SPECIAL VALUES OF SYMMETRIC HYPERGEOMETRIC FUNCTIONS.

by

Francesco Baldassarri

§0 Introduction.

In a recent article P.Th.Young, on the line of previous work by N.Koblitz [Ko] and J.Diamond [D], used some combinatorial identities and a principle of p-adic continuity in all variables to compute special values of a certain function F(a, b, c; λ) related to the classical Gauss hypergeometric function F (a, b, c; λ). As in [p-DE IV], the function F(a, b, c; λ), for a, b, c ∈ Z_{p}, c ∉ Z_{≤0}, is defined to be the maximal p-adic analytic extension of the ratio

(0.1)  \frac{F (a, b, c; λ)}{F (a′, b′, c′; λ^{p})} ,

where for a ∈ Z_{p}, a′ ∈ Z_{p} is uniquely defined by the condition that

(0.2)  pa′ − a = μ_{a} ∈ {0, 1, …, p − 1} .

(We also recursively define a^{(0)} = a, and a^{(i+1)} = (a^{(i)})′, for i = 0, 1, …). In particular, Young obtained the formula (see [Y1, Th. 3.1] and (5.3.1) below),

(0.3)  F(\frac{2α}{p − 1}, \frac{β}{p − 1}, 1 + \frac{2α − β}{p − 1} ; −1) = (−1)^{α} \frac{Γ_{p}(\frac{α}{p−1})Γ_{p}(\frac{β−α}{p−1})}{Γ_{p}(\frac{2α}{p−1})Γ_{p}(\frac{β−2α}{p−1})}

(where Γ_{p} denotes the Morita p-adic gamma function), for any α, β ∈ Z such that

(0.4)  0 < 2α ≤ β < p − 1 and 2(β − α) < p − 1 .

(Actually, the previous assumptions could be relaxed to be

(0.4)′  0 ≤ 2α ≤ β < p − 1 and 2(β − α) ≤ p − 1 ,

This paper is in final form and no version of it will be submitted for publication elsewhere.

AMS Subject Classification:11T23,11S31,12H25,14F30.

(cf. Theorem 5.3), for p ≥ 7). When (a, b, c) are rational, the differential equation satisfied by F (a, b, c; λ) was known, since the time of Euler, to express the variation w.r.t. to the parameter λ of two cohomology classes on a variable algebraic curve y = f_{a,b,c}(x, λ). For more general (a, b, c), the analogues of the previous classes live in a twisted cohomology space for the x-projective line with a ramification point varying with λ, the cohomology being taken relative to the λ-line. The present paper originated from our attempt to gain a full cohomological understanding of the results of [Y1] and [Y2]. We work, naturally, in the framework of Dwork’s theory of generalized hypergeometric equations ([GHF], [LDE]), which encompasses both the classical algebraic and the p-adic cohomological theories. The main points to clarify were:

1) the relevance of the Frobenius structure of the hypergeometric system to a statement like (0.3);

2) the relevance of the “admissible” factorization of a linear differential operator according to the growth of solutions ([DR], [Ch, Prop. 4.2.1]) to the same statement;

3) more generally, the question of uniformity of the previous factorization when applied, say, to the family of Gauss hypergeometric operators, with respect to the variables (a, b, c);

4) the notion of supersingularity of a singular fiber [Dw2];

5) the possibility, in certain cases, of explicitly computing the eigenvalues of Frobenius acting on a special fiber.

In particular, point 1 should clarify the role played in the proof of (0.3) by the Boyarsky principle, asserting the continuous dependence of the Frobenius matrix upon (a, b, c; λ) [GHF, Introduction and Thm. 4.7.1].

Point 2 plays an important role here in that the main condition under which we work is

(0.5)  μ_{c^{(i)}} > max(μ_{a^{(i)}}, μ_{b^{(i)}}) ,

for any i = 0, 1, . . .. This condition guarantees that the holomorphic solution of the hypergeometric operator at 0 (that is F (a, b, c; λ)) is a bounded holomorphic function on D(0, 1^{−}) = {λ ∈ C_{p} | |λ| < 1} [p-DE IV, Lemma 2.2]. When a, b, c ∈ \frac{1}{p−1}Z condition (0.5) (in this case independent of i) guarantees that the (rank 1) unit root F -subcrystal of the hypergeometric F -crystal, extends to the singular class D(0, 1^{−}). We will come back to this point in [Ba], where the more general notion of logarithmic (realization of an overconvergent) F - (iso)crystal is introduced [Sch], and where (0.5) is seen as the condition for the unit root logarithmic F -subcrystal of the hypergeometric logarithmic F -crystal, to actually be an F -crystal in the usual sense in a p-adic formal neighborhood of 0.

In the present article, we will not be concerned with point 4 above. The reason is that that problem does not appear for the (singular) class of 0 unless one deals with an iterate of Frobenius, λ ⟼ λ^{q}, q = p^{f}, f ∈ Z_{>1}. This might be the best approach (cf. [LDE, Chap. 9], [Dw2]) when a, b, c are in \frac{1}{q−1}Z ∩ (0, 1). In that case one would consider

(0.6)  F^{(f)}(a, b, c; λ) = \frac{F (a, b, c; λ)}{F (a^{(f)}, b^{(f)}, c^{(f)}; λ^{q})} .

The condition for F^{(f)}(a, b, c; λ) to extend to an admissible domain of analyticity, would then consist of two separate assumptions. The first is (0.5) for i = f − 1, which already guarantees 1-dimensionality of the space of bounded solutions of the system at a generic point. A second condition is needed to guarantee nonsupersingularity of the class of 0, that is the existence of a bounded solution of the system in D(0, 1^{−}). The condition says that the order of zero at λ = 0 of the lower-right entry of the Frobenius matrix shouldn’t exceed c(q − 1) [Dw2, Def. 1.11]. This is certainly the case if we insist that condition (0.5) holds for all i = 0, …, f −1. This is the viewpoint of this article and of [p-DE IV]. A separate question is the one of non-supersingularity of the non-singular point λ_{0} at which we want to evaluate F(a, b, c; λ). In the case considered here λ_{0} = −1 and c ± (a − b) ∈ Z, this is automatic (cf. (2.31) and (2.36)). In the case considered by [Ko] and [D], λ_{0} = 1 is singular, so condition (0.5) appears combined with a similar condition [D, (iii) of Th.1.1] at 1, to guarantee that the above mentioned unit root F -crystal also extends to a p-adic formal neighbourhood of 1. Again, non-supersingularity of the class of 1 follows from that condition.

While problem 5 also becomes more interesting in the case of a singular fiber, where Frobenius operates on the “eigenvectors of local monodromy” (cf. [GHF, Lemmas 24.3 and 24.5.8]) in this paper, motivated by formula (0.3), we only discuss the non-singular fiber at λ_{0} = −1. We carefully analyze the relation between the value

F(\frac{2α}{p − 1}, \frac{β}{p − 1}, 1 + \frac{2α − β}{p − 1} ; −1) = F(\frac{β}{p − 1}, \frac{2α}{p − 1}, 1 + \frac{2α − β}{p − 1} ; −1)

and the unit root of the L-function

(0.7)  L(t) = exp ∑_{s=1}^{∞} S_{α,β,s} \frac{t^{s}}{s}

where

(0.8)  S_{α,β,s} = ∑_{x∈F_{p^{s}} \{0,1,−1}} ω^{−2α}(Tr_{F_{p^{s}} /F_{p}}(x))ω^{−β}(Tr_{F_{p^{s}} /F_{p}}(1 − x^{2}))

and ω : F_{p}^{×} ⟶ Q(ζ_{p−1})^{×} denotes the Teichmüller character. An elementary calculation, taking into account the classical relation between Gauss and Jacobi sums, the Hasse-Davenport relation and the Gross-Koblitz formula [GrKo], gives

(0.9)  L(t) = (1 − σ_{1}(α, β)t)(1 − σ_{2}(α, β)t) ,

with

(0.10)  \begin{aligned}σ_{1}(α, β) & = (−1)^{α} \frac{Γ_{p}(\frac{α}{p−1})Γ_{p}(\frac{β−α}{p−1})}{Γ_{p}(\frac{β}{p−1})} \\ σ_{2}(α, β) & = −p(−1)^{α+ \frac{p−1}{2}} \frac{Γ_{p}(\frac{α+ \frac{p−1}{2}}{p−1} )Γ_{p}(\frac{β−α+ \frac{p−1}{2}}{p−1} )}{Γ_{p}(\frac{β}{p−1})} . \end{aligned}

In the notation of (2.37.1,2)

(0.11)  \begin{aligned}B(\frac{β}{p − 1}, \frac{2α}{p − 1}, 1 & + \frac{2α − β}{p − 1} ; \frac{β}{p − 1}, \frac{2α}{p − 1}, 1 + \frac{2α − β}{p − 1} ; −1) = \\ & = ( \begin{matrix}σ_{2}(α, β) & 0 \\ ∗ & σ_{1}(α, β) \end{matrix} ) , \end{aligned}

so that

L(t) = det(I_{2} − tB(\frac{β}{p − 1}, \frac{2α}{p − 1}, 1 + \frac{2α − β}{p − 1} ; \frac{β}{p − 1}, \frac{2α}{p − 1}, 1 + \frac{2α − β}{p − 1} ; −1)) .

The fact that the evaluation of F(a, b, c; λ) takes place at λ = −1, which is not a singular point of the differential equation satisfied by F (a, b, c; λ), reduces the calculation of the eigenvalues of Frobenius operating on local solutions of the differential equation to a problem of character sums on the special fiber, which, in our particular case, is elementary. At a singular fiber λ_{0}, the eigenvalues of Frobenius operating on the eigenfunctions of local monodromy, have only been determined when λ_{0} = 0 ([LDE, Chap. 25-26]), via their interpretation on the cohomology of the special fiber. The calculation when λ_{0} = 1, ∞, may then be performed with the help of the theorem in [Ku, §4]. We plan to give full details in [Ba].

The speciality of the fiber of the relative cohomology bundle at λ = −1 in the case under consideration, namely c ± (a − b) ∈ Z, lies in the fact that it is “symmetric”, by which we mean that it is equipped with an involution τ (see [Y1, 3.15] and diagram 1.5 below), commuting with the action of Frobenius. This involution splits the cohomology space at λ = −1 rationally in (a, b, c) and offers the possibility of obtaining nice formulas separately for the two roots of Frobenius (see formulas (2.30) and (2.35) below). That these formulas should involve products of p-adic Γ-functions is then a-priori clear because of the modular properties of the Frobenius matrix w.r.t. integral translation on (a, b, c) (cf. [GHF, Lemma 4.8.1] and [Sel]). These modular properties of the Frobenius matrix simply express the fact that Frobenius represents an intrinsic horizontal transformation of relative cohomology sheaves depending rationally upon (a, b, c), whose matrix may be computed w.r.t. a lattice of natural bases. The same modular properties hold for the matrices representing the action of global monodromy on classical solutions, i.e. relating the eigenvectors of local monodromy at two distinct singular points [Po, Chap. VI, §26], but their full potential does not seem to have been classically exploited.

We point out once again that the mere fact that the two eigenvalues of Frobenius at λ = −1 have p-adic absolute values 1 and p^{−1}, respectively, implies the existence of a rank 1 unit root F -subcrystal [Ka, §4] of the hypergeometric system over an admissible open subset of the λ-line, containing the class of -1. Under condition (0.5), the unit root F -crystal extends to the singular class of 0. This is in the end the key ingredient that permits one to relate F(a, b, a−b+1; −1) to an entry of the Frobenius matrix at -1.

The contribution of this paper to the previous problems is as follows. Section 1 analyzes the Frobenius matrix at the special symmetric fiber λ = −1, via the modular relations. This approach represents an alternative way to the direct elementary computation of the associated L-function. Section 2 reviews some of the results of Dwork contained, in two slightly different forms, in the two books [GHF] and [LDE]: we explicitly describe the dictionary between them. Section 3 represents an improvement on previous results of Dwork. Namely, we describe the Frobenius matrix uniformly for (a, b, c; λ) varying in a region of the form H × S, where H represents an affinoid domain in the (a, b, c)-space (C_{p})^{3}, and S the complement in the rigid λ-line of the residue classes at 1 and at ∞. In particular, S contains the singular residue class D(0, 1^{−}). This is new, since the theory of [LDE] and [Dw2] is not uniform in (a, b, c), while the theory of [GHF] and [Dw1] does not analyze any full singular disk. Our main result is Corollary 3.37, a result which should at least be compared with the theorem in section 4 of [Dw1] and with Lemmas 24.4 and 24.5.8 of [LDE]. By the methods of [Ku] these results may be extended to the other singular classes as well. (In the class of 1 the correct Frobenius map to use on the λ-space should be close to λ ⟼ 1 − (1 − λ)^{p}.) For this we refer to [Ba]. We also wish to point out that, at least when p ≥ 7, our list of distinct cases is exhaustive, which also represents an improvement upon the classification of [LDE, Chap. 6]. In section 4 we examine the question of uniformity in (a, b, c) of the growth decomposition. Here we examine hypergeometric equations in the so-called “split” cases, when at the generic point there are both bounded and unbounded solutions. We prove that the factorization according to the order of growth extends to the residue class of 0. In section 5, we finally show how to recover formula (0.3) from our discussion.

The study of the singular classes of 1 (and infinity), and its application yielding the formulas of Koblitz [Ko] and Diamond [D], is deferred to the previously mentioned article [Ba].

The author is grateful to Professor Dwork for his generous help in the revision of the original manuscript.

List of symbols.

B = ( \begin{matrix}B_{1} & B_{2} \\ B_{3} & B_{4} \end{matrix} ) (2.20) and [LDE, 4.5.1]

d(A, B) = distance between the points or subsets A and B of (C_{p})^{n}, any n; D = D(0, 1^{−}) (2.3) D(0, ρ^{−}) = {λ ∈ C_{p} | |λ| < ρ } D_{a⃗,i} = x_{i}∂/∂x_{i} + πx_{i}∂/∂x_{i}(g(x, λ)) + a_{i}, for i = 1,2,3,4, Chap. 1 of [GHF] D_{a⃗,i}^{∗}, for i = 1,2,3,4, (3.22.1) of [GHF] f = f(x, λ) Chap. 1 of [LDE] f̃ = f̃(x, λ) (2.9) F (a, b, c; λ) (4.16.1) F (x, λ) = exp π(g(x, λ) − g(x^{p}, λ^{p})) Prop. 4.2.2 of [GHF] g(x, λ) = x_{3}(x_{1} + x_{2}) + x_{4}(λx_{1} + x_{2}) §13.2 of [GHF] g(x) = g(x, −1) §1 G_{f,h}(x, λ) (3.7.1) of [LDE] G_{f̃,h̃}(x, λ) (2.13) h = h(x, λ) (2.13) h̃ = h̃(x, λ) (2.13) H= hyperplane t_{1} + t_{2} = t_{3} + t_{4} in affine 4-space Chap. 1 of [GHF] H^{+}(Z) = H(Z) ∩ (Z_{≥0})^{4}; K = a finite extension of Q_{p}(π); k(u⃗) = u_{3} + u_{4}, for u ∈ H(Z); L (1.1) ord(x) = − log_{p} |x|, for x ∈ C_{p}^{×}; p = an odd prime number; R = C_{p}(λ)[x_{1}x_{3}, x_{1}x_{4}, x_{2}x_{3}, x_{2}x_{4}] [GHF, Chap.1] R′ = C_{p}(λ)[x_{1}x_{3}, x_{1}x_{4}, x_{2}x_{3}, x_{2}x_{4}, (x_{1}x_{2}x_{3}x_{4})^{−1}] §2.2 of [GHF] R^{′∗} (3.16) of [GHF] S (2.3) S_{ϵ}, ϵ ≥ 0 (2.3) Sμ⃗ (3.14) and (3.27) S_{i}^{H}, i = 1, 2, (4.2.1) T_{I} (2.17) T_{II} (2.23) U^{(0)}(a, b, c; λ) (4.16.2) u_{i}^{(0)}(a, b, c; λ), i = 1, 2, 3, 4, (4.16.2) B_{K} (4.5) B_{K}^{(1)} (4.5) C_{K} (4.5.1)

C_{K}^{(1)} (4.5.2) H (2.3) Hμ⃗ (2.3) H′ (2.7) H′_{i}, i = 1, 2, (4.2.2)

H′ ; ν⃗ (2.7)

L (1.14) Lμ⃗ (1.14) K′_{a⃗,λ} = ⋂_{i=1}^{4} Ker_{R^{′∗}}D_{a⃗,i}^{∗} §3.2 of [GHF] S′_{i}(−1), i = 1, 2, (5.1.1) S Lemma 2.4 S′ 2.8 T_{i}, i = 1, 2, (4.1,2) T_{i}(−1), i = 1, 2, (5.1.2) W′ ; a⃗ = W′_{a⃗,λ} = R′/ ∑_{i=1}^{4} D_{a⃗,i}R′ Lemma 2.5.2 and §3.2 of [GHF] (C_{p}, | − |) = completion of the algebraic closure of Q_{p}, |p| = p^{−1}; K_{f} = K_{f,λ} Chap. 2 and 4 of [LDE] L = C_{p}(λ)[x, \frac{1}{x(1−x)(1−λx)}] Chap. 1 of [LDE] L̃ = C_{p}(λ)[x, \frac{1}{x(1+x)(1+λx)}] (2.7) W_{f} = W_{f,λ} = L/(xf)^{−1} ∘ x\frac{∂}{∂x} ∘ xfL Chap. 1 of [LDE] W̃f̃ = L̃/(xf)^{−1} ∘ x\frac{∂}{∂x} ∘ xfL̃ (2.11) \overline{α}_{f,h} Cor.3.8.8 of [LDE] α_{a⃗,b⃗,λ}^{∗} (2.2) and §4.7 of [GHF] α_{f,h}^{∗} Lemma 4.4.3 (ii) of [LDE] β(a, b, c; λ) (3.37.0) Γ_{p}(x) = the Morita p-adic gamma function; γ_{p}(a, b) (10.11) of [GHF] and Chap. 21 of [LDE] γ_{g}(a⃗, b⃗; λ) (2.6.1) and §4.7 of [GHF] γ_{i,j}(a⃗, b⃗; λ) (2.6.2) η(a, b, c; λ) (4.9), (4.14) θ(t) = exp π(t − t^{p}) (3.5.1) π = a fixed element of C_{p} such that π^{p−1} = −p; ρ_{i}, i = 1, 2, (1.15) σ = a Frobenius automorphism of K, σ(π) = π; τ (1.2) ϕ (4.6) Φ : x_{i} ⟼ x_{i}^{p}, for i = 1,2,3,4, §4.1 of [GHF] ξ_{1}^{(0)}(a, b, c) = ξ_{1}^{(0)}(a, b, c; A, B, C), (3.33) ξ_{2}^{(0)}(a, b, c) = ξ_{2}^{(0)}(a, b, c; A, B, C), (3.34) χ Lemma 3.4.1 ([1]_{a⃗,λ}^{∗}, [πx_{2}x_{3}]_{a⃗,λ}^{∗}) = basis of K′_{a⃗,λ} dual to the basis ([1]_{a⃗,λ}, [πx_{2}x_{3}]_{a⃗,λ}) of W′_{a⃗,λ} (a⃗ and λ are sometimes understood); ([1]^{∗}, [\frac{1}{1−x}]^{∗}) = basis of K′_{f,λ} dual to the basis ([1], [\frac{1}{1−x}]) of W′_{f,λ} (f and λ are usually understood);

[r] = class of r ∈ R′ in W′ ; a⃗ or of r ∈ L in W_{f}, or of r ∈ L̃ in W̃f̃; [r]a⃗ = class of r ∈ R′ in W′ ; a⃗, when specification of a⃗ is necessary; [r]_{a⃗,λ} = class of r ∈ R′ in W′_{a⃗,λ}, when specification of both a⃗ and λ is necessary; || − ||μ⃗ = supnorm on Hμ⃗ × S_{0}; || − ||_{T} = || − || = supnorm on T §4 (x)_{s} = \frac{Γ(x+s)}{Γ(x)} ∈ Q(x), for the classical gamma function Γ, and s ∈ Z;

a⃗ = (a_{1}, a_{2}, a_{3}, a_{4}), b⃗, μ⃗, μ′ = vectors in 4-space; ν⃗ = (ν_{1}, ν_{2}, ν_{3}), ν′ = vectors in 3-space. ∼ indicates multiplicative congruence modulo units.

§1 Frobenius matrix on a symmetric fiber of exponential cohomology spaces.

We refer here to the computations of §13.2 of [GHF], with a_{3} = a_{4} and λ = −1. We consider the linear Z-subvariety L of the affine 4-space over Z defined by the equations:

(1.1)  \begin{matrix}t_{1} + t_{2} = t_{3} + t_{4} \\ t_{3} = t_{4} \end{matrix}

Notice that L is a subvariety of the Z-hyperplane H of [GHF, Chap.1]. We will consider pairs (a⃗, b⃗) ∈ L(C_{p}) × L(C_{p}), with μ⃗ = pb⃗ − a⃗ ∈ Z^{4}. We consider the automorphism τ of R^{′∗}, stable on R′, such that

(1.2)  \begin{matrix}τ(x_{1}) = −x_{1} & , & τ(x_{2}) = x_{2} \\ τ(x_{3}) = x_{4} & , & τ(x_{4}) = x_{3} \end{matrix}

It’s clear that τ : R^{′∗} ⟶ R^{′∗} is adjoint to τ : R′ ⟶ R′ in the natural pairing [GHF, 3.19] between R′ and R^{′∗}. Here g = x_{3}(x_{1} + x_{2}) + x_{4}(−x_{1} + x_{2}), so that g^{τ} = g. Then

τD_{a⃗,i}τ = D_{a⃗,i}, for i = 1, 2,
τD_{a⃗,i}^{∗}τ = D_{a⃗,i}^{∗}, for i = 1, 2,

(1.3)  τD_{a⃗,3}τ = D_{a⃗,4}
τD_{a⃗,3}^{∗}τ = D_{a⃗,4}^{∗}

τΦτ = Φ .

Now

(1.4)  α_{a⃗,b⃗,−1}^{∗} : K′_{b⃗,−1} ⟶ K′_{a⃗,−1}

is induced by δ_{g,μ⃗} = x^{−μ⃗}F (x, λ)Φ. But τδ_{g,μ⃗}τ = (−1)^{μ_{1}}δ_{g,μ⃗}. So, if μ_{1} is even we have commutative diagrams:

(1.5)  \begin{matrix}K′_{b⃗,−1} & ⟶^{α_{a⃗,b⃗,−1}^{∗}} & K′_{a⃗,−1} \\ τ ↓ & & ↓ τ \\ K′_{b⃗,−1} & ⟶^{α_{a⃗,b⃗,−1}^{∗}} & K′_{a⃗,−1} \end{matrix}

We will therefore assume

(1.6)  μ_{1} ∈ 2Z

(so in fact also μ_{2} ∈ 2Z).

Since τ[1]a⃗ = [1]a⃗ and τ[πx_{2}x_{3}]a⃗ = −a_{2}[1]a⃗−[πx_{2}x_{3}]a⃗, we see that τ[πx_{2}x_{3}]∗ ; a⃗ = −[πx_{2}x_{3}]∗ ; a⃗. From this and the commutative diagram (1.5), we deduce that γ_{g}(a⃗, b⃗; −1) is lower-triangular:

(1.7)  α_{a⃗,b⃗,−1}^{∗} ([1]^{∗}, [πx_{2}x_{3}]^{∗}) = ([1]^{∗}, [πx_{2}x_{3}]^{∗}) γ_{g}(a⃗, b⃗; −1)

(cf. §4.7 of [GHF]).

Remark 1.7.1. The commutative diagram 1.5 shows that

( \begin{matrix}1 & 0 \\ −a_{2} & −1 \end{matrix} ) γ_{g}(a⃗, b⃗; −1) = γ_{g}(a⃗, b⃗; −1) ( \begin{matrix}1 & 0 \\ −b_{2} & −1 \end{matrix} )

and hence writing γ_{g} = ( \begin{matrix}γ_{11} & γ_{12} \\ γ_{21} & γ_{22} \end{matrix} ), we find γ_{12} = 0 (as stated) but also

γ_{21} = \frac{1}{2}(γ_{22}b_{2} − γ_{11}a_{2}), i.e. γ_{g} is determined by γ_{11} and γ_{22}.

For u⃗ ∈ H(Z), we put

k(u⃗) = u_{3} + u_{4} .

We examine the matrix of the mapping multiplication by π^{k(u⃗)}xu⃗, for u⃗ = u′, u′′

u′ = (2, 0, 1, 1) , u′′ = (0, 2, 1, 1)

(1.8)  W′_{a⃗+u⃗} ⟶^{π^{u_{3}+u_{4}} xu⃗}W′ ; a⃗

( \begin{matrix}[1]_{a⃗+u⃗} \\ [πx_{2}x_{3}]_{a⃗+u⃗} \end{matrix} ) ⟼ M(a⃗ + u⃗, a⃗; −1)( \begin{matrix}[1]a⃗ \\ [πx_{2}x_{3}]a⃗ \end{matrix} )

The computations in [GHF, loc. cit.] (after the harmless substitution x_{i} ⟼ πx_{i}, for i = 3, 4), give

M(a⃗ + u′, a⃗; −1) = ( \begin{matrix}−\frac{a_{1}a_{3}}{2} & 0 \\ −\frac{a_{2}a_{3}}{4} & −a_{3} \frac{a_{1}+1}{2} \end{matrix} )

(1.9)

M(a⃗ + u′′, a⃗; −1) = ( \begin{matrix}\frac{a_{2}a_{3}}{2} & 0 \\ −\frac{a_{2}a_{3}}{4} & a_{3} \frac{a_{2}+1}{2} \end{matrix} )

For the convenience of the reader, we review here how formulas (1.9) could be proven. We put x_{i}x_{j} = x^{u^{(i,j)}}, M(a⃗+u^{(i,j)}, a⃗; −1) = A_{i,j}(a⃗), for i = 1, 2, j = 3, 4 and consider

(1.10.1)  W′_{a⃗+u^{(i,j)}+u^{(h,k)}} ⟶^{πx_{h}x_{k}}W′_{a⃗+u^{(i,j)}} ⟶^{πx_{i}x_{j}}W′ ; a⃗

More precisely:

(1.10.2)  \begin{matrix}W′_{a⃗+u^{(i,j)}+u^{(h,k)}} & ⟶^{πx_{h}x_{k}}W′_{a⃗+u^{(i,j)}} \\ ( \begin{matrix}[1] \\ [πx_{2}x_{3}] \end{matrix} ) & ⟼ A_{h,k}(a⃗ + u^{(i,j)}) ( \begin{matrix}[1] \\ [πx_{2}x_{3}] \end{matrix} ) \end{matrix}

(1.10.3)  \begin{matrix}W′_{a⃗+u^{(i,j)}} & ⟶^{πx_{i}x_{j}}W′ ; a⃗ \\ ( \begin{matrix}[1] \\ [πx_{2}x_{3}] \end{matrix} ) & ⟼ A_{i,j}(a⃗) ( \begin{matrix}[1] \\ [πx_{2}x_{3}] \end{matrix} ) \end{matrix} ,

so that

(1.10.4)  \begin{matrix}W′_{a⃗+u^{(i,j)}+u^{(h,k)}} & ⟶^{π^{2}x_{h}x_{k}x_{i}x_{j}}W′ ; a⃗ \\ ( \begin{matrix}[1] \\ [πx_{2}x_{3}] \end{matrix} ) & ⟼ A_{h,k}(a⃗ + u^{(i,j)})A_{i,j}(a⃗) ( \begin{matrix}[1] \\ [πx_{2}x_{3}] \end{matrix} ) \end{matrix} ,

Then

(1.10.5)  \begin{aligned}M(a⃗ + u′, a⃗; −1) & = A_{1,3}(a⃗ + (1, 0, 0, 1))A_{1,4}(a⃗) \\ M(a⃗ + u′′, a⃗; −1) & = A_{2,3}(a⃗ + (0, 1, 0, 1))A_{2,4}(a⃗) , \end{aligned}

and the formulas for A_{i,j}(a⃗) are explicitly given in [GHF,(13.2)] (A_{i,j}(a⃗) = z_{ij}, in the notation of loc. cit.), for (i, j) = (2,3), (2,4), (1,4), (1,3). Of course, we must put a_{3} = a_{4} and λ = −1 in those formulas.

So, in terms of the system of bases ( \begin{matrix}[1]a⃗ \\ [πx_{2}x_{3}]a⃗ \end{matrix} ) used in [GHF], we obtain for

the matrices of multiplication by π^{k(u⃗)}xu⃗, for u⃗ = u′, u′′, formulas (1.9).

Remark 1.10.6. The fact that M(a⃗ + u⃗, a⃗; −1), for u⃗ in the Z-span of u′ and u′′, is lower-triangular may be seen directly as in the previous remark 1.7.1. In fact, for such a u⃗, the map τ commutes with multiplication by xu⃗. Hence writing

M(a⃗ + u⃗, a⃗; −1) = ( \begin{matrix}M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix} ) we deduce M_{12} = 0 and M_{21} = \frac{1}{2}(a_{2}(M_{22} −

M_{11}) − M_{11}u_{2}). Once again M(a⃗ + u⃗, a⃗; −1) is determined by M_{11} and M_{22}.

We recall the determinant formula (§13.6 and Cor.14.1.2 of [GHF])

(1.11)  det γ_{g}(a⃗, b⃗; −1) = p2^{μ_{1}+μ_{2}}(2^{p−1})^{−b_{1}−b_{2}} ∏_{i=1}^{4} γ_{p}(a_{i}, b_{i}) .

The modular behaviour of γ_{g}(a⃗, b⃗; −1) w.r.t. integral translations in a⃗ is described by the formula (§4.8 of [GHF])

(1.12)  γ_{g}(a⃗ + u⃗, b⃗; −1) = M(a⃗ + u⃗, a⃗; −1)γ_{g}(a⃗, b⃗; −1)π^{−k(u⃗)} .

It is apparent that these formulas are consistent with our previous calculations.

Formula (1.12) is of interest to us when u⃗ belongs to the Z-span of u′, u′′. For such a u⃗, the formula involves lower-triangular matrices and spells out in term of the system of bases ([1]∗ ; a⃗, [πx_{2}x_{3}]∗ ; a⃗) the commutative diagram

K′_{b⃗,−1}  ⟶^{1}  K′_{b⃗,−1}
τ ↗  |  τ ↗  |
K′_{b⃗,−1}  ⟶^{1}  K′_{b⃗,−1}  |
|  |  |  |
α_{a⃗,b⃗,−1}^{∗}  α_{a⃗,b⃗,−1}^{∗}  α_{a⃗+u⃗,b⃗,−1}^{∗}  α_{a⃗+u⃗,b⃗,−1}^{∗}
|  ↓  |  ↓
|  K′_{a⃗,−1}  ⟶^{π^{k(u⃗)}xu⃗}  K′_{a⃗+u⃗,−1}
↓ τ ↗  ↓ τ ↗
K′_{a⃗,−1} ⟶^{π^{k(u⃗)}xu⃗} K′_{a⃗+u⃗,−1}
Diagram 1.13

We put ρ = p^{− \frac{1}{p−1} − \frac{1}{p}} and define for μ⃗ ∈ Z^{4} ∩ L with μ_{1} ∈ 2Z:

(1.14)  \begin{matrix}Lμ⃗ = \{(a, b) & ∈ L(C_{p}) × L(C_{p}) | pb⃗ − a⃗ = μ⃗ , d(a⃗, −μ⃗) < ρ\} \\ L = ⋃_{μ⃗∈L(Z) , μ_{1}∈2Z} Lμ⃗ \end{matrix}

We define functions on L by

(1.15) \begin{aligned}ρ_{1}(a⃗, b⃗) & = (−1)^{\frac{μ_{1}}{2}} γ_{p}(\frac{a_{1}}{2} , \frac{b_{1}}{2} ) γ_{p}(\frac{a_{2}}{2} , \frac{b_{2}}{2} ) γ_{p}(a_{3}, b_{3}) \\ ρ_{2}(a⃗, b⃗) & = (−1)^{\frac{μ_{1}+p−1}{2}} γ_{p}(\frac{a_{1} + 1}{2}, \frac{b_{1} + 1}{2}) γ_{p}(\frac{a_{2} + 1}{2}, \frac{b_{2} + 1}{2}) γ_{p}(a_{3}, b_{3}) \end{aligned}

Then:

(1.16)  ρ_{1}(0, 0) = 1 , ρ_{2}(0, 0) = (−1)^{\frac{p−1}{2}} γ_{p}(\frac{1}{2}, \frac{1}{2})^{2} = p .

Since (§10.12 of [GHF]), for s, t ∈ Z,

(1.17)  γ_{p}(b + s, b′ + t) = γ_{p}(b, b′)\frac{(b)_{s}}{(−π)^{s}} \frac{(−π)^{t}}{(b′)_{t}} ,

we have:

(1.18)  \begin{aligned}ρ_{1}(a⃗ + u′, b⃗) & = −\frac{a_{1}a_{3}}{2π^{2}} ρ_{1}(a⃗, b⃗) \\ ρ_{1}(a⃗ + u′′, b⃗) & = −\frac{a_{2}a_{3}}{2π^{2}} ρ_{1}(a⃗, b⃗) \\ ρ_{2}(a⃗ + u′, b⃗) & = −\frac{(a_{1} + 1)a_{3}}{2π^{2}} ρ_{2}(a⃗, b⃗) \\ ρ_{2}(a⃗ + u′′, b⃗) & = −\frac{(a_{2} + 1)a_{3}}{2π^{2}} ρ_{2}(a⃗, b⃗) \end{aligned}

Therefore ρ_{1}, ρ_{2} are precisely the diagonal terms of γ_{g}(a⃗, b⃗; −1):

(1.19)  γ_{g}(a⃗, b⃗; −1) = ( \begin{matrix}ρ_{1}(a⃗, b⃗) & 0 \\ ∗ & ρ_{2}(a⃗, b⃗) \end{matrix} ) .

For fixed μ ∈ Z, the function γ_{p}(b, c) with pc − b = μ is a meromorphic function of b ∈ C_{p} in the disk |b + μ| < ρ. We recall for future use from [GHF,10.7]

(1.20)  \begin{aligned}γ_{p}(b, c)γ_{p}(b + \frac{1}{2}, c + \frac{1}{2}) & = γ_{p}(2b, 2c)4^{μ}(4^{1−p})^{c}γ_{p}(\frac{1}{2}, \frac{1}{2}) \\ γ_{p}(b, c)γ_{p}(1 − b, 1 − c) & = p(−1)^{μ} \\ γ_{p}(\frac{1}{2}, \frac{1}{2})^{2} & = (−1)^{\frac{p−1}{2}} p \end{aligned}

for b, c ∈ C_{p}, pc − b = μ ∈ Z, |b + μ| < ρ. In the same notation

(1.21)  π^{−μ}γ_{p}(b, c) ∼ \{ \begin{matrix}1 & if μ ∈ [0, p − 1], \\ \frac{1}{p(c−1)} & if μ ∈ [p, 2p − 1], \\ pc & if μ ∈ [−(p − 1), 0), \end{matrix}

where ∼ indicates multiplicative congruence modulo units. We also record for completeness that for n ∈ Z

(1.22)  γ_{p}(n, n) = \{ \begin{matrix}1 & if n ≤ 0, \\ p & if n > 0. \end{matrix}

Therefore, the meromorphic function γ_{p}(b, c) has a zero if and only if b ∈ Z_{>0} and c ∈ Z_{≤0}, while it has a pole if and only if c ∈ Z_{>0} and b ∈ Z_{≤0}.

§2 Frobenius: comparison between the results of [GHF] and of [LDE].

We generalize g(x) to

(2.1) g(x, λ) = x_{3}(x_{1} + x_{2}) + x_{4}(λx_{1} + x_{2}) = x_{1}(x_{3} + λx_{4}) + x_{2}(x_{3} + x_{4})

and recall first the general notation of [GHF]. The Frobenius map :

(2.2)  α_{a⃗,b⃗,λ}^{∗} : K′_{b⃗,λ^{p}} ⟶ K′_{a⃗,λ}

is defined under some mild conditions on (a⃗, b⃗, λ) ∈ H(C_{p}) × H(C_{p}) × C_{p}, pb⃗ − a⃗ = μ⃗ ∈ H(Z), which we do not state explicitly [GHF, Lemma 4.5.1]. We consider instead the following subregions of H(C_{p}), indexed by μ⃗ ∈ H(Z), and of C_{p}, indexed by ϵ ≥ 0:

Hμ⃗ = {a⃗ ∈ H(C_{p}) | d(a⃗, −μ⃗) ≤ p^{−1}}

H = ⋃_{μ⃗∈H(Z)} Hμ⃗ = {a⃗ ∈ H(C_{p})) | d(a⃗, H(Z)) ≤ p^{−1}}

S_{ϵ} = {λ ∈ C_{p} | ϵ ≥ ord(λ) ≥ 0 = ord(λ − 1)}

S = D(0, 1^{+}) \ D(1, 1^{−}) = {λ ∈ C_{p} | ord(λ) ≥ 0 = ord(λ − 1)}

D = D(0, 1^{−}) = {λ ∈ C_{p} | ord(λ) > 0}

Definition 2.3

The following lemma is easily checked.

Lemma 2.4. Let

S = {μ⃗ ∈ H(Z) | −p < μ_{i} < p , i = 1, 2, 3, 4, μ_{1} ≥ 0 , μ_{4} ≥ 0 , 0 ≤ μ_{4}−μ_{2} = μ_{1}−μ_{3} < p } .

Then, if μ⃗, μ′ ∈ S, μ⃗ ≠ μ′,

Hμ⃗ ∩ H_{}_{μ′} = ∅ ,

and

H = ⋃_{μ⃗∈S} Hμ⃗ .

We know from [GHF, Lemma 4.5.1] that α_{a⃗,b⃗,λ}^{∗} = α_{a⃗, \frac{a⃗+μ⃗}{p} ,λ}^{∗} is defined for

(a⃗, λ) ∈ Hμ⃗ × S_{ϵ}, for any μ⃗ ∈ H(Z), 0 ≤ ϵ ≤ p^{−1}(1 − \frac{1}{p} − \frac{1}{p−1}) and

(2.5)  a_{i}, b_{i} ∉ Z , for i = 1, 2, 3, 4 .

(The last non-resonance condition is unnecessary for the existence of α_{a⃗,b⃗,λ}^{∗} [A-D]).

We recall the matrix representation (cf. §4.7 of [GHF]; a weakened form of condition (2.5), namely condition (3.1) below, is here necessary)

(2.6.1)  α_{a⃗,b⃗,λ}^{∗} ([1]_{b⃗,λ^{p}}^{∗}, [πx_{2}x_{3}]_{b⃗,λ^{p}}^{∗}) = ([1]_{a⃗,λ}^{∗}, [πx_{2}x_{3}]_{a⃗,λ}^{∗}) γ_{g}(a⃗, b⃗; λ)

We also write

(2.6.2)  γ_{g}(a⃗, b⃗; λ) = ( \begin{matrix}γ_{11}(a⃗, b⃗; λ) & γ_{12}(a⃗, b⃗; λ) \\ γ_{21}(a⃗, b⃗; λ) & γ_{22}(a⃗, b⃗; λ) \end{matrix} ) = ( \begin{matrix}γ_{11} & γ_{12} \\ γ_{21} & γ_{22} \end{matrix} )

The results of [GHF, Th. 4.7.1], strengthened in [A-D, Th. 3.3], assert that for a given μ⃗ ∈ H(Z), there exists a polynomial P (z) ∈ Z[z_{1}, z_{2}, z_{3}, z_{4}] and a natural number N such that P (a⃗)λ^{N}γ_{g}(a⃗, \frac{a⃗+μ⃗}{p}; λ) extends to an analytic function of (a⃗, λ) in Hμ⃗ × S. By equation (5.40) of [GHF] one may check that the order of pole of γ_{g}(a⃗, \frac{a⃗+μ⃗}{p}; λ) at λ = 0 is bounded by sup(0, μ_{3} − μ_{1}). If we insist that μ_{i} ≤ p − 1 for i = 1, 2, 3, 4, then by [GHF, Prop. 6.13.2 and §13.2] we may take P (z) = 1 in the previous assertion.

We are especially interested in estimates for the Frobenius matrix similar to those of Chapter 6 of [LDE], but based on formula 4.40 of [GHF]. We will discuss them after having compared the results of [GHF] and [LDE] as far as the Frobenius structure is concerned.

The theory of [LDE] is related to that of [GHF] by the so-called Laplace transform (Chapters 10 and 11 of [GHF]). This may be done in 2 ways: with respect to the variables (x_{1}, x_{2}) and with respect to the variables (x_{3}, x_{4}).

We define

H′ ; ν⃗ = {(a, b, c) ∈ C_{p}^{3} | d((a, b, c), −ν⃗) ≤ p^{−1}} , for ν⃗ ∈ Z^{3} ,

(2.7)  S′ = {ν⃗ = (ν_{1}, ν_{2}, ν_{3}) | ν_{i} ∈ {0, …, p − 1} , i = 1, 2, 3} ,

H′ = {(a, b, c) ∈ C_{p}^{3} | d((a, b, c), Z^{3}) ≤ p^{−1}} = ⋃_{ν⃗∈S′} H′ ; ν⃗ .

We first consider the algebraic Laplace transform with respect to the variables (x_{1}, x_{2}) (2.8)

R = C_{p}(λ)[x_{1}x_{3}, x_{1}x_{4}, x_{2}x_{3}, x_{2}x_{4}] ⟶L̃ = C_{p}(λ)[x, \frac{1}{x(1 + x)(1 + λx)}]

x^{u} ⟼\frac{x_{3}^{u_{3}}x_{4}^{u_{4}}}{(x_{3} + λx_{4})^{u_{1}}(x_{3} + x_{4})^{u_{2}}} \frac{(a_{1})_{u_{1}}(a_{2})_{u_{2}}}{(−π)^{u_{1}+u_{2}}} =

= \frac{x^{u_{4}}}{(1 + λx)^{u_{1}}(1 + x)^{u_{2}}} \frac{(a_{1})_{u_{1}}(a_{2})_{u_{2}}}{(−π)^{u_{1}+u_{2}}}

In the previous equation we put \frac{x_{4}}{x_{3}} = x. Let

(2.9)  xf̃ = x^{a_{4}}(1 + x)^{−a_{2}}(1 + λx)^{−a_{1}} = x^{b}(1 + x)^{c−b}(1 + λx)^{−a}

and

(2.10)  xf = x^{b}(1 − x)^{c−b}(1 − λx)^{−a}

We define

(2.11) W_{f} = L/(xf)^{−1} ∘ x\frac{∂}{∂x} ∘ xfL , W̃f̃ = L̃/(xf)^{−1} ∘ x\frac{∂}{∂x} ∘ xfL̃ .

For λ transcendental over C_{p}, we obtain isomorphic C_{p}(λ)/C_{p}-differential modules via

(2.12)  \begin{matrix}L & ⟶ L̃ \\ g(x) & ⟼ g(−x) \end{matrix}

The space considered in [LDE] is W_{f}. If (A, B, C) ∈ C_{p}^{3} are such that (pA − a, pB − b, pC − c) = (μ_{a,A}, μ_{b,B}, μ_{c,C}) ∈ Z^{3} and

xh = x^{B}(1 − x)^{C−B}(1 − λx)^{−A}

(2.13)  xh̃ = x^{B}(1 + x)^{C−B}(1 + λx)^{−A}

G_{f,h}(x, λ) = \frac{xf(x, λ)}{x^{p}h(x^{p}, λ^{p})} , G_{f̃,h̃}(x, λ) = \frac{xf̃(x, λ)}{x^{p}h̃(x^{p}, λ^{p})}

then

(2.14)  G_{f̃,h̃}(x, λ) = (−1)^{μ_{b,B}}G_{f,h}(−x, λ)

By [GHF, Lemma 11.1], if a_{1}, a_{2} ∉ Z, the algebraic Laplace isomorphism induces isomorphisms of algebraic cohomology spaces

(2.15)  \begin{matrix}W′ ; a⃗ & ⟶^{∼} & W̃f̃ & ⟶^{∼} & W_{f} \\ [1] & ⟼ & [1] & ⟼ & [1] \\ [πx_{2}x_{3}] & ⟼ & −a_{2}[\frac{1}{1+x}] & ⟼ & −a_{2}[\frac{1}{1−x}] \end{matrix}

In particular, a C_{p}(λ)-basis of W̃f̃ is given by ([1], [\frac{1}{1+x}]) .

We now set

(2.16.1)  a = a_{1} , b = a_{4} , c = a_{4} − a_{2} = a_{1} − a_{3} ,

(2.16.2)  A = b_{1} , B = b_{4} , C = b_{4} − b_{2} = b_{1} − b_{3} ,

so that

(2.16.3)  μ_{a,A} = μ_{1} , μ_{b,B} = μ_{4} , μ_{c,C} = μ_{4} − μ_{2} = μ_{1} − μ_{3} .

So the Laplace transform with respect to the variables (x_{1}, x_{2}) leads us to consider, via (2.16.1), a linear isomorphism

T_{I} : H(C_{p}) ⟶ (C_{p})^{3}

(such that T_{I}(H(Z)) = (Z)^{3} and T_{I}(S) = S′), and, for any μ⃗ ∈ H(Z), an analytic isomorphism

(2.17)  \begin{matrix}T_{I} : Hμ⃗ & ⟶ H′_{T_{I}(μ⃗)} \\ (a_{1}, a_{2}, a_{3}, a_{4}) & ⟼ (a, b, c) = (a_{1}, a_{4}, a_{4} − a_{2}) . \end{matrix}

From [LDE] we know the existence of dual maps, defined for a specialized λ ∈ S \ {0} and

(2.18.1)  a, b, c ∈ Q ∩ Z_{p}

such that

(2.18.2)  a, b, a − c, b − c ∉ Z ,

namely

(2.18.3)  \overline{α}_{f,h} : W_{f,λ} ⟶ W_{h,λ^{p}}

and

(2.18.4)  α_{f,h}^{∗} : K_{h,λ^{p}} ⟶ K_{f,λ} .

We observe that conditions (2.18.1,2) are equivalent, via (2.17), to

(2.18.5)  a_{i} ∈ Q ∩ Z_{p} \ Z , for i = 1, 2, 3, 4 .

If also (2.6.1) is defined, we get commutative diagrams

(2.19.1)  \begin{aligned}& W′_{a⃗,λ} & ⟶^{∼} & W_{f,λ} \\ α_{a⃗,b⃗,λ} & ↓ & & ↓ & (−1)^{μ_{4}}γ_{p}(a_{1}, b_{1})γ_{p}(a_{2}, b_{2})\overline{α}_{f,h} \\ & W′_{b⃗,λ^{p}} & ⟶^{∼} & W_{h,λ^{p}} \end{aligned}

Dually, and under the same assumptions (2.18.5), we have the commutative diagrams

(2.19.2)  \begin{aligned}& K_{f,λ} & ⟶^{∼} & K′_{a⃗,λ} \\ (−1)^{μ_{4}}γ_{p}(a_{1}, b_{1})γ_{p}(a_{2}, b_{2})α_{f,h}^{∗} & ↑ & & ↑ & α_{a⃗,b⃗,λ}^{∗} \\ & K_{h,λ^{p}} & ⟶^{∼} & K′_{b⃗,λ^{p}} \end{aligned} ,

Then, while in the notation of [GHF] we have (2.6.1), in that of [LDE, 4.5.1] (upon renaming B as B) we have

(2.20)

α_{f,h}^{∗} ([1]^{∗}, [\frac{1}{1 − x}]^{∗}) = ([1]^{∗}, [\frac{1}{1 − x}]^{∗}) B^{t} = ([1]^{∗}, [\frac{1}{1 − x}]^{∗}) ( \begin{matrix}B_{1} & B_{3} \\ B_{2} & B_{4} \end{matrix} ) ,

for λ ∈ S \ {0} and (a, b, c) ∈ H′, provided conditions (2.18.1,2) hold. We then have

(2.21)  ( \begin{matrix}B_{1} & B_{2} \\ B_{3} & B_{4} \end{matrix} ) = (−1)^{μ_{4}}γ_{p}(a_{1}, b_{1})^{−1}γ_{p}(a_{2}, b_{2})^{−1} ( \begin{matrix}γ_{11} & −\frac{γ_{21}}{a_{2}} \\ −b_{2}γ_{12} & \frac{b_{2}γ_{22}}{a_{2}} \end{matrix} )

We now use formula (2.21) to extend, for fixed μ⃗ ∈ H(Z), the matrix B to a meromorphic function of (a⃗, λ) ∈ Hμ⃗×S_{ϵ}, where a⃗ stands for (a_{1}, a_{2}, a_{3}, a_{4}), and ϵ ≥ 0 is small, or of (a, b, c, λ), in the region H′_{T_{I}(μ⃗)}×S_{ϵ}. We will take the liberty of writ-

ing indifferently B = B(a⃗, b⃗; λ) = B(a, b, c; A, B, C; λ) (= B(a⃗, λ) = B(a, b, c; λ)) according to the context. When we do so, we understand, unless otherwise specified, that we are using the transformation T_{I} (and that μ⃗ is somehow specified). We will investigate in the next section the maximal meromorphic extension of the previous functions in Hμ⃗ × S.

Remark 2.21.1. The assumption a, b, c ∈ Q ∩ Z_{p} is needed in [LDE] to prove that the algebraic cohomology space W_{f} (and its dual K_{f}), for a specialized λ ≠ 0, 1, ∞, is isomorphic to its Monsky-Washnitzer counterpart W_{f,λ} (resp. K_{f,λ}) (cf. Chap. 1 to 4 of [LDE]). But the theory of [GHF], where only the dual spaces K′_{a⃗,λ} are effectively used, is independent of non-Liouville-type assumptions on the a_{i}’s. This permits us to use freely diagram (2.19.2) for (a, b, c) ∈ H′, under only the assumption (2.18.2) (corresponding to (2.5) via T_{I}), where K_{f,λ}, K_{h,λ} now indicate the algebraic dual cohomology spaces, for a specialized λ ≠ 0, 1, ∞. This seems to suggest that more generally the dual algebraic cohomology spaces of Dwork should be endowed with a Frobenius action, even when they do not coincide with their analytic counterpart, as defined in chapter 4 of [LDE]. We finally point out that the map

K_{f,λ}⟶^{∼}K′_{a⃗,λ}

of diagram (2.19.2), “transpose of Laplace transform”, in a purely algebraic or in an analytic setting, does not appear in the literature.

The Laplace transform with respect to the variables (x_{3}, x_{4}) would yield analogous results with

(2.22.1)  a = a_{4} , b = a_{1} , c = a_{1} − a_{3} = a_{4} − a_{2} ,
(2.22.2)  A = b_{4} , B = b_{1} , C = b_{1} − b_{3} = b_{4} − b_{2} ,

so that

(2.22.3)  μ_{a,A} = μ_{4} , μ_{b,B} = μ_{1} , μ_{c,C} = μ_{1} − μ_{3} = μ_{4} − μ_{2} .

As before we get, via (2.22.1), a linear isomorphism

(2.23)  \begin{matrix}T_{II} : Hμ⃗ & ⟶ H′_{T_{I}(μ⃗)} \\ (a_{1}, a_{2}, a_{3}, a_{4}) & ⟼ (a, b, c) = (a_{4}, a_{1}, a_{4} − a_{2}) . \end{matrix}

(such that T_{II}(H(Z)) = (Z)^{3} and T_{II}(S) = S′), and, for any μ⃗ ∈ H(Z), an analytic isomorphism

T_{II} : Hμ⃗ ⟶ H′_{T_{II}(μ⃗)} .

We then obtain commutative diagrams

(2.24)  \begin{aligned}& W′_{a⃗,λ} & ⟶^{∼} & W_{f,λ} \\ α_{a⃗,b⃗,λ} & ↓ & & ↓ & (−1)^{μ_{1}}γ_{p}(a_{3}, b_{3})γ_{p}(a_{4}, b_{4})\overline{α}_{f,h} \\ & W′_{b⃗,λ^{p}} & ⟶^{∼} & W_{h,λ^{p}} \end{aligned}

and

(2.25)  \begin{aligned}& K′_{a⃗,λ} & ⟶^{∼} & K_{f,λ} \\ α_{a⃗,b⃗,λ}^{∗} & ↑ & & ↑ & (−1)^{μ_{1}}γ_{p}(a_{3}, b_{3})γ_{p}(a_{4}, b_{4})α_{f,h}^{∗} \\ & K′_{b⃗,λ^{p}} & ⟶^{∼} & K_{h,λ^{p}} \end{aligned} ,

As before, for fixed μ⃗ ∈ H(Z), the matrix B could be viewed as a function of (a⃗, λ) ∈ Hμ⃗ × S_{ϵ}, where a⃗ stands for (a_{1}, a_{2}, a_{3}, a_{4}), and ϵ > 0 is small, or of (a, b, c, λ), in the region H′_{T_{II}(μ⃗)} × S_{ϵ}, via the transformation T_{II}. We then have

(2.26)  ( \begin{matrix}B_{1} & B_{2} \\ B_{3} & B_{4} \end{matrix} ) = (−1)^{μ_{1}}γ_{p}(a_{3}, b_{3})^{−1}γ_{p}(a_{4}, b_{4})^{−1} ( \begin{matrix}γ_{11} & −\frac{γ_{21}}{a_{3}} \\ −b_{3}γ_{12} & \frac{b_{3}γ_{22}}{a_{3}} \end{matrix} )

We now point out that the positions (2.17) and (2.23) define a commutative diagram

(2.27)  \begin{aligned}& T_{I} & H′ \\ & ↗ \\ H & & ↓ \\ & ↘ \\ & T_{II} & H′ \end{aligned} ,

where the vertical arrow interchanges a and b, keeping c fixed. From (2.21) and (2.26) we obtain the following formula, valid for any ν⃗ = (μ_{a,A}, μ_{b,B}, μ_{c,C}) ∈ Z^{3},

(a, b, c) ∈ H′ ; ν⃗, and (A, B, C) = p^{−1}((a, b, c) + ν⃗):

(2.28)  \begin{matrix}B(b, a, c; & B, A, C; λ) = (−1)^{μ_{a,A}−μ_{b,B}} \frac{γ_{p}(a, A)γ_{p}(b − c, B − C)}{γ_{p}(b, B)γ_{p}(a − c, A − C)} · \\ · ( \begin{matrix}1 & 0 \\ 0 & \frac{C−A}{C−B} \end{matrix} ) B(a, b, c; A, B, C; λ) ( \begin{matrix}1 & 0 \\ 0 & \frac{c−b}{c−a} \end{matrix} ) \end{matrix}

Formula (2.28) is proved by two different methods in section 5 of [Ku].

We may specialize the results of this section to the situation considered in §1, that is a_{3} = a_{4}, b_{3} = b_{4}, μ_{1} even. More precisely: μ⃗ ∈ L(Z), μ_{1} even, a⃗ ∈ Hμ⃗, a⃗, b⃗ = \frac{a⃗+μ⃗}{p} ∈ L(C_{p}).

We first use T_{I}. The previous assumptions then correspond for (a, b, c) to assuming c = a − b, C = A − B and μ_{a,A} even. We obtain, for (a, b, a − b) ∈ H′ and p(A, B, A − B) − (a, b, a − b) = (μ_{a,A}, μ_{b,B}, μ_{a,A} − μ_{b,B}) ∈ Z^{3}, μ_{a,A} even:

(2.29)  B(a, b, a − b; A, B, A − B; −1) = ( \begin{matrix}τ_{1}(a, b; A, B) & ∗ \\ 0 & τ_{2}(a, b; A, B) \end{matrix} )

where

\begin{aligned}τ_{1}(a, b; A, B) & = (−1)^{μ_{b,B}− \frac{μ_{a,A}}{2}} \frac{γ_{p}(\frac{a}{2}, \frac{A}{2}) γ_{p}(b − \frac{a}{2}, B − \frac{A}{2}) γ_{p}(b, B)}{γ_{p}(a, A)γ_{p}(2b − a, 2B − A)} = \\ & = (−1)^{\frac{μ_{2}}{2}} \frac{γ_{p}(\frac{a_{1}}{2} , \frac{b_{1}}{2} ) γ_{p}(\frac{a_{2}}{2} , \frac{b_{2}}{2} ) γ_{p}(a_{3}, b_{3})}{γ_{p}(a_{1}, b_{1})γ_{p}(a_{2}, b_{2})} \end{aligned}

(2.30)

\begin{aligned}τ_{2}(a, b; A, B) & = (−1)^{μ_{b,B}− \frac{μ_{a,A}}{2} + \frac{p−1}{2}} \frac{γ_{p}(\frac{a+1}{2}, \frac{A+1}{2}) γ_{p}(b − \frac{a}{2} + \frac{1}{2}, B − \frac{A}{2} + \frac{1}{2}) γ_{p}(b, B)}{γ_{p}(a, A)γ_{p}(1 + 2b − a, 1 + 2B − A)} = \\ & = (−1)^{\frac{μ_{2}+p−1}{2}} \frac{γ_{p}(\frac{a_{1}+1}{2}, \frac{b_{1}+1}{2}) γ_{p}(\frac{a_{2}+1}{2}, \frac{b_{2}+1}{2}) γ_{p}(a_{3}, b_{3})}{γ_{p}(a_{1}, b_{1})γ_{p}(a_{2} + 1, b_{2} + 1)} , \end{aligned}

where (a_{1}, a_{2}, a_{3}, a_{4}) and (a, b, c) (resp. (b_{1}, b_{2}, b_{3}, b_{4}) and (A, B, C)) are related via T_{I}. We point out that if we assume μ_{a,A}, μ_{b,B} ∈ {0, …, p − 1}, then the entries of the matrix

( \begin{matrix}τ_{1}(a, b; A, B) & 0 \\ 0 & τ_{2}(a, b; A, B) \end{matrix} )

have the same p-adic orders as those of the matrix given in the following table:

(2.31)

\begin{matrix}( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) & if − (p − 1) ≤ 2μ_{b,B} − μ_{a,A} ≤ 0 \\ ( \begin{matrix}1 & 0 \\ 0 & p(2B − A) \end{matrix} ) & if 0 < 2μ_{b,B} − μ_{a,A} ≤ p − 1 \\ ( \begin{matrix}p(2B − A − 1) & 0 \\ 0 & p(2B − A) \end{matrix} ) & if p − 1 < 2μ_{b,B} − μ_{a,A} ≤ 2(p − 1) \end{matrix}

(To check the last statement, recall that μ_{a,A} is even, so that 2μ_{b,B} − μ_{a,A} ≠ p).

In our applications, we will be especially interested in the formula

(2.32) \begin{matrix}B(a, b, 1 + a − b; A, B, 1 + A − B; −1) & = \\ = ( \begin{matrix}0 & \frac{1−B}{4B−2A} \\ 1 & \frac{4B−2A−1}{4B−2A} \end{matrix} ) B(a, b, a − b; A, B, A − B; −1) & ( \begin{matrix}\frac{4b−2a−1}{b−1} & 1 \\ \frac{4b−2a}{1−b} & 0 \end{matrix} ) , \end{matrix}

that is easily deduced from [Dw1, (3.0.3)], where the matrix γ that is used is the transpose of our B. As a consequence

(2.33)  \begin{matrix}B(a, b, 1 + a − b; & A, B, 1 + A − B; −1) = \\ = & ( \begin{matrix}\frac{1−B}{1−b} \frac{2b−a}{2B−A}τ_{2}(a, b; A, B) & 0 \\ ∗ & τ_{1}(a, b; A, B) \end{matrix} ) , \end{matrix}

so that (2.33.1)

\begin{matrix}B_{1}(a, b & , 1 + a − b; A, B, 1 + A − B; −1) = \\ (−1)^{μ_{b,B}− \frac{μ_{a,A}}{2} + \frac{p−1}{2}} & \frac{γ_{p}(\frac{a+1}{2}, \frac{A+1}{2}) γ_{p}(b − \frac{a}{2} + \frac{1}{2}, B − \frac{A}{2} + \frac{1}{2}) γ_{p}(b − 1, B − 1)}{γ_{p}(a, A)γ_{p}(2b − a, 2B − A)} , \end{matrix}

(2.33.2)  \begin{matrix}B_{4}(a, & b, 1 + a − b; A, B, 1 + A − B; −1) = \\ (−1)^{μ_{b,B}− \frac{μ_{a,A}}{2}} & \frac{γ_{p}(\frac{a}{2}, \frac{A}{2}) γ_{p}(b − \frac{a}{2}, B − \frac{A}{2}) γ_{p}(b, B)}{γ_{p}(a, A)γ_{p}(2b − a, 2B − A)} . \end{matrix}

In particular, if 0 ≤ 2μ_{b,B} − μ_{a,A} ≤ p − 1,

(2.33.3)  B_{2}(a, b, 1 + a − b; A, B, 1 + A − B; −1) = 0 ,

(2.33.4)  |B_{4}(a, b, 1 + a − b; A, B, 1 + A − B; −1)| = 1 ,

(2.33.5) |B_{1}(a, b, 1+a−b; A, B, 1+A−B; −1)| \{ \begin{matrix}= p^{−1}|B − 1| & if μ_{b} < p − 1, \\ = 1 & if μ_{b} = p − 1. \end{matrix}

By use of T_{II} or of (2.28), we similarly obtain, for (a, b, b − a) ∈ H′ and p(A, B, B − A) − (a, b, b − a) = (μ_{a,A}, μ_{b,B}, μ_{b,B} − μ_{a,A}) ∈ Z^{3}, μ_{b,B} even:

(2.34) B(a, b, b − a; A, B, B − A; −1) = ( \begin{matrix}\overline{τ}_{1}(a, b; A, B) & ∗ \\ 0 & \overline{τ}_{2}(a, b; A, B) \end{matrix} )

where

\begin{aligned}\overline{τ}_{1}(a, b; A, B) & = (−1)^{\frac{μ_{b,B}}{2}} \frac{γ_{p}(\frac{b}{2}, \frac{B}{2} ) γ_{p}(a − \frac{b}{2}, A − \frac{B}{2} )}{γ_{p}(a, A)} = \\ & = (−1)^{\frac{μ_{1}}{2}} \frac{γ_{p}(\frac{a_{1}}{2} , \frac{b_{1}}{2} ) γ_{p}(\frac{a_{2}}{2} , \frac{b_{2}}{2} )}{γ_{p}(a_{3}, b_{3})} \end{aligned}

(2.35)

\begin{aligned}\overline{τ}_{2}(a, b; A, B) & = (−1)^{\frac{μ_{b,B}}{2} + \frac{p−1}{2}} \frac{γ_{p}(\frac{b+1}{2}, \frac{B+1}{2}) γ_{p}(a − \frac{b}{2} + \frac{1}{2}, A − \frac{B}{2} + \frac{1}{2})}{γ_{p}(1 + a, 1 + A)} \\ & = (−1)^{\frac{μ_{1}+p−1}{2}} \frac{γ_{p}(\frac{a_{1}+1}{2}, \frac{b_{1}+1}{2}) γ_{p}(\frac{a_{2}+1}{2}, \frac{b_{2}+1}{2})}{γ_{p}(1 + a_{3}, 1 + b_{3})} , \end{aligned}

where (a_{1}, a_{2}, a_{3}, a_{4}) and (a, b, c) (resp. (b_{1}, b_{2}, b_{3}, b_{4}) and (A, B, C)) are related via T_{II}. Again, if we assume μ_{a,A}, μ_{b,B} ∈ {0, …, p − 1}, then the entries of the matrix

( \begin{matrix}\overline{τ}_{1}(a, b; A, B) & 0 \\ 0 & \overline{τ}_{2}(a, b; A, B) \end{matrix} )

have the same p-adic orders as those of the matrix given in the following table: (2.36)

( \begin{matrix}p(A − \frac{B}{2} ) & 0 \\ 0 & pA \end{matrix} )  if − (p − 1) ≤ 2μ_{a,A} − μ_{b,B} < 0 , μ_{a,A} ≠ 0
( \begin{matrix}p(A − \frac{B}{2} ) & 0 \\ 0 & 1 \end{matrix} )  if μ_{a,A} = 0 , μ_{b,B} ≠ 0
( \begin{matrix}1 & 0 \\ 0 & pA \end{matrix} )  if 0 ≤ 2μ_{a,A} − μ_{b,B} ≤ p − 1 , μ_{a,A} ≠ 0
( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} )  if p − 1 < 2μ_{a,A} − μ_{b,B} ≤ 2(p − 1) or μ_{a,A} = μ_{b,B} = 0

As before, we deduce

(2.37)  \begin{matrix}B(a, b, 1 + b − a; & A, B, 1 + B − A; −1) = \\ = & ( \begin{matrix}\frac{1+B−2A}{1+b−2a} \frac{a}{A} \overline{τ}_{2}(a, b; A, B) & 0 \\ ∗ & \overline{τ}_{1}(a, b; A, B) \end{matrix} ) , \end{matrix}

so that

(2.37.1)  \begin{matrix}B_{1}(a, b & , 1 + b − a; A, B, 1 + B − A; −1) = \\ = (−1)^{\frac{μ_{b,B}}{2} + \frac{p−1}{2}} & \frac{γ_{p}(\frac{b+1}{2}, \frac{B+1}{2}) γ_{p}(a − \frac{b}{2} − \frac{1}{2}, A − \frac{B}{2} − \frac{1}{2})}{γ_{p}(a, A)} , \end{matrix}

(2.37.2)  \begin{matrix}B_{4}(a, & b, 1 + b − a; A, B, 1 + B − a; −1) = \\ = (−1)^{\frac{μ_{b,B}}{2}} & \frac{γ_{p}(\frac{b}{2}, \frac{B}{2} ) γ_{p}(a − \frac{b}{2}, A − \frac{B}{2} )}{γ_{p}(a, A)} . \end{matrix}

§3 Approximations of the Frobenius matrix (p ≥ 7).

We now consider simultaneously the theory of [GHF] and of [LDE] as related by the Laplace transform with respect to (x_{1}, x_{2}). For μ⃗ ∈ S, we examine the Frobenius matrix B(a⃗; λ) = B(a⃗, \frac{a⃗+μ⃗}{p}; λ) of (2.20) as a function of (a⃗, λ) ∈ Hμ⃗×S, and, via T_{I} as a function of (a, b, c; λ) ∈ H′_{T_{I}(μ⃗)} × S, T_{I}(μ⃗) ∈ S′. We abuse notation on setting T_{I}(μ⃗) = (μ_{a}, μ_{b}, μ_{c}) ∈ S′, and aim at making more precise the classification in terms of (μ_{a}, μ_{b}, μ_{c}) given in [LDE, Chap. 6]. We are mainly interested in the so-called split cases, namely cases 1 and 2 below. We subdivide S (resp. S′) into 9 disjoint sets:

Case 0 :

μ_{a} = μ_{b} = μ_{c}

i.e.

μ_{1} = μ_{4} = μ_{4} − μ_{2}

Case 1 :

μ_{c} < min(μ_{a}, μ_{b})

i.e.

μ_{4} − μ_{2} < min(μ_{1}, μ_{4})

Case 1′ :

μ_{c} = μ_{a} < μ_{b}

i.e.

μ_{4} − μ_{2} = μ_{1} < μ_{4}

Case 1′′ :

μ_{c} = μ_{b} < μ_{a}

i.e.

μ_{4} − μ_{2} = μ_{4} < μ_{1}

Case 2 :

μ_{c} > max(μ_{a}, μ_{b})

i.e.

μ_{4} − μ_{2} > max(μ_{1}, μ_{4})

Case 2′ :

μ_{c} = μ_{b} > μ_{a}

i.e.

μ_{4} − μ_{2} = μ_{4} > μ_{1}

Case 2′′ :

μ_{c} = μ_{a} > μ_{b}

i.e.

μ_{4} − μ_{2} = μ_{1} > μ_{4}

Case 3 :

μ_{b} < μ_{c} < μ_{a}

i.e.

μ_{4} < μ_{4} − μ_{2} < μ_{1}

Case 4 :

μ_{a} < μ_{c} < μ_{b}

i.e.

μ_{1} < μ_{4} − μ_{2} < μ_{4}

We observe that another characterization of the previous cases is Case 0 :

μ_{2} = 0 , μ_{3} = 0

Case 1 :

μ_{2} > 0 , μ_{3} > 0

Case 1′ :

μ_{2} > 0 , μ_{3} = 0

Case 1′′ :

μ_{2} = 0 , μ_{3} > 0

Case 2 :

μ_{2} < 0 , μ_{3} < 0

Case 2′ :

μ_{2} = 0 , μ_{3} < 0

Case 2′′ :

μ_{2} < 0 , μ_{3} = 0

Case 3 :

μ_{2} < 0 , μ_{3} > 0

Case 4 :

μ_{2} > 0 , μ_{3} < 0

We fix a constant ϵ, 0 ≤ ϵ ≤ p^{−1}(1 − \frac{1}{p} − \frac{1}{p−1}). The results of [LDE, Chap. 24], together with the previously mentioned [GHF, Th. 4.7.1] and [A-D, Th. 3.3], ensure that, for i, j = 1, 2, γ_{i,j}(a⃗, λ), as defined by formulas (2.6.1,2) for (a⃗, λ) ∈ Hμ⃗ × S_{ϵ} and

(3.1)  a_{i}, b_{i} ∉ Z_{>0}

for i = 1, 2, 3, 4, extends to an analytic function on Hμ⃗ × S defined over the field Q(π). This matrix does not need to represent the Frobenius map in the sense of formula (2.6.1) for all (a⃗, λ) ∈ Hμ⃗ × S, in particular because the basis with respect to which that matrix is generically calculated fails to specialize to a basis of K′_{a⃗,λ}, for values of (a⃗, λ) outside of Hμ⃗ × S_{ϵ}, or when condition (3.1) is violated.

We recall formula 4.40 of [GHF] in our special case, insisting that λ ∈ S_{ϵ} and that (3.1) holds. We set

(3.2)  \begin{matrix}w_{1} = (0, 0, 0, 0) , w_{2} = (0, 1, 1, 0) \\ ω̂_{1} = 1 = x^{w_{1}} , ω̂_{2} = πx_{2}x_{3} = π^{k(w_{2})}x^{w_{2}} , \end{matrix}

which represent a basis of W′_{a⃗,λ}, and

(3.3)  ω_{i,a⃗,λ}^{∗} = ∑_{v⃗∈H(Z)} Λ_{i,v⃗}(a⃗, λ)x^{−v⃗} ,

for i = 1, 2, the dual basis of K′_{a⃗,λ}, under the assumption (3.1). According to [GHF, 3.39], we have for i = 1, 2 and v⃗ ∈ H^{+}(Z)

(3.3.1)  Λ_{i,v⃗}(a⃗, λ) ∈ (πλ(1 − λ))^{−k(v⃗)}Z[a_{1}, a_{2}, a_{3}, a_{4}, λ]

In particular we may compute:

(3.4)

Λ_{1,(0,0,0,0)}(a⃗, λ) = 1
Λ_{1,(0,1,1,0)}(a⃗, λ) = 0
Λ_{1,(1,0,1,0)}(a⃗, λ) = −\frac{a_{3}}{π}
Λ_{1,(1,0,0,1)}(a⃗, λ) = \frac{a_{3} − a_{1}}{πλ}
Λ_{1,(0,1,0,1)}(a⃗, λ) = −\frac{a_{2}}{π}
Λ_{2,(0,0,0,0)}(a⃗, λ) = 0
Λ_{2,(0,1,1,0)}(a⃗, λ) = \frac{1}{π}
Λ_{2,(1,0,1,0)}(a⃗, λ) = −\frac{1}{π}
Λ_{2,(1,0,0,1)}(a⃗, λ) = \frac{1}{πλ}
Λ_{2,(0,1,0,1)}(a⃗, λ) = −\frac{1}{π}

Lemma 3.4.1. Let χ be the step function (i.e. the characteristic function of the subset (0, ∞) of R). For any v⃗ ∈ H^{+}(Z), we have

Λ_{1,v⃗}(a⃗, λ) ∈ a_{2}^{χ(v_{2})}a_{2}^{χ(v_{3})}(πλ(1 − λ))^{−k(v⃗)}Z[a_{1}, a_{2}, a_{3}, a_{4}, λ] .

Proof. When k(v⃗) = 1, our result is contained in (3.4). We use induction on k(v⃗). Let v⃗ , w⃗ ∈ H^{+}(Z), k(v⃗) = d − 1, k(w⃗) = 1. Formula (3.40) of [GHF] reads

Λ_{1,v⃗+w⃗}(a⃗, λ) = ∑_{j=1}^{2} M(a⃗ + w⃗, a⃗; λ)_{j,1}Λ_{j,v⃗}(a⃗ + w⃗, λ)π^{−1} ,

where the matrices M(a⃗ + w⃗, a⃗; λ) are explicitly described in [GHF,13.2] (see also 1.10.5)). We are left to prove that M(a⃗+w⃗, a⃗; λ)_{j,1}Λ_{j,v⃗}(a⃗+w⃗, λ) is divisible by a_{2}^{χ(v_{2}+w_{2})}a_{3}^{χ(v_{3}+w_{3})}, for j = 1, 2. Now, a_{2}a_{3} divides M(a⃗ + w⃗, a⃗; λ)_{2,1} by [GHF,13.2]. We are left to consider M(a⃗ + w⃗, a⃗; λ)_{1,1}Λ_{1,v⃗}(a⃗ + w⃗, λ). Let i be either 2 or 3. We may assume that χ(v_{i} + w_{i}) = 1. If w_{i} = 0, then χ(v_{i}) = 1 and so by hypothesis Λ_{1,v⃗}(a⃗ + w⃗, λ) is divisible by a_{i} + w_{i} = a_{i}. Hence we may assume w_{i} ≠ 0, and it is enough to check that a_{i}^{χ(w_{i})} divides M(a⃗ + w⃗, a⃗; λ)_{1,1}, if k(w⃗) = 1. The possibilities are:

1) w⃗ = (0, 1, 1, 0). Here M(a⃗ + w⃗, a⃗; λ)_{1,1} = 0.

2) w⃗ = (1, 0, 1, 0). Here a_{3} divides M(a⃗ + w⃗, a⃗; λ)_{1,1}.

3) w⃗ = (1, 0, 0, 1). The assertion is trivial.

4) w⃗ = (0, 1, 0, 1). Here a_{2} divides M(a⃗ + w⃗, a⃗; λ)_{1,1}.

Q.E.D.

We also put

(3.5.1)  θ(t) = exp π(t − t^{p}) = ∑_{j=0}^{∞} c_{j}t^{j}

(3.5.2)  F (x, λ) = θ(x_{1}x_{3})θ(x_{2}x_{3})θ(x_{2}x_{4})θ(λx_{1}x_{4}) = ∑_{u⃗∈H^{+}(Z)} Au⃗(λ)xu⃗ .

We recall that

(3.6)  ord c_{j} ≥ \{ \begin{matrix}\frac{j}{p−1} & if j = 0, …, p^{2} − 1 \\ j \frac{p−1}{p^{2}} & if j = p^{2}, p^{2} + 1, … \end{matrix}

Then

(3.7)  \begin{aligned}γ_{j,i}(a⃗, λ) & = ⟨α_{a⃗, \frac{a⃗+μ⃗}{p} ,λ}^{∗}(ω_{i, \frac{a⃗+μ⃗}{p} ,λ^{p}}^{∗}), π^{k(w_{j})}x^{w_{j}}⟩ = \\ & = π^{k(w_{j})} ∑ Au⃗(λ)Λ_{i,v⃗}(\frac{a⃗ + μ⃗}{p}, λ^{p}) \end{aligned}

where the sum runs over the set of u⃗, v⃗ ∈ H^{+}(Z) such that u⃗ = μ⃗ − w_{j} + pv⃗, for i = 1, 2. We can now ameliorate, in our special case, formula 4.40 of [GHF]

(3.8)

ord π^{k(w_{j})}Au⃗(λ)Λ_{i,v⃗}(\frac{a⃗ + μ⃗}{p}, λ^{p}) ≥

\{ \begin{matrix}\frac{k(μ⃗)}{p−1} + k(v⃗) & if k(v⃗) ≤ p − 2 \\ \frac{k(μ⃗)}{p−1} + k(v⃗)(1 − \frac{1}{p} − \frac{1}{p−1}) − \frac{(2p−1)(2p−2)}{p^{2}(p−1)} & if k(v⃗) ≥ p − 1 \end{matrix}

for all (a⃗, λ) ∈ Hμ⃗ × S_{0}. To simplify our coming calculations, we then observe that if p ≥ 7 and h = 0, 1, 2, 3, any term π^{k(w_{j})}Au⃗(λ)Λ_{i,v⃗}(b⃗, λ^{p}), with k(v⃗) ≥ h, of the sum appearing in (3.7) has in Hμ⃗ × S_{0} at least the p-adic order \frac{k(μ⃗)}{p−1} + h. We will then assume for simplicity from now on that p ≥ 7. We then evaluate the dominant term γ_{j,i}^{0} of γ_{j,i}, for i, j = 1, 2, in the sense of the supnorm || − ||μ⃗ on Hμ⃗ × S_{0}.

Calculation of γ_{1,1}^{0} in cases 0, 1, 1′ and 1′′.

We take

(3.9)  γ_{1,1}^{0}(a⃗, λ) = Aμ⃗(λ)

We solve

(μ_{1}, μ_{2}, μ_{3}, μ_{4}) = (u_{13} + u_{14}, u_{23} + u_{24}, u_{13} + u_{23}, u_{14} + u_{24})

(3.10)  \begin{aligned}u_{14} & = t \\ u_{24} & = μ_{4} − t \\ u_{13} & = μ_{1} − t \\ u_{23} & = μ_{2} − μ_{4} + t \end{aligned}

So:

γ_{1,1}^{0}(a⃗, λ) = ∑_{t=μ_{4}−μ_{2}}^{min(μ_{1},μ_{4})} c_{t}c_{μ_{4}−t}c_{μ_{1}−t}c_{μ_{2}−μ_{4}+t}λ^{t} =

= ∑_{t=μ_{4}−μ_{2}}^{min(μ_{1},μ_{4})} \frac{π^{μ_{1}+μ_{2}}λ^{t}}{t!(μ_{4} − t)!(μ_{1} − t)!(μ_{2} − μ_{4} + t)!} =

(3.11)  = \frac{π^{μ_{1}+μ_{2}}}{μ_{1}!μ_{2}!} ∑_{t=μ_{4}−μ_{2}}^{min(μ_{1},μ_{4})} \binom{μ_{1}}{t}\binom{μ_{2}}{μ_{4} − t}λ^{t} =

= γ_{p}(−μ_{1}, 0)γ_{p}(−μ_{2}, 0) ∑_{t=μ_{4}−μ_{2}}^{min(μ_{1},μ_{4})} \binom{μ_{1}}{t}\binom{μ_{2}}{μ_{4} − t}λ^{t} =

= ((−1)^{μ_{4}}γ_{p}(−μ_{1}, 0)γ_{p}(−μ_{2}, 0)) (−1)^{μ_{b}} ∑_{s+t=μ_{b}} \binom{μ_{b} − μ_{c}}{s}\binom{μ_{a}}{t}λ^{t} .

Taking into account the previous section, this give precisely the value expected from the calculation of χ_{1} in [LDE, 6.4.8]. We write the previous result in the form

(3.12)  γ_{1,1}^{0}(a⃗, λ) = π^{k(μ⃗)}λ^{μ_{c}}P μ⃗ ; 1,1(λ)

where P μ⃗ ; 1,1(λ) is a polynomial in Z_{p}[λ] of Gauss norm 1 whose zeros all have absolute value 1. We know from [LDE, Lemma 24.4], that γ_{1,1}(a⃗, λ)λ^{−μ_{c}} is analytic in Hμ⃗ × S. Our calculation following (3.8) then shows

(3.13)  ||γ_{1,1} − γ_{1,1}^{0}||μ⃗ < p^{− \frac{k(μ⃗)}{p−1}} .

Let Sμ⃗ denote the region

(3.14)  Sμ⃗ = {λ ∈ S | |P μ⃗ ; 1,1(λ)| = 1} .

We notice here that Sμ⃗ is stable under the map λ ⟼ λ^{p}. We call Sμ⃗ the Hasse domain of type μ⃗. If we are in cases 0 or 1′′, then in fact Sμ⃗ = S. In case 0 this definition will conflict with the one given in (3.27), related to the entry γ_{2,2}. Case 0 is not too interesting for us; rather than introducing a heavier notation, we will agree that a result on γ_{1,1} (resp. γ_{2,2}) involves Sμ⃗ as defined in (3.14) (resp. (3.27)). It is clear that in any case D ⊂ Sμ⃗. We conclude that in cases 0, 1, 1′ and 1′′

(3.15)  γ_{1,1}(a⃗, λ) = λ^{μ_{c}}π^{k(μ⃗)}u_{1,1}(a⃗, λ)

where u_{1,1}(a⃗, λ) is an analytic element bounded by 1 on Hμ⃗ × S, defined over the field Q(π), which is of constant absolute value 1 all over Hμ⃗ × Sμ⃗.

Estimates for γ_{1,1} in case 2.

Formula (3.7) indicates that only terms with k(v⃗) ≥ 1 need to be considered. But in fact, since μ_{2} and μ_{3} are both negative, v⃗ = (0, 1, 1, 0) is the only term with k(v⃗) = 1 that may appear. That term gets cancelled, since Λ_{1,(0,1,1,0)}(a⃗, λ) = 0. On the other hand, lemma 3.4.1 shows divisibility by b_{2}b_{3}. In the end, considering [LDE, Lemma 24.4], we have in case 2

(3.16)  γ_{1,1}(a⃗, λ) = p^{2}π^{k(μ⃗)}b_{2}b_{3}λ^{μ_{c}}u_{1,1}(a⃗, λ)

where u_{1,1}(a⃗, λ) represents an analytic element defined over the field Q(π) and bounded by 1 on Hμ⃗ × S.

Estimates for γ_{1,1} in cases 2′, 2′′, 3 and 4.

We put

(3.17)  γ_{1,1}^{0}(a⃗, λ) = \{ \begin{matrix}−\frac{b_{2}}{π} A_{μ⃗+p(0,1,0,1)}(λ) & in cases 2′′ and 3 \\ −\frac{b_{3}}{π} A_{μ⃗+p(1,0,1,0)}(λ) & in cases 2′ and 4. \end{matrix}

We see that

||γ_{1,1}^{0}(a⃗, λ)||μ⃗ ≤ |π^{k(μ⃗)}p| .

Formula (3.8), in which only terms with k(v⃗) ≥ 2 need now to be considered, shows that

||γ_{1,1}(a⃗, λ) − γ_{1,1}^{0}(a⃗, λ)||μ⃗ ≤ |π^{k(μ⃗)}p^{2}| .

In the end, considering [LDE, Lemma 24.4] and lemma 3.4.1, we have in cases 2′, 2′′, 3 and 4

(3.18)  γ_{1,1}(a⃗, λ) = pπ^{k(μ⃗)}λ^{μ_{c}} \{ \begin{matrix}b_{2}u_{1,1}(a⃗, λ) & in cases 2′′, 3 \\ b_{3}u_{1,1}(a⃗, λ) & in cases 2′, 4 \end{matrix}

where u_{1,1}(a⃗, λ) represents an analytic element defined over the field Q(π) and bounded by 1 on Hμ⃗ × S.

Estimates for γ_{2,1} in all cases.

We refer to formula (3.7), where in any case v⃗ − w_{2} ∈ H^{+}(Z). We take (3.19)

γ_{2,1}^{0}(a⃗, λ) = \{ \begin{matrix}πA_{μ⃗−w_{2}}(λ) , & if μ_{2} > 0 and μ_{3} > 0 (case 1); \\ −b_{2}A_{μ⃗−w_{2}+p(0,1,0,1)}(λ) , & if μ_{2} ≤ 0 and μ_{3} > 0 (cases 1′′, 3); \\ π ∑_{k(v⃗)=2} A_{μ⃗−w_{2}+pv⃗}(λ)Λ_{1,v⃗}(b⃗, λ^{p}) , & if μ_{3} ≤ 0 (cases 0, 1′, 2, 2′, 2′′, 4). \end{matrix}

We now easily estimate ||γ_{2,1}^{0}(a⃗, λ)||μ⃗, while, with the help of formula (3.8), we can bound the remaining terms in formula (3.7). We obtain

(3.20)  ||γ_{2,1}(a⃗, λ)||μ⃗ ≤ \{ \begin{matrix}|π^{k(μ⃗)}| & in case 1; \\ |pπ^{k(μ⃗)}| & in cases 1′′ and 3; \\ |p^{2}π^{k(μ⃗)}| & in cases 0, 1′, 2, 2′, 2′′, 4. \end{matrix}

We know from [LDE, Lemma 24.4], that γ_{2,1}(a⃗, λ)λ^{−μ_{c}−1} is analytic in Hμ⃗ × S. Lemma 3.4.1 shows that that that analytic function is divisible by b_{2} (resp. b_{3}) if μ_{2} < 0 (resp. μ_{3} < 0). We then put

(3.21)  γ_{2,1}(a⃗, λ) = λ^{μ_{c}+1}u_{2,1}(a⃗, λ) \{ \begin{matrix}π^{k(μ⃗)} & in cases 0, 1, 1′, 1′′, \\ p^{2}π^{k(μ⃗)}b_{2}b_{3} & in case 2, \\ pπ^{k(μ⃗)}b_{2} & in cases 2′′, 3, \\ pπ^{k(μ⃗)}b_{3} & in cases 2′, 4, \end{matrix}

where u_{2,1}(a⃗, λ) is an analytic element bounded by 1 on Hμ⃗ × S, defined over the field Q_{p}(π). Notice that in cases 0, 1′ (resp. 1′′, 2′, 2′′, 4), u_{2,1}(a⃗, λ) is in fact bounded by p^{−2} (resp. p^{−1}).

Estimates for γ_{1,2} in all cases.

We refer again to formula (3.7), where in any case v⃗ − w_{2} ∈ H^{+}(Z), so that only terms A_{μ⃗+pv⃗}(λ)Λ_{2,v⃗}(b⃗, λ^{p}) with k(v⃗) ≥ 1 appear. Therefore we obtain

(3.22)  ||γ_{1,2}(a⃗, λ)||μ⃗ ≤ |pπ^{k(μ⃗)}| .

The conclusion is that

(3.23)  γ_{1,2}(a⃗, λ) = pπ^{k(μ⃗)}u_{1,2}(a⃗, λ)

where u_{1,2}(a⃗, λ) is an analytic element bounded by 1 on Hμ⃗ × S, defined over the field Q_{p}(π).

Calculation of γ_{2,2}^{0} in cases 0, 2, 2′ and 2′′.

We take

(3.24)  γ_{2,2}^{0}(a⃗, λ) = A_{μ⃗−w_{2}+pw_{2}}(a⃗, λ) .

We check from (3.8) that

(3.25)  ||γ_{2,2}(a⃗, λ) − γ_{2,2}^{0}(a⃗, λ)||μ⃗ ≤ |p^{2}π^{k(μ⃗)}| .

We explicitly calculate

(3.26)  \begin{matrix}γ_{2,2}^{0}(a, λ) = A_{μ⃗−w_{2}+pw_{2}}(λ) = \\ = \frac{π^{μ_{1}+μ_{2}+p−1}}{μ_{1}!(μ_{2} + p − 1)!} ∑_{t=0}^{min(μ_{1},μ_{4})} \binom{μ_{1}}{t}\binom{μ_{2} + p − 1}{μ_{4} − t}λ^{t} = \\ = ((−1)^{μ_{4}}γ_{p}(−μ_{1}, 0)γ_{p}(−μ_{2} − p, −1)) \\ (−1)^{μ_{b}}(μ_{b} − μ_{c} + p) ∑_{s+t=μ_{b}} \binom{μ_{b} − μ_{c} + p − 1}{s}\binom{μ_{a}}{t}λ^{t} . \end{matrix}

We put

P μ⃗ ; 2,2(λ) = ∑_{s+t=μ_{b}} \binom{μ_{b} − μ_{c} + p − 1}{s}\binom{μ_{a}}{t}λ^{t} ,

and define again, in cases 2, 2′ and 2′′, the Hasse domain Sμ⃗ of type μ⃗ to be the region

(3.27)  Sμ⃗ = {λ ∈ S | |P μ⃗ ; 2,2(λ)| = 1} .

We recall that the notation in case 0 conflicts with the one of (3.14): see the convention we agreed after that formula. Once again D ⊂ Sμ⃗ and Sμ⃗ is stable under the map λ ⟼ λ^{p}. The conclusion is that

(3.28)  γ_{2,2}(a⃗, λ) = pπ^{k(μ⃗)}u_{2,2}(a⃗, λ)

where u_{2,2}(a⃗, λ) represents an analytic element bounded by 1 on Hμ⃗ ×S, defined over the field Q_{p}(π), which is of constant absolute value 1 on Hμ⃗ × Sμ⃗.

Estimates for γ_{2,2} in cases 1, 1′, 1′′, 3, 4. We refer again to formula (3.7), where v⃗ − w_{2} ∈ H^{+}(Z). A calculation similar to the one of (3.23) shows that

(3.29)  γ_{2,2}(a⃗, λ) = pπ^{k(μ⃗)}u_{2,2}(a⃗, λ) ,

where u_{2,2}(a, λ) represents an analytic element bounded by 1 on Hμ⃗ ×S, defined over the field Q_{p}(π).

We summarize the results of this section. Theorem 3.30. Let p ≥ 7 and 0 ≤ ϵ ≤ p^{−1}(1 − \frac{1}{p} − \frac{1}{p−1}). Let us fix μ⃗ ∈ S. The matrix γ_{g}(a⃗, \frac{a⃗+μ⃗}{p}; λ), a priori defined by (2.6.1) when (3.1) is verified and (a⃗, λ) ∈ Hμ⃗ × S_{ϵ}, extends to a matrix of analytic functions on Hμ⃗ × S. We still denote this extension by

γ_{g}(a⃗, \frac{a⃗ + μ⃗}{p}; λ) = ( \begin{matrix}γ_{11} & γ_{12} \\ γ_{21} & γ_{22} \end{matrix} )

We have, depending on the case of μ⃗:

Cases 0, 1, 1′, 1′′.

γ_{g}(a⃗, \frac{a⃗ + μ⃗}{p}; λ) = π^{k(μ⃗)} ( \begin{matrix}u_{1,1} & u_{1,2} \\ qλu_{2,1} & u_{2,2} \end{matrix} ) ( \begin{matrix}λ^{μ_{c}} & 0 \\ 0 & p \end{matrix} )

Case 2.

γ_{g}(a⃗, \frac{a⃗ + μ⃗}{p}; λ) = pπ^{k(μ⃗)} ( \begin{matrix}u_{1,1} & u_{1,2} \\ qλu_{2,1} & u_{2,2} \end{matrix} ) ( \begin{matrix}pλ^{μ_{c}} & 0 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}b_{2}b_{3} & 0 \\ 0 & 1 \end{matrix} )

Cases 2′′, 3.

γ_{g}(a⃗, \frac{a⃗ + μ⃗}{p}; λ) = pπ^{k(μ⃗)} ( \begin{matrix}u_{1,1} & u_{1,2} \\ qλu_{2,1} & u_{2,2} \end{matrix} ) ( \begin{matrix}λ^{μ_{c}} & 0 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}b_{2} & 0 \\ 0 & 1 \end{matrix} )

Cases 2′, 4.

γ_{g}(a⃗, \frac{a⃗ + μ⃗}{p}; λ) = pπ^{k(μ⃗)} ( \begin{matrix}u_{1,1} & u_{1,2} \\ qλu_{2,1} & u_{2,2} \end{matrix} ) ( \begin{matrix}λ^{μ_{c}} & 0 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}b_{3} & 0 \\ 0 & 1 \end{matrix} )

where q is given by the following table

\begin{matrix}Case & 0 & 1 & 1′ & 1′′ & 2 & 2′ & 2′′ & 3 & 4 \\ q & p^{2}b_{2}b_{3} & 1 & pb_{3} & pb_{2} & 1 & pb_{2} & pb_{3} & 1 & 1 \end{matrix}

Table 3.30.1

In all cases the functions u_{i,j}, for i, j = 1, 2, are analytic elements on Hμ⃗×S, defined over Q_{p}(π) and bounded by 1, and

(3.30.2)  u(a⃗, λ) = det ( \begin{matrix}u_{1,1} & u_{1,2} \\ qλu_{2,1} & u_{2,2} \end{matrix} )

is an analytic element of constant absolute value 1 on Hμ⃗ × S. In cases 0, 1, 1′, 1′′ (resp. in cases 0, 2, 2′, 2′′) u_{1,1} (resp. u_{2,2}) has constant absolute value 1 on Hμ⃗ × Sμ⃗, where Sμ⃗ denotes the Hasse domain of type μ⃗. (We recall the convention following (3.14) for case 0).

Proof. Only the assertion concerning u(a⃗, λ) needs to be proved. We have

(3.31)  det γ_{g}(a⃗, b⃗, λ) = π^{2k(μ⃗)}λ^{μ_{c}}u(a⃗, λ) \{ \begin{matrix}p & in cases 0, 1, 1′, 1′′ \\ p^{3}b_{2}b_{3} & in case 2 \\ p^{2}b_{2} & in cases 2′′, 3 \\ p^{2}b_{3} & in cases 2′, 4 \end{matrix}

where

(3.32)  u(a⃗, λ) = det ( \begin{matrix}u_{1,1} & u_{1,2} \\ qλu_{2,1} & u_{2,2} \end{matrix} )

represents an analytic element defined over the field Q_{p}(π) on Hμ⃗ × S. The determinant formula [GHF, 13.6] and the modular properties of the p-adic gamma function (1.17) show that u(a⃗, λ) has absolute value 1 on Hμ⃗ × S. Q.E.D.

Remark 3.32.1. The constant terms of the u_{i,j}’s, i.e. the values u_{i,j}(0), are all units except possibly for u_{2,1}(0) when μ_{c} = p − 1 and u_{1,2}(0) when μ_{c} = 0. This may be most easily deduced from formulas (3.37.5) and (3.37.7) below, taking into account (3.37.10). Professor Dwork has informed us that in terms of the logarithmic derivative of the Morita p-adic gamma function G = \frac{Γ_{p}′}{Γ_{p}} one has the following relevant formulas. For μ_{c} = p − 1

(3.32.1.1)  u_{2,1}(0) ∼ [G(0) − G(a)] + [G(0) − G(b)] + p(1 − c) ;

for μ_{c} = 0

(3.32.1.2)  u_{1,2}(0) ∼ [G(0) − G(a)] + [G(0) − G(b)] + pc′ .

This is also a consequence of formulas (3.37.6) and (3.37.8) below, but requires some analysis of the Taylor expansion of Γ_{p}(z).

We now rephrase Lemma 24.4 of [LDE]. We first introduce the functions

of (a, b, c; A, B, C), defined for ν⃗ = (μ_{a,A}, μ_{b,B}, μ_{c,C}) ∈ Z^{3}, (a, b, c) ∈ H′ ; ν⃗, and

(A, B, C) = p^{−1}((a, b, c) + ν⃗):

(3.33)  ξ_{1}^{(0)}(a, b, c; A, B, C) = \frac{γ_{p}(b, B)γ_{p}(c − b, C − B)}{γ_{p}(1 + c, 1 + C)} ,

(3.34) ξ_{2}^{(0)}(a, b, c; A, B, C) = (−1)^{μ_{b,B}−μ_{c,C}} \frac{γ_{p}(c − 1, C − 1)γ_{p}(1 − a, 1 − A)}{γ_{p}(1 + c − a, 1 + C − A)} .

Under the same assumptions, formula (2.20) defines the matrix B(a, b, c; A, B, C; λ), provided λ ∈ S \ {0} and condition (2.18.2) is satisfied. If ν⃗ = (μ_{a}, μ_{b}, μ_{c}) ∈ S′ = {0, …, p − 1}^{3}, we set in particular

(A, B, C) = (a′, b′, c′) ,

(3.35)  ξ_{i}^{(0)}(a, b, c; a′, b′, c′) = ξ_{i}^{(0)}(a, b, c) , for i = 1, 2 ,

and

(3.36)  B(a, b, c; a′, b′, c′; λ) = B(a, b, c; λ) .

Corollary 3.37. Let p ≥ 7, μ⃗ ∈ S and ν⃗ = T_{I}(μ⃗) = (μ_{a}, μ_{b}, μ_{c}) ∈ S′. The matrix B(a, b, c; λ), defined by (2.20) for (a, b, c; λ) ∈ H′ ; ν⃗ × (S \ {0}) if condition (2.18.2) holds, is related to γ(a⃗, b⃗; λ) by (2.16.1), (2.16.2), (2.16.3) and (2.21), where b⃗ = \frac{a⃗+μ⃗}{p}. The matrix B(a, b, c; λ) extends to a matrix of analytic functions on the region H′ ; ν⃗ × S. We have, depending on the case of ν⃗ (i.e. of μ⃗):

Cases 1, 1′.

B(a, b, c; λ) = ( \begin{matrix}1 & 0 \\ 0 & p(c′ − b′) \end{matrix} ) ( \begin{matrix}λ^{μ_{c}} & 0 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}β_{1} & qλβ_{2} \\ q′β_{3} & β_{4} \end{matrix} )

Case 2, 2′.

B(a, b, c; λ) = ( \begin{matrix}p(a′ − c′) & 0 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}λ^{μ_{c}} & 0 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}β_{1} & qλβ_{2} \\ q′β_{3} & β_{4} \end{matrix} )

Case 0, 1′′, 2′′, 3.

B(a, b, c; λ) = ( \begin{matrix}λ^{μ_{c}} & 0 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}β_{1} & qλβ_{2} \\ q′β_{3} & β_{4} \end{matrix} )

Case 4.

B(a, b, c; λ) = ( \begin{matrix}p(a′ − c′) & 0 \\ 0 & p(c′ − b′) \end{matrix} ) ( \begin{matrix}λ^{μ_{c}} & 0 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}β_{1} & λβ_{2} \\ β_{3} & β_{4} \end{matrix} )

where q, q′ are given by the following table

\begin{matrix}Case & 0 & 1 & 1′ & 1′′ & 2 & 2′ & 2′′ & 3 & 4 \\ q & p(a′ − c′) & 1 & p(a′ − c′) & 1 & 1 & 1 & p(a′ − c′) & 1 & 1 \\ q′ & p(b′ − c′) & 1 & 1 & p(b′ − c′) & 1 & 1 & 1 & 1 & 1 \end{matrix}

Table 3.37.1

In all cases the functions β_{i}, for i = 1, 2, 3, 4 are analytic elements defined over Q_{p}(π) and bounded by 1 on Hν⃗ × S. The analytic element on H′ ; ν⃗ × S

(3.37.2)  β(a, b, c; λ) = det ( \begin{matrix}β_{1} & qλβ_{2} \\ q′β_{3} & β_{4} \end{matrix} )

has constant absolute value 1. In cases 0, 1, 1′, 1′′ (resp. 0, 2, 2′, 2′′) β_{1} (resp. β_{4}) has constant absolute value 1 on H′ ; ν⃗ × Sν⃗ where Sν⃗ = Sμ⃗ denotes the Hasse domain of type μ⃗. (We recall the convention following (3.14) for case 0). At λ = 0, we have, as meromorphic functions on H′ ; ν⃗:

(3.37.3)  (λ^{−μ_{c}}B_{1}(a, b, c; λ))_{λ=0} = \frac{1 − c}{1 − c′} ξ_{2}^{(0)}(a, b, c) ,

(3.37.4)  B_{4}(a, b, c; 0) = \frac{c}{c′} ξ_{1}^{(0)}(a, b, c) .

If μ_{c} < p − 1, we have

(3.37.5)  (λ^{−μ_{c}−1}B_{2}(a, b, c; λ))_{λ=0} = \frac{c − a}{1 − c′} ξ_{2}^{(0)}(a, b, c) ,

a p-adic unit. If μ_{c} = p − 1, we have

(3.37.6) (λ^{−p}B_{2}(a, b, c; λ))_{λ=0} = \frac{c − a}{1 − c′} ξ_{2}^{(0)}(a, b, c) − \frac{c}{c′} \frac{c′ − a′}{1 − c′} ξ_{1}^{(0)}(a, b, c) ,

If μ_{c} > 0, we have

(3.37.7)  B_{3}(a, b, c; 0) = \frac{c − b}{c′}ξ_{1}^{(0)}(a, b, c) ,

a p-adic unit. If μ_{c} = 0, we have

(3.37.8)  B_{3}(a, b, c; 0) = \frac{c − b}{c′}ξ_{1}^{(0)}(a, b, c) − \frac{1 − c}{c′} \frac{c′ − b′}{1 − c′} ξ_{2}^{(0)}(a, b, c) .

N.B. 3.37.9. One may easily verify by direct calculation that the functions in (3.37.3,4,5,6,7,8) are analytic in H′ ; ν⃗, under the indicated assumptions.

Proof. We rewrite formula 2.21:

(3.37.10)  \begin{matrix}( \begin{matrix}B_{1} & B_{2} \\ B_{3} & B_{4} \end{matrix} ) = (−1)^{μ_{4}}γ_{p}(a_{1}, b_{1})^{−1}γ_{p}(a_{2}, b_{2})^{−1} \\ ( \begin{matrix}1 & 0 \\ 0 & −b_{2} \end{matrix} ) ( \begin{matrix}γ_{11} & γ_{21} \\ γ_{12} & γ_{22} \end{matrix} ) ( \begin{matrix}1 & 0 \\ 0 & −\frac{1}{a_{2}} \end{matrix} ) . \end{matrix}

We recall that a_{1} = a, a_{2} = b − c, a_{3} = a − c, a_{4} = b. The result, except for formulas (3.37.4) and (3.37.6), follows from simultaneous consideration of the theorem and [LDE, Lemma 24.4 and Chapter 25]. The statement about B_{2}(a, b, c; λ)) (resp. B_{2}(a, b, c; λ)) ) in [LDE, Lemma 24.4] is in error (resp. is incomplete) when μ_{c} = p − 1 (resp. μ_{c} = 0). The correct statement is (3.37.6) (resp. (3.37.8)). To prove (3.37.6) one follows the proof of [LDE, Lemma 24.4], but needs to calculate (in the notation of loc. cit.

V (λ^{p})^{−1} ≡ −\frac{1}{c′(1 − c′)} ( \begin{matrix}0 & −c′ \\ −(1 − c′) & c′ − b′ \end{matrix} ) + λ^{p} ( \begin{matrix}\frac{c′−a′}{c′(c′−1)} & ∗ \\ ∗ & ∗ \end{matrix} ) mod λ^{2p} ,

where ∗ denotes an unspecified constant. The conclusion follows as in loc. cit.. As for (3.37.8), one simply needs to carry on one step the calculation in loc. cit.. Q.E.D.

N.B. 3.37.11. Our corollary extends the results of [Dw1, §4] into the singular disk at λ = 0.

§4 Bounded solutions (p ≥ 7).

For a = a^{(0)} ∈ Z_{p}, we define a′ = a^{(1)} ∈ Z_{p} and μ_{a} ∈ {0, …, p − 1} by the condition that μ_{a} = pa′ − a. For j = 2, 3, …, we then put a^{(j)} = (a^{(j−1)})′. Therefore

(4.1)  −a = μ_{a} + μ_{a′}p + … + μ_{a^{(j)}}p^{j} + …,

is the Hensel expansion of −a ∈ Z_{p}. We now consider the subset T_{1} (resp. T_{2}) of Z_{p}^{3} consisting of triples (a, b, c) such that for all j = 0, 1, 2, …, (μ_{a^{(j)}}, μ_{b^{(j)}}, μ_{c^{(j)}}) is in case 1 or 1′ (resp. in case 2 or 2′) of the previous section. We define the strict Hasse domain of case i, for i = 1, 2, as

(4.2.1)  S_{i}^{H} = ∩Sμ⃗ ,

where the intersection is taken for all μ⃗ in case i or i′. Notice that, for i = 1, 2, S_{i}^{H} is the complement in S of a finite number of open disks of radius 1, and D ⊂ S_{i}^{H}. We also define for i = 1, 2,

(4.2.2)  H′_{i} = ∪H′ ; ν⃗ ,

where the union is taken for all ν⃗ in case i.

Definition 4.3.0. We say that c ∈ Z_{p} is strongly of type 1 or anti-Liouville if μ_{c^{(j)}} ≥ 1 for all j ∈ N.

An anti-Liouville number is of type 1 in the sense of [B-C]. If c is anti-Liouville, then c is a limit of rational, non-integral, anti-Liouville numbers. In fact, if −c = ∑_{i=0}^{i=∞} μ_{i}p^{i}, let −γ_{s} = ∑_{i=0}^{s−1} μ_{i}p^{i} +p^{s}(1+2p+p^{2} +2p^{3} +…), which is non-integral since \frac{1+2p}{p^{2}−1} ∉ Z; then lim_{s} γ_{s} = c. Clearly, 1 − c is anti-Liouville iff μ_{c^{(j)}} ≤ p−2 for all j ∈ N. No element of Z_{≤0} is anti-Liouville. If (a, b, c) ∈ T_{1} (resp. T_{2}) then 1 − c (resp. c) is anti-Liouville. In particular, if (a, b, c) ∈ T_{1} (resp. T_{2}), then c ∉ Z_{>0} (resp. c ∉ Z_{≤0}). If 1 − c (resp. c) is anti-Liouville, so in particular if (a, b, c) ∈ T_{1} (resp. T_{2}), then

(4.3.1)  |c − s| > \frac{|p|}{s} , for s ∈ Z_{>0}

(resp.

(4.3.2)  |c + s| > \frac{|p|}{s} , for s ∈ Z_{>0} ) .

Notice that the transformation

(4.4.1)  (a, b, c) ⟼ (1 − b, 1 − a, 1 − c)

induces a bijection

(4.4.2)  T_{1} ⟶ T_{2} .

Let K denote any finite extension of Q_{p}(π), and let σ be a Frobenius automorphism of K, such that σ(π) = π. Let T denote any admissible open subset of the K-rigid analytic projective line X. We may interpret T as an admissible open subset “defined over K” of the extension of X to C_{p}.

Definition 4.5. Let T be as before and T be a subset of (Z_{p})^{3}. We define B_{K}(T × T) to be the K-Banach algebra of continuous bounded C_{p}-valued functions f = f(a, b, c; λ) on T × T, such that for each choice of (a, b, c) in T , λ ⟼ f(a, b, c; λ) is analytic function on T, defined over K. We will denote by || − ||_{T ×T} or simply by || − || the norm of B_{K}(T × T), i.e. the supnorm on T × T and by B_{K}^{(1)}(T × T) the unit ball of B_{K}(T × T).

If 0 ∈ T and T is bounded, the principal ideal λB_{K}(T ×T) (resp. λB_{K}^{(1)}(T × T)) generated by the bounded analytic function (projection onto the last factor) λ : (a, b, c; λ) ⟼ λ is closed and is the kernel of the map:

(4.5.1)  \begin{matrix}B_{K}(T × T) & ⟶ C_{K}(T ) \\ f(a, b, c; λ) & ⟼ f(a, b, c; 0) \end{matrix}

(resp.

(4.5.2)  \begin{matrix}B_{K}^{(1)}(T × T) & ⟶ C_{K}^{(1)}(T ) \\ f(a, b, c; λ) & ⟼ f(a, b, c; 0) ) , \end{matrix}

where C_{K}(T ) (resp. C_{K}^{(1)}(T )) denotes the K-Banach algebra of continuous Kvalued functions on T i (resp. its unit ball). Let us assume that T^{σ} = T and that T is stable under the map λ ⟼ λ^{p}. For i = 1, 2, we consider the map:

(4.6)  \begin{matrix}ϕ : T_{i} × T & ⟶ T_{i} × T \\ (a, b, c, λ) & ⟼ (a′, b′, c′, λ^{p}) . \end{matrix}

For a C_{p}-valued function w ∈ B_{K}(T_{i} × T) we then denote by w^{σϕ} (resp. w^{σ}) the composition \overline{σ} ∘ w ∘ ϕ ∘ \overline{σ}^{−1} (resp. \overline{σ} ∘ w ∘ \overline{σ}^{−1}), where \overline{σ} denotes any isometric extension of σ to C_{p}. Then w ⟼ w^{σϕ} (resp. w ⟼ w^{σ}) induces an isometry of B_{K}(T_{i} × T). The previous assumptions are clearly satisfied by T = D and by T = S_{i}^{H}, for i = 1, 2.

We now distinguish the case (a, b, c) ∈ T_{1} from the case (a, b, c) ∈ T_{2}.

Case 1.

There exists a map

(4.7.1)  C_{1} : λB_{K}^{(1)}(T_{1} × D) ⟶ λB_{K}^{(1)}(T_{1} × D)

that induces

(4.7.2)  C_{1} : λB_{K}^{(1)}(T_{1} × S_{1}^{H}) ⟶ λB_{K}^{(1)}(T_{1} × S_{1}^{H})

defined by

(4.7.32)  \begin{aligned}C_{1}(λw) & = \frac{B_{2} + B_{4}λ^{p}w^{σϕ}}{B_{1} + B_{3}λ^{p}w^{σϕ}} = \frac{λ^{μ_{c}+1}β_{2} + p(c′ − b′)β_{4}λ^{p}w^{σϕ}}{λ^{μ_{c}}β_{1} + p(c′ − b′)β_{3}λ^{p}w^{σϕ}} = \\ & = λ \frac{β_{2}β_{1}^{−1} + p(c′ − b′)β_{4}β_{1}^{−1}λ^{p−1−μ_{c}}w^{σϕ}}{1 + p(c′ − b′)β_{3}β_{1}^{−1}λ^{p−μ_{c}}w^{σϕ}} . \end{aligned}

We observe that, for w_{1} , w_{2} ∈ λB_{K}^{(1)}(T_{1} × D) (resp. λB_{K}^{(1)}(T_{1} × S_{1}^{H})),

C_{1}(λw_{1}) − C_{1}(λw_{2}) = \frac{λ^{p} det B (w_{1}^{σϕ} − w_{2}^{σϕ})}{(B_{1} + B_{3}λ^{p}w_{1}^{σϕ})(B_{1} + B_{3}λ^{p}w_{2}^{σϕ})} =

(4.7.4)  = \frac{p(c′ − b′)λ^{p+μ_{c}}β(a, b, c; λ)(w_{1}^{σϕ} − w_{2}^{σϕ})}{(λ^{μ_{c}}β_{1} + p(c′ − b′)β_{3}λ^{p}w_{1}^{σϕ})(λ^{μ_{c}}β_{1} + p(c′ − b′)β_{3}λ^{p}w_{2}^{σϕ})} =

= p(c′ − b′)λ^{p−μ_{c}}β_{1}^{−2}β(a, b, c; λ)\frac{(w_{1}^{σϕ} − w_{2}^{σϕ})}{(1 + p(c′ − b′)β_{3}β_{1}^{−1}λ^{p−μ_{c}}w_{1}^{σϕ})(1 + p(c′ − b′)β_{3}β_{1}^{−1}λ^{p−μ_{c}}w_{2}^{σϕ})}

We conclude that

(4.8)  \begin{aligned}||\frac{C_{1}(λw_{1})}{λ} − \frac{C_{1}(λw_{2})}{λ}|| & ≤ ||p(c′ − b′)λ^{p−μ_{c}−1}β_{1}^{−2}β|| ||w_{1} − w_{2}|| ≤ \\ & ≤ p^{−1} ||w_{1} − w_{2}|| . \end{aligned}

The map w ⟼ \frac{C_{1}(λw)}{λ} is then a contraction of the complete metric space B_{K}^{(1)}(T_{1} × D) (resp. B_{K}^{(1)}(T_{1} × S_{1}^{H})) and as such it admits a unique fixed point. The map C_{1} therefore also admits a unique fixed point

(4.9)  η(a, b, c; λ) ∈ λB_{K}^{(1)}(T_{1} × S_{1}^{H}) ⊂ λB_{K}^{(1)}(T_{1} × S_{1}^{H}) .

We then have on T_{1} × S_{1}^{H}

(4.10)  η = \frac{B_{2} + B_{4}η^{σϕ}}{B_{1} + B_{3}η^{σϕ}} .

Case 2.

We now consider the map

(4.11.1)  C_{2} : B_{K}^{(1)}(T_{2} × D) ⟶ B_{K}^{(1)}(T_{2} × D)

that induces

(4.11.2)  C_{2} : B_{K}^{(1)}(T_{2} × S_{2}^{H}) ⟶ B_{K}^{(1)}(T_{2} × S_{2}^{H})

and is defined by

(4.11.3)  \begin{aligned}C_{2}(w) & = \frac{B_{3} + B_{1}w^{σϕ}}{B_{4} + B_{2}w^{σϕ}} = \frac{β_{3} + p(a′ − c′)λ^{μ_{c}}β_{1}w^{σϕ}}{β_{4} + p(a′ − c′)λ^{μ_{c}+1}β_{2}w^{σϕ}} = \\ & = \frac{β_{3}β_{4}^{−1} + p(a′ − c′)λ^{μ_{c}}β_{1}β_{4}^{−1}w^{σϕ}}{1 + p(a′ − c′)λ^{μ_{c}+1}β_{2}β_{4}^{−1}w^{σϕ}} . \end{aligned}

We observe that, for w_{1} , w_{2} ∈ B_{K}^{(1)}(T_{2} × D),

C_{2}(w_{1}) − C_{2}(w_{2}) = \frac{det B (w_{1}^{σϕ} − w_{2}^{σϕ})}{(B_{4} + B_{2}w_{1}^{σϕ})(B_{4} + B_{2}w_{2}^{σϕ})} =

(4.12)  = \frac{p(a′ − c′)λ^{μ_{c}}β(a, b, c; λ)(w_{1}^{σϕ} − w_{2}^{σϕ})}{(β_{4} + p(a′ − c′)λ^{μ_{c}+1}β_{2}w_{1}^{σϕ})(β_{4} + p(a′ − c′)λ^{μ_{c}+1}β_{2}w_{2}^{σϕ})} =

= p(a′−c′)λ^{μ_{c}}β_{4}^{−2}β(a, b, c; λ)\frac{(w_{1}^{σϕ} − w_{2}^{σϕ})}{(1 + p(a′ − c′)λ^{μ_{c}+1}β_{2}β_{4}^{−1}w_{1}^{σϕ})(1 + p(a′ − c′)λ^{μ_{c}+1}β_{2}β_{4}^{−1}w_{2}^{σϕ})}

We conclude that

(4.13)  \begin{aligned}||C_{2}(w_{1}) − C_{2}(w_{2})|| & ≤ ||p(a′ − c′)λ^{μ_{c}}β_{4}^{−2}β|| ||w_{1} − w_{2}|| ≤ \\ & ≤ p^{−1} ||w_{1} − w_{2}|| . \end{aligned}

The map C_{2} is then a contraction of the complete metric space B_{K}^{(1)}(T_{2} × D) (resp. B_{K}^{(1)}(T_{2} × S_{2}^{H})) and as such it admits a unique fixed point

(4.14)  η(a, b, c; λ) ∈ B_{K}^{(1)}(T_{2} × S_{2}^{H}) .

We then have on T_{2} × S_{2}^{H}

(4.15)  η = \frac{B_{3} + B_{1}η^{σϕ}}{B_{4} + B_{2}η^{σϕ}} .

We define, for any (a, b, c) ∈ (C_{p})^{3} for which it makes sense, the standard Gauss hypergeometric series

(4.16.1)  F (a, b, c; λ) = ∑_{s=0}^{+∞} \frac{(a)_{s}(b)_{s}}{(c)_{s}s!} λ^{s} ∈ C_{p}[[λ]] ,

where, as usual, (a)_{s} = a(a + 1) … (a + s − 1), and a matrix of power series in C_{p}[[λ]]

U^{(0)}(a, b, c; λ) = ( \begin{matrix}u_{1}^{(0)} & u_{2}^{(0)} \\ u_{3}^{(0)} & u_{4}^{(0)} \end{matrix} ) =

(4.16.2)

= ( \begin{matrix}(c − b)F (a, b, c + 1; λ) & cF (a, b, c; λ) \\ (1 − c)F (b − c, a − c, 1 − c; λ) & (c − a)λF (1 + b − c, 1 + a − c, 2 − c; λ) \end{matrix} ) .

We observe that the first (resp. second) line of U^{(0)}(a, b, c; λ) is formally defined when c ∉ Z_{≤0} (resp. c ∉ Z_{>0}). We recall that, whenever defined, a line of the matrix

(4.17)  ( \begin{matrix}1 & 0 \\ 0 & λ^{−c} \end{matrix} ) U^{(0)}(a, b, c; λ)

formally represents a solution at 0 of the system

(4.18)  \frac{d}{dλ}Y = Y ( −\frac{c}{λ}  \frac{c−a}{1−λ}_{\frac{c−b}{λ} \frac{a+b−c}{1−λ}} ) .

In particular, if (a, b, c) ∈ T_{2}, (u_{1}^{(0)}, u_{2}^{(0)}) is a power-series solution of the system (4.18), while if (a, b, c) ∈ T_{1}, (u_{3}^{(0)}, u_{4}^{(0)}) is a power-series solution of

(4.19)  \frac{d}{dλ}Z = Z ( \begin{matrix}0 & \frac{c−a}{1−λ} + \frac{c}{λ} \\ \frac{2c−b}{λ} & \frac{a+b−c}{1−λ} + \frac{c}{λ} \end{matrix} ) .

Let us recall the formula [LDE, 21.2.4] (but notice an error in that reference: the factor (a − 1) is there missing)

(4.20.1)  \frac{s!}{(a)_{s}} = (a − 1) ∑_{i+j=s}(−1)^{j}\binom{s}{j}\frac{1}{a + j − 1} .

Together with the bound (4.3.1) (resp. (4.3.2)) it guarantees that

(4.20.2)  max(|\frac{(b − c)_{s}(a − c)_{s}}{s!(1 − c)_{s}} |, |\frac{(1 + b − c)_{s}(1 + a − c)_{s}}{s!(2 − c)_{s}} |) < ps

if 1 − c is anti-Liouville and in particular for (a, b, c) ∈ T_{1} (resp.

(4.20.3)  min(|\frac{(a)_{s}(b)_{s}}{s!(c)_{s}} |, |\frac{(a)_{s}(b)_{s}}{s!(c + 1)_{s}})| < ps

if c is anti-Liouville and in particular for (a, b, c) ∈ T_{2}). We conclude that u_{3}^{(0)}, u_{4}^{(0)} (resp. u_{1}^{(0)}, u_{2}^{(0)}) are elements of B_{K}^{(1)}(T_{1}×D(0, (p^{−1})^{+})) (resp. B_{K}^{(1)}(T_{2}× D(0, (p^{−1})^{+}))).

We point out (cf. (3.33,34)) that ξ_{1}^{(0)}(a, b, c) (resp. ξ_{2}^{(0)}(a, b, c)) is an analytic function defined over K of constant absolute value 1 on H′_{2} (resp. H′_{1}), hence an invertible element of B_{K}^{(1)}(T_{2} × D(0, 1^{+})) (resp. B_{K}^{(1)}(T_{1} × D(0, 1^{+}))).

Lemma 4.21. We have the identities

(4.21.1) (u_{3}^{(0)}, u_{4}^{(0)})^{σϕ}B = ξ_{2}^{(0)}λ^{μ_{c}}(u_{3}^{(0)}, u_{4}^{(0)}) in B_{K}^{(1)}(T_{1} × D(0, (p^{−1})^{+}))^{2} ,

(4.21.2)  (u_{1}^{(0)}, u_{2}^{(0)})^{σϕ}B = ξ_{1}^{(0)}(u_{1}^{(0)}, u_{2}^{(0)}) in B_{K}^{(1)}(T_{2} × D(0, (p^{−1})^{+}))^{2} .

Proof. We restrict our attention to (4.21.2). Let (α, β, γ) lie in (Z_{p} ∩ Q)^{3} with γ non-integral. Then γ′ is also non-integral and by Lemma 24.3 and Chapter 25 of [LDE], (4.21.3)

\begin{matrix}(u_{1}^{(0)}, u_{2}^{(0)})(α′, β′, γ′; λ^{p}) & B(α, β, γ; α′, β′, γ′; λ) = \\ = ξ_{1}^{(0)}(α, β, γ; α′, β′, γ′)(u_{1}^{(0)}, u_{2}^{(0)})(α, β, γ; λ) , \end{matrix}

valid for all λ ∈ D, so in particular for λ ∈ D(0, (p^{−1})^{+})).

Now let (a, b, c) ∈ Z_{p}^{3}, with c anti-Liouville. There exists a sequence of elements (α, β, γ) ∈ (Z_{p} ∩ Q)^{3} with γ non-integral and anti-Liouville such that (α, β, γ) ⟶ (a, b, c), and hence (α′, β′, γ′) ⟶ (a′, b′, c′). By taking limits we obtain

(4.21.4) \begin{matrix}(u_{1}^{(0)}, u_{2}^{(0)})(a′, b′, c′; λ^{p}) & B(a, b, c; a′, b′, c′; λ) = \\ = ξ_{1}^{(0)}(a, b, c; a′, b′, c′)(u_{1}^{(0)}, u_{2}^{(0)})(a, b, c; λ) , \end{matrix}

valid for λ ∈ D(0, (p^{−1})^{+})), the point being that the series (u_{1}^{(0)}, u_{2}^{(0)})(α, β, γ; λ) converges termwise to (u_{1}^{(0)}, u_{2}^{(0)})(a, b, c; λ) and since γ is anti-Liouville, we may use the universal estimate (4.20.3) to conclude that

(4.21.5)  \begin{matrix}(u_{1}^{(0)}, u_{2}^{(0)})(α, β, γ; λ) & ⟶ (u_{1}^{(0)}, u_{2}^{(0)})(a, b, c; λ^{p}) \\ (u_{1}^{(0)}, u_{2}^{(0)})(α′, β′, γ′; λ) & ⟶ (u_{1}^{(0)}, u_{2}^{(0)})(a′, b′, c′; λ^{p}) , \end{matrix}

uniformly on D(0, ρ^{+}), for each ρ < 1. In particular this is valid for ρ = \frac{1}{p}, and here the series are bounded by 1. The proof of (4.21.1) is the same except that now 1 − γ and 1 − c are anti-Liouville. Q.E.D.

Corollary 4.22. We have the identities

(4.22.1)  (u_{3}^{(0)}, u_{4}^{(0)})^{σϕ}B = ξ_{2}^{(0)}λ^{μ_{c}}(u_{3}^{(0)}, u_{4}^{(0)}) in B_{K}^{(1)}(T_{1} × D)^{2} ,

(4.22.2)  (u_{1}^{(0)}, u_{2}^{(0)})^{σϕ}B = ξ_{1}^{(0)}(u_{1}^{(0)}, u_{2}^{(0)}) in B_{K}^{(1)}(T_{2} × D)^{2} .

Proof. To prove (4.22.2), we notice that the coefficients of B and the functions ξ_{1}^{(0)}, (ξ_{1}^{(0)})^{−1} are elements of B_{K}^{(1)}(T_{2} × D). Iterating the u ⟼ u^{σϕ} operation s times, we conclude that u_{i}^{(0)} ∈ B_{K}^{(1)}(T_{2} × D(0, (p^{− \frac{1}{s}})^{+})), for i = 1, 2 . The statement follows. To prove (4.22.1) we write B in the form

(4.22.3)  B = ( \begin{matrix}λ^{μ_{c}}Q_{1} & λ^{μ_{c}}Q_{2} \\ pQ_{3} & pQ_{4} \end{matrix} ) ξ_{2}^{(0)} ,

where Q_{1}, Q_{2}, Q_{3}, Q_{4} are all bounded by 1 on T_{1} × D. Hence by (4.21.1) we deduce

(4.22.4)  \begin{matrix}u_{3}^{(0)} & u_{3}^{(0)}^{σϕ}Q_{1} + u_{4}^{(0)}^{σϕ} \frac{p}{λ^{μ_{c}}} Q_{3} \\ u_{4}^{(0)} & u_{3}^{(0)}^{σϕ}Q_{2} + u_{4}^{(0)}^{σϕ} \frac{p}{λ^{μ_{c}}} Q_{4} \end{matrix} .

Using the maximum modulus theorem we conclude that if (u_{3}^{(0)}, u_{4}^{(0)}) is bounded by 1 on T_{1} × D(0, ρ^{+}) and if

(4.22.5)  \frac{|p|}{ρ^{\frac{μc}{p}}} ≤ 1 ,

then (u_{3}^{(0)}, u_{4}^{(0)}) is bounded by 1 on T_{1} ×D(0, ρ^{\frac{1}{p}} ^{+}). Since μ_{c} ≤ p−1, ρ^{μ_{c}} ≥ ρ^{p−1} and so (4.22.5) is certainly valid if ρ ≥ |π|^{p} ( < p^{−1}). This point is not treated properly in [Dw2, §2]. Q.E.D.

The conclusion is (cf. [p-DE IV, Lemma 2.2]):

Corollary 4.23. The functions (c − b)F (a, b, c + 1; λ) and cF (a, b, c; λ) (resp. (1 − c)F (b − c, a − c, 1 − c; λ) and (c − a)λF (1 + b − c, 1 + a − c, 2 − c; λ)) are elements of B_{K}^{(1)}(T_{2} × D) (resp. B_{K}^{(1)}(T_{1} × D)). As a consequence, for (a, b, c) ∈ T_{2} (resp. (a, b, c) ∈ T_{1}) the elements F (a, b, c + 1; λ) and F (a, b, c; λ) (resp. F (b − c, a − c, 1 − c; λ) and F (1 + b − c, 1 + a − c, 2 − c; λ)) of Q_{p}[[λ]], are in fact in 1 + λZ_{p}[[λ]].

Theorem 4.24. For i = 1, 2, the fixed-point η ∈ E_{K}^{(1)}(T_{i}×S_{i}^{H}) of the contraction C_{i}, has the following local expression in T_{i} × D:

(4.24.1)  η(a, b, c; λ) = \{ \begin{matrix}\frac{u_{4}^{(0)}}{u_{3}^{(0)}} = \frac{c−a}{1−c} \frac{λF (1+b−c,1+a−c,2−c;λ)}{F (b−c,a−c,1−c;λ)} & in T_{1} × D, \\ \frac{u_{1}^{(0)}}{u_{2}^{(0)}} = \frac{c−b}{c} \frac{F (a,b,c+1;λ)}{F (a,b,c;λ)} & in T_{2} × D. \end{matrix}

From the identities (4.22.1,2) we deduce

(4.25.1)  (u_{3}^{(0)})^{σϕ}B_{1} + (u_{4}^{(0)})^{σϕ}B_{3} = ξ_{2}^{(0)}λ^{μ_{c}}u_{3}^{(0)} ,

(4.25.2)  (u_{1}^{(0)})^{σϕ}B_{2} + (u_{2}^{(0)})^{σϕ}B_{4} = ξ_{1}^{(0)}u_{2}^{(0)} .
(4.26.1)  B_{1} + \frac{(u_{4}^{(0)})^{σϕ}}{(u_{3}^{(0)})^{σϕ}} B_{3} = ξ_{2}^{(0)}λ^{μ_{c}} \frac{u_{3}^{(0)}}{(u_{3}^{(0)})^{σϕ}} ,

(4.26.2)  \begin{matrix}(u_{1}^{(0)}) & σϕ \\ (u_{2}^{(0)}) & σϕ \end{matrix} B_{2} + B_{4} = ξ_{1}^{(0)} \frac{u_{2}^{(0)}}{(u_{2}^{(0)})^{σϕ}} .

In other words in T_{1} × D we have

(4.27.1)  \begin{matrix}B_{1}(a, b, c; λ)+ & η^{σϕ}(a, b, c; λ)B_{3}(a, b, c; λ) = \\ = & ξ_{2}^{(0)}(a, b, c)\frac{1 − c}{1 − c′} \frac{F (b − c, a − c, 1 − c; λ)}{F (b′ − c′, a′ − c′, 1 − c′; λ^{p})} , \end{matrix}

while in T_{2} × D we have

(4.27.2)  \begin{matrix}η^{σϕ}(a, b, c; λ)B_{2}(a, b, c; λ)+ & B_{4}(a, b, c; λ) = \\ = & ξ_{1}^{(0)}(a, b, c)\frac{c}{c′} λ^{μ_{c}} \frac{F (a, b, c; λ)}{F (a′, b′, c′; λ^{p})} . \end{matrix}

A consequence of the previous formulas is Theorem 4.28. The function locally defined in T_{1} × D (resp. T_{2} × D) by G(a, b, c; λ) = \frac{F (b−c,a−c,1−c;λ)}{F (b′−c′,a′−c′,1−c′;λ^{p})} (resp. F(a, b, c; λ) = \frac{F (a,b,c;λ)}{F (a′,b′,c′;λ^{p})}) extends to an element of E_{K}^{(1)}(T_{1} × S_{1}^{H}) (resp. E_{K}^{(1)}(T_{2} × S_{2}^{H})). We have in T_{1} × S_{1}^{H}

(4.28.1)  \begin{matrix}G(a, b, c; λ) & = (−1)^{μ_{c}−μ_{b}} \frac{γ_{p}(1 + c − a, 1 + c′ − a′)}{γ_{p}(c, c′)γ_{p}(1 − a, 1 − a′)}λ^{−μ_{c}} \\ (B_{1}(a, b, c; λ) + η^{σϕ}(a, b, c; λ)B_{3}(a, b, c; λ)) , \end{matrix}

while in T_{2} × S_{2}^{H} we have

(4.28.2)  \begin{matrix}F(a, b, c; λ) & = \frac{γ_{p}(c, c′)}{γ_{p}(b, b′)γ_{p}(c − b, c′ − b′)} \\ (η^{σϕ}(a, b, c; λ)B_{2}(a, b, c; λ) + B_{4}(a, b, c; λ)) . \end{matrix}

§5 Young’s formula (p ≥ 7).

We plan to put together formulas (2.33) and (4.28.2). This requires a modification of the discussion of the previous section, since as it stands formula (4.28.2) cannot be evaluated at λ = −1, since −1 ∉ S_{2}^{H}. We then define

(5.1.1)

S′_{2}(−1) = {(μ_{a}, μ_{b}, μ_{c}) = T_{I}(μ⃗) ∈ S′ |
μ⃗ is in case 2 and |P μ⃗ ; 2,2(−1)| = 1 } ,

(5.1.2) T_{2}(−1) = {(a, b, c) ∈ T_{2} | (μ_{a^{(j)}}, μ_{b^{(j)}}, μ_{c^{(j)}}) ∈ S′_{2}(−1) , ∀j = 0, 1, …} ,

and

(5.1.3)  S_{2}^{H}(−1) = {λ ∈ S | |P μ⃗ ; 2,2(λ)| = 1 , ∀μ⃗ ∈ T_{I}^{−1}(S′_{2}(−1))} .

The arguments used in section 4 show that

Corollary 5.2. The function locally defined in T_{2}(−1) × D by F(a, b, c; λ) = \frac{F (a,b,c;λ)}{F (a′,b′,c′;λ^{p})}) extends to an element of B_{K}^{(1)}(T_{2}(−1) × S_{2}^{H}(−1)). Formula (4.28.2) holds in T_{2}(−1) × S_{2}^{H}(−1), and in particular at (a, b, c; −1), for any (a, b, c) ∈ T_{2}(−1).

We are now supposed to replace (a, b, c) ∈ T_{2} in formula (4.28.2) by (a, b, 1+ a − b) ∈ T_{2}, assuming μ_{a^{(j)}} even and μ_{a^{(j)}} ≤ μ_{b^{(j)}} for j = 0, 1, . . .. We point out that then automatically (1 + a − b)^{(j)} = 1 + a^{(j)} − b^{(j)}, 2μ_{b^{(j)}} − μ_{a^{(j)}} ≤ p − 1, μ_{b^{(j)}} < p − 1, for j = 0, 1, …, and, by (2.33.4), (a, b, 1 + a − b) ∈ T_{2}(−1). We now evaluate formula (4.28.2) at (a, b, 1 + a − b; −1) ∈ T_{2}(−1) × S_{2}^{H}(−1), as before. We get

Theorem 5.3. Let (a, b) ∈ Z_{p}^{2} be such that μ_{a^{(j)}} ≤ μ_{b^{(j)}} < p−1, 2μ_{b^{(j)}} −μ_{a^{(j)}} ≤ p − 1 and μ_{a^{(j)}} is even, for j = 0, 1, . . .. Then (a, b, 1 + a − b) ∈ T_{2}(−1) × S_{2}^{H}(−1) and

(5.3.1)  F(a, b, 1 + a − b; −1) = (−1)^{\frac{μa}{2}} \frac{γ_{p}(\frac{a}{2}, \frac{a′}{2} ) γ_{p}(b − \frac{a}{2}, b′ − \frac{a′}{2} )}{γ_{p}(a, a′)γ_{p}(b − a, b′ − a′)}

Proof. Direct substitution of formula (2.33.2) in (4.28.2) yields

(5.3.2) \begin{aligned}F(a, b, 1 + a − b; −1) & = (−1)^{μ_{b}− \frac{μa}{2}} \frac{γ_{p}(1 + a − b, 1 + a′ − b′)}{γ_{p}(b, b′)γ_{p}(1 + a − 2b, 1 + a′ − 2b′)} \\ & \frac{γ_{p}(\frac{a}{2}, \frac{a′}{2} ) γ_{p}(b − \frac{a}{2}, b′ − \frac{a′}{2} ) γ_{p}(b, b′)}{γ_{p}(a, a′)γ_{p}(2b − a, 2b′ − a′)} . \end{aligned}

The result follows from the reflection formula for γ_{p} (1.20). Q.E.D.

The previous result, restricted to T_{2} ∩ Q^{3}, is essentially theorem 3.1 of [Y2], and generalizes formula (0.3).

References.

[AD] Adolphson A., Dwork B.: “The non-regular case - Chapter 9 revisited”, preprint Princeton University (1992), 21 pages.

[Ba] Baldassarri F.: “p-adic formulas and unit root F -subcrystals of the hypergeometric system”, article in preparation.

[Bo] Boyarsky M.: “p-adic gamma function and Dwork cohomology”, Trans. Amer. Math. Soc. 257 (1980), 359-369.

[Ch] Christol G.: “Modules Différentiels et Equations Différentielles p-adiques”, Queen’s Papere in Pure and Applied Mathematics 66 (1983), 218 pages.

[D] Diamond J.: “Hypergeometric series with a p-adic variable”, Pacific J. Math. Soc. 94 (1981), 265-276.

[DR] Dwork B., Robba Ph.: “On ordinary linear p-adic differential equations”, Trans. Amer. Math. Soc. 231 (1977), 1-46.

[Dw1] Dwork B.: “On the Boyarsky principle”, Amer. J. of Math. 104 (1983), 115-156.

[Dw2] Dwork B.: “Singular residue classes which are ordinary for F (a, b, c; λ)”, Groupe d’étude d’Analyse ultramétrique (Y.Amice, G.Christol, Ph.Robba), 10ème année, 1982/83, n^{o} 23, 11 pages.

[GHF] Dwork B.: “Generalized Hypergeometric Functions.”, Oxford Mathematical Monographs, Clarendon Press, Oxford (1990), 188 pages.

[GrKo] Gross B., Koblitz N.: “Gauß sums and the p-adic Γ-function”, Annals of Math. 109 (1979), 569-581.

[Ka] Katz N.: “Travaux de Dwork”, Séminaire Bourbaki 24ème année, 1971/72, n^{o} 409, Springer Lecture Notes 317, 34 pages.

[Ko] Koblitz N.: “The hypergeometric function with p-adic parameters”, in Proceedings of the Queen’s (Ontario) Number Theory Conference, 1979”, 319-328.

[Ku] Dwork B.: “On Kummer’s twenty-four solutions of the hypergeometric differential equation”, Trans. Amer. Math. Soc. 285 (1984), 497-521.

[LDE] Dwork B.: “Lectures on p-adic Differential Equations.”, Grundlehren der mathematischen Wissenschaften 253, Springer (1982), 310 pages.

[Mo] Morita Y.: “A p-adic analogue of the Γ-function”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), 255-266.

[p-C] Dwork B.: “p-adic cycles”, Publ. Math. IHES 37 (1969), 327-415.

[p-DE IV] Dwork B.: “On p-adic differential equations IV. Generalized hypergeometric functions as p-adic analytic functions in one variable”, Annales scientifiques de l’École Normale Supérieure, 4^{e} Série, t.6, fasc.3 (1973), 295-315.

[Po] Poole E.G.C.: “Introduction to the Theory of Linear Differential Equations.”, Dover Publications Inc., New York (1960), 199 pages.

[Ro] Robba Ph.: “Une introduction naïve aux cohomologies de Dwork”, Soc. Math. de France, 2^{e} Série, Mémoire n. 23 (1986), 61-105.

[Sch] Scholl A.J.: “A trace formula for F -crystals”, Inv. Math. 79 (1985), 31-48.

[Sel] Baldassarri F.:“ p-adic interpolation of Evans sums and deformation of the Selberg integrals: an application of Dwork’s theory”, in Proceedings of the Taniguchi Symposium on Special Differential Equations. Kyoto, September 2-5, 1991, RIMS Publications.

[Y1] Young P.Th.: “Apéry numbers, Jacobi sums, and special values of generalized p-adic hypergeometric functions”, J. of Number Theory, 41 (1992), 231-255.

[Y2] Young P.Th.: “On Jacobi sums, multinomial coefficients, and p-adic hypergeometric functions”, J. of Number Theory, 52 (1995), 125-144.

Author’s address: Dipartimento di Matematica Università di Padova Via Belzoni 7 I-35131 PADOVA - ITALY E-mail: BALDASSARRI@PDMAT1.MATH.UNIPD.IT Fax: (39-49-)8758596