Iterated line integrals over Laurent series fields of characteristic p
aa r X i v : . [ m a t h . AG ] M a r ITERATED LINE INTEGRALS OVER LAURENT SERIES FIELDSOF CHARACTERISTIC p AMBRUS P ´AL
Abstract.
Inspired by Besser’s work on Coleman integration, we use ∇ -modules to define iterated line integrals over Laurent series fields of char-acteristic p taking values in double cosets of unipotent n × n matrices withcoefficients in the Robba ring divided out by unipotent n × n matrices withcoefficients in the bounded Robba ring on the left and by unipotent n × n matri-ces with coefficients in the constant field on the right. We reach our definitionby looking at the analogous theory for Laurent series fields of characteristic 0first, and reinterpreting the classical formal logarithm in terms of ∇ -moduleson formal schemes. To illustrate that the new p -adic theory is non-trivial, weshow that it includes the p -adic formal logarithm as a special case. Formal iterated line integrals over Laurent series fields ofcharacteristic zero
In order to motivate our investigations over fields of positive characteristic, firstwe will look at a theory which could be justifiably considered as a formal analogueof line integrals over Laurent series fields of characteristic zero. We will start withthe formal analogue of the logarithm, the most basic such contruction. Let k a fieldof characteristic 0. The formal logarithm:log(1 − z ) = − ∞ X n =1 z n n ∈ Q [[ z ]]can be used to define a homomorphism: k [[ t ]] ∗ /k ∗ −→ k [[ t ]]as follows. Every u ∈ k [[ t ]] ∗ can be written uniquely as: u = c (1 − w ) , c ∈ k ∗ , w ∈ tk [[ t ]] . The infinite sum: log(1 − w ) = − ∞ X n =1 w n n converges in the t -adic topology to a power series in k [[ t ]], and the map: k [[ t ]] ∗ → k [[ t ]] , u log(1 − w )is a homomorphism with kernel k ∗ which we will denote by log by slight abuse ofnotation.It is possible to reinterpret this construction using differential algebra. LetΩ k [[ t ]] /k be module of continuous K¨ahler differentials of k [[ t ]] over k , i.e. the free Date : March 16, 2017. module over k [[ t ]] generated by the symbol dt , where the derivation d : k [[ t ]] → Ω k [[ t ]] /k is given by the formula d (cid:0) ∞ X j =0 x j t j (cid:1) = (cid:0) ∞ X j =1 jx j t j − (cid:1) dt. Then the first de Rham cohomology group H dR ( k [[ t ]]) def = Ω k [[ t ]] /k /dk [[ t ]]of k [[ t ]] is trivial. Therefore for every u ∈ k [[ t ]] ∗ there is a unique v ∈ tk [[ t ]] suchthat dv = duu . Note that v = log( u ). Indeed this follows at once by differentiating the infinite sumterm by term and using that d is continuous in the t -adic topology. So the relation: d log( u ) = duu can be used to define the formal logarithm. Next we give a geometric reformulationof this relation using the theory of ∇ -modules. Definition 1.1. A ∇ -module over k [[ t ]] is a pair ( M, ∇ ), where M is a finite, free k [[ t ]]-module, and ∇ is a connection on M , i.e. a k -linear map: ∇ : M → M ⊗ k [[ t ]] Ω k [[ t ]] /k satisfying the Leibniz rule ∇ ( c v ) = c ∇ ( v ) + v ⊗ dc ( ∀ c ∈ k [[ t ]] , v ∈ M ) . The trivial ∇ -module over k [[ t ]] is just the pair ( k [[ t ]] , d ). A horizontal map froma ∇ -module ( M, ∇ ) to another ∇ -module ( M ′ , ∇ ′ ) is just a k [[ t ]]-linear map f : M → M ′ such that the following diagram is commutative: M ∇ / / f (cid:15) (cid:15) M ⊗ k [[ t ]] Ω k [[ t ]] /kf ⊗ k [[ t ]] id Ω1 k [[ t ]] /k (cid:15) (cid:15) M ′ ∇ ′ / / M ′ ⊗ k [[ t ]] Ω k [[ t ]] /k . As usual we will simply denote by M the ordered pair ( M, ∇ ) whenever this isconvenient.These objects form a k -linear Tannakian category, with respect to horizontalmaps as morphisms, and with the obvious notion of directs sums, tensor products,quotients and duals. In fact this Tannakian category is neutral, and the fibre functoris supplied by the lemma below. Definition 1.2.
A horizontal section of a ∇ -module ( M, ∇ ) over k [[ t ]] is an s ∈ M such that ∇ ( s ) = 0. We denote the set of the latter by M ∇ .The following claim is very well-known: Lemma 1.3.
For every ( M, ∇ ) as above M ∇ is a k -linear vector space of dimensionequal to the rank of M over k [[ t ]] . ine integrals over Laurent series fields 3 Proof.
See the proof of Theorem 7.2.1 of [2] on page 121. Note that the recurrence( i + 1) U i +1 = i X j =0 N j U i − j has a solution in our case, too, since k has characteristic zero. (cid:3) Note that for every s ∈ M ∇ there is a unique morphism from the trivial ∇ -module to ( M, ∇ ) such that the image of 1 is s . Therefore the lemma above impliesthat every ∇ -module over k [[ t ]] is trivial , i.e. it is isomorphic to the n -fold directsum of the trivial ∇ -module for some n . In fact we get more: Corollary 1.4.
The functor ( M, ∇ ) M ∇ is a k -linear tensor equivalence of between the Tannakian categories of ∇ -modulesover k [[ t ]] and of finite dimensional k -linear vector spaces.Proof. Since it is hard to find a convenient reference, we indicate the proof for thesake of the reader. Let F be the functor in the claim above, and let G denote thefunctor V ( V ⊗ k k [[ t ]] , id V ⊗ k d )from the category of finite dimensional k -linear vector spaces to the category of ∇ -modules over k [[ t ]]. It is easy to see that F and G are functors of k -linear tensorcategories, so we only need to see that they are equivalences of categories. Notethat the k [[ t ]]-multiplication induces a natural map M ∇ ⊗ k k [[ t ]] −→ M which is an isomorphism by Lemma 1.3. Similarly the natural map V −→ ( V ⊗ k k [[ t ]]) id V ⊗ k d given by the rule v v ⊗ k (cid:3) We will need a slight variant of Lemma 1.3, taking into accounts filtrations, butthis will follow easily from Corollary 1.4.
Notation 1.5.
Let M be a ∇ -module over k [[ t ]] equipped with a filtration:0 = M ⊂ M ⊂ · · · ⊂ M n = M by sub ∇ -modules such that the rank of M i over k [[ t ]] is r + · · · + r i . Set r = r + r + · · · + r n , and equip the trivial ∇ -module T = k [[ t ]] ⊕ r with the filtration:0 = T ⊂ T ⊂ · · · ⊂ T n = T, where T i = k [[ t ]] ⊕ k [[ t ]] ⊕ · · · ⊕ k [[ t ]] | {z } r + ··· + r i ⊕ ⊕ · · · ⊕ | {z } r i +1 + ··· + r n . Lemma 1.6.
There is an isomorphism φ : M → T of ∇ -modules such that φ ( M i ) = T i for every index i = 1 , , . . . , n . Ambrus P´al
Proof.
By taking horizontal sections we get a filtration:0 = M ∇ ⊂ M ∇ ⊂ · · · ⊂ M ∇ n = M ∇ of M ∇ by k -linear subspaces such that the k -dimension of M ∇ i is r + · · · + r i byLemma 1.3. Similarly 0 = T ∇ ⊂ T ∇ ⊂ · · · ⊂ T ∇ n = T ∇ is a filtration of T ∇ such that the k -dimension of T ∇ i is r + · · · + r i . It is a basicfact of linear algebra that there is a k -linear isomorphism f : M ∇ → T ∇ such that f ( M ∇ i ) = T ∇ i . The claim now follows from Corollary 1.4. (cid:3) Let M and T be as in Notation 1.5. Assume now that for every index i =1 , , . . . , n an isomorphism: φ i : M i /M i − −→ k [[ t ]] ⊕ r i is given where k [[ t ]] is equipped with the trivial connection. Lemma 1.7.
There is an isomorphism φ : M → T of ∇ -modules such that φ ( M i ) = T i for every index i = 1 , , . . . , n and the induced isomorphism φ i : M i /M i − −→ T i /T i − ∼ = k [[ t ]] ⊕ r i is φ i for every index i = 1 , , . . . , n .Proof. Let φ ∇ i : ( M i /M i − ) ∇ ∼ = M ∇ i /M ∇ i − −→ T ∇ i /T ∇ i − ∼ = ( T i /T i − ) ∇ ∼ = k ⊕ r i be the k -linear isomorphism induced by φ i on horizontal sections. It is possible tochoose a a k -linear isomorphism f : M ∇ → T ∇ such that f ( M ∇ i ) = T ∇ i and theinduced map: M ∇ i /M ∇ i − −→ T ∇ i /T ∇ i − is φ ∇ i above for every index i = 1 , , . . . , n . The claim now follows from Corollary1.4. (cid:3) Definition 1.8.
Let r = ( r , r , . . . , r n ) be a vector consisting of positive integers,and set r = r + r + · · · + r n . A framed ∇ -module of signature r is a ∇ -module( M, ∇ ) over k [[ t ]] equipped with a k [[ t ]]-basis e , e , . . . , e r of M such that M i = the k [[ t ]]-span of e , e , . . . , e r + ··· + r i is a sub ∇ -module, and the image of e r + ··· + r i − +1 , . . . , e r + ··· + r i in the quotient M i /M i − is a k -basis of ( M i /M i − ) ∇ . There is a natural notion of isomorphism offramed ∇ -modules of signature r , namely, it is an isomorphism of the underlying ∇ -modules which maps the k [[ t ]]-bases to each other (respecting the indexing, too). Definition 1.9.
Let R be a commutative ring with unity. Let U r ( R ) denote thegroup of r × r matrices composed of blocks U ij such that for every pair ( i, j ) ofindices U ij is an r i × r j matrix with coefficients in R , moreover U ii is the identitymatrix for every i and U ij is the zero matrix for every i > j . It is reasonable to call U r ( R ) the group of unipotent matrices of rank r with coefficients in R . ine integrals over Laurent series fields 5 Remark . Note that for every framed ∇ -module ( M, ∇ , e , e , . . . , e r ) of signa-ture r as above there is a unique isomorphism: φ i : M i /M i − −→ k [[ t ]] ⊕ r i which maps the the image of e r + ··· + r i − +1 , . . . , e r + ··· + r i under the quotient mapto the 1st, 2nd,. . . , r i th basis vector of k [[ t ]] ⊕ r i , respectively. Therefore there is anisomorphism φ : M → T of ∇ -modules such that φ ( M i ) = T i and the inducedisomorphism φ i : M i /M i − −→ T i /T i − ∼ = k [[ t ]] ⊕ r i is φ i for every index i = 1 , , . . . , n by Lemma 1.7. The matrix of φ in the basis e , e , . . . , e r is an element of U r ( k [[ t ]]), unique up to multiplication on the rightby a matrix in U r ( k ). We get a well-defined map from the isomorphism classes offramed ∇ -modules of signature r into the set U r ( k [[ t ]]) /U r ( k ) which is obviously abijection. Example . For every u ∈ k [[ t ]] ∗ consider the following framed ∇ -module ofsignature (1 , M = k [[ t ]] ⊕ , let e , e be the 1st, respectively 2nd basisvector of M , and let ∇ be the unique connection of M such that ∇ ( e ) = 0 , ∇ ( e ) = e ⊗ duu . Let φ : M ∼ = k [[ t ]] ⊕ → T ∼ = k [[ t ]] ⊕ be an isomorphism of the type consideredabove. Then the matrix V of φ in the basis e , e is V = (cid:18) v (cid:19) ∈ U (1 , ( k [[ t ]]) such that d ◦ V = (cid:18) dv (cid:19) = V ◦ ∇ = (cid:18) v (cid:19) · (cid:18) duu (cid:19) = (cid:18) duu (cid:19) , and hence dv = duu . So the isomorphism class of the framed ∇ -module ( M, ∇ , e , e ) in U (1 , ( k [[ t ]]) /U (1 , ( k ) ∼ = k [[ t ]] /k is just log( u ) (modulo constants).The point of the construction above is that we can get the family in the exampleabove as a pull-back of a similar type of object on the formal multiplicative groupscheme over the formal spectrum Spf( k [[ t ]]) of k [[ t ]]. This is the description whicheasily generalises, and which we are going to describe next. Definition 1.12.
Let X be a formally smooth t -adic formal scheme of finite typeover Spf( k [[ t ]]). Then X is also a formally smooth formal scheme of finite typeover Spf( k ) via the map Spf( k ) → Spf( k [[ t ]]) induced by the embedding k ֒ → k [[ t ]].Therefore the sheaf of continuous K¨ahler differentials Ω X/k is well-defined, and itis a finite, locally free formal O X -module. A ∇ -module over X is a pair ( M, ∇ ),where M is a finite, locally free formal O X -module, and ∇ is a connection on M ,i.e. a k -linear map of sheaves: ∇ : M → M ⊗ O X Ω X/k
Ambrus P´al satisfying the Leibniz rule ∇ ( c v ) = c ∇ ( v ) + v ⊗ dc for every open U ⊂ X and c ∈ Γ( U, O X ) , v ∈ Γ( U, M ). Definition 1.13.
The trivial ∇ -module over X is just O X equipped with thedifferential d : O X → Ω X/k ∼ = O X ⊗ O X Ω X/k . These notions specialise to thoseintroduced in Definition 1.1 when X is Spf( k [[ t ]]). Moreover horizontal maps of ∇ -modules over X is defined the same way as above. We get a k -linear categorywith the usual notion of direct sums, duals and tensor products. Again we willdenote by M the ordered pair ( M, ∇ ) whenever this is convenient. Finally let M ∇ denote the sheaf of horizontal sections of M :Γ( U, M ∇ ) def = { s ∈ Γ( U, M ) | ∇ ( s ) = 0 } . Note that M is a trivial ∇ -module of rank n , that is, isomorphic to the n -fold directsum of ( O X , d ), if and only if M ∇ is the constant sheaf in n -dimensional k -linearvector spaces. Definition 1.14.
It is possible to define the notion of framed ∇ -modules in thismore general context, too. Let r and r be as in Definition 1.8. A framed ∇ -moduleover X of signature r is a ∇ -module ( M, ∇ ) over X equipped with a O X -frame e , e , . . . , e r of M such that M i = the O X -span of e , e , . . . , e r + ··· + r i is a sub ∇ -module, and the image of e r + ··· + r i − +1 , . . . , e r + ··· + r i in the quotient M i /M i − is a k -frame of ( M i /M i − ) ∇ . Definition 1.15.
The notion of ∇ -modules and framed ∇ -modules are natural in X . Let f : X → Y be a morphism of formally smooth formal schemes of finite typeover Spf( k [[ t ]]). The morphism f induces an O X -linear map df : f ∗ (Ω Y/k ) → Ω X/k .The pull-back f ∗ ( M, ∇ ) of a ∇ -module ( M, ∇ ) with respect to f is f ∗ ( M ) equippedwith the composition: f ∗ ( ∇ ) : f ∗ ( M ) / / f ∗ ( M ⊗ O Y Ω Y/k ) ∼ = f ∗ ( M ) ⊗ O X f ∗ (Ω Y/k ) / / Ω X/k , where the first arrow is the pull-back of ∇ with respect to f , and the second isid f ∗ ( M ) ⊗ O X df . The pull-back of a framed ∇ -module ( M, ∇ , e , . . . , e r ) of signature r on Y with respect to f is the pull-back f ∗ ( M, ∇ ) equipped with the O X -frame f ∗ ( e ) , . . . , f ∗ ( e r ). Since pull-back commutes with quotients and the pull-backof horizontal sections are horizontal, this construction is a framed ∇ -module ofsignature r on X . Definition 1.16.
For every X as above let X ( k [[ t ]]) denote the set of sections f : Spf( k [[ t ]]) → X . Let M = ( M, ∇ , e , . . . , e r ) be a framed ∇ -module of signature r on X . Then for every f ∈ X ( k [[ t ]]) the pull-back of M with respect to f is aframed ∇ -module of signature r over k [[ t ]]. Taking isomorphism classes we get afunction Z M : X ( k [[ t ]]) −→ U r ( k [[ t ]]) /U r ( k )which we will call the line integral of M . ine integrals over Laurent series fields 7 Example . Let X be Spf( k [[ t, x ]]). In order to give a ∇ -module on X , it issufficient to give a k -linear map: ∇ : k [[ t, x ]] ⊕ −→ k [[ t, x ]] ⊕ ⊗ k [[ t,x ]] Ω k [[ t,x ]] /k satisfying the Leibniz rule, whereΩ k [[ t,x ]] /k = k [[ t, x ]] · dt ⊕ k [[ t, x ]] · dx, with differential d : k [[ t, x ]] → Ω k [[ t,x ]] /k given by: d (cid:0) X ij a ij t i x j (cid:1) = X ij ( ia ij t i − x j dt + ja ij t i x j − dx ) . Let e , e be the 1st, respectively 2nd basis vector of k [[ t, x ]] ⊕ , and let ∇ be theunique connection of k [[ t, x ]] ⊕ such that ∇ ( e ) = 0 , ∇ ( e ) = e ⊗ dx x , where (1 + x ) − = P ∞ i =0 ( − i x i . Equipped with the frame e , e this ∇ -module isframed of signature (1 , M denote this object. Note that sections of X → Spf( k [[ t ]]) are exactly continuous k [[ t ]]-algebra homomorphisms ψ : k [[ t, x ]] → k [[ t ]].Every such ψ is determined by ψ (1 + x ) which must be an invertible element of k [[ t ]]. Conversely for every u ∈ k [[ t ]] ∗ there is a unique such ψ u : k [[ t, x ]] → k [[ t ]]with the property ψ u (1 + x ) = u . The pull-back of M with respect to ψ u is just theframed ∇ -module appearing in Example 1.11. We get that the formal line integral: Z M : X ( k [[ t ]]) ∼ = k [[ t ]] ∗ −→ U (1 , ( k [[ t ]]) /U (1 , ( k ) ∼ = k [[ t ]] /k is just the formal logarithm.2. The p -adic logarithm for Laurent series fields of characteristic p The perfect reference for the background material in this section and the nextis Kedlaya’s book [2].
Notation 2.1.
Let k a perfect field of characteristic p > O denote thering of Witt vectors over k . Let v p denote the valuation on O normalised so that v p ( p ) = 1. For x ∈ O , let x denote its reduction in k . Let Γ denote the ring ofbidirectional power series:Γ = (cid:8) X i ∈ Z x i u i | x i ∈ O , lim i →−∞ v p ( x i ) = ∞ (cid:9) . Then Γ is a complete discrete valuation ring whose residue field we could identifywith k (( t )) by identifying the reduction of P x i u i with P x i t i (see page 263 of [2]).Let K = O [ p ] and E = Γ[ p ]; they are the fraction fields of the rings O and Γ,respectively. Definition 2.2.
Let Ω E be the free module over E generated by a symbol du , anddefine the derivation d : E → Ω E by the formula d (cid:0) X j x j u j (cid:1) = (cid:0) X j jx j u j − (cid:1) du. Ambrus P´al
We define the first de Rham cohomology group H dR ( E ) of E as the quotient Ω E /d E .Note that the dlog map: x dxx , E ∗ → Ω E followed by the quotient map Ω E → H dR ( E ) furnishes a homomorphism Γ ∗ → H dR ( E ) which we will denote by dlog by slight abuse of notation. Lemma 2.3.
The homomorphism dlog : Γ ∗ → H dR ( E ) factors through the reduc-tion map · : Γ ∗ → k (( t )) ∗ .Proof. We need to show that for every x ∈ Γ ∗ of the form 1 − py with y ∈ Γ wehave dlog( x ) ∈ d E . Set z = − ∞ X n =1 ( py ) n n . Since 0 ≤ v p ( p n ) − v p ( n ) → ∞ as n → ∞ , the infinite sum above converges in the p -adic topology, and hence z ∈ Γ is well-defined. Differentiation is continuous withrespect to the p -adic topology, so dz = ∞ X n =1 ( py ) n − d ( − py ) = (1 − py ) − d (1 − py ) = dlog( x ) . (cid:3) Let dlog also denote the induced homomorphism k (( t )) ∗ → H dR ( E ). This mapis trivial restricted to k ∗ , for example because dlog : Γ ∗ → H dR ( E ) is trivial on O ∗ .The basic result about this construction is the following Theorem 2.4.
The kernel of dlog : k (( t )) ∗ → H dR ( E ) is k ∗ .Proof. Let deg : k (( t )) ∗ → Z be the discrete valuation on k (( t )) normalised so thatdeg( t ) = 1. We define the residue map on Ω E as follows: X j x j u j du x − , Ω E → K. Since there is no term of degree − dx ∈ d E , we get a well-definedhomomorphism res: H dR ( E ) → K . We will need the following: Lemma 2.5.
The diagram commutes: k (( t )) ∗ dlog / / deg (cid:15) (cid:15) H dR ( E ) res (cid:15) (cid:15) Z (cid:31) (cid:127) / / K. Proof.
Clearly res ◦ dlog( t ) = 1. Now let x ∈ k [[ t ]] ∗ . Then x has a lift to (Γ + ) ∗ ⊂ Γ ∗ ,where Γ + denotes the subringΓ + = (cid:8) X i ∈ N x i u i | x i ∈ O} of Γ. By definition res ◦ dlog((Γ + ) ∗ ) = 0. Since the group k (( t )) ∗ is generated by t and k [[ t ]] ∗ , the claim now follows, as all arrows in the diagram are homomorphisms. (cid:3) ine integrals over Laurent series fields 9 Let us return to the proof of Theorem 2.4. Let x ∈ k (( t )) ∗ be such that dlog( x ) =0, but x k ∗ . By the above x ∈ k [[ t ]] ∗ . We may assume without loss of generalitythat x ∈ tk [[ t ]] by multiplying x with an element of k ∗ . Choose a lift y ∈ (Γ + ) ∗ of x . We may assume that y = 1 − au m − bu m +1 , where m is a positive integer, with a ∈ O ∗ and b ∈ Γ + . Set z = − ∞ X n =1 ( au m + bu m +1 ) n n . The infinite sum above converges with respect to the topology generated by theideal ( u ) ⊳ K [[ u ]], so z is a well-defined element of K [[ u ]].Let R be one of the rings K [[ t ]] and E + = Γ + [ p ], and let Ω R be the free moduleover R generated by a symbol du , and define the derivation d : R → Ω R by theformula d (cid:0) X j x j u j (cid:1) = (cid:0) X j jx j u j − (cid:1) du. Clearly Ω E + ⊂ Ω K [[ t ]] . Let v ∈ E be such that dv = dlog( y ). Since dlog( y ) ∈ Ω E + we have v ∈ E + . Note that differentiation is continuous with respect to the ( u )-adictopology, so dz = ∞ X n =1 ( au m + bu m +1 ) n − d ( − au m − bu m +1 )= (1 − au m − bu m +1 ) − d (1 − au m − bu m +1 ) = dlog( y ) . Therefore dv = dz and hence v − z ∈ K . We get that z ∈ E + , too. But this is acontradiction since, if z = ∞ X i =0 z i u i , then v p ( z mp i ) = − i for every positive integer i . We can see the latter as follows.By definition: z ≡ − p i − X n =1 ( au m + bu m +1 ) n n + ( au m ) p i p i mod ( u mp i +1 ) . In the first summand all coefficients have p -adic valuation ≥ − i , while in thesecond the coefficient of u mp i has valuation − i . (cid:3) Next we are going to give a slightly more convoluted variant of this construction,which nevertheless ties it up better with the general theory of line integrals overLaurent series fields of characteristic p . Definition 2.6.
Let Γ † denote the subring:Γ † = (cid:8) X i ∈ Z x i u i | x i ∈ O , lim inf i →−∞ v p ( x i ) − i > (cid:9) ⊂ Γ . The latter is also a discrete valuation ring with residue field k (( t )), although it isnot complete (see Definition 15.1.2 and Lemma 15.1.3 of [2] on page 263). Let E † = Γ † [ p ]. Then E † is the fraction field of the ring Γ † . Similarly to the above let Ω E † be the module of continuous K¨ahler differentials of E † , i.e. the free module over E † generated by a symbol du , equipped with the derivation d : E † → Ω E † given by d (cid:0) X j x j u j (cid:1) = (cid:0) X j jx j u j − (cid:1) du. We define the first de Rham cohomology group H dR ( E † ) of E † as the quotientΩ E † /d E † . Note that the dlog map: x dxx , ( E † ) ∗ → Ω E † followed by the quotient map Ω E † → H dR ( E † ) furnishes a homomorphism (Γ † ) ∗ → H dR ( E † ) which we will denote by dlog † . Lemma 2.7.
The homomorphism dlog † : (Γ † ) ∗ → H dR ( E † ) factors through thereduction map · : (Γ † ) ∗ → k (( t )) ∗ .Proof. We need to show that for every x ∈ (Γ † ) ∗ of the form 1 − py with y ∈ Γ † wehave dlog( x ) ∈ d E † . It will be sufficient to prove that the element z = − ∞ X n =1 ( py ) n n ∈ Γis actually in Γ † . Note that E † is the ring of the bidirectional (or Laurent) expan-sions of bounded holomorphic functions over K on an open annulus of outer radius1 and inner radius 1 − ǫ , for some ǫ ∈ (0 ,
1) (see page 263 of [2]). If y is such afunction then the infinite sum defining z converges with respect to the supremumnorm and defines a bounded holomorphic function over K on the annulus of outerradius 1 and inner radius 1 − ǫ . The claim is now clear. (cid:3) Let dlog † also denote the induced homomorphism k (( t )) ∗ → H dR ( E † ). This mapis trivial restricted to k ∗ , for example because dlog † : (Γ † ) ∗ → H dR ( E † ) is trivialon O ∗ . Then we have the following variant of Theorem 2.4 above: Theorem 2.8.
The kernel of dlog † : k (( t )) ∗ → H dR ( E † ) is k ∗ .Proof. Note that there is a commutative diagram: k (( t )) ∗ dlog † / / dlog % % ❏❏❏❏❏❏❏❏❏❏ H dR ( E † ) (cid:15) (cid:15) H dR ( E ) , where the right vertical map is induced by the pair of inclusions Ω E † → Ω E and d E † → d E . Now the claim immediately follows from Theorem 2.4. (cid:3) Definition 2.9.
Let R denote the ring of bidirectional power series: R = (cid:8) X i ∈ Z x i u i | x i ∈ O [ 1 p ] , lim inf i →−∞ v p ( x i ) − i > , lim inf i → + ∞ v p ( x i ) i ≥ } . (See Definition 15.1.4 of [2] on page 264.) Let R + denote its subring: R + = R ∩ (cid:8) X i ∈ N x i u i | x i ∈ O [ 1 p ] (cid:9) . ine integrals over Laurent series fields 11 Clearly E + ⊂ R + and E † ⊂ R . Note that we may define the continuous K¨ahler dif-ferentials and the first de Rham cohomology group of the rings R and R + similarlyto the above, and we will use similar notation to denote them, too.The reason we like the ring R + is the following very well-known claim: Lemma 2.10.
The group H dR ( R + ) is trivial.Proof. Simply note that if P ∞ i =0 x i u i ∈ R + then P ∞ i =0 x i i +1 u i +1 also lies in R + . (cid:3) Now we can tie in the contents of this section with the formal logarithm con-struction of the previous section.
Definition 2.11.
Let v ∈ k [[ t ]] ∗ . Then dlog † ( v ) ∈ H dR ( E + ). By the above theimage of this class under the natural map H dR ( E + ) → H dR ( R + ) is trivial, so thereis a w ∈ R + such that dw = dlog † ( v ), unique up to adding an element of E + . It isreasonable to denote the class of this element in R + / E + by log † ( v ) in light of theabove. The resulting map log † : k [[ t ]] ∗ → R + / E + is a homomorphism with kernel k ∗ . Remark . There is an obstruction to extend this construction to the whole k (( t )) ∗ , taking values in R / E † , namely the residue map. Indeed similarly to theconstruction in the proof of Theorem 2.4, there is a residue map on Ω E † given by X j x j u j du x − , Ω E † → K, moreover we have a similar map for Ω R , and these maps are compatible withthe inclusions Ω E † ⊂ Ω E and Ω E † ⊂ Ω R . Since there is no term of degree − H dR ( E † ) → K andres: H dR ( R ) → K . From Lemma 2.5 we get that the diagram commutes: k (( t )) ∗ dlog † / / deg (cid:15) (cid:15) H dR ( E † ) res (cid:15) (cid:15) / / H dR ( R ) res x x rrrrrrrrrrr Z (cid:31) (cid:127) / / K. On the other hand the map res : H dR ( R ) −→ K is an isomorphism by the lemma below, so dlog † ( v ) is integrable if and only if v ∈ k [[ t ]] ∗ . Lemma 2.13.
The map res : H dR ( R ) −→ K is an isomorphism.Proof. The map is obviously surjective. In order to see injectivity, simply note thatif P i ∈ N ,i = − x i u i ∈ R then P i ∈ N ,i = − x i i +1 u i +1 also lies in R . (cid:3) Iterated p -adic line integrals over Laurent series fields ofcharacteristic p Definition 3.1.
Let R be one of the rings E + , E , E † , R + or R . A ∇ -module over R is a pair ( M, ∇ ), where M is a finite, free R -module, and ∇ is a connection on M , i.e. a K -linear map: ∇ : M → M ⊗ R Ω R satisfying the Leibniz rule ∇ ( c v ) = c ∇ ( v ) + v ⊗ dc ( ∀ c ∈ R, v ∈ M ) . The trivial ∇ -module over R is just the pair ( R, d ). A horizontal map from a ∇ -module ( M, ∇ ) to another ∇ -module ( M ′ , ∇ ′ ) is just a R -linear map f : M → M ′ such that the following diagram is commutative: M ∇ / / f (cid:15) (cid:15) M ⊗ R Ω Rf ⊗ R id Ω1 R (cid:15) (cid:15) M ′ ∇ ′ / / M ′ ⊗ R Ω R . As usual we will simply denote by M the ordered pair ( M, ∇ ) whenever this isconvenient. Definition 3.2.
Now let R ⊂ R ′ be two rings from the list above and let ( M, ∇ )be a ∇ -module over R . Let ∇ ′ be the unique connection: ∇ ′ : M ⊗ R R ′ −→ ( M ⊗ R R ′ ) ⊗ R ′ Ω R ′ ∼ = ( M ⊗ R Ω R ) ⊗ R ′ R ′ such that ∇ ′ ( m ⊗ R s ) = ∇ m ⊗ R s + m ⊗ R ds, ( ∀ m ∈ M, ∀ s ∈ R ) . Then the couple ( M ⊗ R R ′ , ∇ ′ ) is a ∇ -module over R ′ which we will denote by M ⊗ R R ′ for simplicity and will call the pull-back of M onto R ′ . Moreover forevery homomorphism h : M → M ′ of ∇ -modules over R the R ′ -linear extension h ⊗ R id R ′ : M ⊗ R R ′ → M ′ ⊗ R R ′ is a morphism of ∇ -modules over R ′ . These objectsform a K -linear Tannakian category, with respect to horizontal maps as morphisms,and with the obvious notion of directs sums, tensor products, quotients and duals.Note that we may define similar notions for the integral rings Γ + , Γ † and Γ bysubstituting K -linearity with O -linearity. Definition 3.3.
A horizontal section of a ∇ -module ( M, ∇ ) over R is an s ∈ M such that ∇ ( s ) = 0. We denote the set of the latter by M ∇ . Note that for every s ∈ M ∇ there is a unique morphism from the trivial ∇ -module to ( M, ∇ ) such thatthe image of 1 is s . Of course a ∇ -module over R is trivial if it is isomorphic to the n -fold direct sum of the trivial ∇ -module for some n (over R ).Note that any reasonable version of Lemma 1.3 is false; in fact there is a ∇ -module over E + whose pull-back to R is not trivial. (In fact the basic counterex-ample is very simple; it corresponds to the differential equation y ′ = y . For afurther explanation see Example 0.4.1 of [2] on page 7.) However the analogue ofthe framed version (Lemma 1.7) is true, at least over R + . We are going to formulatethis claim next. Notation 3.4.
Let r = ( r , r , . . . , r n ) be a vector consisting of positive integers,and set r = r + r + · · · + r n , as in Definition 1.8. Let M be a ∇ -module over R equipped with a filtration:0 = M ⊂ M ⊂ · · · ⊂ M n = M by sub ∇ -modules such that the rank of M i over R is r + · · · + r i . Set r = r + r + · · · + r n , and equip the trivial ∇ -module T = R ⊕ r with the filtration:0 = T ⊂ T ⊂ · · · ⊂ T n = T, ine integrals over Laurent series fields 13 where T i = R ⊕ R ⊕ · · · ⊕ R | {z } r + ··· + r i ⊕ ⊕ · · · ⊕ | {z } r i +1 + ··· + r n . Also assume that for every index i = 1 , , . . . , n an isomorphism of ∇ -modules: φ i : M i /M i − −→ R ⊕ r i is given where R is equipped with the trivial connection. We will call such objects(consisting of ( M, ∇ ), the filtration M ⊂ M ⊂ · · · ⊂ M n , and the isomorphisms φ i ) filtered ∇ -modules of signature r . There is a natural notion of isomorphism offiltered ∇ -modules of signature r , namely, it is an isomorphism of the underlying ∇ -modules which maps the filtrations to each other, and identifies the isomorphisms φ i .Now let ( M, ∇ , M i , φ i ) be a filtered ∇ -module of signature r and let ( T, T i ) beas above. Lemma 3.5.
Assume that R = R + . Then there is an isomorphism φ : M → T of ∇ -modules such that φ ( M i ) = T i and the induced isomorphism φ i : M i /M i − −→ T i /T i − ∼ = ( R + ) ⊕ r i is φ i for every index i = 1 , , . . . , n . It will be simpler to introduce some additional definitions before we give theproof of the lemma above.
Definition 3.6.
Let r = ( r , r , . . . , r n ) be a vector consisting of positive integers,and set r = r + r + · · · + r n . A framed ∇ -module of signature r (over R ) is a ∇ -module ( M, ∇ ) over R equipped with an R -basis e , e , . . . , e r of M such that M i = the R -span of e , e , . . . , e r + ··· + r i is a sub ∇ -module, and the image of e r + ··· + r i − +1 , . . . , e r + ··· + r i in the quotient M i /M i − is a k -basis of ( M i /M i − ) ∇ . There is a natural notion of isomorphism offramed ∇ -modules of signature r in this setting, too. Proof of Lemma 3.5.
We are going to prove the claim by induction on n . The case n = 1 is obvious. Assume now that the claim holds for n −
1. Note that ( M i /M i − ) ∇ spans M i /M i − as an R + -module, since the latter is a trivial ∇ -module. Also notethat M is a free R + -module. Therefore we may choose a R + -basis e , e , . . . , e r of M such that M i is the R + -span of e , . . . , e r + ··· + r i , and ( M, ∇ ) equipped with thisbasis is a framed ∇ -module of signature r . By the induction hypothesis we mayassume that e , . . . , e r + ··· + r n − are horizontal. Let e , e , . . . , e r is the 1st, 2nd,etc. basis vector of T . We may also assume without loss of generality that φ i mapsthe image of e r + ··· + r i − +1 , . . . , e r + ··· + r i under the quotient map to the image of e r + ··· + r i − +1 , . . . , e r + ··· + r i under the quotient map for every i = 1 , . . . , n .Let C be the matrix of the connection ∇ in the R + -basis e , . . . , e r , that is, forevery s , s , . . . , s r ∈ R + we have: ∇ ( s e + · · · + s r e r ) = e ⊗ ds + · · · + e r ⊗ ds + ( s e , · · · , s r e r ) · C, where the · in the last term denotes the row-column multiplication with respect tothe tensor product. Then C is an r × r matrix with coefficients in Ω R + composedof blocks C ij such that for every pair ( i, j ) of indices C ij is an r i × r j matrix withcoefficients in Ω R + , and C ij is the zero matrix unless i = 1 and j = n . By Lemma 2.10 there is a matrix U of rank r with coefficients in R + such that dU = C and U ij is the zero matrix unless i = 1 and j = n . Consider R + -linearmap φ : M → T given by: φ ( λ e + · · · + λ r e r ) = ( λ e , · · · , λ r e r ) · ( I + U )for every λ , . . . , λ r ∈ R + , where I is the r × r identity matrix and · denotes therow-column multiplication here. It is the isomorphism of ∇ -modules we are lookingfor. (cid:3) Definition 3.7.
Now let ( M, ∇ , M , . . . , M r , φ , . . . , φ r ) be a filtered ∇ -moduleof signature r over E + . We may choose an E + -basis e , e , . . . , e r of M such that M i is the E + -span of e , . . . , e r + ··· + r i , and ( M, ∇ ) equipped with this basis is aframed ∇ -module of signature r . By Lemma 3.5 above there is an isomorphism φ : M ⊗ E + R + → T of ∇ -modules over R + such that φ ( M i ⊗ E + R + ) = T i and theinduced isomorphism φ i : M i ⊗ E + R + /M i − ⊗ E + R + ∼ = ( M i /M i − ) ⊗ E + R + −→ T i /T i − ∼ = R ⊕ r i + is φ i ⊗ E + id R + for every index i = 1 , , . . . , n . The matrix of φ in the basis e ⊗ E + , e ⊗ E + , . . . , e r ⊗ E + U r ( R + ), unique up to multi-plication on the right by a matrix in U r ( K ), corresponding to an automorphism ofthe ∇ -module T respecting its filtration and the horizontal bases on the Jordan–H¨older components, and up to multiplication on the left by a matrix in U r ( E + ),corresponding to a change of the basis e , . . . , e r . We get a well-defined map fromthe isomorphism classes of framed ∇ -modules of signature r over E + into the set U r ( E + ) \ U r ( R + ) /U r ( K ) of double cosets. Definition 3.8.
Write O n = O / ( p n +1 ). For a topologically finitely generatedΓ + -algebra A , with reductions A n = A/ ( p n +1 ), we letΩ A/ O def = lim ←− n →∞ Ω A n / O n be the module of p -adically continuous differentials. The limit of the differentialsof A n over O n furnishes a p -adically continuous differential d : A → Ω A/ O . When A = Γ + = O [[ u ]] then Ω O [[ u ]] / O is the free O [[ u ]]-module of rank one generated bythe symbol du . Let X be a formally smooth u -adic formal scheme of finite typeover Spf(Γ + ). Then we may define the p -adically continuous K¨ahler differentialsΩ X/ O by patching, and it is a finite, locally free formal O X -module, equipped witha differential d : O X → Ω X/ O . Definition 3.9.
Let X be as above. A ∇ -module over X is a pair ( M, ∇ ), where M is a finite, locally free formal O X -module, and ∇ is a connection on M , i.e. an O -linear map of sheaves: ∇ : M → M ⊗ O X Ω X/ O satisfying the Leibniz rule ∇ ( c v ) = c ∇ ( v ) + v ⊗ dc for every open U ⊂ X and c ∈ Γ( U, O X ) , v ∈ Γ( U, M ). Definition 3.10.
The trivial ∇ -module over X is just O X equipped with thedifferential d : O X → Ω X/ O ∼ = O X ⊗ O X Ω X/ O . Moreover horizontal maps of ∇ -modules over X is defined the same way as above. We get a K -linear category with ine integrals over Laurent series fields 15 the usual notion of direct sums, duals and tensor products. Again we will denoteby M the ordered pair ( M, ∇ ) whenever this is convenient. Finally let M ∇ denotethe sheaf of horizontal sections of M :Γ( U, M ∇ ) def = { s ∈ Γ( U, M ) |∇ ( s ) = 0 } . Note that M is a trivial ∇ -module of rank n , that is, isomorphic to the n -fold directsum of ( O X , d ), if and only if M ∇ is the constant sheaf in rank n free O -modules.It is possible to define the notion of filtered and framed ∇ -modules in this moregeneral context, too. We will leave the details to the reader. Definition 3.11.
The notion of ∇ -modules and framed ∇ -modules are natural in X . Let f : X → Y be a morphism of formally smooth formal schemes of finite typeover Spf(Γ + ). The morphism f induces an O X -linear map df : f ∗ (Ω Y/ O ) → Ω X/ O .The pull-back f ∗ ( M, ∇ ) of a ∇ -module ( M, ∇ ) with respect to f is f ∗ ( M ) equippedwith the composition: f ∗ ( ∇ ) : f ∗ ( M ) / / f ∗ ( M ⊗ O Y Ω Y/ O ) ∼ = f ∗ ( M ) ⊗ O X f ∗ (Ω Y/ O ) / / Ω X/ O , where the first arrow is the pull-back of ∇ with respect to f , and the second isid f ∗ ( M ) ⊗ O X df . The pull-back of a filtered ∇ -module ( M, ∇ , M , . . . , M r , φ , . . . , φ r )of signature r on Y with respect to f is the pull-back f ∗ ( M, ∇ ) equipped with thefiltration f ∗ ( M ) , . . . , f ∗ ( M r ) , f ∗ ( φ ) , . . . , f ∗ ( φ r ). Since pull-back commutes withquotients and the pull-back of horizontal sections are horizontal, this constructionis a filtered ∇ -module of signature r on X . Definition 3.12.
For every X as above let X (Γ + ) denote the set of sections f :Spf(Γ + ) → X . Let M = ( M, ∇ , M , . . . , M r , φ , · · · , φ r ) be a filtered ∇ -module ofsignature r on X . Then for every f ∈ X (Γ + ) the pull-back of M with respect to f is a filtered ∇ -module of signature r over Γ + . By applying the functor · ⊗ Γ + E + we get a filtered ∇ -module of signature r over E + . By taking isomorphism classesand using the construction in Definition 3.7 we get a function Z M : X (Γ + ) −→ U r ( E + ) \ U r ( R + ) /U r ( K )which we will call the line integral of M . Example . Let X be Spf( O [[ u, x ]]). In order to give a ∇ -module on X , it issufficient to give a O -linear map: ∇ : O [[ u, x ]] ⊕ −→ O [[ u, x ]] ⊕ ⊗ O [[ u,x ]] Ω O [[ u,x ]] / O satisfying the Leibniz rule, whereΩ O [[ u,x ]] / O = O [[ u, x ]] · du ⊕ O [[ u, x ]] · dx, with differential d : O [[ u, x ]] → Ω O [[ u,x ]] / O given by: d (cid:0) X ij a ij u i x j (cid:1) = X ij ( ia ij u i − x j du + ja ij u i x j − dx ) . Let e , e be the 1st, respectively 2nd basis vector of O [[ u, x ]] ⊕ , and let ∇ be theunique connection of O [[ u, x ]] ⊕ such that ∇ ( e ) = 0 , ∇ ( e ) = e ⊗ dx x , where (1 + x ) − = P ∞ i =0 ( − i x i . Equipped with the frame e , e this ∇ -module isframed of signature (1 , M denote this object. Note that sections of X → Spf( O [[ u ]]) are exactly continuous O [[ u ]]-algebra homomorphisms ψ : O [[ u, x ]] →O [[ u ]]. Every such ψ is determined by ψ (1 + x ) which must be an invertible elementof O [[ u ]]. Conversely for every v ∈ O [[ u ]] ∗ there is a unique such ψ v : O [[ u, x ]] →O [[ u ]] with the property ψ v (1 + x ) = v . The pull-back of M with respect to ψ v isthe framed ∇ -module, where M = O [[ u ]] ⊕ , the frame e , e is the 1st, respectively2nd basis vector of M , and ∇ is the unique connection of M such that ∇ ( e ) = 0 , ∇ ( e ) = e ⊗ dvv . Let φ : M ⊗ Γ + R + ∼ = R ⊕ → T ∼ = R ⊕ be an isomorphism of the type consideredin Definition 3.7 above. Then the matrix V of φ in the basis e ⊗ Γ + R + , e ⊗ Γ + R + is V = (cid:18) w (cid:19) ∈ U (1 , ( R + ) such that d ◦ V = (cid:18) dw (cid:19) = (cid:18) dvv (cid:19) , and hence dw = dvv . So the invariant of the framed ∇ -module ( M, ∇ , e , e ) is log † ( v ), i.e. we get thatthe p -adic line integral: Z M : X ( O [[ u ]]) ∼ = O [[ u ]] ∗ −→ U (1 , ( E + ) \ U (1 , ( R + ) /U (1 , ( K ) ∼ = E + \R + is just the p -adic logarithm. Concluding remarks . What we have described is just the beginning of a theory,barely setting up the formalism to state less trivial results. However the simple, butkey idea is already present: we should think of line integrals as fibre functors (orisomorphisms between them), but the functor should take values in a non-trivialTannakian category, such as ∇ -modules over E + . One of the main reasons to carrythis theory further is to study rational points on varieties over k (( t )) which can beseen as follows.Let X denote the special fibre of X , that is, its base change to Spec( k [[ t ]]).It is a smooth scheme of finite type over Spec( k [[ t ]]). We have a reduction map r : X (Γ + ) → X ( k [[ t ]]). Assume that ( M, ∇ ) is integrable , i.e. the curvature of ∇ ,defined completely analogously to the classical construction is trivial. Then themap R M factors through r : X (Γ + ) → X ( k [[ t ]]), that is, there is a map X ( k [[ t ]]) −→ U r ( E + ) \ U r ( R + ) /U r ( K ) , necessarily unique, whose composition with the reduction map r is the line integralof M . Clearly we need to show the following: let s , s ∈ X (Γ + ) be two sectionssuch that r ( s ) = r ( s ). Then the base changes of the filtered ∇ -modules s ∗ ( M )and s ∗ ( M ) to E + are isomorphic. The latter can be proved in the usual way, usingGrothendieck’s equivalence between integrable ∇ -modules and crystals.The natural next step is to study k [[ t ]]-valued points of smooth projective curvesover Spec( k [[ t ]]) via these line integrals. These have smooth, proper formal lifts toΓ + , and we may look at the universal n -unipotent (and integrable) ∇ -modules on ine integrals over Laurent series fields 17 these lifts, similarly to Besser’s work (see [1]). The natural expectation is that themap which we get this way is independent of the formal lift to Γ + , it is injectiveon residue disks, and it is possible to prove a suitable analogue of the main resultof Kim’s article [3] (Theorem 1 on page 93). Combined with the global methods ofthe paper [4], we are set to give a new proof of the Mordell conjecture over globalfunction fields along the lines of Kim’s method. We plan to carry out this programin a forthcoming publication. Finally, let me also add that such a theory shouldexists also for analytic varieties, in the sense of Huber, over the adic spectrum of( E + , Γ + ), and it is perhaps the natural setting, too. Acknowledgement 3.15.
I wish to thank Amnon Besser and Chris Lazda forsome useful discussions related to the contents of this article, and the referee forhis comments. The author was partially supported by the EPSRC grant P36794.
References [1] A. Besser,
Coleman integration using the Tannakian formalism , Math. Ann. (2002),19–48.[2] K. Kedlaya, p -adic differential equations , Cambridge studies in advanced mathematics ,Cambridge University Press, Cambridge, (2010).[3] M.-H. Kim, The unipotent Albanese map and Selmer varieties for curves , Publ. Res. Inst.Math. Sci. (2009), no. 1, 89–133.[4] C. Lazda, Relative fundamental groups and rational points , Rend. Sem. Mat. Univ. Padova (2015) 1–45.
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