INTEGRATION OF ONE-FORMS ON p-adic ANALYTIC SPACES

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1 INTEGRATION OF ONE-FORMS ON p-adic ANALYTIC SPACES VLADIMIR G. BERKOVICH Recall that there is a unique way to define for every comple manifold, every closed analytic one-form ω, and every continuous path : [0, 1], a number ω C, called the integral of ω along, such that the following is true: (a) if ω = df for an analytic function f on, then ω = f((1)) f((0)); (b) ω depends only on the homotopy class of ; (c) given a second path σ : [0, 1] with σ(0) = (1), one has σ ω = ω + σ ω. The definition is based on the facts that each point of has an open neighborhood isomorphic to an open polydisc and each closed form on the latter is eact (Poincaré lemma). Namely, if ([0, 1]) U 1... U n and t 1 = 0 < t 2 <... < t n+1 = 1 are such that each U i is isomorphic to an open polydisc and ([t i, t i+1 ]) U i for all 1 i n, then ω = n i=1 (f i((t i+1 )) f i ((t i ))), where f i is a primitive of ω at U i. (Of course, one checks that the integral depends only on the homotopy class of the path.) Can an integral of a closed one-form along a path be defined for separated smooth analytic spaces over a non-archimedean field k of characteristic zero? That such definition is possible was demonstrated by Robert Coleman for smooth k-analytic curves ([Col], CoSh]) with k a closed subfield of C p, the completion of an algebraic closure Q p of Q p. I am going to eplain the results from [Ber] which etend Coleman s work to smooth k-analytic spaces of arbitrary dimension. First of all, we recall that each point has an associated non-archimedean field H() so that the value f() of a function f analytic in an open neighborhood of lies in H(). The field H() is in general bigger than k, and so an integral along a path : [0, 1] can be defined at least under the assumption that the ends (0) and (1) of are k-rational points, i.e., H() = k. Such a point has a sufficiently small open neighborhood isomorphic to an open polydisc, and the classical Poincaré lemma implies that every closed one-form ω in an open neighborhood of the point has a primitive (in a smaller neighborhood). Moreover, the latter fact holds for a point if and only if has a fundamental system of étale neighbourhoods isomorphic to an open polydisc or, equivalently, the residue field H() is algebraic over k and the group H() / k is torsion. Let st denote the set of points with the above property. Although the set st is dense in, it is much smaller than so that the topology on st induced from that on is totally disconnected. This means that any path connecting two distinct k-rational points contains points at which closed analytic one-forms non necessarily have a primitive in the class of analytic functions. Thus, in order to define an integral of a closed one-form along a path, one should define a primitive of that form in an open neighborhood of each point of in a bigger class of functions. 1

2 2 VLADIMIR G. BERKOVICH Furthermore, if this is done, one may ask how to define a primitive of a closed oneform with coefficients in the module over the ring of analytic functions generated by the previous primitives, and so on. What is clear is that all primitive constructed in such a way should be analytic in an open neighsorhood of each point from st. To make things more precise, let us consider the simplest eample of a closed one-form that has no a primitive in the class of analytic functions. Let be the analytification G m = A 1 \{0} of the one dimensional split torus. Then the oneform dt T has no primitive at any open neighborhood of each point of the skeleton S(G m ), i.e., the maimal point of a closed disc with centre at zero. This oneform has an analytic primitive at the open disc with centre at any k-rational point a G m (k) = k of radius a. For eample, if a = 1, the power series of the usual logarithm (1 T ) i i=1 i is such a primitive. The natural requirement for a global primitive f that coincides with the latter at the open disc of radius one with centre at one is that it should behave functorially with respect to dt T. If m and p i, i = 1, 2, denote the multiplication and i-th projection G m G m G m, then m ( dt T ) = p 1( dt T ) = p 2( dt T ). Thus, the primitive f should satisfy the relation m (f) = p 1(f) + p 2(f) which in the usual form is written as f(ab) = f(a) + f(b). Such a primitive is determined by its value at p. Fiing this value λ, we get a primitive Log λ (T ) (called a branch of the logarithm) that satisfies the above relation and is analytic at the complement of the skeleton S(G m ). In [Ber], we consider a general situation that allows one to consider all possible branches of the logarithm. Namely, we fi a filtered k-algebra K, i.e., a commutative k-algebra provided with an ehausting filtration by k-vector subspaces K 0 K 1... with K i K j K i+j, and an element λ K 1 which is assigned to be the value of the logarithm at p. For eample, a branch of the logarithm considered in Coleman s work is obtained for K = k. If K is the ring of polynomials k[log(p)] in the variable log(p) with K i = k[log(p)] i, the subspace of polynomials of degree at most i, and λ = log(p), the corresponding primitive is denoted by Log(T ) and called the universal logarithm. We now introduce the maimal class of functions that contains all possible primitives. For i 0, we set O K,i = O k K i and denote by N K,i the étale sheaf on defined as follows: for an étale morphism Y, N K,i (Y ) = lim O K,i (V), where the inductive limit is taken over all open neighborhoods V of Y st in Y. The inductive limit N K = lim N K,i is a sheaf of filtered O -algebras, called the sheaf of naive analytic functions. The sheaves N K,i are functorial with respect to and, in particular, for a morphism ϕ : Y and a function f N K,i (), there is a well defined function ϕ (f) N K,i (Y ). Furthermore, for q 0, the sheaf of N K,i -differential q-forms Ω i is the N K,i, tensor product N K,i O Ω q. The differentials d : Ωq Ωq+1 induce differentials d : Ω q N K,i, Ωq+1 N. Notice that the kernel of the first differential K,i, d : N K,i Ω1 is much bigger than the sheaf N K,i CK,i, = c k K i, where d c = Ker(O Ω 1 ). (The latter is called the sheaf of constant analytic functions; if k is algebraically closed, it is the constant sheaf associated to k.) A D -module is an étale O -module F provided with an integrable connection : F Ω 1 F = F O Ω 1. A D -algebra is an étale commutative O -algebra

3 INTEGRATION OF ONE-FORMS ON p-adic ANALYTIC SPACES 3 A which is also a D -module whose connection satisfies the Leibniz rule (fg) = fdg + gdf. If in addition A is a filtered O -algebra such that all A i are D - submodules of A, then A is said to be a filtered D -algebra. For eample, N is a filtered D -algebra. Here is the main result of [Ber]. Theorem 1. Given a closed subfield k C p, a filtered k-algebra K and an element λ K 1, there is a unique way to provide every separated smooth k-analytic space with a filtered D -subalgebra S λ NK such that the following is true: (a) S λ,0 = O k K 0 ; (b) Ker(S λ,i d Ω 1 S λ,i, ) = CK,i ; (c) Ker(Ω 1 d S λ,i, Ω 2 S λ,i, ) dsλ,i+1 ; (d) S λ,i+1 is generated by local sections f such that df Ω 1 S λ,i, ; (e) Log λ (T ) S λ,1 (G m ); (f) for any morphism ϕ :, one has ϕ (S λ,i ) Sλ,i. The sheaves S λ possess many additional properties. We mention only that, if is connected, then for any nonempty open subset U the homomorphism S λ () S λ (U) is injective. The sheaves S λ are functorial with respect to k,, K and λ. For eample, given a homomorphism of filtered k-algebras K K : λ λ, one has S λ λ K K S. Furthermore, the sheaves Sλ are much bigger than the sheaf O k K. For eample, let s i denote the subsheaf of Slog(p),i (for the universal logarithm) consisting of the functions f that do not depend on log(p), i.e., the restriction of f to some open neighborhood U of each point st belongs to O(U). Then for every i 1, the quotient s i (P 1 )/s i 1 (P 1 ) is of infinite dimension over k. Theorem 1 is used to construct the required integrals of closed one-forms along a path. For a k-analytic space, we set = k C p, and denote by H 1 (, Q) and H 1 (, Q) the singular homology of and with rational coefficients. Theorem 2. Given (k, K, λ) as in Theorem 1, there is a unique way to construct, for every separated smooth k-analytic space with H 1 (, Q) H 1 (, Q), every closed one-form ω Ω 1 S () and every path : [0, 1] with ends in (k), an λ,i integral ω Ki+1 such that the following is true: (a) if ω = df for f S λ,i+1 (), then ω = f((1)) f((0)); (b) ω depends only on the homotopy class of ; (c) given a second path σ : [0, 1] with ends in (k) and σ(0) = (1), one has σ ω = ω + σ ω. One shows that the integral ω is linear on ω, functorial on, and depends nontrivially on the homotopy class of. Moreover, if ([0, 1]) Y, where Y is an analytic subdomain of with good reduction, then ω K0. The net natural question is as follows. What are the analytic differential equations which have a full set of solutions in the bigger class of functions S? Let F be an O -coherent D -module. (Recall that such F is always a locally free O -module.) F is said to be trivial, if it is isomorphic to a direct sum of copies of the D -module O. It is said to be unipotent if there is a sequence of D -submodules F 0 = 0 F 1... F n = F such that all quotients F i /F i 1 are trivial D -modules. The minimal n with this property is called the level of F.

4 4 VLADIMIR G. BERKOVICH Finally, F is said to be locally (resp. étale locally) unipotent, if every point has an open (resp. étale) neighborhood U such that F U is unipotent. Theorem 3. Given and n 1, the following are equivalent: (a) there is an étale neighborhood U of such that F U is unipotent of level at most n; (b) there is an étale neighborhood U of such that, for some m 1, there is an embedding of D U -modules F U (S λ,n 1 U ) m. Let F S λ be the D -module F O S λ. It is a D Sλ-module, i.e., a sheaf of D- modules over the D -algebra S λ. The sheaf of horizontal sections F S is an étale λ sheaf of C K-modules, where CK = c k K. Corollary 4. The following properties of F are equivalent: (a) F is étale locally unipotent; (b) the étale C K-module F S is locally free; λ (c) there is an isomorphism of D S λ-modules F S λ C K S F λ S λ. Notice that the stalk of FS at a point st coincides with F k K. This allows one to define a parallel transport T F : F k K Fy k K along a path : [0, 1] with ends in, y st (see [Ber, 9.4]). Let now be a smooth k-analytic curve of the form an \ n i=1 E i, where is a smooth projective curve over k with good reduction, and E i are affinoid subdomains isomorphic to a closed disc in A 1 with center at zero and lying in pairwise distinct residue classes of the reduction. (It is what Coleman calls a basic wide open curve.) Such is simply connected and H 1 (, c ) = 0. It follows that each one-form ω Ω 1 S () has a primitive in S λ,i+1 (). We set A λ,0 () = O() λ,i k K 0 and, for i 1, define A λ,i () as the O()-submodule of S λ,i () generated by primitives of one-forms from A λ,i 1 () O() Ω 1 (). The filtered O()-algebra of locally analytic functions constructed by Coleman is A λ () = i=0 Aλ,i () for K = k. Since is simply connected, the integral of a one-form with coefficients in A() along a path : [0, 1] with ends in (k) depends only on the ends, and it coincides with the integral constructed by Coleman. The proof of Theorem 1 follows ideas of Coleman s construction. The Frobenius endomorphism, which is used even in the formulation of his result, is sewn here in the proof. Among main ingredients are a local description of smooth k-analytic spaces, based on de Jong s alteration results, and the fact that each point of a smooth k-analytic space has a fundamental system of open neighborhoods U such that H 1 (U, c U ) = 0. Theorem 1 implies that the de Rham comple 0 C K Sλ Ω1 S λ,... is eact at Ω q for q = 0, 1, and we conjecture that it is eact for all q. S λ, One may ask if something like Theorems 1 and 2 holds for other non-archimedean fields of characteristic zero. Of course, for this one should impose additional properties on the integral. Here is a simple eample for the field of comple numbers C provided with the trivial valuation. Let be an irreducible separated smooth scheme of finite type over C. Then the (non-archimedean) analytification an of is a contractible topological space. This implies that a possible integral ω of a closed one-form ω Ω1 ( ) along a path : [0, 1] an with ends, y (C) should depend only on the ends, y, and so it can be denoted by y ω. Let us require that the map (C)

5 INTEGRATION OF ONE-FORMS ON p-adic ANALYTIC SPACES 5 C : y y ω is continuous in the comple topology of both spaces. This imposes continuity on the logarithm homomorphism C C. Any such homomorphism corresponds to a comple number λ C with λ = 1. Namely, if λ = re iϕ 0, the corresponding homomorphism log λ : C C takes z C to log r ( z )e iϕ, and if λ = 0, it takes every z to zero. An analog of Theorem 2 for algebraic one-forms states that, given λ C with λ = 1, there is a unique way to construct for every irreducible separated smooth scheme of finite type over C, any closed one-forms ω Ω 1 ( ) and any pair of points, y (C), an integral y ω C such that the following is true: (a) if ω = df for f O( ), then y ω = f(y) f(); (b) for a third point z (C), one has z ω = y ω + z y ω; (c) ω is linear on ω; (d) for any point z C, one has z dt 1 T = logλ (z); (e) the map (C) C : y y ω is continuous in the comple topology; (f) for any morphism ϕ : and any pair of points, y (C), one has y ϕ (ω) = ϕ(y ) ϕ( ) ω. If is proper, the integral y ω is always zero, which is not surprising since it depends on the ends of a path only. Furthermore, let be considered as a formal scheme. Its generic fiber η is a closed analytic domain in an (which coincides with an if is proper), and let π be the reduction map η s =. Since is smooth, the preimage π 1 () of each point (C) is isomorphic to the (non-archimedean) unit open polydisc with centre at zero with the only C-rational point. Each closed one-form ω Ω 1 ( ) has a primitive f at π 1 () defined up to a constant. If we fi a point 0 (C), the analytic function f on π 1 ( (C)), whose restriction to π 1 () is the local primitive f with f () = 0 ω, is a global primitive of ω which does not depend of 0 up to a constant. It would be interesting to know if it is possible to iterate this procedure, i.e., to construct a primitive of a closed one-form with coefficients in the O( )-module generated by the above primitives f s, and so on. References [Ber] Berkovich, V. G.: Integration of one-forms on p-adic analytic spaces, Annals of Mathematics Studies 162, Princeton University Press, Princeton and Oford, 2007, 156 pp. [Col] Coleman, R.: Dilogarithms, regulators, and p-adic L-functions, Invent. Math. 69 (1982), no. 2, [CoSh] Coleman, R.; de Shalit, E.: p-adic regulators on curves and special values of p-adic L- functions, Invent. Math. 93 (1988), no. 2, Department of Mathematics, The Weizmann Institute of Science, P.O.B. 26, Rehovot, ISRAEL address: vladimir.berkovich@weizmann.ac.il