Geometric theory of G-functions.

by

Yves André^{1} and Francesco Baldassarri

§0 Summary.

Our object of study are G-connections i.e. linear partial differential equations satisfied by G-functions in several variables [A], [DGS]; typical examples are non-confluent generalized hypergeometric connections with rational parameters [GHF, Chap. 12]. Our main result is the stability of this notion under higher direct images, for any smooth morphism. As a by-product we obtain a purely p-adic proof of the open local monodromy theorem which does not use resolution of singularities.

§1 G-connections.

1.1. Notations. \overline{Q} = the field of algebraic numbers; K = a number field of finite degree; V_{K} = the ring of integers of K; S = a non-empty open affine subscheme of Spec V_{K}; V_{S} = O(S); Σ_{S} = { finite places of K having center on V_{S} } = { closed points of S }. For any v ∈ Σ_{S}, v dividing the rational prime p = p(v), we put: K_{v} = the v-completion of K, V_{v} = the ring of integers of K_{v}, π_{v} = a uniformizing parameter of V_{v}, chosen in K, k(v) = the residue field of V_{v}. We normalize the absolute value | |_{v} corresponding to v by

(1.1.1)  |p|_{v} = p^{− \frac{[Kv:Qp]}{[K:Q]}} .

We denote by | |_{v} also the unique extension of | |_{v} to the completion C_{p(v)} of the algebraic closure of K_{v} and by Γ_{v} its value group |C_{p(v)}^{×}|_{v}.

Partially sponsored by EC contract CHRXCT-93-0403 (DG 12 COMA). AMS Subject Classification:11Gxx,12H25.

1 The first author thanks the University of Padova for hospitality and support during the preparation of this paper.

For an element λ ∈ C_{p} , let us consider the series

(1.1.2)  g_{λ}(x) = ∑_{{s=0 \atop λ−s≠0}}^{∞} \frac{x^{s}}{λ − s} .

An element λ ∈ C_{p} will be said to be non-Liouville if the radius of convergence of both g_{λ}(x) and g_{−λ}(x) is 1. For \underline{α} = (α_{1}, …, α_{n}) ∈ (R_{≥0})^{n}, any n, we set |\underline{α}|_{R} = ∑_{i=1}^{i=n} α_{i}.

1.2. Let X ⟶^{f} S be a smooth surjective morphism of finite type with geometrically connected fibers. We say that X/S is a smooth S-model of the function field F = κ(X_{K}) over K. Analogously, we define the notion of a smooth K-model of F. For X/S as before, let η (resp. η_{v}, v ∈ Σ_{S}) be the generic point of X (resp. of the closed fiber X_{k(v)}). The choice of a smooth S-model X of F privileges one extension of the absolute value | |_{v} to F. Namely, the local ring O_{X,η_{v}} is a DVR of F, with residue field k(v) and uniformizing parameter π_{v}. Let | |_{X,v} denote the absolute value corresponding to O_{X,η_{v}}, normalized so as to extend | |_{v}. We extend | |_{v} to matrices of elements of F, by taking the maximum of the values of the entries. We introduce the K_{v}-rigid analytic space X_{v}^{rig} = the Raynaud generic fiber of the formal completion of X along X_{k(v)}. The functor X ⟼ X_{v}^{rig} is extended to a functor E ⟼ E_{v}^{rig} from O_{X_{K}} -modules to O_{X_{v}^{rig}}-modules in a well-known way.

Similar definitions can obviously be given when X is a smooth geometrically connected V_{v}-scheme; they will be understood in the sequel. In that case, for a locally closed subvariety T_{0} of X_{k(v)}, ]T_{0}[=]T_{0}[_{X} will denote the tube (of radius 1) of T_{0}, that is the set of points of X_{v}^{rig} that specialize to points of T_{0} [Be]. For ρ ∈ Γ_{v} ∩ [0, 1), and y ∈ X_{v}^{rig}, we will denote by [y]_{X,ρ}, or simply [y]_{ρ}, the closed tube of y in X_{v}^{rig}. So, in terms of an étale coordinate neighborhood in X (U, \underline{x}) = (U, x_{1}, …, x_{d}) of the specialization y_{0} of y in X_{k(v)}, with x_{i}(y) = 0, for i = 1, …, n,

(1.2.1)  [y]_{ρ} = {t ∈]y_{0}[ | |x_{i}(t)|_{v} ≤ ρ , i = 1, …, n} .

If ρ = 1 and X is affine, we abusively set [y]_{1} = X_{v}^{rig}. We introduce also M(y)_{ρ} = F rac(O([y]_{ρ})) = the field of meromorphic functions on [y]_{ρ}.

1.3. Let (M, ∇) be an F/K-differential module of finite rank μ, i.e. M ≅ F^{μ} and

(1.3.1)  ∇ : M ⟶ M ⊗_{F} Ω_{F/K}^{1}

is an integrable F/K-connection. A model of (M, ∇) on X/S is a locally free O_{X}-module (of rank μ) M, with an integrable X/S-connection

(1.3.2)  ∇ : M ⟶ M ⊗_{O_{X}} Ω_{X/S}^{1}

such that (M, ∇)_{η} = (M, ∇). Analogous definitions for a model of (M, ∇) on V/K, for a smooth K-model V of F. For v ∈ Σ_{S}, we define the generic radius of convergence R_{X,v}(M) of (M, ∇) on X/S, as follows. We pick an étale coordinate neighborhood (U, \underline{x}) = (U, x_{1}, …, x_{d}) of η_{v} in X and an F-basis \underline{e} = (e_{1}, …, e_{μ}) of M. We set, for any \underline{α} = (α_{1}, …, α_{d}) ∈ N^{d}, ∇(\frac{∂}{∂\underline{x}})\underline{α} = ∏_{i=1}^{d} ∇(\frac{∂}{∂x_{i}})^{α_{i}} and

(1.3.3)  \frac{1}{\underline{α}!}∇(\frac{∂}{∂\underline{x}})\underline{α}\underline{e} = \underline{e}G\underline{α} , G\underline{α} ∈ M_{μ×μ}(F) .

We put

(1.3.4)  \frac{1}{R_{X,v}(M)} = max(1, lim sup_{|\underline{α}|_{R}→∞} (|G\underline{α}|_{X,v})^{\frac{1}{|\underline{α}|_{R}}} ) .

An easy generalization to several variables of [CD, Prop. 1.3] shows that R_{X,v}(M) is independent of the choice of the F-basis \underline{e} of M. If (M, ∇) is a model of (M, ∇) on X_{K}/K, we write R_{X,v}(M) or R_{X,v}(M), indifferently. If (M, ∇) admits a model (M, ∇) on an S-model X/S, then

(1.3.5)  R_{X,v}(M) ≥ |p|v^{\frac{1}{p−1}} ,

where p = p(v), for any v ∈ Σ_{S}.

Once again we observe that the previous definitions can be given when X is a smooth geometrically connected V_{v}-scheme, and (M, ∇) is an X_{K_{v}}/K_{v}- connection, model of (M, ∇) = (M, ∇) ⊗ κ(X_{K_{v}}) on X_{K_{v}}/K_{v}: analogous results hold. In that situation, a generalization of a standard transfer theorem, shows the following facts. Let y be any point of X_{v}^{rig}; then the Taylor series of (M, ∇) at y converges in the closed disk of radius ρ, for any ρ ∈ Γ_{v}∩]0, R_{X,v}(M)[. On the other hand, if y_{0} is a closed point of X_{k(v)}, and if for each y ∈]y_{0}[, (M, ∇)⊗M(y)_{ρ} admits a basis of horizontal sections over [y]_{ρ}, for a fixed ρ ∈ Γ_{v}∩]0, 1[, then R_{X,v}(M) ≥ ρ.

If X ⟶ Y is a birational morphism of smooth S-models of F, then R_{X,v}(M) ≥ R_{Y,v}(M) for each v ∈ Σ_{S}. Let K′ be a finite extension of K and S′ be the normal closure of S relative to K′. Then X′ = X ×_{S} S′ is a smooth S′-model of the function field F′ = F ⊗_{K} K′. Given an F/K-differential module (M, ∇) as before, (M′, ∇′) = (M ⊗_{K} K′, ∇ ⊗_{K} K′) is an F′/K′-differential module. If v′ ∈ Σ_{S′} is valuation of K′, dividing v ∈ Σ_{S},

(1.3.6)  R_{X′,v′}(M′) = R_{X,v}(M)^{\frac{[K{′ \atop v′}:Kv]}{[K′:K]}} .

Hence,

(1.3.7)  ∏_{v′|v} R_{X′,v′}(M′) = R_{X,v}(M) .

We define the global inverse radius of (M, ∇) on X/S as

(1.3.8)  ρ_{X/S}(M) = ∑_{v∈Σ_{S}} log \frac{1}{R_{X,v}(M)} ∈ [0, +∞] .

In the previous situation, ρ_{X/S}(M) = ρ_{X′/S′}(M′).

1.4. We say that the F/K-differential module (M, ∇) is of type G or a G-module if ρ_{X/S}(M) < ∞, for one (hence for all possible) choice(s) of X/S, as before. This happens in particular if for one (hence for all possible) choice(s) of X/S, as before, and for almost all v ∈ Σ_{S}, R_{X,v}(M) = 1. In that particular case, we will say that the F/K-differential module (M, ∇) is globally convergent. We say instead that (M, ∇) is almost everywhere (a.e.) nilpotent if, for one (i.e., as it turns out, for all) choice(s) of X/S and of the X/S-model (M, ∇), as before, the set of rational primes p such that the connection (M, ∇) ⊗ k(v) induced by (M, ∇) on X_{k(v)}/k(v) is nilpotent for any v ∈ Σ_{S}, v|p, [Ka1, sect. 5], has Dirichlet density one. Motivated by [Ka1], we will say that (M, ∇) is globally nilpotent if for one choice of X/S and of the X/S-model (M, ∇), as before, X_{k(v)}/k(v) is nilpotent for all v ∈ Σ_{S}.

By [DGS, prop. 5.1], (M, ∇) is globally nilpotent (resp. almost everywhere nilpotent) iff strict inequality holds in (1.3.5) for almost all v ∈ Σ_{S} (resp. for the v ∈ Σ_{S} dividing the rational primes p in a set of density one). So, a G-module is almost everywhere nilpotent.

A G-connection (resp. a globally convergent connection, resp. an almost everywhere (a.e.) nilpotent connection, resp. a globally nilpotent connection) on a smooth geometrically connected K-variety V is a model on V/K of a differential module of type G (resp. globally convergent, resp. a.e. nilpotent, resp. globally nilpotent) on κ(V)/K.

Similarly let V′ be a smooth \overline{Q}-variety and (M′, ∇′) be an object of the category MIC(V′/\overline{Q}), of quasi-coherent O_{V′}-modules with integrable V′/\overline{Q}- connection [Ka1]; we say that (M′, ∇′) is a G-connection (resp. a globally convergent connection, resp. an a.e. nilpotent connection, resp. a globally nilpotent connection) on V′ if there exist data K, V, as before, and a G-connection (resp. a globally convergent connection, resp. an a.e. nilpotent connection, resp. a globally nilpotent connection) (M, ∇) on V, such that V′ and (M′, ∇′) are obtained by the base change K ⟶ \overline{Q} from V and (M, ∇), respectively.

For a smooth \overline{Q}-variety V, we denote by G(V) (resp. GC(V), resp. Nilp^{ae}(V), resp. Nilp(V)) the full subcategory of MIC(V/\overline{Q}), whose objects are G-connections (resp. globally convergent connections, resp. a.e. nilpotent connections, resp. globally nilpotent connections). In particular, objects of GC(V), of G(V), of Nilp^{ae}(V), and of Nilp(V) are meant to be O_{V} -coherent.

Then GC(V), G(V), Nilp^{ae}(V), and Nilp(V), are thick tannakian subcategories of MIC(V/\overline{Q}); in particular they contain (O_{V} , d) and are stable under − ⊗_{O_{V}} −, Hom_{O_{V}} (−, −), duality and extensions. Moreover, these subcategories of MIC(V/\overline{Q}) are stable by taking subquotients.

Let V be a smooth variety over K (resp. \overline{Q}). An O_{V} -coherent object (E, ∇) of MIC(V/K) (resp. MIC(V/\overline{Q})) is said to be a regular connection on V/K (resp. V/\overline{Q}), if its pullback along any smooth \overline{Q}-curve in V has regular singularities at infinity. The group of exponents of (E, ∇) is the additive subgroup of \overline{Q} generated by 1 and the exponents at any point at infinity on (the smooth proper model of) any smooth \overline{Q}-curve in V. An argument based on the Turrittin theory of irregular singularities [Ka1, section 11], shows that if (E, ∇) is a regular connection on V/K, its pullback along any smooth curve defined over any extension field Ω of K, is regular (cf. [AB]). Its exponents still belong to the group of exponents of (E, ∇) [AB]. Let us denote by Reg(V) the full subcategory of MIC(V/\overline{Q}) consisting of regular connections. We sometimes fix an additive subgroup Θ ⊃ Z of \overline{Q} and define Reg^{(Θ)}(V) as the full subcategory of Reg(V) consisting of the objects whose group of exponents is contained in Θ. Then, Reg(V) and Reg^{(Θ)}(V) are also thick tannakian subcategories of MIC(V/\overline{Q}), stable by taking subquotients. Let (E, ∇) be an O_{V} -coherent object of MIC(V/\overline{Q}), and let U be an open subvariety of V. Then (E, ∇) is an object of Reg(V) (resp. Reg^{(Θ)}(V)) iff its restriction to U (E_{|U}, ∇_{|U}) is an object of Reg(U) (resp. Reg^{(Θ)}(U)). So, regularity (and, say, rationality of the exponents) is again a property of the generic fiber (E, ∇) ⊗ κ(V) of (E, ∇).

A basic result, “Katz criterion” [Ka2, thm. 8.1, (2)], implies that the pullback of any a.e. nilpotent connection (E, ∇) on a smooth variety V over K (resp. \overline{Q}), along any smooth \overline{Q}-curve in V, has regular singularities at infinity and rational exponents. For an elementary explanation see [DGS, theorem 6.1 and remark 6.3]. Therefore we have

(1.4.1)  GC(V) ⊂ G(V) ⊂ Nilp^{ae}(V) ⊂ Reg^{(Q)}(V) ⊂ Reg(V) ,

and

(1.4.2)  GC(V) ⊂ Nilp(V) ⊂ Nilp^{ae}(V) .

It is conjectured that GC(V) = Nilp^{ae}(V).

1.5. Example: non-confluent generalized hypergeometric differential modules in several variables with rational parameters [GHF], [Ki], [DL]. The book of Dwork [GHF] shows, for a fixed p, the existence of a strong Frobenius structure in the differential modules associated to certain “nondegenerate” generalized hypergeometric series. This is known to imply R_{p} = 1 for those differential modules. On the other hand, the fact that R_{p} = 1, for almost all p, follows from our theorem 3.1.2 below and the existence of an algebraic integral representation for all generalized non-confluent hypergeometric series with rational parameters [Ki], [DL] (independently of the non-degeneracy requirements of [GHF]). There are speculations that all G-connections arise from pull-backs of such hypergeometric connections.

We collect a few examples in which the theory of [GHF] applies. For ν ∈ Z let (x)_{ν} = Γ(x + ν)/Γ(x). In each of the following examples, we give a power series F (\underline{x}) = F (\underline{a}; \underline{x}) ∈ Q[[\underline{x}]], \underline{a} = (a_{1}, …, a_{N}) ∈ Q^{N}, \underline{x} = (x_{1}, …, x_{d}), where the vector \underline{a} is meant to be “generic” (i.e. it should not belong to a certain finite union of hyperplanes) and the differential module we have in mind, that happens to be of type G, is the irreducible Q(\underline{x})/Q-differential module that has F (\underline{x}) as a solution.

1.5.1. The hypergeometric function _{k}F_{k−1}(a_{1}, …, a_{k}; b_{1}, …, b_{k−1}; x) [GHF, 12.3], [DL, p.126]. Here a_{1}, …, a_{k}, b_{1}, …, b_{k−1} ∈ Q and

kF_{k−1}(a_{1}, …, a_{k}; b_{1}, …, b_{k−1}; x) = ∑_{i=0}^{∞} \frac{(a_{1})_{i} … (a_{k})_{i}}{(b_{1})_{i} … (b_{k−1})_{i}i!}x^{i} .

This example includes, as the particular case k = 2, the classical hypergeometric function of Gauss.

1.5.2. Lauricella’s F_{D} [GHF, 12.1], [Ki, §7], [DL, p.123]. Here

F_{D}(α; β_{1}, …, β_{n}; γ; x_{1}, …, x_{n}) = ∑_{\underline{s}} \frac{(α)_{s_{1}+...+s_{n}}(β_{1})_{s_{1}} … (β_{n})_{s_{n}}}{(γ)_{s_{1}+...+s_{n}}s_{1}! … s_{n}!} \underline{x}\underline{s} .

This example again includes, as the particular case n = 1, the classical hypergeometric function of Gauss. For n = 2, this example specializes to Picard’s F_{1} function.

1.6. Stability. For any morphism f : V ⟶ W of smooth \overline{Q}-varieties, f^{∗} (in the sense of O-modules) induces functors

(1.6.1)

f^{∗} : GC(W) ⟶ GC(V) ,
f^{∗} : G(W) ⟶ G(V) ,
f^{∗} : Nilp(W) ⟶ Nilp(V) ,
f^{∗} : Nilp^{ae}(W) ⟶ Nilp^{ae}(V) ,
f^{∗} : Reg^{(Q)}(W) ⟶ Reg^{(Q)}(V) ,
f^{∗} : Reg(W) ⟶ Reg(V) .

If f is étale, then an object (M, ∇) of MIC(W/\overline{Q}) is an object of GC(W) (resp. G(W), resp. Nilp(W), resp. Nilp^{ae}(W), resp. Reg(W), resp. Reg^{(Q)}(W)) as soon as f^{∗}(M, ∇) is an object of GC(V) (resp. G(V), resp. Nilp(V), resp. Nilp^{ae}(V), resp. Reg(V), resp. Reg^{(Q)}(V)).

If f is an étale covering, f_{∗} in the sense of Zariski sheaves also induces functors

(1.6.2)

f_{∗} : GC(V) ⟶ GC(W) ,
f_{∗} : G(V) ⟶ G(W) ,
f_{∗} : Nilp(V) ⟶ Nilp(W) ,
f_{∗} : Nilp^{ae}(V) ⟶ Nilp^{ae}(W) ,
f_{∗} : Reg^{(Q)}(V) ⟶ Reg^{(Q)}(W) ,
f_{∗} : Reg(V) ⟶ Reg(W) .

An object (M, ∇) of MIC(V/\overline{Q}) is an object of GC(V) (resp. G(V), resp. Nilp(V) , resp. Nilp^{ae}(V), resp. Reg(V), resp. Reg^{(Q)}(V)) as soon as f_{∗}(M, ∇) is an object of GC(W) (resp. G(W), resp. Nilp(W), resp. Nilp^{ae}(W), resp. Reg(W), resp. Reg^{(Q)}(W)).

Let now f : V ⟶ W be any smooth morphism of smooth \overline{Q}-varieties and let (M, ∇) be a G-connection (resp. a globally convergent connection) on V/\overline{Q}. We consider the direct image of (M, ∇) via f in the sense of D_{V/\overline{Q}}-modules^{2}

(1.6.3) f_{+}M = f_{+}(M, ∇) = Rf_{∗}DR_{V/W} M[dimV −dimW] ∈ ob(D^{b}(D_{W/K})) ,

as far as the O_{W} -module structure is concerned. It is known that there exists an open subset W′ of W such that h^{i}(f_{+}M)_{|W′} is a coherent O_{W′}-module for all i. This follows from the stability of holonomicity under f_{+}^{3}.

Our main result is the following theorem.

2 For lack of a good reference concerning equality (1.6.3), i.e. identification of the direct image via the smooth morphism f in the sense of D-modules and in the sense of Katz-Oda [Ka1, 2.0.1], we sketch the following argument. As in the proof of [Bo, VI, 5.3.2], what has to be proven is that, for d = dimV − dimW, DR_{V/W} (D_{V})[d] is a locally free left resolution of D_{W ←V} in the category of right D_{V} -modules. Local freedom is obvious. For an étale map f = j : V ⟶ W, D_{W ←V} = D_{V →W} = j^{∗}(D_{W}) = D_{V} = DR_{V/W} (D_{V}), by definition ([Bo, VI, 4.2] and [Bo, VI, 5.1 (4)]). For a projection f = p : A_{W}^{d} ⟶ W the assertion is proven in [Bo, VI, 5.3.1 (i)]. Consider the case of f = p ∘ j : V ⟶^{j}V′ = A_{W}^{d} ⟶^{p}W, composite of a projection and of an étale map j. Then, by [Bo, VI, 6.3 (2)], D_{W ←V} =

j^{−1}(D_{W ←V′}) ⊗_{j^{−1}(D_{V ′})} D_{V′←V} = j^{−1}(D_{W ←V′})⊗^{L}_{j^{−1}(D_{V ′})}D_{V′←V}, and D_{V′←V} =

D_{V}. So, D_{W ←V} = j^{∗}(D_{W ←V′}) = Lj^{∗}(D_{W ←V′}) = j^{∗}(DR_{V′/W} (D_{V′})[d]) = DR_{V/W} (D_{V})[d]. Since any smooth morphism f : V ⟶ W is locally for the Zariski topology on V of the type p ∘ j, the assertion follows.

3 As was pointed out to us by Mebkhout, this last statement is purely “formal” and does not require resolution of singularities, [Bo, VII,10.1 (ii) and VII, 10.4 (ii)]. We shall give a more elementary proof of this fact in the forthcoming paper [AB].

Main Theorem. In the previous situation, assume moreover that h^{i}(f_{+}M) is O_{W} -coherent. Then it is a G-connection (resp. a globally convergent connection) on W/\overline{Q}.

This theorem shows a remarkable stability of the notions of G-connection and of globally convergent connection, and supports the conjecture that all such connections should arise from geometry, i.e. be equations of variation of cohomology in families of algebraic varieties defined over an algebraic number field. A consequence of theorem 5.10 of [Ka1] is the analogon of our main theorem for globally and almost everywhere nilpotent connections. The methods of the present paper differ considerably from those of [Ka1]. A straightforward consequence of our main theorem^{4} and of the criterion of Katz is a purely p-adic proof of the main result, theorem 14.3, of [Ka1], namely the famous

Open Local Monodromy Theorem. The differential equations arising from the cohomology of families of complex algebraic varieties have regular singularities and quasi-unipotent local monodromy.

It might be of interest to point out that our proof does not make use of the theorem of Hironaka on resolution of singularities.

We will prove our main theorem for G-connections. The case of globally convergent connections requires only minor changes.

§2 Dévissage: reduction to a rational elementary fibration.

2.1. We are given a smooth morphism of smooth connected \overline{Q}-varieties V ⟶^{f}W and an object (M, ∇) of G(V). We consider the following statements.

E(M, f) : There exists a non-empty open subvariety W′ of W such that, for all i, h^{i}(f_{+}M)_{|W′} is an object of G(W′).

E_{d} : E(M, f) holds for any smooth morphism V ⟶^{f}W of smooth connected \overline{Q}-varieties, of relative dimension ≤ d, and for any G-connection M on V.

We first notice that E_{0} holds as a consequence of the second assertion in (1.6.2). Obviously, h^{i}(f_{+}M) = 0 if i < dimW − dimV. By ascending induction on the relative dimension of f, we shall reduce the problem to the case when V ⟶^{f}W is a “rational elementary fibration”, M is O_{X}-free and i = −1, 0. We recall that f is called an elementary fibration if it can be embedded in a commutative diagram

(2.1.1)  \begin{aligned}V & ↪^{j} & \overline{V} & ↩ & Z \\ & ↘^{f} & ↓ \overline{f} & ↙^{g} \\ & & W \end{aligned}

4 together with a standard “thickening” argument which allows to descend from C to a finitely generated smooth \overline{Q}-algebra. where

(i) j is an open immersion dense on each fiber and V = \overline{V} \ Z; (ii) \overline{f} is projective and smooth with geometric fibers irreducible of dimension 1; (iii) g is an étale covering.

It is easy to see that f is an affine morphism.

We say that an elementary fibration is rational if \overline{V} = P_{W}^{1} , \overline{f} is the projection and Z is a disjoint union of images of sections of \overline{f}, one of which is the section W ⟶ Z_{∞} = {∞} × W.

Definition 2.1.2. Let

V′
j ↓ ↘^{f′}
V_{W′} ⟶ W′
σ ↓  π ↓
V ⟶^{f} W

be a commutative diagram of \overline{Q}-varieties and morphisms of \overline{Q}-varieties, with π étale, j an open immersion, and where the square is fibered. We then say that f′ is smaller than f, and write f′ ≤ f.

Notation 2.1.3. In the situation of definition 2.1.2, let M be an O_{V} -module. We denote the inverse image (σ ∘ j)^{∗}M by M_{|V′}.

Lemma 2.2. Let f : V ⟶ W be a smooth morphism of smooth connected \overline{Q}-varieties of relative dimension d, M an object of G(V), and f′ : V′ ⟶ W′ be a morphism smaller than f. Let us assume E_{d−1} and E(M_{|V′}, f′). Then E(M, f) holds.

Proof. We discuss separately the case where W′ = W and the case where V_{W′} = V′.

Case W′ = W. We first discuss the special case of V′ = V \ Z↪^{j}V, where

Z↪^{i}V is a smooth closed irreducible subvariety of codimension c in V, such that the morphism g = f ∘ i : Z ⟶ W is smooth. We have the exact triangle in D^{b}(D_{V}) [Bo, VI, 8.3]

(2.2.1)  \begin{aligned}i_{+}i^{!}(M) = RΓ_{[Z]}(M) & & ⟶ & & M \\ & ↖^{[1]} & & ↙ \\ & & j_{+}(M_{|V′}) \end{aligned}

where M is viewed as an object of D^{b}(D_{V}). By application of the functor f_{+} to the triangle, we obtain a triangle in D^{b}(D_{W})

(2.2.2)  \begin{aligned}g_{+}i^{!}(M) & & ⟶ & & f_{+}M \\ & ↖^{[1]} & & ↙ \\ & & f′_{+}(M_{|V′}) \end{aligned}

We recall [Bo, VI, 4.2 (7)] that, since M is locally free over O_{V}, i^{!}(M) = i^{∗}(M)[−c]. By the second assertion in (1.6.1), i^{∗}(M) is a G-connection. The exact cohomology sequence of the triangle (2.2.2) gives the result in this particular case. For the general case, we decompose Z = ⋃_{i=1}^{r} Z_{i} in a union of irreducible closed subsets Z_{i}; we set, for s = 0, …, r, V′_{s} = V \ ⋃_{s<i≤r} Z_{i}, f′_{s} = f_{|V′_{s}} : V′_{s} ⟶ W so that V′ = V′_{0}, f′ = f′_{0}, V = V′_{r}, f = f′_{r}. It’s enough to prove that the conjonction of E(M_{|V′_{s}}, f′_{s}) and of E_{d−1} implies E(M_{|V′_{s+1}}, f′_{s+1}). We have then reduced ourselves to the case r = 1. So, Z will be assumed to be irreducible. If the morphism g = f ∘ i : Z ⟶ W is not dominant, we replace W by a non-empty open subvariety W′ such that f^{−1}W′ ⊂ V′. Then, by base change [Bo, VI, 8.4], (h^{i}f_{+}M)_{|W′} = h^{i}f′_{+}M_{|f^{−1}W′}, so E(M, f) coincides with E(M_{|V′}, f′) in this case. If g = f ∘ i : Z ⟶ W is dominant, there exists a nonempty open subvariety W′ of W such that, after restriction to W′, g becomes smooth. We are then in the special case we discussed first.

Case V_{W′} = V′. We consider the fibered square

(2.2.3)  \begin{matrix}V′ & ⟶^{f′} & W′ \\ ↓ σ & & ↓ π \\ V & ⟶^{f} & W \end{matrix}

where both π and σ are étale. Assume we can prove E(M′, f′), where M′ denotes the G-connection σ^{∗}M. We claim that the assertion for E(M, f) follows. We are free to replace W by a dense open subset and to restrict diagram (2.2.3) accordingly: we may assume that h^{i}(f′_{+}M′) is an object of G(W′), for all i. By étale base-change (proven as [Bo, VI, 8.4]), f′_{+}M′ = π^{∗}f_{+}M; then h^{i}(f′_{+}M′) = π^{∗}h^{i}(f_{+}M) and h^{i}(f_{+}M) is a G-connection for all i, by the remarks following (1.6.1). Q.E.D.

2.3. Let f : V ⟶ W be a smooth morphism of smooth connected \overline{Q}- varieties of relative dimension d ≥ 1. We distinguish 2 possibilities.

Case d = 1. Then there exists an étale neighborhood π : W′ ⟶ W, such that after base change by π

(2.3.1)  f′ = f ×_{W} 1_{W′} : V′ = V ×_{W} W′ ⟶ W′ ,

is an elementary fibration. By lemma 2.2, we are then reduced to the case where f itself is an elementary fibration.

Case d > 1. Then, up to replacing f by a smaller morphism, we may assume that f is a product of two smooth morphisms of smooth connected \overline{Q}-varieties

(2.3.2)  V ⟶^{a}U⟶^{b}W ,

of positive relative dimensions (both less than d).

We prove that E_{d−1} ⇒ E_{d}. E_{d−1} implies the existence of an open subset U′′

of U such that for all i, (h^{i}a_{+}M)_{|U′′} is an object of G(U′′). Let us set U′′⟶^{b′′}W, b′′ ≤ b.

Again by E_{d−1}, there exists an open subset W′ of W such that

(h^{j}b′′_{+}((h^{i}a_{+}M)_{|U′′}))_{|W′}

is an object of G(W′), for all i, j. Let us set:

U′ = U′′ ∩ b^{−1}W′ , U′⟶^{b′}W′ , b′ ≤ b′′ ≤ b ,

and

V′ = a^{−1}U′ ⊂ V , V′⟶^{a′}U′ , a′ ≤ a , M′ = M_{|V′} .

We have

(2.3.3)  (h^{i}a_{+}M)_{|U′} = h^{i}a′_{+}M′ ,

for all i, by base change with respect to an open immersion, and

(2.3.4)  h^{j}b′_{+}((h^{i}a_{+}M)_{|U′}) = (h^{j}b′′_{+}((h^{i}a_{+}M)_{|U′′}))_{|W′} ,

for all i, j, for the same reason. So,

(2.3.5)  h^{j}b′_{+}(h^{i}a′_{+}M′) ,

for all i, j, are objects of G(W′), and by exactness of the property of being a G-connection in MIC(W′), h^{i}(b′ ∘ a′)_{+}M′ is an object of G(W′), for all i. We have verified E(M′, b′∘a′). But b′∘a′ ≤ f, and lemma 2.2 then implies E(M, f).

We are reduced to proving our main theorem in the case of an elementary fibration.

2.4. We now reduce the case when f is an elementary fibration to the special case of a rational elementary fibration. So, looking at (2.1.1), we enlarge Z to a bigger divisor Z′ and set V′ = \overline{V} \ Z′; we then change the base by an étale morphism W′ ⟶ W, so as to obtain an elementary fibration

(2.4.1)  \begin{aligned}V′ & ↪^{j′} & \overline{V′} = \overline{f′}^{−1}W′ & ↩ & Z′ \\ & ↘^{f′} & ↓ \overline{f′} & ↙^{g′} \\ & & W′ \end{aligned}

with f′ ≤ f and \overline{f′} ≤ \overline{f}. We may enlarge Z′ and shrink V′, W′ so much that

M_{|V′} is free and \overline{f′} factors through a finite map \overline{V′}⟶^{ϕ′}P_{W′}^{1} unramified on V′ and that U′ = ϕ′(V′) ⊂ A_{W′}^{1}. Then (ϕ′_{|V′})_{∗}(M_{|V′}) is free on U′.

Definition 2.4.2. An elementary fibration, as in (2.1.1), is explicitly coordinated if it can be embedded in a commutative diagram

V ↪^{j}  \overline{V} ↩ Z
↓ π  ↓ \overline{π}  ↓ π′
U ↪^{j′} P_{W}^{1} ↩ Z′
↘^{f′} ↓^{pr} ↙^{g′}
W

where

(i) \overline{π} is finite and π is an étale covering; (ii) f = f′ ∘ π, \overline{f} = pr ∘ \overline{π}, g = g′ ∘ π′; (iii) the lower part of the diagram is a rational elementary fibration with j′(Y) ⊂ A_{W}^{1} .

A rational elementary fibration is ipso facto explicitly coordinated. We just proved the following lemma.

Lemma 2.4.3. For any elementary fibration, there exists a smaller morphism which is an explicitly coordinated elementary fibration.

Let f be an explicitly coordinated elementary fibration, as in definition 2.4.2. We have f_{+} = f′_{+} ∘ π_{+}, and π_{+} = π_{∗} induces a functor G(V) ⟶ G(U), since π is an étale covering. To complete the proof of our main theorem, we are then left to prove E(M, f), when f : V ⟶ W is a rational elementary fibration and M is free. By further base change, we can also assume that W is affine, étale over Ad ; \overline{Q}. On the other hand, one has h^{i}(f_{+}M) = 0 if i ≠ −1, 0, because f is affine of relative dimension one.

The purely p-adic proof will be discussed in the next section.

§3. p-adic estimates.

3.1. Let us fix a finite place v of K, v|p and consider a diagram

(3.1.1)  \begin{aligned}X & ↪^{j} & \overline{X} = P_{Y}^{1} & ↩ & Z = ⨿_{ν∈S}Z_{ν} \\ & ↘^{f} & ↓ \overline{f} & ↙^{g} \\ & & Y \end{aligned}

where S is a finite set containing the symbol ∞, j is an open immersion, \overline{f} is the projection, g induces isomorphisms Z_{ν} ⟶^{∼}Y, and where the following simple coordinate description holds: Y is an affine étale neighborhood of A_{V_{v}}^{d} with coordinates λ = (λ_{1}, …, λ_{d}); x = the standard “vertical” coordinate on P_{Y}^{1} ; Z_{ν} = V (x − θ_{ν}(λ)), θ_{ν}(λ) ∈ O(Y), for ν ∈ S \ {∞}; θ_{ν}(λ) − θ_{ν′}(λ) ∈ O(Y)^{×}, for ν ≠ ν′ ∈ S \ {∞}; Z_{∞} = V (\frac{1}{x}).

Let M_{K_{v}} be a free O_{X_{Kv}} -module of finite rank μ, endowed with an integrable X_{K_{v}}/K_{v}-connection ∇_{K_{v}}. We assume that M_{K_{v}} is regular and that its group of exponents consists of non-Liouville elements of Z_{p}. We assume moreover that h^{i}(f_{∗}DR_{X_{Kv} /Y_{Kv}} M_{K_{v}}) is free of finite type over O_{Y_{Kv}} for i = 0, 1. (Since X and Y are affine, will often disregard the specification f_{∗} in these formulas).

Under these assumptions we shall prove:

Theorem 3.1.2. The generic radius convergence of h^{i}(f_{∗}DR_{X_{Kv} /Y_{Kv}} M_{K_{v}}) is at least R_{v} = R_{X,v}(M_{K_{v}}) if i = 0, and R_{v}^{μ^{2}+1} if i = 1.

3.2. Let us explain at once how this result implies our main theorem. Let us start with a rational elementary fibration f : V ⟶ W, and with a Gconnection (M, ∇) on V/\overline{Q}. We need to prove that R^{i}f_{∗}(DR_{V/W} M) is again a G-connection, after localization on W. We may assume that both M and R^{i}f_{∗}(DR_{V/W} M) = h^{i}(f_{∗}DR_{V/W} M), i = 0, 1, are free and that the base W is étale over Ad ; \overline{Q}. One can then choose K, S as in section 1, in such a way that our elementary fibration is the geometric generic fiber of a diagram of smooth S-schemes

(3.2.1)  \begin{aligned}X_{S} & ↪ & \overline{X}_{S} = P_{Y_{S}}^{1} & ↩ & Z_{S} \\ & ↘^{f_{S}} & ↓ \overline{f}_{S} & ↙ \\ & & Y_{S} \end{aligned}

which induces a diagram of the type of (3.1.1), for every v ∈ Σ_{S}. We may assume that (M, ∇) is obtained by base change from the G-connection (M_{K}, ∇_{K}) on X_{K}/K and that its group of exponents is a subgroup of Q ∩ V_{S} (hence consists of non-Liouville elements of Z_{p(v)}, for every v ∈ Σ_{S}). Since

(3.2.2)  ∑_{v∈Σ_{S}} log \frac{1}{R_{v}^{μ^{2}+1}} = (μ^{2} + 1) ∑_{v∈Σ_{S}} log \frac{1}{R_{v}} < ∞ ,

our main theorem follows from theorem 3.1.2.

3.3. We now prove (3.1.2). Let us choose an isomorphism M ≅ O_{X_{Kv}}^{μ}; we may write the connection in the form

(3.3.1)  ∇_{K_{v}} = d_{X_{Kv} /K_{v}} − G_{1,\underline{0}}d_{X_{Kv} /K_{v}}x − ∑_{i=1}^{d} G_{0,\underline{1}_{i}}d_{X_{Kv} /K_{v}}λ_{i} ,

with G_{1,\underline{0}}, G_{0,\underline{1}_{i}} ∈ M_{μ×μ}(O(X_{K_{v}})), for i = 1, …, d. The de Rham complex relative to Y_{K_{v}} is

(3.3.2)  \begin{matrix}DR_{X_{Kv} /Y_{Kv}} M_{K_{v}} = \\ = 0 ⟶ O_{X_{Kv}}^{μ} ⟶^{d_{X_{Kv} /Y_{Kv}} −G_{1,\underline{0}}d_{X_{Kv} /Y_{Kv}} x}(Ω^{1}_{X_{Kv} /Y_{Kv}})^{μ} ⟶ 0 ≅ \\ ≅ 0 ⟶ O_{X_{Kv}}^{μ} ⟶^{\frac{∂}{∂x} −G_{1,\underline{0}}}O_{X_{Kv}}^{μ} ⟶ 0 . \end{matrix}

3.4. We first settle the easy case of

H^{0}(DR_{X_{Kv} /Y_{Kv}} M_{K_{v}}) ≅ ker_{O(X_{Kv})^{μ}}(\frac{∂}{∂x} − G_{1,\underline{0}}) ,

viewed as an O(Y_{K_{v}})-module endowed with the connection ℵ defined by

(3.4.1)  ℵ(\frac{∂}{∂λ_{i}}) = \frac{∂}{∂λ_{i}} − G_{0,\underline{1}_{i}} .

If we pick a κ(Y_{K_{v}})-basis (e_{1}, …, e_{μ′}) of ker_{κ(X_{Kv})^{μ}}(\frac{∂}{∂x} − G_{1,\underline{0}})), the e_{j}’s remain linearly independent over the field of functions of X_{K_{v}}. Hence we may complete (e_{1}, …, e_{μ′}) into a basis of sections (e_{1}, …, e_{μ}) of M_{K_{v}}. The fact that

R_{Y,v}(H^{0}(DR_{X_{Kv} /Y_{Kv}} M_{K_{v}})) ≥ R_{v}

then follows from the calculation of the two generic radii of convergence, by using the bases (e_{1}, …, e_{μ′}) and (e_{1}, …, e_{μ}).

3.5. We now turn to the case of

H^{1}(DR_{X_{Kv} /Y_{Kv}} M_{K_{v}}) ≅ O(X_{K_{v}})^{μ}/(\frac{∂}{∂x} − G_{1,\underline{0}})O(X_{K_{v}})^{μ} ,

endowed with the connection ℵ induced by

(3.5.1)  ℵ(\frac{∂}{∂λ_{i}}) = \frac{∂}{∂λ_{i}} − G_{0,\underline{1}_{i}}

on the quotient space (this makes sense because of the integrability of ∇). Let Y(x, λ) be a solution of the system

(3.5.2)  \begin{matrix}\frac{∂}{∂λ_{i}} Y(x, λ) = & G_{0,\underline{1}_{i}}Y(x, λ) , i = 1, …, d \\ \frac{∂}{∂x}Y(x, λ) = & G_{1,\underline{0}}Y(x, λ) \end{matrix}

We write the Taylor series of Y(x, λ) at a closed point (x^{0}, y^{0}) of X_{K_{v}} (with λ(y^{0}) = λ^{0}) in the form

(3.5.3)  Y(x, λ) = ∑ G_{i,\underline{α}}(x^{0}, λ^{0})(x − x^{0})^{i}(λ − λ^{0})\underline{α}Y(x^{0}, λ^{0})

Proposition 3.5.4. (cf. [AB]) There is a constant C_{1} such that

(3.5.4.1)  G_{i,\underline{α}} = \frac{1}{(∏_{ν≠∞}(x − θ_{ν}))^{i+|\underline{α}|_{R}+C_{1}}} H_{i,\underline{α}} ,

with H_{i,\underline{α}} ∈ M_{μ×μ}(O(Y_{K_{v}})[x]), and deg_{x}H_{i,\underline{α}} ≤ card(S)(i + |\underline{α}|_{R}) − 2i + C_{1}.

Proof. We pick β_{j} ∈ K_{v}^{×}, for j = 1, …, d, and change the variables (x, λ_{1}, …, λ_{d}) into (x̃, λ̃_{1}, …, λ̃_{d}) = (x + ∑_{j=1}^{d} β_{j}λ_{j}, λ_{1}, …, λ_{d}). We compute

\frac{∂}{∂x̃} = \frac{∂}{∂x} , \frac{∂}{∂λ̃_{j}} = \frac{∂}{∂λ_{j}} − β_{j} \frac{∂}{∂x} .

We set

(3.5.4.2)  (i!)^{−1}(\underline{α}!)^{−1}(\frac{∂}{∂x̃})^{i} ∏_{j}(\frac{∂}{∂λ̃_{j}})^{α_{j}}Y = G̃_{i,\underline{α}}Y .

Let κ_{j} be the algebraic closure of K_{v}(x̃, λ_{1}, …, λ̂_{j}, …, λ_{d}) in κ(X_{K_{v}}), for j = 1, …, d, and κ_{0} = κ(Y_{K_{v}}). For j = 0, 1, …, d, we consider the regular κ(X_{K_{v}})/κ_{j}- connection ∇_{j} induced by ∇. For j = 0, Der(κ(X_{K_{v}})/κ_{j}) is spanned by \frac{∂}{∂x̃} = \frac{∂}{∂x}, while for j ≠ 0 it is spanned by \frac{∂}{∂λ̃_{j}} = \frac{∂}{∂λ_{j}} − β_{j} \frac{∂}{∂x}. Besides, x ∉ κ_{j}, (x − θ_{ν}) ∉ κ_{j} for any ν ∈ S, any j = 0, 1, …, d. By regularity in one variable, we deduce that there exists a constant C_{2} such that for any j = 0, 1, …, d, when (i, α_{1}, …, α_{d}) is a vector with all components except the (j + 1)-st one equal to 0,

(3.5.4.3)  G̃_{i,\underline{α}} = \frac{1}{(∏_{ν≠∞}(x − θ_{ν}))^{i+|\underline{α}|_{R}+C_{2}}} H̃_{i,\underline{α}} ,

with H̃_{i,\underline{α}} ∈ M_{μ×μ}(κ_{j}[x]), and deg_{x}H̃_{i,\underline{α}} ≤ card(S)(i+|\underline{α}|_{R})−2i+C_{2}. From this one deduces that there exists a constant C_{1} such that for arbitrary (i, α_{1}, …, α_{d})

(3.5.4.4)  G̃_{i,\underline{α}} = \frac{1}{(∏_{ν≠∞}(x − θ_{ν}))^{i+|\underline{α}|_{R}+C_{1}}} H̃_{i,\underline{α}} ,

with H̃_{i,\underline{α}} ∈ M_{μ×μ}(O(Y_{K_{v}})[x]), and deg_{x}H̃_{i,\underline{α}} ≤ card(S)(i + |\underline{α}|_{R}) − 2i + C_{1}. Formula (3.5.4.1) then follows from the fact that the jacobian matrix relating the two bases of tangent vectors is in GL(d + 1, K_{v}^{×}). Q.E.D.

We observe that, formally,

(3.5.5)  \begin{matrix}O(X_{K_{v}})^{μ}/(\frac{∂}{∂x} − G_{1,\underline{0}})O(X_{K_{v}})^{μ} ≅ \\ ≅ & \frac{Y(x, λ)^{−1}O(X_{K_{v}})^{μ}d_{X_{Kv} /Y_{Kv}} x}{Y(x, λ)^{−1}(\frac{∂}{∂x} − G_{1,\underline{0}})O(X_{K_{v}})^{μ}d_{X_{Kv} /Y_{Kv}} x} \\ ≅ \frac{Y(x, λ)^{−1}O(X_{K_{v}})^{μ}d_{X_{Kv} /Y_{Kv}} x}{d_{X_{Kv} /Y_{Kv}} Y(x, λ)^{−1}O(X_{K_{v}})^{μ}} . \end{matrix}

Therefore, if we indicate by z the λ-coordinates of a closed point y of Y_{K_{v}}, by X_{y} the fiber of X_{K_{v}} over y, and by X̂_{{y}} the formal completion of X_{K_{v}} along X_{y}, the horizontal displacement for ℵ is given by multiplication by

(3.5.6)  χ(x; z, λ) = Y(x, λ)Y(x, z)^{−1} = ∑_{\underline{α}} G_{0,\underline{α}}(x, z)(λ − z)\underline{α}

and we have a commutative diagram

(3.5.7)  \begin{aligned}& O(X_{y})^{μ} & ⟶^{χ(x;z,λ)} & O(X̂_{\{y\}})^{μ} \\ ( \frac{∂}{∂x} −G_{1,\underline{0}}(x,z)) & ↓ & & ↓ & ( \frac{∂}{∂x} −G_{1,\underline{0}}(x,λ)) \\ & O(X_{y})^{μ} & ⟶^{χ(x;z,λ)} & O(X̂_{\{y\}})^{μ} \end{aligned}

3.6. Let us consider the convergence of χ. By a generalization of the Dwork-Robba theorem [DGS, IV.3; see in particular remark 3.2], that can easily be deduced from loc. cit. via restriction to a generic line, there exists a constant C_{3} such that

(3.6.1)  |G_{i,\underline{α}}|_{X,v} ≤ C_{3}(i + |\underline{α}|_{R})^{μ−1}R_{v}^{−(i+|\underline{α}|_{R})} .

When specialized at a point (x, y) ∈ \overline{X}_{v}^{rig} (λ(y) = z), the previous formula for i = 0, together with (3.5.4), gives

(3.6.2)  |G_{0,\underline{α}}|_{v} ≤ C_{3}(|\underline{α}|_{R})^{μ−1}R_{v}^{−|\underline{α}|_{R}}(\frac{|x|_{v}^{card(S)}}{∏_{ν≠∞} |x − θ_{ν}(z)|_{v}})^{|\underline{α}|_{R}+C_{1}} .

We conclude from this that, for fixed y ∈ Y_{v}^{rig} (hence fixed z = λ(y)), the series of rational functions on X_{K_{v}} (3.5.6) for χ(x; y, λ), converges (in the sense of rigid analytic functions on (P_{Y}^{1} )_{v}^{rig}) as soon as

(3.6.3)  |λ − z|_{v} < R_{v} \frac{1}{|x|_{v}} ∏_{ν≠∞} |\frac{x − θ_{ν}(z)}{x}|_{v} .

The reciprocal role played by λ and z in (3.5.6) and (3.6.3) shows that under the assumption (3.6.3), χ is an invertible matrix of analytic functions. We have in fact

(3.6.4)  χ(x; z, λ)χ(x; λ, z) = I_{μ} .

3.7. We choose here a section τ : C_{p}/Z ⟶ C_{p} of the canonical projection. The formal fuchsian theory over κ(Y_{K_{v}})((x − θ_{ν})) [DGS, III.8], for ν ∈ S \ {∞}, shows that a formal solution matrix of

(3.7.1)  \frac{∂Z}{∂x} = G_{1,\underline{0}}Z

at Z_{ν}, for ν ∈ S \ {∞}, has the form

(3.7.2)  Z = P_{ν}(x)U_{ν}(x − θ_{ν})(x − θ_{ν})^{A_{ν}} ,

for a μ by μ Jordan matrix A_{ν} with eigenvalues in the image of τ,

(3.7.3)  U_{ν}(x − θ_{ν}) ∈ GL(μ, κ(Y_{K_{v}})[[x − θ_{ν}]]) ,

and

(3.7.4)  P_{ν}(x) ∈ GL(μ, κ(Y_{K_{v}})(x)) .

A similar statement holds when ν = ∞. After replacing Y by an affine neighborhood in Y of the generic point of its closed fiber Y_{k(v)}, we may and will assume that the Taylor coefficients at θ_{ν} of the matrices U_{ν}, P_{ν} and of their inverses belong to M_{μ×μ}(O(Y_{K_{v}})). We recall that both the eigenvalues and the differences of eigenvalues of each of the matrices A_{ν}, ν ∈ S, are non-Liouville elements of Z_{p}.

Let k_{1}(Y) denote the completion of the field κ(Y_{K_{v}}) with respect to the absolute value | |_{Y,v}. The transformation Z ⟼ P_{ν}Z transforms (3.7.1) into an ordinary system of linear differential equations in the open disk |x − θ_{ν}|_{Y,v} < 1 of the k_{1}(Y)-rigid analytic line, with a logarithmic singularity at x = θ_{ν}, and generic radius of convergence R_{v}, of the type considered in [BC]. The theorem of transfer [BC, 3.5] tells us that the entries of the matrices U_{ν}(x − θ_{ν}) converge for

(3.7.5)  |x − θ_{ν}|_{Y,v} < R_{v}^{μ^{2}} ,

Similarly the entries of U_{∞}(\frac{1}{x}) converge for |x|_{Y,v} > R_{v}^{−μ^{2}}.

3.8. Application of Adolphson’s comparison theorem [Ad]. Let us first introduce some notation. We fix y ∈ Y_{v}^{rig}, ρ ∈ Γ_{v}∩]0, 1], both to be specified later. We recall that the supnorm on [y]_{ρ} defines an absolute value | |_{y}(ρ) on the field M(y)_{ρ} of meromorphic functions on [y]_{ρ}. For ρ = 1, | |_{y}(1) coincides with | |_{Y,v} on κ(Y_{K_{v}}). Let k_{ρ}(y) = M̂(y)_{ρ} be the | |_{y}(ρ)-completion of M(y)_{ρ}; for ρ = 1, (k_{1}(y), | |_{y}(1)) is the completion of (κ(Y_{K_{v}}), | |_{Y,v}). Then κ(Y_{K_{v}}) ⊂ M(y)_{ρ} ⊂ k_{ρ}(y), and in particular θ_{ν} ∈ k_{ρ}(y). Let us point out that, for h ∈ k_{ρ}(y) and \underline{α} ∈ N^{d},

(3.8.1)  |\frac{1}{\underline{α}!}(\frac{∂}{∂λ})\underline{α}(h)|_{y}(ρ) ≤ ρ^{−|\underline{α}|_{R}}|h|_{y}(ρ) .

We may regard (3.7.1) as a k_{ρ}(y)(x)/k_{ρ}(y)-connection and observe that, since |h|_{ρ} ≤ |h|_{1} = |h|_{Y,v}, for all h ∈ O(Y_{K_{v}}), the generic radius of convergence of (3.7.1) with respect to the absolute value

(3.8.2)  | ∑ a_{i}x^{i}|_{X,v,ρ} = max |a_{i}|_{y}(ρ) , a_{i} ∈ k_{ρ}(y)

(so that |h|_{X,v,ρ} ≤ |h|_{X,v,1} = |h|_{X,v}, for all h ∈ O(Y_{K_{v}})) is

(3.8.3)  R_{X,v,ρ} ≥ R_{X,v,1} = R_{v} .

As in section 3.7, we use the theorem of transfer [BC, 3.5] to conclude that the entries of the matrices U_{ν}(x − θ_{ν}) converge for

(3.8.4)  |x − θ_{ν}|_{y}(ρ) < R_{X,v,ρ}^{μ^{2}} ,

hence for

(3.8.5)  |x − θ_{ν}|_{y}(ρ) < R_{v}^{μ^{2}} ,

Similarly the entries of U_{∞}(\frac{1}{x}) converge for |x|_{y}(ρ) > R_{v}^{−μ^{2}}. This will allow us to apply Adolphson’s comparison theorem [Ad] to the k_{ρ}(y)(x)/k_{ρ}(y)-connection (3.7.1).

For any ϵ ∈]0, 1[, let us put

(3.8.6)  \begin{matrix}A_{ν,ρ,ϵ} = \{x ∈ k_{ρ}(y) | |x − θ_{ν}|_{y}(ρ) > R_{v}^{μ^{2}}ϵ \} , for ν ∈ S \ \{∞\} , \\ A_{∞,ρ,ϵ} = \{x ∈ k_{ρ}(y) | |x|_{y}(ρ) < R_{v}^{−μ^{2}}ϵ^{−1} \} , \\ A_{ρ,ϵ} = ⋂_{ν∈S} A_{ν,ρ,ϵ} , \end{matrix}

all rigid analytic spaces over k_{ρ}(y), and

(3.8.7) \begin{matrix}X_{ν,ρ,ϵ} = & \{(x, t) ∈ P_{K_{v}}^{1rig} × [y]_{ρ} ⊂ \overline{X}_{v}^{rig} | |x − θ_{ν}(λ(t))|_{v} > R_{v}^{μ^{2}}ϵ \} , \\ for ν ∈ S \ \{∞\} , \\ X_{∞,ρ,ϵ} = & \{(x, t) ∈ P_{K_{v}}^{1rig} × [y]_{ρ} ⊂ \overline{X}_{v}^{rig} | |x|_{v} < R_{v}^{−μ^{2}}ϵ^{−1} \} , \\ X_{ρ,ϵ} = & ⋂_{ν∈S} X_{ν,ρ,ϵ} , \end{matrix}

all rigid analytic spaces over K_{v}. We have natural inclusions

(3.8.8)  \begin{matrix}O(X_{ν,ρ,ϵ}) ↪ O(X_{ν,ρ,ϵ}) ⊗_{O([y]_{ρ})} k_{ρ}(y) ↪ O(A_{ν,ρ,ϵ}) , ∀ν ∈ S \\ O(X_{ρ,ϵ}) ↪ O(X_{ρ,ϵ}) ⊗_{O([y]_{ρ})} k_{ρ}(y) ↪ O(A_{ρ,ϵ}) . \end{matrix}

There are natural maps

(3.8.9)  \begin{matrix}(O(X_{K_{v}})^{μ}/(\frac{∂}{∂x} − G_{1,\underline{0}})O(X_{K_{v}})^{μ}) ⊗_{O(Y_{Kv})} k_{ρ}(y) ⟶ \\ (O(X_{ρ,ϵ})^{μ}/(\frac{∂}{∂x} − G_{1,\underline{0}})O(X_{ρ,ϵ})^{μ}) ⊗_{O([y]_{ρ})} k_{ρ}(y) ⟶ \\ O(A_{ρ,ϵ})^{μ}/(\frac{∂}{∂x} − G_{1,\underline{0}})O(A_{ρ,ϵ})^{μ} \end{matrix}

Then

\begin{matrix}(O(X_{K_{v}})^{μ}/(\frac{∂}{∂x} − G_{1,\underline{0}})O(X_{K_{v}})^{μ}) ⊗_{O(Y_{Kv})} k_{ρ}(y) = \\ = O(X_{k_{ρ}(y)})^{μ}/(\frac{∂}{∂x} − G_{1,\underline{0}})O(X_{k_{ρ}(y)})^{μ} , \end{matrix}

so that the result of Adolphson [Ad, Thm. 2] over the field k_{ρ}(y) gives:

Proposition 3.8.10. Under the assumptions of (3.1) and (3.7), for any ρ ∈ Γ_{v} ∩ [0, 1], any ϵ ∈]0, 1[, and any y ∈ Y_{v}^{rig}, the natural map

\begin{matrix}(O(X_{K_{v}})^{μ}/(\frac{∂}{∂x} − G_{1,\underline{0}})O(X_{K_{v}})^{μ}) ⊗_{O(Y_{Kv})} k_{ρ}(y) ⟶ \\ ⟶ (O(A_{ρ,ϵ})^{μ}/(\frac{∂}{∂x} − G_{1,\underline{0}})O(A_{ρ,ϵ})^{μ}) \end{matrix}

is an isomorphism of k_{ρ}(y)-vector spaces.

3.9. Radius of convergence of ℵ. We now determine the generic radius of convergence of (H^{1}(DR_{X_{Kv} /Y_{Kv}} M_{K_{v}}), ℵ). Let us fix ϵ ∈]0, 1[ and choose ρ = R_{v}^{μ^{2}+1}ϵ. Then (3.6.3) is satisfied for λ = λ(t), z = λ(y), for any (x, t) ∈ X_{ρ,ϵ}. Let Ω = (ω_{i,j}), i = 1, …, μ, j = 1, …, N, be a matrix with entries in O(X_{K_{v}}) such that its columns represent an O(Y_{K_{v}})-basis of H^{1}(DR_{X_{Kv} /Y_{Kv}} M_{K_{v}}) ≅ O(X_{K_{v}})^{μ}/(\frac{∂}{∂x} −G_{1,\underline{0}})O(X_{K_{v}})^{μ}. Since H^{1} is torsion-free, its formation commutes with passage to the closed fiber {y} ⟶ Y_{K_{v}} (cf. [Bo, VI, 8.4], taking into account the relation between i^{!} and i^{∗} in the torsion-free case [Bo, VI, 4.2]). So, the columns of Ω(y) represent a basis of H^{1}(DR_{X_{y}/κ(y)}M_{y}) over κ(y). By the definition of the connection ℵ the columns of the matrix χ(x; z, λ)Ω(y) are formal horizontal vectors for ℵ in O(X̂_{\{y\}})^{μ}/(\frac{∂}{∂x}−G_{1,\underline{0}})O(X̂_{\{y\}})^{μ}. Since (3.6.3) is satisfied, χ(x; z, λ) ∈ GL(μ, O(X_{ρ,ϵ})), and the comparison isomorphism (3.8.10) shows that

(3.9.1)  χ(x; z, λ)Ω(y) = ΩR(λ) ,

for a matrix R(λ) ∈ GL(N, k_{ρ}(y)). But R(λ) is precisely a fundamental solution matrix of ℵ at λ = z. Therefore we have

(3.9.2)  \frac{1}{\underline{α}!}(\frac{∂}{∂λ})\underline{α}(R(λ)) = H\underline{α}R(λ) ,

with H\underline{α} ∈ M_{N}(O(Y_{K_{v}})) and, on the other hand,

(3.9.3)  |\frac{1}{\underline{α}!}(\frac{∂}{∂λ})\underline{α}(R(λ))|_{y}(ρ) ≤ ρ^{−|\underline{α}|_{R}}|R(λ)|_{y}(ρ) ,

hence

(3.9.4)  |H\underline{α}|_{y}(ρ) ≤ ρ^{−|\underline{α}|_{R}}|R(λ)|_{y}(ρ)|R(λ)^{−1}|_{y}(ρ) .

Therefore the Taylor series for the solution matrix of ℵ at λ = z

∑ H\underline{α}(z)(λ − z)\underline{α}

converges to an analytic function on the open disk of radius ρ at y ∈ Y_{v}^{rig}. This implies by the transfer theorems quoted in (1.3) that

(3.9.5)  R_{Y,v}(H^{1}(DR_{X_{Kv} /Y_{Kv}} M_{K_{v}})) ≥ R_{v}^{μ^{2}+1}ϵ , ∀ϵ ∈ ]0, 1[ ,

as required.

3.10. Remark. It seems that proposition 3.8.10 may be improved into a relative comparison theorem between algebraic and rigid cohomologies [Be]. Let notations be as in (1.1) and (3.8). We set here R = R_{v}^{μ^{2}}, for brevity. For any ϵ ∈]0, 1[, we denote by j_{ϵR} the open embedding of K_{v}-rigid analytic spaces

(3.11.1)  j_{ϵR} : X_{1,ϵ} ↪ \overline{X}_{v}^{rig} .

For any O_{\overline{X}}_{v}^{rig}-module F, we set

(3.11.2)  F_{ϵR} = j_{ϵR∗}j_{ϵR}^{−1}F

and

(3.11.3)  j†_{R}F = lim_{ϵ→1^{−}} j_{ϵR∗}j_{ϵR}^{−1}F

In particular, when R = R_{v} = 1, we get the usual †-completion of [Be]. We define

(3.11.4)  DR^{alg}(M) = j_{K_{v}∗}DR_{X_{Kv} /Y_{Kv}} (M_{K_{v}}) ,

(3.11.5)  DR_{ϵR}^{rig}(M) = j_{ϵR∗}j_{ϵR}^{−1}(DR^{alg}(M)_{v}^{rig}) ,

(3.11.6)  DR†_{R}(M) = j†_{R}(DR^{alg}(M)_{v}^{rig}) .

We have natural arrows of O_{\overline{X}}_{v}^{rig}-modules

(3.11.7)  (R\overline{f}_{K_{v}∗}(DR^{alg}(M)))_{v}^{rig} ⟶ R\overline{f}_{v∗}^{rig}(DR_{ϵR}^{rig}(M)) .

(3.11.8)  (R\overline{f}_{K_{v}∗}(DR^{alg}(M)))_{v}^{rig} ⟶ R\overline{f}_{v∗}^{rig}(DR†_{R}(M)) .

We believe that, under the assumptions of (3.1), both arrows are isomorphisms. We plan to come back to this question in the next future.

References.

[A] André Y.: “G-functions and Geometry”, Aspects of Mathematics E13, Vieweg, Braunschweig/Wiesbaden, 1989.

[AB] André Y., Baldassarri F.: “A new approach to some fundamental results on direct images of differential modules. Application to complex and p-adic comparison theorems”, in preparation.

[Ad] Adolphson A.: “An index theorem for p-adic differential operators”, Trans. Amer. Math. Soc., 216 (1976), 279-293.

[BC] Baldassarri F., Chiarellotto B.:“ On André’s transfer theorem”, in p-adic Methods in Number Theory and Algebraic Geometry, A.Adolphson, S.Sperber, M.Tretkoff Eds., Contemporary Mathematics, Volume 133, AMS Publications 1992, 25-37.

[Be] Berthelot P.: “Cohomologie rigide et cohomologie rigide à support propre”, to appear.

[Bo] Borel A.: “Algebraic D-modules”, Perspectives in Mathematics, Vol. 2, Academic Press 1987.

[CD] Christol G., Dwork B.:“Modules différentiels sur des couronnes”, Annales de l’Insitut Fourier, Tome 44 (1994), 663-701.

[De] Deligne P.:“Équations Différentielles à Points Singuliers Réguliers”, Lecture Notes in Math. 163, Springer-Verlag, 1970.

[DGS] Dwork B., Gerotto G., Sullivan F.:“An Introduction to G-functions”, Annals of Mathematical Studies 133, Princeton University Press, Princeton N.J., 1994.

[DL] Dwork B., Loeser F.:“Hypergeometric series”, Japanese J. of Math., Vol. 19 (1993), 81-129.

[GHF] Dwork B.: “Generalized Hypergeometric Functions.”, Oxford Mathematical Monographs, Clarendon Press, Oxford 1990.

[Ka1] Katz N.:“ Nilpotent connections and the monodromy theorem. Applications of a result of Turrittin”, Publ. Math. IHES 39 (1970), 175-232.

[Ka2] Katz N.:“A conjecture in the arithmetic theory of differential equations”, Bull. Soc. math. France 110 (1982), 203-239.

[Ki] Kita M.:“On hypergeometric functions in several variables 1. New integral representations of Euler type”, Japanese J. of Math., Vol. 18 (1992), 25-74.

Author’s address:

Yves André Institut de Mathématiques 4 place Jussieu Tour 46-56, 5ème étage, case 247 F-75252 PARIS CEDEX 05 - FRANCE E-mail: ANDRE@MATHP6.JUSSIEU.FR Fax: (33-1-)44274844

Francesco Baldassarri Dipartimento di Matematica Università di Padova Via Belzoni 7 I-35131 PADOVA - ITALY E-mail: BALDASSA@MATH.UNIPD.IT Fax: (39-49-)8758596