or l-adic etale cohomology when S t is smooth and projective).It follows by the product formulaL( S, s ) = Y t ∈ T cl L( S t , s ) that the L-function L( S, s ) has a ”cohomological description”. Hence the Soul´econjecture may be stated as follows: There is for every integer i an equality X m ∈ Z ( − m +1 dim Q ( E ( m, i )) = ord s = i (L( S, s )) . Hence we may argue that the Soul´e conjecture can be formulated ”using cohomol-ogy and homology” groups associated to the affine scheme S . There are preciseconjectures on the existence of an ”arithmetic cohomology theory” that simulta-neously generalize the algebraic K-theory of S (or homology of the infinite generallinear group GL( A )) and p-adic cohomology of the fibers S t for all closed points t ∈ T , and what properties such a theory must have in order to prove the Soul´econjecture (see [6])4. Conjectures on L-functions for projective bundles and flagbundles on Dedekind domains.
In this section we prove the Beilinson-Soul´e vanishing conjecture 3.3.1 and Soul´econjecture 3.3.3 for any partial flag bundle F ( N, E ) on O K where K is an algebraicnumber field (see Corollary 4.16). Hence we get for each number field K an infinitenumber of non-trivial examples where Conjecture 3.3.1 and 3.3.3 hold (see Example4.17). Example 4.1.
Conjecture 3.3.1 and 3.3.3 for the complete flag bundle F ( E ) . Theorem 4.2.
Let K be an algebraic number field and let S := Spec( O K ) . Let P ( E ∗ ) be a P d -bundle on S . It follows Conjecture 3.3.1 and 3.3.3 holds for P ( E ∗ ) .Proof. By induction there is the following result: χ ( P ( E ∗ ) , k ) = d X i =0 χ ( S, k − i )and since χ ( S, j ) is an integer for all integers j it follows χ ( P ( E ∗ ) , k ) is an integerfor all integers k . By Lemma 3.4 it follows Conjecture 3.3.1 holds for P ( E ∗ ). Let P nS := Proj( O K [ x , .., x n ])be projective n -space over O K . Let E be a rank n +1 projective O K -module and let P ( E ∗ ) be the P n -bundle of E . It follows by Lemma 3.9 that L( P ( E ∗ ) , s ) = L( P nS , s )and χ ( P ( E ∗ ) , j ) = χ ( P nS , j ), hence Conjecture 3.3.3 holds for P ( E ∗ ) if and only ifit holds for P nS . Let n = 1 and let P S := Proj( O K [ x , x ]). Let S := V ( x ) ∼ =Spec( O K ) := S and let D ( x ) = ∼ = A S = Spec( O K [ x x ]). It follows χ ( P S , k ) = χ ( A S , k ) + χ ( S, k )and L( P S , s ) = L( A S , s ) L( S, s ) . Hence ord s = k (L( P S , s )) = ord s = k (L( A S , s )) + ord s = k (L( S, s )) = χ ( A S , k ) + χ ( S, k ) = χ ( P S , k ) , and it follows 3.3.3 holds for any P -bundle on S . Assume the conjecture holdsfor any P d − -bundle on S and consider P dS := Proj( O K [ x , .., x d ]). Let Z := V ( x d )and let U := D ( x d ). It follows Z ∼ = P d − S and U ∼ = A dS . Hence Conjecture 3.3.3 ONJECTURES ON L-FUNCTIONS FOR PARTIAL FLAG BUNDLES ON DEDEKIND DOMAINS13 holds for Z and U . By Lemma 3.9 it follows 3.3.3 holds for P dS and P ( E ∗ ) for any E . The Theorem is proved. (cid:3) Note: Theorem 4.2 is a generalization of Borel’s classical result on Spec( O K ) tohigher dimensional schemes. The picard group Pic( O K ) is a finite nontrivial groupin general, and given any set of elements L i for i = 0 , .., d we get a locally trivial O K -module E := ⊕L i of rank d + 1 and a P d -bundle P ( E ∗ ). Corollary 4.3.
Let U be a scheme over Z where Conjecture 3.3.1 and 3.3.3 holdsand let E be a rank d + 1 locally trivial O U -module. It follows Conjecture 3.3.1 and3.3.3 holds for the projective bundle P ( E ∗ ) of E .Proof. The proof is similar to the proof of Theorem 4.2 and is left to the reader.The Corollary is proved. (cid:3)
Example 4.4.
Sequences of projective bundles.
Let U be a scheme over Z and construct X as follows: Let E be a locally trivial O U -module of rank d + 1. Let X := P ( E ∗ ). Let E be a locally trivial O X -module of rank d + 1 and let X := P ( E ∗ ). Continue this process to arrive at ascheme X := X e := P ( E ∗ e ) with a projection morphism π : X e → U. (4.4.1) Example 4.5.
An explicit construction of the partial flag bundle F ( N, E ) . Recall the following construction of the partial flag bundle F ( N, E ) of a locallyfree sheaf E using grassmannian bundles. Let U be a scheme over Z and let E bea locally trivial O U -module of rank n . Let N := { n , .., n l } be a set of positiveintegers with P i n i = n . Let G := G ( n , E ) be the grassmannian bundle of rank n subbundles of E . There is a tautological rank n sub-bundle S ⊆ π ∗ E (4.5.1)where π : G → U is the projection morphism. We get an exact sequence of locallytrivial O G -modules 0 → S → π ∗ E → Q → . For any morphism of schemes f : V → U there is a canonical isomorphism V × U G ( n , E ) ∼ = G ( n , f ∗ E ) . As a particular case let s ∈ U be a point with residue field κ ( s ). We get an inclusionmap i : Spec( κ ( s )) → U. There is by construction a one to one correspondence between maps of schemesover
U g : Spec( κ ( s )) → G and inclusions of κ ( s )-vector spaces S ( g ( s )) ⊆ E ( s )where S ( g ( s )) is the fiber of S at g ( s ) ∈ G . The κ ( s )-vector space S ( g ( s )) hasby definition dimension n . We get a one-to-one correspondence between the κ ( s )-rational points in the fiber π − ( s ), and subspaces W ⊆ E ( s ) of dimension n . Byfunctoriality there is an isomorphism of schemes over κ ( s ) π − ( s ) ∼ = Spec( κ ( s )) × U G ( n , E ) ∼ = G ( n , i ∗ E ) ∼ = G ( n , E ( s )) , where G ( n , E ( s )) is the classical grassmannian scheme parametrizing n -dimensionalsubspaces of the fiber E ( s ) of E at s . Hence ( G , π ) is a fibration over U withfibers grassmannian schemes.Let G := G ( n , Q ). We get a canonical projection map π : G → U withthe property that the fiber π − ( s ) is isomorphic to the flag variety F ( n , n , E ( s ))parametrizing flags W ⊆ W ⊆ E ( s )of κ ( s )-vector spaces with dim κ ( s ) ( W i ) = n + · · · + n i for i = 1 ,
2. Continue thisprocess to get a scheme F ( N, E ) := G ( n l − , Q l − ) and a projection morphism π : F ( N, E ) → U. it follows that for any point s ∈ U it follows the κ ( s )-rational points of the fiber π − ( s ) corresponds to flags0 = W ⊆ W ⊆ · · · ⊆ W l − ⊆ E ( s )with dim κ ( s ) ( W i ) = n + · · · + n i for i = 1 , .., l . It follows there is an isomorphismof schemes π − ( s ) ∼ = F ( N, E ( s )) , where F ( N, E ( s )) is the flag scheme of flags of type N in E ( s ). The scheme( F ( N, E ) , π ) is the flag bundle of E of type N . Let E n := π ∗ E . It follows E n is a locally trivial O -module on F ( N, E ). There is a sequence of locally free sheaves0 = E ⊆ E ⊆ · · · ⊆ E l − ⊆ E n (4.5.2)on F ( N, E ) and E i is locally trivial of rank n i . The sequence 4.5.2 is the universalflag on F ( N, E ). There is a stratification of closed subschemes ∅ = X − ⊆ X ⊆ · · · ⊆ X n := F ( N, E )with the following property: dim ( X i ) = i + dim ( U ), and there is a decompositon X i − X i +1 = ∪ j =1 ,..,n i U i,j into a finite disjoint union of open subschemes U i,j ⊆ X i with U i,j ∼ = A iU an isomorphism of schemes over U for i = 1 , , .., n i . Theconstruction and basic properties of the partial flag bundle is done in completegenerality in [12]. Example 4.6.
The complete flag bundle and projective bundles. If l = n and n i = 1 for all i it follows F ( N, E ) is the complete flag bundle of E . By the above construction we may realize F ( N, E ) as a ”sequence of projectivebundles”.
Example 4.7.
The partial flag variety of a vector space over a field. If U := Spec( k ) with k a field, and E an n -dimensional k -vector space and let N := { n , .., n l } with P i n i = n . It follows F ( N, E ) is the classical partial flagvariety of E of type N , parametrizing flags0 = W ⊆ W ⊆ · · · ⊆ W l − ⊆ E (4.7.1)in E . Here W i is a k -vector subspace of E of dimension n + · · · + n i . This meansthere is a one-to-one correspondence between the set of k -rational points F ( N, E )( k )of the flag variety F ( N, E ) and the set of flags { W i } of type N in E . If SL( E ) is thespecial linear group on E and P ⊆ SL( E ) ie the subgroup of elements fixing a flag { W i } in E of type N , it follows we may use geometric invariant theory to construct ONJECTURES ON L-FUNCTIONS FOR PARTIAL FLAG BUNDLES ON DEDEKIND DOMAINS15 the quotient variety SL( E ) /P . It follows SL( E ) /P is canonically isomorphic to theflag variety F ( N, E ). Hence there is a canonical left action of SL( E ) on F ( N, E ).Hence for the partial flag bundle π : F ( N, E ) → U with E a locally trivial O U -module of rank n , it follows the fiber π − ( s ) may be realized as a quotientSL( E ( s )) /P ( s ) where P ( s ) ⊆ SL( E ( s )) is a parabolic subgroup. Corollary 4.8.
Let U be a scheme over Z such that Conjecture 3.3.1 and 3.3.3holds for U and let E be a locally trivial O U -module of rank n . Let X e be thescheme constructed in 4.4.1. It follows Conjecture 3.3.1 and 3.3.3 holds for X e .In particular it follows Conjecture 3.3.1 and 3.3.3 holds for the full flag bundle F ( N, E ) of E .Proof. The first part of the Corollary follows from 4.3, 3.10 and an induction. Thefull flag bundle F ( N, E ) is by 4.5 constructed using projective bundles and theCorollary follows. (cid:3)
Theorem 4.9.
Let S := Spec( A ) where A is a finitely generated and regular Z -algebra. Let X ⊆ P nS be a quasi projective regular scheme of dimension d . Ifconjecture 3.3.1 holds for all affine regular schemes of finite type over Z , it followsConjecture 3.3.1 holds for X .Proof. By Lemma 3.11 there is an affine torsor p : W → X with W := Spec( B ) with dim ( W ) = d + l . The map p induce an isomorphism atK-theory p ∗ : K ′ m ( X ) → K ′ m ( W )and weight spaces p ∗ : K ′ m ( X ) ( i ) → K ′ m ( W ) ( i + l ) . Since W is affine and finite dimensional it follows for a fixed i + l the groupK ′ m ( W ) ( i + l ) = 0 for almost all m by assumption. Hence the same holds forK ′ m ( X ) ( i ) . The Theorem follows. (cid:3) Corollary 4.10.
Let A be a finitely generated and regular Z -algebra and let X ⊆ P nS be a quasi projective and regular scheme with S := Spec( A ) . Assume Conjecture3.3.1 holds for all affine regular schemes of finite type over Z . It follows χ ( X, i ) isan integer for all i ∈ Z .Proof. This follows from Theorem 4.9, since in this case χ ( X, i ) is a finite sum ofintegers. (cid:3)
Example 4.11.
The projective bundle formula and the Adams operation.
In the following we calculate the K-theory of any finite rank projective bundleon S := Spec( O K ) using the projective bundle formula and Borel’s calculation ofK ′ m ( O K ).The projective bundle formula says the following. There is a canonical pull backmorphism π ∗ : K ∗ ( S ) → K ∗ ( P ( E ∗ ))inducing maps π ∗ : K m ( S ) ( i ) Q → K m ( P ( E ∗ )) ( i ) Q and an isomorphism K ∗ ( P ( E ∗ )) ∼ = K ∗ ( S ) ⊗ K ( S ) K ( P ( E ∗ )) ∼ = K ∗ ( S ) ⊗ Z Z [ t ] / ( t d +1 ) . (4.11.1)with t := 1 − L and L := [ O P ( E ∗ ) ( − ∈ K ( P ( E ∗ )). The Adams operation ψ k actsas follows: ψ k ( t ) := 1 − ψ k ( L ) = 1 − L k . We get for any element zt j ∈ K m ( P ( E ∗ )) ∼ = K m ( S ) { , t, .., t d } the following formula: ψ k ( zt j ) = ψ k ( z )(1 − L k ) j ∈ K m ( P ( E ∗ )) . The isomorphism K m ( P ( E ∗ )) ∼ = K m ( S ) { , t, .., t d } is an isomorphism of K ( S )-modules. In Theorem 4.12 we use formula 4.11.1 andTheorem 2.3 to calculate K m ( P ( E ∗ )) for all integers m . Theorem 4.12.
Let P ( E ∗ ) be a P d -bundle on S . The following holds: K ( P ( E ∗ )) Q ∼ = Q d +1 (4.12.1) K m ( P ( E ∗ )) Q ∼ = 0 for m = 2 i, i = 0(4.12.2) K m ( P ( E ∗ )) Q ∼ = Q r + r ⊗ Q d +1 for m ≡ mod m ( P ( E ∗ ) Q ∼ = Q r ⊗ Q d +1 for m ≡ mod . (4.12.4) Proof.
The Theorem follows from Theorem 2.3 and the formula 4.11.1. (cid:3)
Corollary 4.13.
Let T be a scheme of finite type over Z with the property thatConjecture 3.3.2 holds for T . Let P ( E ∗ ) be a P d -bundle on T . It follows Conjecture3.3.2 holds for P ( E ∗ ) . In particular it follows Conjecture 3.3.2 holds for any P d -bundle on O K .Proof. By the projective bundle formula there is an isomorphism of abelian groupsK ′ m ( P ( E ∗ )) ∼ = K ′ m ( T ) { , t, .., t d } . Let R := Q [ t ] / ( t d +1 ) with t := 1 − L . It follows ψ k acts on R as follows: ψ k ( t ) = ψ k (1 − L ) = 1 − L k . Let v ∈ Z be an integer and let R ( v ) denote the vector space ofelement x ∈ R with ψ k ( x ) = k v x . It follows there is an inclusion of vector spacesover Q : K ′ m ( P ( E ∗ )) ( j ) ⊆ ⊕ u + v = j K ′ m ( T ) ( u ) ⊗ R ( v ) and since by asumption dim Q ( ⊕ u + v = j K ′ m ( T ) ( u ) ⊗ R ( v ) ) < ∞ for all m, j it follows dim Q (K ′ m ( P ( E ∗ )) ( j ) ) < ∞ for all m, j and the Corollaryfollows. (cid:3) The aim of this section is to prove Conjecture 3.3.1 and 3.3.3 for all flag bundles F ( N, E ) on S := Spec( O K ). Let T be a fixed regular and quasi projective schemeof finite type over S := Spec( O K ) with K a number field and let X be a scheme offinite type over T . Assume there is a stratification ∅ = X − ⊆ X ⊆ X ⊆ · · · ⊆ X n = X (4.13.1)of X by closed subschemes X i ⊆ X with dim ( X i ) = i + dim ( T ). ONJECTURES ON L-FUNCTIONS FOR PARTIAL FLAG BUNDLES ON DEDEKIND DOMAINS17
Definition 4.14.
We say the stratification { X i } i =0 ,..,n is a cellular decomposition of X if the following holds: For each i there is an isomorphism (as subschemes of X i ) X i − X i − = ∪ j U i,j where ∪ j U i,j is a finite disjoint union of open subschemes U i,j ⊆ X i , with isomor-phisms f i,j : U i,j ∼ = A iT where A iT is affine i -space over T . The map f i,j is anisomorphism of schemes over T . Theorem 4.15.
Let T be a regular quasi projective scheme of finite type over O K such that Conjecture 3.3.1 and 3.3.3 holds for T . Let X be a scheme of finite typeover T with a cellular decomposition ∅ = X − ⊆ X ⊆ · · · ⊆ X n − ⊆ X n := X with X i − X i +1 = ∪ j =1 ,..,n i A iT . It follows Conjecture 3.3.1 holds for X . Moreover ord s = k (L( X, s )) = χ ( X, k ) , hence Conjecture 3.3.3 holds for X .Proof. The proof is by induction. We will repeatedly use the following Lemma: Let X be a scheme of finite type over T and let U ⊆ X be an open subscheme with Z := X − U . If Conjecture 3.3.1 and 3.3.3 holds for U and Z it follows 3.3.1 and3.3.3 holds for X .Since X = T it follows Conjecture 3.3.1 holds for X . Let X − X = ∪ j U ,j be a finite disjoint union of X − X into affine open subschemes U ,j ∼ = A T . Weget the following calculation: χ ( X − X , k ) = χ ( ∪ j U ,j , k ) = X j χ ( U ,j , k ) = X j χ ( A T , k ) , and since 3.3.1 holds for T it holds for A T . Hence χ ( A T , k ) is an integer for allintegers k . It follows the finite sum χ ( X − X , k ) = X j χ ( U ,j , k )is an integer for all integers k . Hence Lemma 3.4 implies that Conjecture 3.3.1holds for X − X . Since the conjecture holds for X = T by assumption, if follows3.3.1 holds for X . By induction it follows 3.3.1 holds for X = X n .Assume 3.3.3 holds for X = T and let X − X = ∪ j U ,j a finite disjoint unioninto affine bundles U ,j ∼ = A T . We get ord s = k (L( X − X , s )) = ord s = k ( Y j L( A T , s )) = X j ord s = k (L( A T , s )) = X j χ ( A T , k ) = χ ( ∪ j U ,j , k ) = χ ( X − X , k )hence conjecture 3.3.3 holds for X − X . It follows Conjecture 3.3.3 holds for X . By induction it follows Conjecture 3.3.3 holds for X n = X and the Theoremfollows. (cid:3) Corollary 4.16.
Let E be a finite rank projective O K -module and let F ( N, E ) aflag bundle for E of type N . It follows Conjecture 3.3.1 and 3.3.3 holds for F ( N, E ) .In particular it holds for the projective bundle P ( E ∗ ) and the grassmannian bundle G ( m, E ) with ≤ m < rk ( E ) . The same holds for any finite rank locally trivial O T -module F : If it holds for T it holds for F ( N, F ) .Proof. By Example 4.5 it follows any partial flag bundle F ( N, E ) has a cellulardecomposition over O K , hence the Corollary follows from Theorem 4.15 (cid:3) Example 4.17.
Non trivial examples for O K . Let K be an algebraic number field with S := Spec( O K ) and Pic( O K ) non-trivial. Let L , .., L n ∈ Pic( O K ) and let E := ⊕ i L i . It follows E is a non triviallocally trivial O K -module of rank n . Hence the flag bundle π : F ( N, E ) → S isa non-trivial partial flag bundle on S , with the property that the Beilinson-Soul´evanishing conjecture and the Soul´e conjecture on L-functions holds for F ( N, E ). Example 4.18.
Generalized cellular decompositions.
In this example we prove Theorem 4.15 for a larger class of schemes: Schemesequipped with a cellular decomposition of type { T i } . Definition 4.19.
Let T , . . . , T n be schemes of finite type over S := Spec( O K ) andLet X be a scheme of finite type over T n . Assume there is a stratification ∅ = X − ⊆ X ⊆ X ⊆ · · · ⊆ X n = X of X , where X i ⊆ X is a closed subscheme for every i with the following property:For any i it follows E i := X i − X i − is a vector bundle over T i with fiber A d i . Wesay { X i } i =0 ,..,n is a cellular decomposition of X of type { T i } . We also say X has ageneralized cellular decomposition .Note: It is clear a cellular decomposition is a generalized cellular decomposition:From Definition 4.14 it follows the scheme X i − X i − is an affine vector bundleover T i with fiber A i , since X i − X i − has an open cover X i − X i − = ∪ j U i,j with U i,j ∼ = A iT . Let T i := ∪ j T for all i . Hence if ∪ j U i,j = U i, ∪ · · · ∪ U i,l it follows T i := T ∪ · · · ∪ T : The disjoint union of T taken l times. It follows X i − X i − is anaffine finite rank vector bundle over T i . Lemma 4.20.
Assume X has a generalized cellular decomposition X i ⊆ X of type { T i } and assume Conjecture 3.3.1 and 3.3.3 holds for T i . It follows Conjecture3.3.1 and 3.3.3 holds for X .Proof. Since X := X − X − = E is a finite rank affine vector bundle over T and Conjecture 3.3.1 holds for T it follows Conjeture 3.3.1 holds for E := X . Bydefinition X − X := E is a finite rank affine vector bundle over T . Conjecture3.3.1 holds for T hence it holds for E . It follows Conjecture 3.3.1 holds for X .By induction it follows Conjecture 3.3.1 holds for X . Conjecture 3.3.3 is provedsimilarly and the Lemma follows.Assume Conjecture 3.3.3 holds for T and let E be a finite rank vector bundle on T of rank d . Since χ ( E, k ) = χ ( A dT , k ) and L( E, s ) = L( A dT , s ) it follows Conjecture3.3.3 holds for T if and only if it holds for E . Since E := X − X is a finite rankvector bundle over T it follows Conjecture 3.3.3 holds for E . Since 3.3.3 holds for X and X − X it holds for X . By induction it follows Conjecture 3.3.3 holds for X n = X and the Lemma follows. (cid:3) Let π : X → T be a scheme of finite type over T with the following property:There is a zero dimensional closed subscheme S ⊆ X with U := X − S a vector ONJECTURES ON L-FUNCTIONS FOR PARTIAL FLAG BUNDLES ON DEDEKIND DOMAINS19 bundle over T of rank l . Since Conjecture 3.3.1 and 3.3.3 hold for T it followsby Lemma 4.20 Conjecture 3.3.1 and 3.3.3 hold for X . The scheme X does notneccessarily have a cellular decomposition but it has by definition a generalizedcellular decomposition. Example 4.21.
A generalized cellular decomposition for abelian schemes
Let S := Spec( A ) where A is a finitely generated and regular over Z and let A ⊆ P nS be an abelian scheme over S . Let i : Z → A be a closed sub-scheme withopen complement j : U → A , and consider the localization sequenceCH ∗ ( Z ) → CH ∗ ( A ) → CH ∗ ( U ) → ∗ ( A ) is the Chow- group of A . The Chow-group CH ∗ ( A ) is non-trivialin general and assume Im ( i ∗ ) = (0) and CH ∗ ( U ) = (0). Since CH ∗ ( A ) is highlynon-trivial, it follows many closed subshchemes Z have this property. One wantto construct a scheme T of finite type over Z with the property that there ismorphism π : U → T and such that U is a finite rank vector bundle over T . Itfollows L( U, s ) = L(
T, s − d ) and χ ( U, k ) = χ ( T, k − i ). Hence the study of theSoule conjecture for U is reduced to the study of the same conjecture for T . Itis a natural question to ask if there is a generalized cellular decomposition of theabelian scheme A . This is a non-trivial open problem. Example 4.22.
Explicit formulas of L-functions and Euler characteristics.
We get explicit formulas for the L-function and Euler characteristic for a scheme X with a cellular decomposition of type { T i } . Lemma 4.23.
Let X be a scheme of finite type over O K with a cellular de-composition of type { T i } i =0 ,...,n , where T i satisfy Conjecture 3.3.1 and 3.3.3. Let E i := X i − X i be a rank d i trivial vector bundle on T i for i = 0 , . . . , n . It follows L( X, s ) = n Y i =0 L( T i , s − d i )(4.23.1) χ ( X, k ) = n X i =0 χ ( T i , k − d i ) . (4.23.2) Proof.
Let X be a scheme of finite type over Z with U ⊆ X an open subschemewith complement Z := X − U . Let E → X be a vector bundle of rank d . Usingmethods from Lemma 3.7 and 3.9, it follows L( X, s ) = L(
U, s ) L(
Z, s ) and L(
E, s ) =L(
X, s − d ). Moreover χ ( X, k ) = χ ( U, k )+ χ ( Z, k ) and χ ( E, k ) = χ ( X, k − d ). Usingthis, the Lemma follows by induction. (cid:3) Let E be a rank n locally trivial O S -module with S := Spec( O K ), and let F ( N, E )be the partial flag bundle of E of type N . There is a cellular decomposition ∅ = X − ⊆ X ⊆ · · · ⊆ X n := F ( N, E )(4.23.3)with X i − X i − := E i a rank i trivial vector bundle on T i for i = 0 , . . . , n . Here T i := Q l i j =1 S . Hence dim ( T i ) = l i . We get explicit formulas for the L-function andEuler characteristic of F ( N, E ): Lemma 4.24.
The following holds: L( F ( N, E ) , s ) = n Y i =0 L( S, s − d i ) l i (4.24.1) χ ( X, k ) = n X i =0 l i χ ( S, k − d i ) . (4.24.2) Proof.
Since any partial flag bundle F ( N, E ) has a cellular decomposition of type { T i } , the Lemma follows from Lemma 4.23. (cid:3) Example 4.25.
An alternative proof of Corollary 4.16 using induction.
Given a locally trivial finite rank O K -module E and a flag bundle F ( E ), we mayask if it is possible to give a proof of Conjecture 3.3.3 using an induction similar toExample 4.1. One wants a stratification of closed subschemes ∅ = X n +1 ⊆ X n ⊆ · · · ⊆ X ⊆ X = F ( E )with X i − X i +1 = ∪ i,j A d i is a disjoint union of affine spaces, and where the sub-schemes X i are flag schemes of dimension smaller than F ( E ) with the property thatConjecture 3.3.1 hold for X i . This is done in Example 4.1 for P d -bundles on O K .In Example 4.11.1 the schemes X i are projective spaces over S of dimension lessthan d .Let k be a field, E an n -dimensional vector space over k and let N := { n , .., n l } with P i n i = n . Let E have a flag of k -vector spaces E ⊆ E ⊆ · · · ⊆ E l ⊆ E with dim k ( E i ) = n + · · · + n i . Let F ( E ) be the complete flag variety of E . It followsthere is a Borel subgroup B ⊆ SL( E ) and an isomorphism SL( E ) /B ∼ = F ( E ). Thereis moreover a parabolic subgroup P ⊆ SL( E ) with SL( E ) /P ∼ = F ( N, E ), and acanonical surjective map π : F ( E ) → F ( N, E ) . The map π is locally trivial in the Zariski topology with fibers π − ( s ) ∼ = F × · · · × F l , where F i is the complete flag variety of an n i -dimensional k -vector space. Sometimesthis fibration is used to reduce the study of the partial flag variety to the studyof the complete flag variety. There are similar constructions valid in the relativesituation for flag bundles. Example 4.26.
Special values of L-functions and Beilinson’s conjectures.
Let X be a quasi projective scheme of finite type over Z . In [16], Section 6 thenotion of a regulator map r X : K ′ j − i ( X ) ( j ) Q → H iMX R ( X R , j )is defined, where H iMX R ( X R , j ) is motivic cohomology of X R := X ⊗ Z R . In Conjec-ture 6 . − . X, s )at integers to the map r X . These conjectures are referred to as the Beilinsonconjectures . ONJECTURES ON L-FUNCTIONS FOR PARTIAL FLAG BUNDLES ON DEDEKIND DOMAINS21
In [10] the author defines for any cohomology theory H ∗ α ( − , Z ( i )) satisfying aset of axioms, and any quasi projective scheme X of finite type over Z Chern classmaps c i : K ′ m ( X ) ( i ) Q → H i − mα ( X, Z ( i )) . When α := D and H ∗ D is Deligne cohomology we get Chern character maps ch i : K ′ m ( X ) ( i ) Q → H i − mD ( X ⊗ C , R ( i )) + . The Chern character map ch i is a regulator map for Deligne-Beilinson cohomology,and the map ch i has been used by Borel in [4] to prove the Beilinson conjecturesfor the ring O K when K is any number field. There are the well known formulasfor the values of the Riemann zeta function included in any elementary course incalculus and integration: L( Z ,
2) := ∞ X n =1 n = 16 π , L( Z ,
4) := ∞ X n =1 n = 190 π and L( Z ,
6) := ∞ X n =1 n = 1945 π . In general there are the following results:L( Z , − k ) = − B k k (4.26.1)with k > Z , m ) = ( − m − (2 π ) m B m m )!(4.26.2)where m ≥ B i is the i ′ th Bernoulli number. The formulasin 4.26.1 and 4.26.2 go back to Euler and Riemann (see the introductory book [17]Section VII.1 for more information).One would like to check if the Chern character map ch i can be used to cal-culate special values of the L-function L( F ( N, E ) , s ) where F ( N, E ) is any flagbundle on O K , generalizing of Borels formula 4.27.1 to arithmetic flag schemes inany dimension. The Beilinson conjectures are known for rings of integers in al-gebraic number fields, Dirichlet L-functions, some elliptic curves, Shimura curvesand Hilbert-Blumenthal surfaces. See Section 8 in the paper [16] for more preciseinformation and references. Example 4.27.
Values of L-functions of flag bundles over O K at integers. Let E be a free Z -module of rank n and let F ( N, E ) be the flag bundle of type N on S := Spec( Z ). It followsL( F ( N, E ) , s ) = n Y i =0 L( S, s − d i ) l i = n Y i =0 L( Z , s − d i ) l i . Hence L( F ( N, E ) , − k ) = n Y i =0 L( Z , − k − d i ) l i = ( − n +1 n Y i =0 ( B k + d i k + d i ) l i for k > F ( N, E ) , s ) at positive integers is by Lemma 4.24 determined bythe values of L( Z , s ) at positive integers.If K is a number field with ring of integers O K and E a locally trivial rank n O S -module where S := Spec( O K ) it follows again by Lemma 4.24 the values ofL( F ( N, E ) , s ) is completely determined by the values of L( O K , s ).Borel discovered in [4] regulator maps r : K ′ k − ( Z ) ⊗ Z R ∼ = R . A non-zero element a ∈ K ′ k − ( Z ) ⊗ Z Q maps to a well defined element R k := r ( a ) ∈ R ∗ / Q ∗ . This gives a formulaL( Z , k − ≡ R k mod Q ∗ . (4.27.1)Formulas similar to 4.27.1 exist for any algebraic number field K and its ring ofintegers O K . Borel’s formula for the special values of L( O K , s ) is defined up tomultiplication with a non-zero rational number. Hence if we view the values in4.26.1 and 4.26.2 as elements in R ∗ / Q ∗ , it follows 4.26.1 and 4.26.2 are recoveredby the formula from [4].Recent work of Bloch and Kato give an explicit formula with values in the realnumbers. Lemma 4.24 and Borel’s formula gives an explicit formula for an elementL( F ( N, E ) , m ) := n Y i =0 L( O K , m − d i ) l i ∈ R ∗ / Q ∗ (4.27.2)for any partial flag bundle F ( N, E ) on O K . In [2] the authors conjecture a formulafor an element L( M, m ) ∈ R ∗ where M is a ”motive”, generalizing the formula in4.27.2. The formula conjectured in [2] is known to hold for some number fields andelliptic curves with complex multiplication. By Lemma 4.24 it follows the study ofthe Bloch-Kato conjecture for partial flag bundles is reduced to the study of ringsof integers in number fields.5. Appendix A: The weight space decomposition for algebraicK-theory of projective bundles
In this section we calculate explicitly the weight spaces K ′ m ( P ( E ∗ )) ( i ) for any P d -bundle on S to illustrate that it is easy to make explicit calculations for projectivebundles. The calculation is not neccessary for the main results of the paper, but itshows how to perform such calculations using elementary methods.Let in the following X := Proj( Z [ x , .., x n ]) be projective n-space over the ringof integers Z . By the projective bundle formula for algebraic K-theory we getK m ( X ) Q = K m ( Z ) Q ⊗ Q [ t ] / ( t n +1 ) = K m ( Z ) Q ⊗ Q { , t, .., t n } , where t = 1 − L = 1 − [ O ( − L = [ O ( − O ( −
1) is the tautologicalbundle on projective space X := P ( V ). Let R := Q [ t ] / ( t n +1 ) = Q { , t, t , .., t n } .Let ONJECTURES ON L-FUNCTIONS FOR PARTIAL FLAG BUNDLES ON DEDEKIND DOMAINS23 x := ln(1 − t ) = − ( t + (1 / t + (1 / t + + (1 /n ) t n )(5.0.1)in the ring R = Q { , t, t , .., t n } . Lemma 5.1.
Let ψ k be the kth Adams operator acting on R . The following holdsfor all integers k ≥ : ψ k ( x ) = kx. (5.1.1) For every integer i ≥ we get ψ k ( x i ) = k i x i . (5.1.2) Proof.
By definition L = [ O ( − ( X ) of the tautological linebundle O ( −
1) on projective space, hence the Adams operator ψ k acts as follows: ψ k ( L ) = L k . We get since t = 1 − L the following calculation: ψ k ( x ) = ψ k ( − ( t + (1 / t + (1 / t + + (1 /n ) t n ) = ψ k ( − ((1 − L ) + (1 / − L ) + (1 / − L ) + . + (1 /n )(1 − L ) n ) = − ((1 − ψ k ( L )) + (1 / − ψ k ( L )) + + (1 /n )(1 − ψ k ( L )) n ) =ln( ψ k ( L )) = ln( L k ) = k ln( L ) = kx by Corollary A2 in the Appendix. Claim 1 is proved. Claim 2: We get ψ k ( x i ) = ψ k ( x ) i = ( kx ) i = k i x i and Claim 2 is proved. (cid:3) Note: Formal properties of exponential power series and logarithm power seriesvalid in the formal power series ring Q [[ t ]] implies similar properties for exponen-tials and logarithms in the quotient ring R := Q [[ t ]] / ( t n +1 ). Formula 5.0.1 wascommunicated to me by Charles Weibel.If we defineln( L ) := ln(1 − t ) = − ( t + (1 / t + (1 / t + + (1 /i ) t i + . ) , (5.1.3)where ln( L ) lives in the formal power series ring Q [[ t ]], one proves there is anequality of formal power series ln( L k ) = k ln( L ) for all integers k ≥ Q [[ t ]]. Fora proof of this property see the Appendix. It follows the vector x i is an eigen vectorfor ψ k with eigen value k i . It follows the inclusion of vector spaces Q { , x, x , .., x n } ⊆ Q { , t, t , .., t n } (5.1.4)Is an isomorphism of vector spaces: The vectors { , x, x , .., x n } are linearlyindependent over Q since they have different eigenvalues with respect to ψ k - thekth Adams operator. Hence 5.1.4 gives a decomposition of R := Q [ t ] / ( t n +1 ) intoeigen spaces for the Adams operations ψ k for k ≥
0. We get an isomorphism ofabelian groups K ∗ ( X ) Q ∼ = K ∗ ( Z ) Q ⊗ Q Q { , x, x , .., x n } . (5.1.5)We get the following formula for K m ( X ) Q : K m ( X ) Q = 0 if m < . (5.1.6) K m ( X ) Q = Q { , x, x , .., x n } if m = 0 . (5.1.7) K m ( X ) Q = 0 if m = 1 or m = 2 k, k ≥ . (5.1.8) K m ( X ) Q = Q { , x, x , .., x n } if m = 4 k + 1 , k > . (5.1.9) K m ( X ) Q = 0 if m = 4 k + 3 , k ≥ . (5.1.10)For the field of rational numbers Q we have r = 1 and r = 0. Lemma 5.2.
The following holds for K m ( X ) ( i ) Q and i = 0 , .., n : K m ( X ) ( i ) Q = 0 if m < . (5.2.1) K m ( X ) ( i ) Q = Q if m = 0 . (5.2.2) K m ( X ) ( i ) Q = 0 if m = 1 or m = 2 k with k ≥ . (5.2.3) K m ( X ) ( i ) Q = Q if m = 4 k + 1 with k > . (5.2.4) K m ( X ) ( i ) Q = 0 if m = 4 k + 3 with k ≥ . (5.2.5) Proof.
The Lemma follows from the discussion above: The basis { , x, x , .., x n } gives a decomposition of R := Q [ t ] / ( t n +1) into eigen spaces for the Adams operation ψ k and the Lemma follows from the projective bundle formula and the calculationof K m ( Z ) Q given above. (cid:3) Corollary 5.3.
For all m = 4 k + 1 with k > and all i = 0 , .., n it follows K m ( X ) ( i ) Q = Q = 0 .Proof. This follows from Lemma 1 above. (cid:3)
Algebraic K-theory K m ( O K ) Q is well known, the Adams eigen space K m ( O K ) ( i ) Q is well known by [8], Volume 1, Theorem 47 and the projective bundle formulaholds for P ( E ∗ ): K ∗ ( P ( E ∗ )) Q ∼ = K ∗ ( O K ) Q ⊗ Q [ t ] / ( t n +1 ) . Hence the study of the eigen space K m ( P ( E ∗ )) ( i ) Q is by the above calculationreduced to the study of K m ( O K ) ( i ) Q which is well known by Theorem 2.3. We getthe following Theorem: Theorem 5.4.
Let Q ⊆ K be an algebraic number field with ring of integers O K .Let r , r be the real and complex places of K . Let P ( E ∗ ) be a rank e projectivebundle on S := Spec( O K ) and let K m ( P ( E ∗ )) Q denote the mth algebraic K-theoryof the category of algebraic vector bundles on P ( E ∗ ) with rational coefficients. The ONJECTURES ON L-FUNCTIONS FOR PARTIAL FLAG BUNDLES ON DEDEKIND DOMAINS25 following holds: Let j ≥ be an integer. K m ( P ( E ∗ )) ( j ) Q = 0 for all m < and m = 2 k with k ≥ an integer . (5.4.1) K ( P ( E ∗ )) ( j ) Q = Q if j = 0 , , , .., e. (5.4.2) K ( P ( E ∗ )) ( j ) Q = 0 if j > e. (5.4.3) K a +1 ( P ( E ∗ )) ( j ) Q = Q r + r if j is in I := 2 a + 1 , a + 2 , .., a + 1 + e. (5.4.4) K a +1 ( P ( E ∗ )) ( j ) Q = 0 if j is not in I . (5.4.5) K a +3 ( P ( E ∗ )) ( j ) Q = Q r if j is in J := 2 a + 2 , a + 3 , .., a + 2 + e . (5.4.6) K a +3 ( P ( E ∗ )) ( j ) Q = 0 if j is not in J . (5.4.7) Here a ≥ is an integer.Proof. This follows from the calculation of K m ( O K ) ( j ) Q , the projective bundle for-mula and the eigen space decomposition R := Q [ t ] / ( t e +1 ) = Q { , x, x , .., x e } ofthe ring R , with x := ln( L ) := ln(1 − t ) ∈ R , as described above. (cid:3) Example 5.5.
Example of Theorem 5.4 for terms m = 0 , , , . m = 0 : K ( P ( E ∗ )) ( l ) Q = Q if l = 0 , , , .., e. K ( P ( E ∗ )) ( l ) Q = Q if l > e.m = 1: K ( P ( E ∗ )) ( l ) Q = Q r + r − if l = 1 , , , .., e + 1 . K ( P ( E ∗ )) ( l ) Q = 0 if l = 0 or l > e + 1 .m = 2: K ( P ( E ∗ )) ( l ) Q = 0 .m = 3: K ( P ( ∗ )) ( l ) Q = Q r if l = 2 , , , .., e + 2 . K ( P ( E ∗ )) ( l ) Q = 0 if I = 0 , l > e + 2 . Example 5.6.
Schubert calculus for algebraic K-theory.
In a future paper a similar theory and calculation will be developed for the al-gebraic K-theory K ∗ ( G ( m, E )) of the grassmannian G ( m, E ) of E . The aim of thisstudy is to introduce and study Schubert calculus for the K-theory of the grassman-nian and flag schemes F ( E ) of a bundle E over S := Spec( O K ), and to relate thisstudy to Bloch’s higher Chow groups. In [11], Proposition 3.1 (Berthelot’s talk)the following formula is proved: Let S be a noetherian scheme, E a locally trivial O S -module of rank n and P := ( p , .., p k ) a set of positive integers with P i p i = n and F P ( E ) := F ( P, E ) the flag bundle of E of type P , it follows the canonicalmorphism K ∗ ( S ) ⊗ K ∗ ( S ) K ∗ ( F P ( E )) ∼ = K ∗ ( F P ( E ))is an isomorphism. Hence a formula similar to the projective bundle formula isknown for flag bundles. One wants to calculate weight space decompositionK m ( F P ( E )) Q ∼ = ⊕ i ∈ Z K m ( F P ( E )) ( i ) for all integers m . Corollary 5.7.
Let X be a scheme of finite type over Spec( O K ) . There are nointegers M, L >> with the property that K m ( X ) ( l ) Q = 0 for m ≥ M and l ≥ L .Proof. Choose an integer a such that 2 a +1 ≥ max { M, L } . It follows from Theorem5.4 that K a +1 ( P ( E ∗ )) (2 a +1) Q = Q r + r = 0. By choice 4 a + 1 ≥ M and 2 a + 1 ≥ L . (cid:3) Appendix B: Some general properties of formal power series
In this section we recall some well known elementary facts on formal powerseries,logarithm power series and maps of abelian groups.Recall the following results from [Bour], page A.IV.39 on formal power series:Let l ( g ( x )) := X n ≥ ( − n − (1 /n )( g ( x )) n ∈ Q [[ x ]] . For any g ( x ) ∈ Q [[ x ]]. Define the following formal power series:Log( g ( x )) := l ( g ( x ) − g ( x ) ∈ Q [[ x ]]. It followsLog(1 − x ) = l ( − x ) = − ( x + (1 / x + (1 / x + (1 / x + · · · ) ∈ Q [[ x ]] . Let A be a commutative unital ring containing the field Q of rational numbers.Let nil ( A ) be the nilradical of A . Let 1 − nil ( A ) denote the set of elements on theform 1 − u with u ∈ nil ( A ). It follows 1 − u is a multiplicative unit in A . Theset 1 − nil ( A ) has a multiplication: (1 − u )(1 − v ) = 1 − u − v + uv = 1 − z with z = − u − v + uv , and the element z is again in nil ( A ). Hence (1 − u )(1 − v ) = 1 − z is in 1 − nil ( A ). It follows 1 − nil ( A ) is a subgroup of the multiplicative group ofunits in A . Lemma 6.1. (A1) Let u ∈ nil ( A ) be an element with u k +1 = 0 . Define thefollowing map: ln : 1 − nil ( A ) → nil ( A ) by ln(1 − u ) := − ( u + (1 / u + (1 / u + · · · + (1 /k ) u k ) ∈ nil ( A ) . It follows ln is a morphism of groups: For any two elements − u, − v ∈ − nil ( A ) it follows ln((1 − u )(1 − v )) = ln(1 − u ) + ln(1 − v ) . Proof.
From [5], page A.IV.40 we getLog(1 − x ) = l (1 − x −
1) = l ( − x ) in Q [[ x ]] . The following holds for the powerseries l ( x ): l ( x + y + xy ) = l ( x ) + l ( y ) in thering Q [[ x, y ]]. We may for any two elements u, v in nil ( A ) define a map f : Q [[ x, y ]] → A by f ( x ) = u, f ( y ) = v . It follows f induce a well defined map of rings f ′ : Q [[ x, y ]] /I → A where I = ker ( f ). In the ring Q [[ x, y ]] we get the following formula: ONJECTURES ON L-FUNCTIONS FOR PARTIAL FLAG BUNDLES ON DEDEKIND DOMAINS27
Log((1 − x )(1 − y )) = Log(1 − x − y + xy ) := l ( − x − y + xy ) =(6.1.1) l ( − x − y + ( − x )( − y ) = l ( − x ) + l ( − y ) = Log(1 − x ) + Log(1 − y ) . It follows the same formula 6.1.1 holds in the quotient ring Q [[ x, y ]] /I . Hencewe get the following formula for the map ln (viewing u and v as elements in thequotient Q [[ x, y ]] /I ):ln((1 − u )(1 − v )) = Log((1 − x )(1 − y )) = Log(1 − x )+Log(1 − y ) = ln(1 − u )+ln(1 − v ) . Hence the map ln is a map of groups. (cid:3)
Note: Lemma A1 may also be proved using Bell polynomials.
Corollary 6.2. (A2) Use the notation from Lemma A1. If − u ∈ − nil ( A ) thefollowing holds for any integer k ≥ : ln((1 − u ) k ) = k ln(1 − u ) . Proof.
This follows from Lemma A1 and an induction. (cid:3)
Example: Let A := Q [ t ] / ( t e +1 ) with nil ( A ) = ( t ) define the following logarithmmap ( u ∈ nil ( A )):ln(1 − u ) := − ( u + (1 / u + (1 / u + · · · + (1 /e ) u e ) ∈ A. It follows ln((1 − u ) k ) = k ln(1 − u )(6.2.1)for any integer k ≥
1. The property 6.2.1 is well known when we consider thelogarithm function defined for real numbers, and the above section proves it holdsfor formal power series.Note: Formal properties of exponentials and logarithms in Q [[ t ]] can also beproved using Bell polynomials. Acknowledgements : Thanks to Shrawan Kumar, Chris Soul´e and CharlesWeibel for answering questions and providing references on algebraic K-theory andflag varieties. Thanks also to Alexander Beilinson and Christopher Deninger foranswering questions on L-functions and the Beilinson and Bloch-Kato conjectures.
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