Exponentially twisted de Rham cohomology and rigid cohomology

We prove a comparison theorem between exponentially twisted de Rham cohomology and rigid cohomology with coefficients in a Dwork crystal.

Introduction 0.1.Cohomology groups with exponential twists.Let k be a field.Let f : X → A1 k be a morphism of algebraic varieties over k.Depending upon what k is, one can consider the following realizations of the "exponential motive" (in the sense of Fresán-Jossen [18]) associated with the function f .
(1) Betti realization.When k is the field C of complex numbers, one can consider the relative singular cohomology H • (X an , f −1 (t) an ) (here t ∈ C and |t| is sufficiently large, in fact any typical value 1 of f will do).The Betti realization has an integral structure.
(2) De Rham realization.For an arbitrary k, one can consider the exponentially twisted de Rham cohomology H • DR (X, ∇ f ), where ∇ f is the integrable connection on the trivial module O X defined by ∇ f (h) = dh + hdf .It should be brought to the reader's attention that the connection ∇ f has irregular singularity, thus does not fit into the picture of [10].
(3) Rigid analytic de Rham realization.When k is a field equipped with a complete ultrametric, one can consider the rigid analytic version of the twisted de Rham cohomology H • DR (X an , ∇ f ).When k is of characteristic 0, it follows from the André-Baldassarri comparison theorem [2, Theorem 6.1] that (2) and (3) are isomorphic2 .Note that the complex analytic version of this result is false even in the simplest situation X = A 1 , f = Id, since ∇ f has irregular singularity at infinity (indeed, the complex analytification of ∇ f is isomorphic to the trivial connection: (4) ℓ-adic realization.Assume that k is a finite field of characteristic p > 0. Fix a nontrivial additive character ψ : k → C * , and an algebraic closure k a of k.Let k m be the subfield of k a such that [k m : k] = m.One can consider the L-series associated with the exponential sums defined by f : By a theorem of Grothendieck, this L-series is the (super) determinant of the Frobenius operation on a twisted Q ℓ -étale cohomology theory.
(5) Crystalline realization.When k is a perfect field of characteristic p > 0, one can consider the rigid cohomology (or rigid cohomology with compact support) Here, L π is a certain overconvergent isocrystal on A 1 k called "Dwork isocrystal".
By a theorem of Berthelot, the rigid cohomology admits a Frobenius operation which, when k is finite, could determine the L-series as in Item (4).
In this note, we shall prove a comparison theorem between (0.1/2) and (0.1/5), thus building a bridge between topology and arithmetic.
To state the result, let us set up some notation.
• Let X be a smooth scheme over a finitely generated Z-algebra R of characteristic 0 which is an integral domain.Let f : X → A 1 R be a morphism.Let σ : R → C be any embedding of R into the field C of complex numbers.
• For each maximal p of R, let κ(p) be the residue field of p, let K p be the field of fractions of W (κ(p))[ζ p ], the ring of Witt vectors of κ(p) with p th roots of unity adjoined.
• For an R-algebra R ′ , we still use f to denote the morphism X R ′ = X × R Spec(R ′ ) → A 1 R ′ .The most accessible statement of our result is the following.Theorem 0.2.There is a dense Zariski open subset U of Spec(R) such that for any p ∈ U , any integer m, the K p -dimension of the rigid cohomology H m rig (X κ(p) /K p , f * L π ) equals the complex dimension of the complex vector space H m DR (X × R,σ Spec(C), ∇ f ).In the main text, we shall give a precise condition on which p is good for the comparison theorem to hold based on ramifications of f at infinity.See Theorem 3.15.
We shall also prove a version of Theorem 0.2 comparing the algebraic Higgs cohomology associated with f and an overconvergent Higgs cohomology.See Proposition 5.4.This latter comparison theorem has significant weaker restrictions on the shape of f .When the X is an open subspace of P 1 , the theorem is due to Phillepe Robba [25].When X is a curve, the theorem is a simple corollary of Joe Kramer-Miller's theorem [21,Theorem 1.1], see Example 4.3.
Theorem 0.2 (or rather its stronger version, Theorem 3.15) is desirable, because it seems that in the literature, the methods used to study exponential sums are either toric in nature, or only applicable to "tame" functions (e.g., Newton-nondegenerate, convenient Laurent polynomials), whereas Theorem 3.15 is unconditional.In practice, Theorem 3.15 allows us to calculate the dimension of the rigid cohomology, hence the degree of the L-series, using topological methods.See §4 for some concrete examples.Here we only explain one general procedure for producing examples on which Theorem 0.2 is applicable.
Example 0.3 ("Standard situation").Let P be a smooth projective variety of pure dimension n over a number field K. Let L 1 , . . ., L r be invertible sheaves on P .Suppose we are given sections s i ∈ H 0 (X, L i ) of these invertible sheaves such that the zero loci D i = {s i = 0} form a divisor with strict normal crossings. Let Then g(x) = s 0 (x)/s ∞ (x) is a well-defined regular function on X.We assume in addition the divisor X 0 = {s 0 = 0} is transverse to all the intersections ) the logarithmic pairs (P, X ∞ ) and (X 0 , X 0 ∩ X ∞ ) all have good reductions at p, and (2) the residue characteristic of p does not divide e 1 • • • e r , then Theorem 0.2 (for the function g : X → A 1 ) is valid at p.Moreover, if r i=1 L ⊗ei is ample, then the rigid cohomology is nonzero in degree n only.For more details, we refer the reader to Corollary 4.10.

Previously known theorems about degrees of L-series.
(a) ℓ-adic theorems.When k is a finite field, and when X = G n m , Denef and Loeser studied the étale cohomology appeared in (0.1/4).They showed that [13,Theorem 1.3] if f is "nondegenerate with respect to its Newton polyhedron at infinity", then the twisted étale cohomology is acyclic except in degree n, and the Frobenius eigenvalues are pure of weight n.In general, they were able to show that the Euler characteristic of the étale cohomology agrees with the Euler characteristic of the algebraic de Rham cohomology (0.1/2) defined by a Teichmüller lift of f (the combinatorial formulas for both theories match).
In the standard situation (0.3), assuming the invertible sheaves L i are ample, the étale cohomology associated with the exponential sums of the function g was studied by Katz, see [19,Theorem 5.4.1].In this case, he proved the L-series is a polynomial or a reciprocal of a polynomial, whose degree can be calculated using Chern classes.We could also deduce these results from Theorem 0.2.
Katz also proved the Frobenius eigenvalues are pure.Our method is not capable of proving this purity result.
(b) p-adic theorems.When k is a finite field and X = G n m ×G m a , the p-adic properties of the L-series were studied by E. Bombieri [5], and later greatly expanded by A. Adolphson and S. Sperber [1].The studies of Adolphson and Sperber are based on Dwork's works [15,16], and methods from singularity theory and toric geometry.
The upshot is that Adolphson and Sperber introduced a complex of p-adic Banach spaces, and an operator α with trace acting on the complex, such that the hyper-determinant of α gives rise to the L-series of the exponential sum.Moreover, when the function f is "nondegenerate and convenient", Adolphson and Sperber proved that the cohomology spaces of this complex are finite dimensional, and concentrated in a single cohomological degree.The dimensions of these cohomology spaces are the same as the algebraic de Rham cohomology spaces.
Even when the exponential sum is defined on G n m × G m a , Theorem 3.15 could imply results that cannot be deduced from the classical theorems, as it could handle Newton-degenerate functions.See Example 4.1 for a (trivial) illustration and Example 4.2 for two more complicated cases.
Remark 0.5.The Dwork-Bombieri-Adolphson-Sperber complex is rather different from the complex computing the rigid cohomology (0.1/5) in two respects: • the Dwork-Bombieri-Adolphson-Sperber complex is a complex of Banach spaces, whereas the complex computing rigid cohomology is a complex of ind-Banach spaces, and is never Banach; • the Dwork-Bombieri-Adolphson-Sperber complex should be thought of as a twisted de Rham complex on a rigid analytic subspace of a toric variety, but the rigid cohomology is defined via a complex on the rigid analytic torus (as a dagger space).Nevertheless, a comparison theorem between a dagger variant of Adolphson-Sperber cohomology and rigid cohomology has been proven by Peigen Li [23].0.6.About the proof.The strategy is to reduce the problems to A 1 via taking direct images, and then use the theory of p-adic ordinary differential equations to deal with the problems on A 1 .There are two major inputs, namely Christol and Mebkhout's characterization of "p-adic regular singularity" [7], and Robba's index computation using radii of convergence [25].It should also be obvious that many of the arguments we present below are influenced by Baldassarri [3], and Chiarellotto [6].
In Section 1 we recall the notion of radius of convergence of a differential module.In Section 2 we explain how to use Robba's index theorem to make local calculations.In Section 3 we globalize the results of Section 2 and prove the main theorem.Section 4 contains some examples.The last section discusses the Higgs variant of Theorem 0.2.
Acknowledgment.We are grateful to Daqing Wan for communications on several examples of exponential sums and for pointing out Katz's theorem.

Radii of convergence
This section reviews the notion of radius of convergence of differential module.We also recall a few basic results, well-known to experts, that we will be using later.
1.1.Notation.In this section we fix the following notation.
• Let K be a complete ultrametric field of characteristic 0. Assume that the residue field of K is of characteristic p > 0. Let π be an element of K satisfying π p−1 + p = 0.The field K is the "base field" where spaces are defined.
• Let Ω be an algebraically closed complete ultrametric field containing K, such that |Ω| = R ≥0 .Assume that the residue field Ω is a transcendental extension of the residue field of K.The field Ω plays an auxiliary role which will give the so-called "generic points" to geometric objects.
• Let I be a connected subset of R ≥0 .Let ∆ I be the rigid analytic space whose underlying set is {x ∈ K alg : |x| ∈ I}.
Let D ± (x; r) be the rigid analytic space whose underlying set is the open/closed disk of radius r centered at x ∈ K.We use O to denote the sheaf of rigid analytic functions on these spaces.
• By a "differential module" over ∆ I or D ± (a; r), we shall mean a finite free O-module E over ∆ I or D ± (a; r) equipped with an integrable connection.
• The ρ-Gauss norm on K It extends to the field K(x) of rational functions naturally.For ρ ∈ R >0 , denote by F ρ the completion K(x) with respect to the ρ-Gauss norm | • | ρ .It turns out that F ρ is also a complete ultrametric field, and carries a continuous extended derivation d/dx.
• A differential module over F ρ is a finite dimensional F ρ -vector space V equipped with an K-linear map D : V → V , such that for any a ∈ F ρ , any v ∈ V , the Leibniz rule D(av) = da dx v + aD(v) holds.It follows that D is automatically continuous.Remark 1.2 (From O-modules to F ρ -modules).Let I ⊂ R ≥0 be an interval.Let ρ > 0 be an element in I. Then there is a natural continuous homomorphism To construct the homomorphism, first assume I = [a, b] is a closed interval.Then each rigid analytic function f on ∆ I could be written as such that c n b n → 0 as n → +∞, and c n a n → 0 as n → −∞.In particular, c n ρ n → 0 as n → ±∞.Set P N (x) = |n|≤N c n x n ∈ K(x).Then P N → f with respect to the supremum norm of O(∆ I ).The condition that c n ρ n → 0 implies that (P N ) ∞ N =1 is a Cauchy sequence with respect to the ρ-Gauss norm on K(x).Hence lim N →+∞ P N exists in F ρ .We define this element to be ϕ ρ (f ).
In general, we choose an interval [a, b] ⊂ I containing ρ and define ).One checks that this definition satisfies the required properties.
Thus, if N is a free O-module on ∆ I for some connected I ⊂ R ≥0 , ρ ∈ I, then the pullback V = N ⊗ O(∆I ),ϕρ F ρ gives rise to a differential module over F ρ , with D(n) = ∇ d dx n for any n ∈ N .For simplicity we shall write this tensor product simply by V = N ⊗ O F ρ .Definition 1.3.Let V be a vector space over F ρ equipped with a norm | • |.Recall that the operator norm of an operator T on V is defined to be |T | V = sup v∈V {0} |T (v)|/|v|; and the spectral radius of T to be the quantity The operator norm of T certainly depends upon the norm, but two equivalent norms determine the same spectral radius [20,Proposition 6.1.5].
Let V be a differential module over F ρ .Then the radius of convergence of V is Example 1.4.The spectral radius of the trivial differential module (F ρ , d/dx) equals |π|ρ −1 .Thus its radius of convergence equals ρ.As the spectral radius of a differential module V is bigger than or equal to that of d/dx [20, Lemma 6.2.4], we know that the radius of convergence of any differential module over F ρ is in the range ]0, ρ].
The terminology "radius of convergence" comes from the so-called "Dwork transfer theorem", which we record below.
Theorem 1.5 (Dwork).Let M be a differential module over ∆ I of rank n.Let ρ ∈ I. Then the following two conditions are equivalent.
(1) The radius of convergence of M ⊗ O F ρ is R.
Proof.The proof of (1) ⇒ ( 2) is [20,Theorem 9.6.1].Here the variable t used by Kedlaya is t − ξ in our context, and the differential module considered by Kedlaya is the restriction of M to the open disk (thus the connection matrix automatically has entries in the ring of analytic elements, fulfilling the hypothesis of the cited theorem).
The proof of ( 2) ⇒ ( 1) is [20,Proposition 9.7.5].Here it is important to note that we should consider the field Ω instead of K itself, so that we have enough "generic points" available.
The "most" convergent differential module over ∆ I are those who satisfy the Robba condition.
Definition 1.6.Let M be a differential module over ∆ I .M is said to satisfy the Robba condition if for any ρ ∈ I, the differential module M ⊗ O F ρ has radius of convergence equal to ρ. Lemma 1.7.Let M be a differential module over ∆ I satisfying the Robba condition.Then any subquotient differential module of M satisfies the Robba condition.
Proof.Let x 0 be a point of ∆ I,Ω .
Let M ′′ be a quotient of M .Then the horizontal basis of Finally, we quote a theorem due to Christol and Mebkhout [7].See also [17] and [20,Theorem 13.7.1].
Theorem 1.8 (Christol-Mebkhout).Let M be a differential module over ∆ ]0,1[ .Assume that there exists a basis e 1 , . . ., e n of M such that • the entries of the matrix representation η of ∇ t d dt with respect to this basis belong to O(D − (0; 1)), • M is overconvergent (see Definition 1.3), and Then there exists a basis of M under which the matrix of ∇ t d dt has entries in Z p .Moreover, M satisfies the Robba condition.

Indices of differential modules
In this section, we use the notion of radii of convergence and Robba's index theorem to prove some cohomology groups are zero.The notation and conventions made in Paragraph 1.1 are still enforced in this section.
Lemma 2.1.Let N be a differential module over ∆ [a,b] .Assume that there exists a ≤ ρ ≤ b such that the radius of convergence of any differential submodule of ) is noetherian, N ′ is finitely generated, and is equipped with a trivial connection.It follows that N ′ is a finite free differential module over ∆ [a,b] , say of rank r.In particular, it is flat over O.It follows that The hypothesis then implies that r = 0, in other words, We give a simple calculation of the radius of convergence.
Proof (cf.[25, 5.4.2]).Let t ∈ Ω be a any point of radius ρ.Let x = t + y.A horizontal section of the differential system is given by exp −π 1 t+y − 1 t .The Taylor series for For each r < ρ, the r-Gauss norm of this Taylor series equals Thus exp π 1 t+y − 1 t converges for y in the open disk D − (0; r) where r < ρ 2 .
Indeed, write  The above vanishing of cohomology groups implies the vanishing of the cohomology groups of some special differential modules over the Robba ring.
Definition 2.5.The Robba ring is the colimit Let s be a horizontal section of M ⊗ L over the Robba ring, then there must exist r < 1 such that the section is defined on the annulus ∆ ]r,1[ , and thus on the annulus ∆ [a,b] for any [a, b] ⊂ ]r, 1[.By Lemma 2.4, we know the section has to be zero on all such ∆ [a,b] , and hence the section must be globally zero.This implies that H 0 DR (R, (M ⊗ L) ⊗ R) = {0}.Thanks to the vanishing of H 0 , the vanishing of H 1 will follow if we can show the Euler characteristic of M ⊗ L ⊗ R is zero.By Theorem 1.8, we can make a Q p -linear change of bases to put the matrix of ∇ x d dx in a upper triangular form.Thus there is a filtration of M by differential submodules such that the subquotients M i /M i−1 are of rank 1, necessarily satisfy the Robba condition (Lemma 1.7).Thus, the vanishing of the Euler characteristic of M ⊗ L ⊗ R is implied by the vanishing of the Euler characteristic of M i /M i−1 ⊗ L. Thus we may assume M has rank 1 and is defined by a differential equation Choose a sequence of numbers a n (r) such that a n (r) ↑ 1 − .The above complex reads The transition maps in the inverse system having dense images, one knows that R 1 lim n equals zero (Kiehl's Theorem B).Thus it suffices to prove the Euler characteristic of (2.7) is zero.Note that M ⊗L is defined by a differential operator of order one with coefficients in Ω(x).Thus we can use Robba's index theorem [25,Proposition 4.11], which in our situation asserts that, if Since the radius of convergence of M ⊗ L at F ρ always equals ρ 2 , both quantities of the right hand side of the displayed equation are equal to 2. Thus the Euler characteristic is zero.This completes the proof of the lemma.
Lemma 2.9.Let M be differential module over ∆ ]0,1[ satisfying the hypothesis of Theorem 1.8.Let L be as in Example 2.2.Then Proof.The de Rham cohomology groups are computed by the inverse limit where Again, R 1 lim is zero since the transition maps have dense image (Kiehl's Theorem B).Thus, it suffices to prove the vanishing of H * DR for each ∆ I .For each closed interval I ⊂ ]0, 1[, the vanishing of zeroth cohomology follows from Lemma 2.4.To show that the first cohomology groups are zero, it suffices to prove the Euler characteristic of is zero.By Theorem 1.8, there is a filtration of M by differential submodules such that the subquotients M i /M i−1 are of rank 1, necessarily satisfy the Robba condition.By induction, it suffices to prove the assertion assuming M has rank 1.In the rank 1 case, Robba's index theorem (2.8) applies.Arguing as in the proof of Lemma 2.6 shows that the Euler characteristic is zero.
3. Rigid cohomology associated with a regular function 3.1.In this section, we continue using the notation made in Section 1. Thus k is a perfect field of characteristic p > 0, K is a complete discrete valued field of characteristic 0 containing an element π satisfying π p−1 + p = 0, and the residue field of K is k.The ring of integers of K is denoted by O K Our policy is to use Gothic letters to denote schemes over O K .For a O K -scheme S, let S 0 = S ⊗ OK k, S = S ⊗ OK K.For a finite type K-scheme T , let T an be the rigid analytic space associated with T .

3.2.
Let conventions be as in Paragraph 3.1.Let f : X → P 1 OK be a proper morphism between smooth O K -schemes.We make the following assumption.
( * ) Let S ⊂ P 1 K be the non-smooth locus of f : X → P 1 K .Then the intersection of P 1 k with the nonsmooth locus of f : X → P 1 OK is contained in the Zariski closure of S. In addition, we assume S ∩ D − (∞; 1) is a subset of {∞}.The condition ( * ) will be used to ensure the Gauss-Manin connection on D − (∞; 1) is overconvergent.
Suppose that X has relative pure dimension n over O K .Let X = f −1 (A 1 OK ).Assume that the polar divisor P = f * (∞ OK ) is a relative Cartier divisor with relative strict normal crossings over O K .Write P = m i D i , where D i are smooth proper O K -schemes.We assume p ∤ m i for any i.The morphisms X 0 → P 1 k and X → P 1 K induced by f are still denoted by f .This abuse of notation is unlikely to cause confusions.

3.3.
In order to calculate rigid cohomology, we need to set up some notation for tubular neighborhoods.For r < 1, set , and V r := X [P 0 ] X,r .Denote by j the inclusion map ]X 0 [ X → X, and by j r the inclusion map V r → X.

(The Dwork isocrystal). In this paragraph we explain what the Dwork isocrystal is. The affine line A 1
k sits in the frame (A 1 k ⊂ P 1 k ⊂ P 1 OK ) where P 1 OK is the formal completion of the projective line over O K with respect to the maximal ideal of O K .Therefore to describe this crystal we only need to write down a presentation of it (as an integrable connection) on P 1,an K .On the rigid analytic projective line, the tubular neighborhood of ∞ k ∈ P 1 k is the complement of the closed unit disk D + (0, 1) of radius 1, i.e., D − (∞; 1).The analytification of the algebraic integrable connection k , which could be taken as a coefficient system for the rigid cohomology.We denote it by L π , and call it the Dwork isocrystal [22, §4.2.1].

3.5.
Let notation and conventions be as in Paragraph 3.2.Consider the morphism f : X → A 1 K .Then we can define an algebraic integrable connection on the struc- By analytification, we obtain an analytic connection, still denoted by ∇ πf , on the rigid analytic space X an .Proposition 3.6.Let notation and conventions be as in Paragraphs 3.2 -3.5.Then for each integer m, the natural maps are isomorphisms of K-vector spaces.
Proof that the right hand side arrow is an isomorphism.Without loss of generality we could assume X is irreducible.When f | X : X → A 1 K is a constant morphism, the proposition is trivial.In the sequel we shall assume f | X : X → A 1 K is surjective.To begin with, let us write down the complex that computes the rigid cohomology.Let The analytification of the connection ∇ πf (still denoted by ∇ πf ) gives rise to an integrable connection an , which extends to a dagger version of the de Rham complex DR(X an , ∇ † ) : For an admissible open subspace V of X an , let DR(V, ∇ πf | V ) be the de Rham A priori, this complex depends upon the formal completion of the scheme X along the maximal ideal of O K .However, since X is proper and the integrable connection (O X an , ∇ πf ) is overconvergent (being the inverse image of an overconvergent integrable connection), [22, Corollary 8.2.3] asserts that it only depends upon X 0 and f | X0 : X 0 → A 1 k .We have then DR(X an , ∇ † ) = RΓ rig (X 0 , f * 0 L π ).Next we explain how to compute the cohomology of the de Rham complex.For any r sufficiently close to 1, set V r = f −1 (D − (0; r −1 )).Then X an has an admissible open covering In the bounded derived category of K-vector spaces, the Mayer-Vietoris theorem gives an isomorphism between the de Rham complex DR(X, ∇ πf ) and the homotopy fiber of the map Since the colimit, as r → 1 − , of DR(V r , ∇ πf ) equals RΓ rig (X 0 , f * 0 L π ), in order to prove the comparison between rigid and de Rham cohomology it suffices to prove the natural morphism (3.7) is an isomorphism in the derived category of vector spaces over K.
Let S be the finite subspace of A 1,an K containing all the critical values of f .Let X ′ = X f −1 (S).Then the direct image sheaf is equipped with a Gauss-Manin connection ∇ GM .By projection formula, we have R Using the Leray spectral sequence, we see that in order to prove the morphism (3.7) is an isomorphism, it suffices to prove that for any i, the map (3.9) is an isomorphism.We shall show that both sides of (3.9) are acyclic.For convenience, we shall use a coordinate x around ∞ ∈ P 1,an K , thus swap ∞ and 0. Let M be the restriction of the analytification of E = R i f * (Ω • X ′ /K ) to the disk D − (0; 1) {0}.In Lemma 3.10 below, we shall explain that M satisfies the hypothesis of Theorem 1.8.On the other hand, the differential module L considered in Example 2.2 is precisely the restriction of (O A 1 K , ∇ D ) an in the vicinity of ∞.The right hand side of (3.9) now reads which is precisely the de Rham complex of M ⊗ L restricted to the Robba ring: Thus, Lemma 2.6 implies that the right hand side of (3.9) is trivial.The acyclity of the left hand side of (3.9) follows from Lemma 2.9.This completes the proof that the analytic twisted de Rham cohomology is isomorphic to the rigid cohomology.Lemma 3.10.Let notation be as above.Then the differential module M satisfies the hypotheses of Theorem 1.8.
Proof.By the regularity of the Gauss-Manin system, the differential module M is a restriction of an algebraic integrable connection which is regular singular around 0. In particular, there exists a basis of M such that the derivation ∇ d dx is given by where η is a rational function which has at worst a simple pole at x = 0 (for example, take the restriction of an algebraic basis and restrict to the analytic open D − (∞; 1)).
As we have assumed that the multiplicities of the polar divisor are prime to p, the algebraic calculation of exponents of the Gauss-Manin system around infinity implies that the eigenvalues of (xη)| x=0 belong to Z p ∩ Q. (One can embed field of definition of the variety into C, then use [11, Exposé XIV, Proof of Proposition 4.15] to show that the eigenvalue of η(0) with respect to the algebraic basis are rational numbers whose denominators are not divisible by p.) Finally the overconvergence of the Gauss-Manin system in the lifted situation under the hypothesis ( * ) is proved by P. Berthelot [4, Théorème 5] and N. Tsuzuki [28,Theorem 4.1.1].
Proof that the left hand side arrow is an isomorphism.This is a theorem of André and Baldassarri [2,Theorem 6.1].Since the precise hypothesis of their theorem is not met in the present situation (the connection is not defined over a number field), we shall nevertheless provide a proof.The key point is a theorem of Clark, which states that in a good situation, the analytic index of a differental operator equals its formal index.
By GAGA, the algebraic de Rham complex is computed by complex which is a subcomplex of DR(X an , ∇ πf ). Here an by X an = f −1 (A 1,an ) and Taking colimit with respect to ǫ → 0, we see it suffices to prove the colimit, as ǫ → 0, of the following map is a quasi-isomorphism.Again, we shall show both items are acyclic.
Let E be the algebraic Gauss-Manin system on A 1 S as in (3.8).Let ι : A 1 K S → P 1 K be the inclusion.Let E( * S) = ι * E. Using Leray spectral sequence, it suffices to prove the de Rham complex of (3.11) colim ("de Rham complex with essential singularities") and that of (3.12) ("de Rham complex with moderate singularities") are acyclic.
A similar argument as in the proof of Lemma 2.9 using Robba's index theorem yields that (3.11) has zero Euler characteristic and vanishing H 0 , thus acyclic.
Since the de Rham complex of (3.12) is a subcomplex of that of (3.11), its H 0 is also trivial.
We proceed to prove that (3.12) has zero Euler characteristic.Let O 0 be the local ring O P 1,an K ,∞ with the uniformizer x defined by a coordinate around ∞. Let O 0 ∼ = K[[x]] be its x-adic completion.Then the de Rham complex of (3.12) is that of Choose a cyclic vector [10,II 1.3] for the differential module Thus we obtain a differential operator u = i a i dx i with a i ∈ O 0 , and the Euler characteristic equals the index of On the other hand, Malgrange's index theorem [24, Théorème 2.1b] implies that the index of x] equals zero.Thus, it suffices to prove the index of Since E is regular singular, ∇ D is rank one and irregular, the indicial polynomial of u is zero.Thus the hypothesis on non-Liouville difference in Clark's theorem [8] as stated in [6, Théorème 2.12] is satisfied, and this theorem implies the index of (3.13) is zero.This completes the proof of Proposition 3.6.
Next we prove the main result of this note by removing the properness hypothesis from Proposition 3.6.

3.14.
We follow the conventions made in Paragraph 3.1.Let f : X → P 1 OK be a proper morphism between smooth O K -schemes, such that the condition ( * ) in Paragraph 3.2 is satisfied.Assume X has relative pure dimension n over O K .
Let X be a Zariski open subscheme of X, such that Proof.For each subset J of {1, 2, . . ., s}, denote by H J the intersection j∈J H j .Then each H J is a smooth proper K-scheme, and the restriction of f to H J is denoted by f J , which is a proper morphism into P 1 K .The polar divisor of f J is the restriction of P to H J .Note that ∇ πf restricts to a connection on H J , which is ∇ πfJ .By Proposition 3.6, the natural maps are isomorphisms.By general results in rigid cohomology, there exists a second-quadrant spectral sequence with which abuts to H −i+j rig (X 0 /K, f * L π ).(We are unable to locate a precise reference, but this could be deduced from [22,Proposition 7.4.13].)We have similar spectral sequences for the algebraic and analytic de Rham cohomology groups.The E 1differentials of these spectral sequence are all Gysin maps in the various theories.Thus the natural maps between the theories give rise to maps of these E 1 -spectral sequences.Proposition 3.6 implies that these maps are isomorphisms on the E 1stage.Thus they induce isomorphisms on the abutments.
Proof of Theorem 0.2.Let notation be as in the statement of Theorem 0.2.Let K be the field of fractions of R. Using resolution of singularities, upon making a finite extension of K (both rigid and de Rham cohomology are compatible with extensions, so performing a finite extension does not change the result), we can embed X K into a smooth proper K-scheme X K such that X K X K has strict normal crossings, and f extends to a morphism f : • X K as well as all the intersections of the boundary divisors have good reduction at primes in U , and • the multiplicities of the polar divisor are not divisible by residue characteristics of U .We may assume U is affine, and by abusing notation still denote its associated ring by R. For each closed point p in U , let p be the characteristic of the residue field κ(p).Then there exists a p-adically complete discrete valuation ring R ′ containing ζ p and a surjective ring homomorphism R → R ′ , such that the closed point of Spec(R ′ ) is mapped to p. Let K ′ be the field of fractions of R ′ and let k ′ be its residue field.Then we have We can then apply Theorem 3.15 to X R ′ , obtaining an isomorphism between the rigid cohomology H • rig (X k ′ /K ′ , L π ) and a twisted de Rham cohomology of X K ′ .To conclude, we use the following two facts: (i) ∇ cf and ∇ f have isomorphic de Rham cohomology group over K ′ , for any c ∈ K ′× ; and (ii) the formation of twisted algebraic de Rham cohomology is compatible with extension of scalars.The fact (i) is proved below as a lemma.Lemma 3.16.Let K be a field of characteristic 0. Let f : X → A 1 be a morphism of smooth K-schemes.Then for all c ∈ K × , the dimensions of the K-vector spaces H i (X, ∇ cf ) are the same.
Proof.It suffices to prove ∇ cf and ∇ f have isomorphic twisted de Rham cohomology groups.The standard argument (extracting coefficients defining X and f , choosing a finitely generated subfield, embedding the field into C) allows us to assume K = C. Then we can use the isomorphism provided by (0.1/1) and (0.1/2) to conclude that the two twisted de Rham cohomology groups are isomorphic to the relative cohomology groups H i (X an , f −1 (t) an ), and H i (X an , (cf ) −1 (t) an ) (|t| > > 0) respectively.When |t| is large, these two groups are isomorphic, since f is topologically a fibration away from finitely many points.

Examples
Example 4.1.As a sanity check, here is a trivial example calculating the degree of the L-series of a Newton degenerate polynomial.
Let k be a finite field of characteristic > 2. Let f : A 2 k → A 1 k be the regular function defined by the polynomial f (x, y) = x 2 y − x.(The projective completion of the curve f (x, y) = t, t = 0, has a cuspidal singularity at [0, 1, 0], which can be simultaneously resolved by blowing this point up.) The morphism is smooth, but the polynomial f is neither Newton non-degenerate nor convenient, thus the theorems of Adolphson-Sperber [1] and Denef-Loeser [13] do not give the degree of the L-series.
A direct computation is easy: It follows that the L-series is (1 Here is a topological calculation.It is easy to see f (x, y) = t for t = 0 is isomorphic to G m as schemes over k.Thus one can still use the Teichmüller lift of f to conclude that the rigid cohomology H * (A 2 k , f * L π ) is 1-dimensional in degree 2, and zero otherwise.In particular, L f (t) −1 is a linear polynomial.
Next, we recall the degree and total degree of the L-series of exponential sums.Let notation be as in Paragraph (0.1/4).In view of Grothendieck's theorem, we can write L f (t) = P (t)/Q(t), where By Berthelot's trace formula for rigid cohomology, we know that the degree of L f is equal to the Euler characteristic (−1) i dim H i rig,c (X, f * L π ), and the total degree of L f is no bigger than dim H i rig,c (X, f * L π ).(Since X is smooth, Poincaré duality identifies the dimension of the compactly supported cohomology with the dimension of the rigid cohomology up to a dimension shift.)Example 4.2.Hyperplane arrangements give rise to many interesting exponential sums whose L-series cannot be computed using traditional methods.But topological and combinatorial methods sometimes can deduce useful information about the Milnor fiber of the arrangement, which, through Theorem 3.15, can determine the dimension of the rigid cohomology for large primes.
Here are two concrete examples.
(1) Let f (x, y, z) = xyz(x − y)(y − z)(z − x) be the polynomial defining the so-called A 3 plane arrangement.Then f is not convenient, but Cohen and Suciu [9,Example 5.1] have shown that the Betti numbers of the "Milnor fiber" f = 1 are: b 0 = 1, b 1 = 7, b 2 = 18.Since f is homogeneous, all the fibers f −1 (t), t = 0, are homeomorphic.It follows that the relative cohomology H • (C 3 , f −1 (t)) equals the reduced singular cohomology of f −1 (1), up to a shift of cohomology degree.Using the isomorphism between relative cohomology (0.1/1) and twisted de Rham cohomology (0.1/2), we deduce that the nonzero Betti numbers of the twisted de Rham cohomology are b 2 = 7, b 3 = 18.For p large, Theorem 0.2 implies that the rigid cohomology groups of f * L π with compact support are of dimensions 0, 7, 18 respectively.In particular, the degree of the L-series of the exponential sum associated with f has degree 11; and total degree ≤ 25.
( As in (1), when p is large, we get the dimension of the rigid cohomology of f * L π , and we conclude that the degree of the L-series of f is 71, and the total degree is ≤ 87.
Example 4.3 (Exponential sum on a curve).Let X be a smooth irreducible projective curve of genus g over a finite field k.Let f : X → P 1 k be a generically étale, degree d morphism with ramification indices not divisible by p.We assume there is a good lift of f (i.e., satisfies Hypothesis 3.14).Let Z = f −1 (∞) = {τ 1 , . . ., τ m }.Let V be a nonempty Zariski open subset of X such that f (V ) ⊂ A 1 k .We assume that the ramification points of f as well as the points X V are rational over k (which is harmless in considering the degree of L-series).Let c = Card(Z V ).
Then a topological calculation implies that the twisted algebraic de Rham cohomology of a lift of f has cohomology in degree 1 only, and the dimension equals 2g + c + m + d − 2. In this case Theorem 0.2 applies and we conclude that L −1 f |C (t) is a polynomial of degree 2g + c + m + d − 2. This matches the length of the "Hodge polygon" considered by Kramer-Miller [21,Theorem 1.1].
Example 4.4 (Exponential sum on SL 2 ).Let k be a finite field of characteristic p > 0. Let V = k 2 be the standard representation of SL 2 .For A ∈ SL 2 (k), let is a regular function on SL 2 .Then the rigid cohomology H * rig (SL 2 , f * L π ) is related to the exponential sum (4.5) where k m is a fixed degree m extension of k, ψ 1 is a nontrivial additive character on k, ψ m = ψ 1 • Tr km/k .If p is sufficiently large, and if (a 1 , . . ., a N ) is sufficiently general, we can calculate the dimension of the rigid cohomology using topology.It is not difficult to see f −1 (t) is N disjoint union of P 1 × P 1 ∆.Using the long exact sequence for relative cohomology one sees that H * (SL 2 (C), f −1 (t)) is nonzero only in degree 1, and 3 and of dimension N − 1, N + 1 respectively.Thus the L-series L f (t) of the exponential sums (4.5) is the reciprocal of a degree 2N polynomial.We begin with some rather trivial topological discussions.In (4.7) -(4.8), complex algebraic varieties are equipped with analytic topology, and singular cohomology groups are assumed to have coefficients in Q.
Let f : X → A 1 be a proper, generically smooth morphism of algebraic varieties over C. Let X be an open subvariety of X, and E = X X. Suppose that there is a neighborhood T of E such that f | T and f | T E are locally topologically trivial fibrations (hence are trivial as A 1 is contractible).Let F be a generic fiber of f and Proof.This is a simple application of excision.To begin with, since T → A 1 is a trivial fibration, there is a deformation retract from T onto T ∩ F .This induces a deformation retract from T ∪ F onto F .Thus the pair (X, T ∪ F ) and the pair (X, F ) have the same cohomology.
Since E is contained in the interior of T ∪ F , excision implies that the pair (X, T ∪ F ) and the pair (X, (T E) ∪ F ) have the same cohomology.Using the fact that T E → A 1 is a trivial fibration, we find a deformation retract from (T E) ∪ F onto F .Thus the pair (X, (T E) ∪ F ) and the pair (X, F ) have the same cohomology.This completes the proof.
Let P be a smooth projective variety.Let L 1 , . . ., L r be invertible sheaves on P .Fix sections s i ∈ H 0 (P, L i ), whose associated Cartier divisor is D i .Assume that D i is smooth and irreducible.Let e 1 , . . ., e r be non-negative integers.Let s ∈ H 0 P, r i=1 L ⊗ei i .Denote by X 0 the Cartier divisor defined by s, and assume that X 0 is smooth and irreducible.Let X ∞ = e i D i .Assume that X 0 + X ∞ is a divisor with strict normal crossings.Let X = Bl Z P X ∞ , where Z is the base locus of the pencil generated by X 0 and X ∞ .Then there is a generically smooth morphism f : X → A 1 and a projection π : X → P .Now we are in the situation considered in Lemma 4.7.Here E is the intersection of the exceptional divisor in Bl Z P with X.By construction, it has a "tubular neighborhood" T (e.g., the preimage of a tubular neighborhood of Z under the blowing up) within X such that the restriction of f to T is a topologically trivial fibration.
Retain the above notation.Let s ∞ = s ei i .Let X = P X ∞ .Then via the morphism π, we can regard X as an open subscheme of X.The restriction of f to X is given by the ratio g = s 0 /s ∞ .Then by Lemma 4.7, for t generic we have In practice, we are more interested in considering the function g on X; and the construction above allows us to construct a proper function which is ready for taking reduction modulo p.
The next lemma tells us that in a certain preferable situation, the calculation of cohomology groups reduces to the calculation of the Euler characteristics.Lemma 4.8.Notation as above, assume in addition that X and P X 0 are affine (e.g., when the invertible sheaf . See for example [19,Theorem 5.4.1].If the hypothesis of Lemma 4.8 is fulfilled, then the absolute value of this number is also the dimension of H dim X (X, g −1 (0)).
Proof of Lemma 4.8.Since X and g −1 (0) are smooth affine varieties, the relative cohomology H i (X, g −1 (0)) vanishes if i > dim X.It suffices to prove the relative cohomology also vanishes when i < dim X.
For any subset J of {1, 2, . . ., r}, let D J = j∈J D j .Let D (p) = Card J=p D J , and write D (0) = P .The scheme D (p) is smooth proper of dimension n − p, and the natural morphism D (p) → P is affine.
There exists a spectral sequence (4.9) Granting the existence of this spectral sequence, let us finish the proof.Since P X 0 is affine, D (p) D (p) × P X 0 are affine for all p.If i < dim X, then i − p < dim D (p) .By Artin's vanishing theorem, we have This implies that H i (X, g −1 (0)) vanishes when i < dim X, as desired.The construction of the spectral sequence (4.9) will be recalled in Paragraph 4.11 at the end of this section.
where v : P X 0 → P is the open immersion.
] can be understood explicitly using the fact that D J intersects X 0 transversely.Indeed, if a J is the inclusion morphism from D J into P , then a i = Card J=i a J , and (we use ∨ for Verdier dual) We claim that a * J Rv * Q ∼ = Rv J * Q.Indeed, consider the following fiber diagram, X 0 ∩ D J X 0 D J P ιJ ι aJ , and the distinguished triangle Since X 0 is a smooth divisor in P , the Thom isomorphism theorem implies that . Pulling back the above distinguished triangle by a J yields a distinguished triangle Since ι is proper, and since X 0 ∩ D J is smooth of codimension 1, we have By adjunction, (Hom being taken in the derived category) Hom(ι Thus up to a nonzero constant multiple on each connected component of X 0 ∩ D J , there exists only one nonzero morphism from ι J * ι !J Q to Q in the derived category.(In fact, using the existence of integral structure, the term "up to a nonzero constant" could be replaced by "up to sign".)Therefore the homotopy cofiber of the morphism is necessarily isomorphic to the homotopy cofiber of the canonical morphism ι J * ι !J Q DJ → Q DJ which is Rv J * Q.This justifies the claim.
Thus, letting p = Card J be the codimension of D J , we have The complex a J * (v J! Q) is precisely the derived incarnation of the relative cohomology H • (D J , D J ∩ X 0 ).The spectral sequence (associated with the "filtration bête") of the complex a * a !v !Q → Q gives the spectral sequence (4.9).

A remark on Higgs cohomology
In the theory of twisted algebraic de Rham cohomology, one can associate an irregular field to the connection ∇ f , i.e., the Higgs field defined by ∧df .A celebrated theorem of Barannikov-Kontsevich (the first published proof is due to Sabbah [27]) asserts that the algebraic Higgs cohomology and the twisted algebraic de Rham cohomology have the same dimension.
In this section we explain how to transplant this to the rigid analytic world.We shall prove a comparison between a "dagger" Higgs cohomology and the algebraic Higgs cohomology.Then in the nice situation, Theorem 0.2 allows us to relate twisted rigid cohomology and the dagger Higgs cohomology.
In the work of Adolphson-Sperber [1] on exponential sums, one also finds the use of Higgs cohomology.In fact their finiteness theorem is deduced from the finiteness of the Higgs cohomology of the reduction.In contrast, the result of this section happens completely on the generic fiber, thereby does not really concern whether the pole divisor has good reduction or not.

Notation.
• Let K be a discrete valuation field of characteristic 0 whose residue characteristic is p.Let X be a scheme smooth over a discrete valuation ring O K .
• Let f : X → A 1 R be a proper function admitting a compactification f : X → P 1  OK , in which X is smooth over R.
• Also denote by f : X → P 1 K and f : X → P 1 K be the restriction of f to the generic fibers X and X of X and X respectively.
• Let V r be the inverse image f −1 (D + (0; r)) of the rigid analytic disk under f .Thus V r is an rigid analytic subspace of X. 5.2.Hypothesis.We assume that f : X → A 1 K has no critical values in D − (∞; 1).
Note that we do not enforce the hypothesis that the components of the pole divisor have good reduction anymore.

5.3.
We can consider three types of Higgs cohomology.
It remains to show that the "moderate Higgs complex" on Sp(A) is acyclic.With respect to the coordinate system, this is the Koszul complex of the partial derivatives of f on the ring A[1/f ] (the localization of A).Again, since f is smooth, the partials of f form a regular sequence and has no common zeros.Thus the moderate Higgs complex is also acyclic.This completes the proof.
We leave the apparent generalization to non-proper functions (using logarithmic forms) to the reader.It should be mentioned that one cannot use algebraic forms in the non-proper case to calculate the Higgs cohomology, as non-isolated critical points will contribute infinite dimensional cohomology.

Lemma 2 . 4 .
Suppose [a, b] is an interval contained in ]0, 1[.Let M be a differential module on ∆ [a,b] satisfying the Robba condition.Then we have H 0 DR (∆ [a,b] , M ⊗ L) = 0. Proof.By Lemma 1.7, any submodule of M satisfies the Robba condition.By [20, Lemma 9.4.6(c)] and Example 2.2, the radii of convergence of all differential submodules of M ⊗ L are equal to the radius of convergence of L, which is < ρ at F ρ .Thus Lemma 2.1 implies the desired result.

1 .
is easily seen to have ρ as its radius of convergence on D − (a; ρ) for any |a| ≤ 1, ρ < Its restriction to the tubular neighborhood of ∞ is precisely the connection we dealt with in Example 2.2, hence has radius of convergence equal to ρ 2 at F ρ , which converges to 1 as ρ → 1.Thus (O A 1,an K , ∇ D ) is overconvergent for any open disk.By the theory of rigid cohomology ([22, Definition 7.2.14, Proposition 7.2.15]), the differential module determined by (O A 1,an K , ∇ D ) gives rise to an overconvergent isocrystal on A 1 is a relative Cartier divisor with relative strict normal crossings over O K .The support of the polar divisor P = f * (∞ OK ) is therefore contained in D. We could write P = r i=1 m i D i .Again, we assume p ∤ m i for any i.Finally we could write D = r i=1 D i + s j=1 H i .Thus, for any subset I of {1, 2, . . ., r} and any subset J of {1, 2, . . ., s}, the intersection i∈I D i ∩ j∈J H j is a smooth proper O K -scheme.Theorem 3.15.Let notation and conventions be as in Paragraph 3.14.Then for each integer m, the natural maps ) Let f (x, y, z) = xyz(x+y)(x−y)(x+z)(x−z)(y +z)(y −z) be the polynomial defining the so-called B 3 plane arrangement.Cohen and Suciu [9, Example 5.2] has shown that the Betti numbers of the Milnor fiber f = 1 are: b 0 = 1, b 1 = 8, b 2 = 79.

Example 4 . 6 .
Below we shall prove Corollary 4.10, confirming the assertions made in Example 0.3.
r,1[ ).It is equipped with a derivation d/dx.As in Paragraph 1.1, one can define the notion of a differential module over R. Suppose M is a differential module over R with derivation D. Define H 0 DR (R, M ) = Ker D, and H 1 DR (R, M ) = Coker D. Lemma 2.6.Let M be a differential module on the space D − (0; 1) {0}.Assume that M satisfies the hypothesis of Theorem 1.8.Let L be as in Example 2.2.Then we have