MULTIVARIATE p-ADIC FORMAL CONGRUENCES AND INTEGRALITY OF TAYLOR COEFFICIENTS OF MIRROR MAPS

C. KRATTENTHALER^{†} AND T. RIVOAL

Abstract. We generalise Dwork’s theory of p-adic formal congruences from the univariate to a multivariate setting. We apply our results to prove integrality assertions on the Taylor coefficients of (multivariable) mirror maps. More precisely, with z = (z_{1}, z_{2}, …, z_{d}), we show that the Taylor coefficients of the multivariable series q(z) = z_{i} exp(G(z)/F (z)) are integers, where F (z) and G(z) + log(z_{i})F (z), i = 1, 2, …, d, are specific solutions of certain GKZ systems. This result implies the integrality of the Taylor coefficients of numerous families of multivariable mirror maps of Calabi–Yau complete intersections in weighted projective spaces, as well as of many one-variable mirror maps in the “Tables of Calabi–Yau equations” [arχiv:math/0507430] of Almkvist, van Enckevort, van Straten and Zudilin. In particular, our results prove a conjecture of Batyrev and van Straten in [Comm. Math. Phys. 168 (1995), 493–533] on the integrality of the Taylor coefficients of canonical coordinates for a large family of such coordinates in several variables.

1. Introduction and statement of the results

In [7, 8, 9, 10, 11], Dwork developed a sophisticated theory for proving analytic and arithmetic properties of solutions to (p-adic) differential equations. In [7, 11], he focussed on the case of hypergeometric differential equations. In particular, the article [11] contains a “formal congruence” criterion that enabled him to address the analytic continuation of quotients of certain solutions and to establish arithmetic properties satisfied by exponentials of such quotients. These exponentials of ratios of solutions to hypergeometric differential equations (in fact, of Picard–Fuchs equations) have recently received great attention in mathematical physics and algebraic geometry under the name of canonical coordinates. Their compositional inverses, known as mirror maps, are an important ingredient in the computation of the Yukawa coupling in the theory of mirror symmetry. It is conjectured that the coefficients in the Lambert series expansion of the Yukawa coupling produce Gromov–Witten invariants of classes of rational curves.

2000 Mathematics Subject Classification. Primary 11S80; Secondary 11J99 14J32 33C70. Key words and phrases. Calabi–Yau manifolds, integrality of mirror maps, p-adic analysis, Dwork’s

theory in several variables, harmonic numbers, hypergeometric differential equations.

†Research partially supported by the Austrian Science Foundation FWF, grants Z130-N13 and S9607-N13, the latter in the framework of the National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.

This paper was written in part during the authors’ stay at the Erwin Schrödinger Institute for Physics and Mathematics, Vienna, during the programme “Combinatorics and Statistical Physics” in Spring 2008.

It is only relatively recent that Dwork’s theory has been systematically applied to obtain general arithmetic results on the Taylor coefficients of mirror maps. Partial results in this direction were found by Lian and Yau [18, 19], by Zudilin [22], and by Kontsevich, Schwarz and Vologodsky [13, 21]. The (so far) strongest and most general results are contained in [6, 14, 15], where, in particular, numerous integrality results for the Taylor coefficients of univariate mirror maps of Calabi–Yau complete intersections in weighted projective spaces are proven, improving and refining the afore-mentioned results by Lian and Yau, and by Zudilin. However, all these results do not touch the case of multivariable mirror maps, upon which they are not able to say anything. The goal of this paper is to set the basis of a theory which is capable to address questions of integrality of Taylor coefficients of multivariable mirror maps, and to apply this theory systematically to large classes of such mirror maps.

1.1. Multivariate theory of formal congruences. The proof strategy in [6, 14, 15, 18, 19, 22] for obtaining integrality assertions on the Taylor coefficients of one-variable mirror maps is crucially based on a series of reductions and results, of which the corner stones are:

(D1) the conversion of the integrality problem to a p-adic problem; (D2) a lemma due to Dieudonné and Dwork (cf. [17, Ch. 14, p. 76]) providing a criterion

for deciding whether a power series with coefficients over ℚ_{p} has coefficients in ℤ_{p}; (D3) a reduction lemma for harmonic numbers due to the authors (cf. [14, Lemma 1,

respectively Lemma 5] and [15, Lemma 3]); (D4) a combinatorial lemma due to Dwork [11, Lemma 4.2] for rearranging sums that

appear in this context in a way tailor-made for p-adic analysis; (D5) Dwork’s theorem on formal congruences (cf. [11, Theorem 1.1]).

We point out that Lian and Yau, and Zudilin do not need item (D3) due to the nature of the special families of mirror maps that they were considering. Indeed, item (D3) is the decisive novelty which enabled the authors to arrive at their general sets of results in [14, 15]. We remark that Zudilin also condenses (D4) and (D5) into one step in the proof of his main result in [22]. However, in order to arrive at the general results in [14, 15], it turned out to be necessary to follow the full path outlined by (D1)–(D5) above, as attempts to lift Zudilin’s variation to this generality failed.

With the exception of (D1), which trivially extends to the multivariable case, for none of the above items there exist multivariate extensions in the current literature. In particular, no approach for attacking integrality questions for multivariable mirror maps has been available so far.

In this paper, we present multivariate versions for all of (D2)–(D5); all of them seem to be new. Our multivariate extension of (D2) is the content of Lemma 1 in Section 2, our multivariate version of (D3) can be found in Lemma 3 in Section 2, while Lemma 5 in Section 6 provides our multivariate extension of (D4). On the other hand, we state our multivariate extension of item (D5) in Theorem 1 below. Since its one-variable special case enabled Dwork to address the question of analytic extension of certain ratios of generalised p-adic hypergeometric series in one variable, we expect our result below to be the appropriate tool for analogous studies of multivariable p-adic hypergeometric series.

For the statement of our multivariate theorem on formal congruences, we need some standard multi-index notation. Namely, given a positive integer d, a real number λ, and vectors m = (m_{1}, m_{2}, …, m_{d}) and n = (n_{1}, n_{2}, …, n_{d}) in ℝ^{d}, we write m + n for (m_{1} + n_{1}, m_{2} + n_{2}, …, m_{d} + n_{d}), λm for (λm_{1}, λm_{2}, …, λm_{d}), we write m ≥ n if and only if m_{i} ≥ n_{i} for i = 1, 2, …, d, and we write 0 for (0, 0, …, 0) ∈ ℤ^{d} and 1 for (1, 1, …, 1) ∈ ℤ^{d}.

Theorem 1. Let A : ℤ_{≥0}^{d} → ℤ_{p} \ {0} and g : ℤ_{≥0}^{d} → ℤ_{p} \ {0} be maps satisfying the following three properties:

(i) v_{p}(A(0)) = 0; (ii) A(n) ∈ g(n)ℤ_{p}; (iii) for all non-negative integers s and all integer vectors v, u, n ∈ ℤ_{≥0}^{d} with 0 ≤ v_{i} < p

and 0 ≤ u_{i} < p^{s}, i = 1, 2, …, d,

\frac{A(v + pu + np^{s+1})}{A(v + pu)} − \frac{A(u + np^{s})}{A(u)} ∈ p^{s+1} \frac{g(n)}{g(v + pu)} ℤ_{p}.

Then, for all non-negative integers s and all integer vectors m, K, a ∈ ℤ_{≥0}^{d} with 0 ≤ a_{i} < p, i = 1, 2, …, d, we have

∑_{p^{s}m≤k≤p^{s}(m+1)−1} (A(a + pk)A(K − k) − A(a + p(K − k))A(k)) ∈ p^{s+1}g(m)ℤ_{p},

where we extend A to ℤ^{d} by A(n) = 0 if there is an i such that n_{i} < 0.

While the proofs of Lemmas 1 and 5 (corresponding to items (D2) and (D4)) are relatively straightforward extensions of the one-variable proofs given in [17, Ch. 14, p. 76] and [11, proof of Lemma 4.2], respectively, the proofs of Lemma 3 and Theorem 1 (corresponding to items (D3) and (D5)) need new ideas. The proof of Lemma 1 is given in Section 3. Section 5 is devoted to the proof of Lemma 3. This proof was kindly provided by an anonymous referee. (For our original proof, see [16, Sec. 5].) Even in the one-dimensional case, this proof (as well as the one in [16]) is new, as it simplifies the earlier proofs [14, proofs of Lemma 1, respectively Lemma 5] and [15, proof of Lemma 3]. In fact, it turned out that these earlier proofs could not be extended to the multivariate case. The proof of Lemma 5 can be found in Section 6. Finally, in Section 7 we prove Theorem 1.

The main application of our multivariate theory of formal congruences that we present in this paper concerns the proof that, for a large class of multivariable mirror maps, their Taylor coefficients are integers. We state the corresponding general theorem in the next subsection. The subsequent subsection collects some particularly interesting special cases and consequences.

1.2. A family of GKZ functions and their associated mirror maps. In order to state the results in this section conveniently, we need to further enlarge our set of multiindex notations given before Theorem 1. Given vectors m = (m_{1}, m_{2}, …, m_{d}) and n = (n_{1}, n_{2}, …, n_{d}) in ℝ^{d}, we write m · n for the scalar product m_{1}n_{1} + m_{2}n_{2} + ⋯ + m_{d}n_{d}, and we write |m| for m_{1} + m_{2} + ⋯ + m_{d}. Furthermore, given a vector z = (z_{1}, z_{2}, …, z_{d}) of variables and n = (n_{1}, n_{2}, …, n_{d}) ∈ ℤ^{d}, we write z^{n} for the product z_{1}^{n_{1}}z_{2}^{n_{2}} ⋯ z_{d}^{n_{d}}. On the other hand, if n is an integer, we write z^{n} for the vector (z_{1}^{n}, z_{2}^{n}, …, z_{d}^{n}).

Given k vectors N^{(j)} = (N_{1}^{(j)}, N_{2}^{(j)}, …, N_{d}^{(j)}) ∈ ℤ^{d}, j = 1, …, k, with N^{(j)} ≥ 0, let us define the series

F_{N}(z) = ∑_{m≥0} z^{m} ∏_{j=1}^{k} \frac{(N^{(j)} · m)!}{∏_{i=1}^{d} m_{i}!^{N_{i}^{(j)}}} = ∑_{m≥0} z^{m} ∏_{j=1}^{k} \frac{( ∑_{i=1}^{d} N_{i}^{(j)}m_{i})!}{∏_{i=1}^{d} m_{i}!^{N_{i}^{(j)}}}.

Since the Taylor coefficients of F_{N}(z) are products of multinomial coefficients, it follows that F_{N}(z) ∈ 1 + ∑_{i=1}^{d} z_{i}ℤ[[z]], where ℤ[[z]] denotes the set of all (formal) power series in the variables z_{1}, z_{2}, …, z_{d} with integer coefficients.

This series is a GKZ hypergeometric function (^{1}) and it is known to “come from geometry,” i.e., it can be viewed as the period of certain multiparameter families of algebraic varieties in a product of weighted projective spaces (see [12] for details). It satisfies a linear differential system {L_{i,N}(F_{N}) = 0 : i = 1, …, d} defined by the operators

L_{i,N} = θ_{i}^{N_{i}^{(1)}+···+N_{i}^{(k)}} − z_{i} ∏_{j=1}^{k} ∏_{r_{j}=1}^{N_{i}^{(j)}} ( ∑_{ℓ=1}^{d} N_{ℓ}^{(j)}θ_{ℓ} + r_{j}), i = 1, …, d,

where θ_{i} = z_{i} \frac{∂}{∂z_{i}}. Amongst the other solutions of this system, we find the d functions log(z_{i})F_{N}(z) + G_{i,N}(z), i = 1, …, d, defined by

G_{i,N}(z) = ∑_{m≥0} z^{m}( ∑_{j=1}^{k} N_{i}^{(j)}H_{N^{(j)}·m} − H_{m_{i}} ∑_{j=1}^{k} N_{i}^{(j)}) ∏_{j=1}^{k} \frac{(N^{(j)} · m)!}{∏_{i=1}^{d} m_{i}!^{N_{i}^{(j)}}} .

Here and in the rest of the article, H_{m} = ∑_{j=1}^{m} 1/j denotes the m-th harmonic number, with the convention H_{0} = 0.

This set of solutions enables us to define d canonical coordinates q_{i,N}(z) by

q_{i,N}(z) = z_{i} exp (G_{i,N}(z)/F_{N}(z)),

which are objects with many fundamental properties for the “mirror symmetry” study of the underlying multiparameter families of varieties. The compositional inverse of the map

z ↦ (q_{1,N}(z), q_{2,N}(z), …, q_{d,N}(z))

defines the vector (z_{1,N}(q), z_{2,N}(q), …, z_{d,N}(q)) of mirror maps. In this paper, by abuse of terminology, we will also use the term “mirror map” for any canonical coordinate. (^{2})

1See [20] for an introduction to these functions, which are a far-reaching generalisation of the classical hypergeometric functions to several variables.

2Canonical coordinates and mirror maps have distinct geometric meanings. However, in the numbertheoretic study undertaken in the present paper, they play strictly the same role, because q_{i,N}(z) ∈ z_{i}ℤ[[z]], i = 1, 2, …, d, implies that z_{i,N}(q) ∈ q_{i}ℤ[[q]], i = 1, 2, …, d, and conversely.

Let us define the series

G_{L,N}(z) = ∑_{m≥0} z^{m}H_{L·m} ∏_{j=1}^{k} \frac{(N^{(j)} · m)!}{∏_{i=1}^{d} m_{i}!^{N_{i}^{(j)}}} ∈ ∑_{i=1}^{d} z_{i}ℚ[[z]],

where L ∈ ℤ^{d} is ≥ 0. For any i = 1, …, d, the function G_{i,N}(z) is a finite linear combination with integer coefficients in the functions G_{L,N}(z), where the summation runs over various vectors L, each one with the property that 0 ≤ L ≤ N^{(j(L))} for some j(L) ∈ {1, …, k}. Therefore, the following theorem concerns as well our mirror maps q_{i,N}(z) ∈ z_{i}ℤ[[z]] (see Corollary 1).

Theorem 2. Let d and k be positive integers. For all vectors L = (L_{1}, L_{2}, …, L_{d}) ∈ ℤ^{d} and N^{(j)} = (N_{1}^{(j)}, N_{2}^{(j)}, …, N_{d}^{(j)}) ∈ ℤ^{d}, j = 1, 2, …, k, with 0 ≤ L ≤ N^{(1)}, N^{(2)} ≥ 0, …, N^{(k)} ≥ 0, we have

q_{L,N}(z) := exp (G_{L,N}(z)/F_{N}(z)) ∈ ℤ[[z]].

Remarks 1. (a) Given the fact that the canonical coordinates q_{i,N}(z) (which, in their turn, define the mirror maps z_{i,N}(q)) can be expressed as products of several series of the form q_{L,N}(z) (with varying L), we call q_{L,N}(z) a mirror-type map.

(b) By carefully going through our arguments, one sees that minor modifications lead to the slightly stronger statement that, under the assumptions of Theorem 2, we have

exp (G_{L,N}(z)/F_{N}(z)) ∈ ∏_{j=2}^{k} ( min{N_{1}^{(j)}, N_{2}^{(j)}, …, N_{d}^{(j)}})! ℤ[[z]].

The statement of the theorem might suggest that N^{(1)} plays a special role amongst the vectors N^{(1)}, N^{(2)}, …, N^{(k)}. Of course, this is not the case: by symmetry, given any j ∈ {1, …, k}, a similar result holds for any L such that 0 ≤ L ≤ N^{(j)}. This remark implies the following result for the mirror maps q_{i,N}(z) ∈ z_{i}ℤ[[z]], proving a conjecture of Batyrev and van Straten [3, Conjecture 7.3.4] for a large family of canonical coordinates in several variables.

Corollary 1. Let d, k be positive integers. For all vectors N^{(j)} = (N_{1}^{(j)}, N_{2}^{(j)}, …, N_{d}^{(j)}) ∈ ℤ^{d}, j = 1, 2, …, k, with N^{(1)} ≥ 0, N^{(2)} ≥ 0, …, N^{(k)} ≥ 0, we have q_{i,N}(z) ∈ ℤ[[z]], i = 1, 2, …, d.

We outline the proof of Theorem 2 in Section 2, thereby showing how the various pieces of our multivariate theory of formal congruences fit together in order to prove integrality assertions for multivariable mirror(-type) maps. The details are deferred to Sections 3–9.

1.3. Consequences of Theorem 2. In order to illustrate the range of applicability of Theorem 2, we collect in this subsection some examples and applications that are of particular interest to multivariable and one-variable mirror-type maps.

(1) A classical multivariate example, studied in detail in [3, Sec. 7] and [20, Sec. 8.4], is the case of the two parameters (w and z say) family of hypersurfaces V of degree (3, 3) in ℙ^{2}(ℂ) × ℙ^{2}(ℂ), which is a family of Calabi–Yau threefolds. The periods of the associated mirror family of Calabi–Yau hypersurfaces can be expressed in terms of the double series

F (w, z) = ∑_{m≥0} ∑_{n≥0} \frac{(3m + 3n)!}{m!^{3}n!^{3}} w^{m}z^{n},  (1.1)

which is symmetric and holomorphic in \{(w, z) ∈ ℂ^{2} : |w|^{1/3} + |z|^{1/3} < \frac{1}{3}\}. It is a solution of the linear differential system {L_{1}(F) = 0, L_{2}(F) = 0} defined by the operators

\begin{cases}L_{1} = θ_{1}^{3} − w(3θ_{1} + 3θ_{2} + 1)(3θ_{1} + 3θ_{2} + 2)(3θ_{1} + 3θ_{2} + 3), \\ L_{2} = θ_{2}^{3} − z(3θ_{1} + 3θ_{2} + 1)(3θ_{1} + 3θ_{2} + 2)(3θ_{1} + 3θ_{2} + 3), \end{cases}

where θ_{1} = w\frac{∂}{∂w} and θ_{2} = z \frac{∂}{∂z}.

Two other solutions of this system are of the form log(w)F (w, z)+G_{1}(w, z) and log(z)F (w, z)+ G_{2}(w, z) where G_{1}(w, z) and G_{2}(w, z) are holomorphic in \{(w, z) ∈ ℂ^{2} : |w|^{1/3}+|z|^{1/3} < \frac{1}{3}\}, and are given explicitly by

G_{1}(w, z) = ∑_{m≥0} ∑_{n≥0} (3H_{3m+3n} − 3H_{m})\frac{(3m + 3n)!}{m!^{3}n!^{3}} w^{m}z^{n},

G_{2}(w, z) = ∑_{m≥0} ∑_{n≥0} (3H_{3m+3n} − 3H_{n})\frac{(3m + 3n)!}{m!^{3}n!^{3}} w^{m}z^{n}.

Let us now define the two variable mirror maps q_{1}(w, z) = w exp (G_{1}(w, z)/F (w, z)) and q_{2}(w, z) = z exp (G_{2}(w, z)/F (w, z)). Here, q_{1}(w, z) = q_{2}(z, w), but this is not the case in general. It was observed in the early developments of mirror symmetry theory that q_{1}(w, z) and q_{2}(w, z) seem to have integral Taylor coefficients (see the end of Section 7.1 in [3] for example). Corollary 1 with d = 2, k = 1, N^{(1)} = (3, 3) now provides a proof for this observation.

(2) Interesting consequences result also by considering the series expansion q_{L,N}(z) for cases where some or all of the variables z_{i} are equal to each other. The obtained series is obviously still a formal power series. Furthermore, since the initial power series has integer coefficients, any such specialisation leads again to a series with integer coefficients. In this way, we can construct many new mirror-type maps, and, for several of them, this leads to proofs of conjectures in the literature on the integrality of their Taylor coefficients.

Here, we provide details for a corresponding example derived from the mirror-type map of item (1). Subsequently, Item (3) will address another family of one-variable examples derived from two-variable series, which, for example, includes the series whose coefficients form the famous sequences that appear in Apéry’s proof of the irrationality of ζ(2) and ζ(3). Finally, in Item (4), we mention briefly certain cases studied in [1, 2, 3].

We put w = z in the example (1.1) considered in Item (1) above and get

f(z) = ∑_{m≥0} ∑_{n≥0} z^{m+n} \frac{(3m + 3n)!}{m!^{3}n!^{3}} = ∑_{k=0}^{∞} z^{k} \frac{(3k)!}{k!^{3}} ∑_{j=0}^{k} \binom{k}{j}^{3}

after rearrangement. This map is studied in [3, Sec. 7.3], where it is shown to be of significance in the theory of mirror symmetry. The function f satisfies a Fuchsian differential equation of order 4 with maximal unipotent monodromy at the origin: it is annihilated by the minimal operator

θ^{4} − 3z(7θ^{2} + 7θ + 2)(3θ + 1)(3θ + 2) − 72z^{2}(3θ + 5)(3θ + 4)(3θ + 2)(3θ + 1).

Another solution is g(z) + log(z)f(z), where g(z) is given by

g(z) = ∑_{k=0}^{∞} z^{k} \frac{(3k)!}{k!^{3}} ∑_{j=0}^{k} \binom{k}{j}^{3}(3H_{3k} − 3H_{k−j}).

The function g(z) is a linear combination with integer coefficients of the functions

g_{L}(z) = ∑_{k=0}^{∞} z^{k} \frac{(3k)!}{k!^{3}} ∑_{j=0}^{k} \binom{k}{j}^{3}H_{L_{1}j+L_{2}(k−j)},

where L = (L_{1}, L_{2}) ∈ ℤ^{2} is such that 0 ≤ L_{1}, L_{2} ≤ 3. For these L, setting z_{1} = z_{2} = z in Theorem 2 leads to

exp (g_{L}(z)/f(z)) ∈ ℤ[[z]],

which, in particular, implies the new result that z exp (g(z)/f(z)) ∈ zℤ[[z]].

(3) For any integers α, β such that 0 ≤ β ≤ α, we consider the function

∑_{m≥0} ∑_{n≥0} (\frac{(m + n)!}{m! n!})^{α−β} (\frac{(2m + n)!}{m!^{2}n!})^{β} w^{m}z^{n}.  (1.2)

The specialisation w = z produces the function

A_{α,β}(z) = ∑_{k=0}^{∞} ( ∑_{j=0}^{k} \binom{k}{j}^{α}\binom{k + j}{j}^{β})z^{k},

to which we associate the function B_{α,β}(z) + log(z)A_{α,β}(z) defined by

B_{α,β}(z) = ∑_{k=0}^{∞} ( ∑_{j=0}^{k} \binom{k}{j}^{α}\binom{k + j}{j}^{β}((α − β)H_{k} − αH_{k−j} + βH_{k+j}))z^{k}.

Let L_{α,β} be the minimal Fuchsian differential operator that annihilates A_{α,β}(z): it does not always have maximal unipotent monodromy at z = 0, as the case (α, β) = (6, 0) shows (cf. [1, Sec. 10]). The operator L_{α,β} also annihilates B_{α,β}(z) + log(z)A_{α,β}(z) and we define the mirror map z exp (B_{α,β}(z)/A_{α,β}(z)). We observe that B_{α,β}(z) is a linear combination with integer coefficients in the functions

B_{L,α,β}(z) = ∑_{k=0}^{∞} ( ∑_{j=0}^{k} \binom{k}{j}^{α}\binom{k + j}{j}^{β}H_{L_{1}j+L_{2}(k−j)})z^{k}.

Here, L = (L_{1}, L_{2}) ∈ ℤ^{2} is such that 0 ≤ L_{1} ≤ 2 and 0 ≤ L_{2} ≤ 1. For these L, setting z_{1} = z_{2} = z in Theorem 2 leads to

exp (B_{L,α,β}(z)/A_{α,β}(z)) ∈ ℤ[[z]],

and this implies that z exp (B_{α,β}(z)/A_{α,β}(z)) ∈ zℤ[[z]]. This example is particularly interesting because it proves that maximal unipotent monodromy at the origin is not a necessary condition to obtain mirror-type maps with integer Taylor coefficients.

It is interesting to note that the Taylor coefficients of A_{2,1}(z) and A_{2,2}(z) form the sequences appearing in Apéry’s proof of the irrationality of ζ(2) and ζ(3), respectively. Beukers [4] showed that A_{2,1}(z) and A_{2,2}(z) are strongly related to modular forms, a fact which also explains the integrality properties of the associated mirror-type maps. (For p-adic properties of A_{2,1}(z), we refer the reader to [5].)

(4) Setting some or all of the variables in Theorem 2 equal to each other can explain the integrality properties of many of the mirror-type maps in [1], many of which have been incorporated in the table [2] of “Calabi–Yau differential equations”. This table contains a list of more than 300 Fuchsian differential equations of order 4 with certain analytic properties, amongst which are maximal unipotent monodromy at the origin and conjectural integrality of the instanton-type numbers. Only the first 29 items are currently known to have a geometric origin, meaning that they have an interpretation in mirror symmetry; for example, the instanton-type numbers in these cases are really instanton numbers. In particular, the table contains the mirror-type maps of geometric origin considered in Sections 8.1, 8.2, 8.3 and 8.4 of [3], which all come from setting variables equal to each other in series covered by Theorem 2.

Although this is not mentioned explicitly in [2], it is plausible that the mirror-type maps associated to each example of the table have integer Taylor coefficients. In this direction, we have checked that the functions whose Taylor coefficients are given in items 15 to 23, 25, 34, 39, 45, 58, 60, 72, 76, 78, 79, 81, 91, 93, 96, 97, 127, 130, 188, 190 and 191, are specialisations of multivariable series that can be treated with Theorem 2. Hence the mirror-type maps associated to these items have integer Taylor coefficients. Incidentally, items 1 to 14 are all covered by the results in [14, 18, 22] and therefore, amongst the “geometric” items 1 to 29, there remains to understand only items 24, 26, 27, 28, 29.

We could use many other ways of specialisation in conjunction with Theorem 2, for example “weighted equating” such as z_{1} = Mz_{2}^{N} for some integer parameters M ≠ 0 and N ≥ 1. However, the range of application of Theorem 2 is limited. For example, it cannot be applied to the functions A_{α,β}(z) when 0 ≤ α < β because the latter have (m + n)! in the denominator of the Taylor coefficients of the function (1.2). This case is not covered by our results, as well as many items in the table [2] for the same reason. Therefore, it is of great interest to find the conditions on the vectors of integers N^{(j)}, D^{(j)} ∈ ℤ^{d} and the integers s_{j}, t_{j} ≥ 0 for which the GKZ function

∑_{m≥0} z^{m} ∏_{j=1}^{k} \frac{(N^{(j)} · m)!^{s_{j}}}{(D^{(j)} · m)!^{t_{j}}}

satisfies a generalisation of Theorem 2.^{3}

2. Outline of the proof of Theorem 2

In this section, we present a decomposition of the proof of Theorem 2 into various assertions, which form our multivariate theory of formal congruences described in Subsection 1.1. The individual assertions will be proved in the later sections.

The starting point (listed as (D1) in Subsection 1.1) is the observation that, given a power series S(z) = S(z_{1}, z_{2}, …, z_{d}) in ℚ[[z]], the series S(z) is an element of ℤ[[z]] if and only if, for all primes p, it is an element of ℤ_{p}[[z]].

Next, we want to get rid of the exponential function in the definition of the mirror-type map q_{L,N}(z). To achieve this, we use a generalisation of a lemma attributed to Dieudonné and Dwork in [17, Ch. 14, p. 76] to several variables, the latter being the univariate case of the following lemma (corresponding to (D2) in Subsection 1.1).

Lemma 1. For S(z) ∈ 1 + ∑_{i=1}^{d} z_{i}ℚ_{p}[[z]], we have

S(z) ∈ 1 + ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]] if and only if \frac{S(z^{p})}{S(z)^{p}} ∈ 1 + p ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]].

This lemma enables us to prove the following reduction of our problem.

Lemma 2. Given two formal series F (z) ∈ 1 + ∑_{i=1}^{d} z_{i}ℤ[[z]] and G(z) ∈ ∑_{i=1}^{d} z_{i}ℚ[[z]], let q(z) := exp(G(z)/F (z)). Then we have q(z) ∈ 1 + ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]] if and only if

F (z)G(z^{p}) − pF (z^{p})G(z) ∈ p ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]].

These two lemmas are proved in Sections 3 and 4, respectively.

We write B_{N}(m) = ∏_{j=1}^{k} B(N^{(j)}, m), where

B(P, m) = \frac{(∑_{i=1}^{d} P_{i}m_{i})!}{∏_{i=1}^{d} m_{i}!^{P_{i}}}  (2.1)

for all vectors P, m ∈ ℤ^{d} with P ≥ 0 and m ≥ 0, while we define B(P, m) = 0 for vectors m for which m_{i} < 0 for some i. (If we interpret factorials n! as Γ(n + 1), where Γ stands for the gamma function, then this convention is in accordance with the behaviour of the gamma function.) Note that, using this notation, we have F_{N}(z) = ∑_{m≥0} z^{m}B_{N}(m) and G_{L,N}(z) = ∑_{m≥0} z^{m}H_{L·m}B_{N}(m).

3This problem is now solved. Indeed, since the first version of this text, Eric Delaygue has found a complete and simple characterisation of multivariate mirror maps with integral Taylor coefficients, which also encompasses his result in one variable [6]. His result is expressed in terms of a multivariate generalisation of the classical Landau step functions associated to quotients of products of factorials. His method is based on the theory presented in the present paper, but he has to prove another multivariate “formal congruences theorem,” our own Theorem 1 being apparently not strong enough to prove his characterisation.

As already mentioned, we have F_{N}(z) ∈ 1 + ∑_{i=1}^{d} z_{i}ℤ[[z]] and thus we can use Lemma 2 with F (z) = F_{N}(z) and G(z) = G_{L,N}(z). The coefficient of z^{a+pK} (with 0 ≤ a_{i} < p for all i) in the Taylor expansion of the formal power series F_{N}(z)G_{L,N}(z^{p}) − pF_{N}(z^{p})G_{L,N}(z) can be written in the form

C(a + pK) = ∑_{0≤k≤K} B_{N}(a + pk)B_{N}(K − k)(H_{L·(K−k)} − pH_{L·a+pL·k}).

Lemma 2 tells us that we have to show that C(a + pK) is in pℤ_{p}.

To prove this, we will proceed step by step. First, because of the congruence (^{4})

pH_{L·a+pL·k} ≡ H_{⌊\frac{1}{p} L·a⌋+L·k} mod pℤ_{p},

we obtain

C(a + pK) ≡ ∑_{0≤k≤K} B_{N}(a + pk)B_{N}(K − k)(H_{L·(K−k)} − H_{⌊\frac{1}{p} L·a⌋+L·k}) mod pℤ_{p}.

Then, the following lemma (corresponding to (D3) in Subsection 1.1) is proved in Section 5.

Lemma 3. For any prime p, vectors a, k, L, N^{(1)} ∈ ℤ^{d} with k ≥ 0, 0 ≤ L ≤ N^{(1)}, and 0 ≤ a_{i} < p for i = 1, 2, …, d, we have

B(N^{(1)}, a + pk)(H_{⌊\frac{1}{p} L·a⌋+L·k} − H_{L·k}) ∈ pℤ_{p},  (2.2)

where B(N^{(1)}, a + pk) is defined in (2.1).

Since B(N^{(1)}, a + pk) is a factor of B_{N}(a + pk), it follows that

C(a + pK) ≡ ∑_{0≤k≤K} B_{N}(a + pk)B_{N}(K − k)(H_{L·(K−k)} − H_{L·k}) mod pℤ_{p}.

For the right-hand side, we obviously have

∑_{0≤k≤K} B_{N}(a + pk)B_{N}(K − k)(H_{L·(K−k)} − H_{L·k})

= − ∑_{0≤k≤K} H_{L·k}(B_{N}(a + pk)B_{N}(K − k) − B_{N}(a + p(K − k))B_{N}(k)). (2.3)

We now use the multivariable extension of the combinatorial lemma of Dwork (corresponding to (D4) in Subsection 1.1; stated here as Lemma 5 in Section 6, with proof in the same section) in order to decompose the sum over k. Namely, if in Lemma 5 we let Z(k) = H_{L·k},

W (k) = B_{N}(a + pk)B_{N}(K − k) − B_{N}(a + p(K − k))B_{N}(k),

4This is an immediate consequence of the identity H_{J} = ∑_{j=1}^{⌊J/p⌋} \frac{1}{pj} + ∑_{j=1,p∤j}^{J} \frac{1}{j}.

and choose an integer r that satisfies p^{r−1} > max{K_{1}, K_{2}, …, K_{d}}, then

C(a + pK) ≡ − ∑_{s=0}^{r−1} ∑_{0≤m≤(p^{r−s}−1)1} (H_{∑_{i=1}^{d} L_{i}m_{i}p^{s}} − H_{∑_{i=1}^{d} L_{i}⌊\frac{m_{i}}{p} ⌋p^{s+1}})

· ∑_{p^{s}m≤k≤p^{s}(m+1)−1} (B_{N}(a + pk)B_{N}(K − k) − B_{N}(a + p(K − k))B_{N}(k))
mod pℤ_{p}. (2.4)

(Since for the first term appearing on the right-hand side of (6.1) we have Z(0)\overline{W}_{r}(0) = H_{0}\overline{W}_{r}(0) = 0, the right-hand sides of (2.3) and (2.4) are in fact equal.)

To deal with the sum over k in (2.4), we invoke Theorem 1 (corresponding to (D5) in Subsection 1.1). (Its proof is given in Section 7). We show in Section 8 that Theorem 1 can be applied with A = g = B_{N}. Using this, we obtain

∑_{p^{s}m≤k≤p^{s}(m+1)−1} (B_{N}(a+pk)B_{N}(K−k)−B_{N}(a+p(K−k))B_{N}(k)) ∈ p^{s+1}B_{N}(m)ℤ_{p}. (2.5)

We now have to deal with the harmonic sums

H_{∑_{i=1}^{d} L_{i}m_{i}p^{s}} − H_{∑_{i=1}^{d} L_{i}⌊\frac{m_{i}}{p} ⌋p^{s+1}}

occurring on the right-hand side of (2.4). In this regard, we prove the following lemma in Section 9. (As we show there, it can be reduced to Lemma 3.)

Lemma 4. For all primes p, vectors m, L, N^{(1)}, N^{(2)}, …, N^{(k)} ∈ ℤ^{d} with m, L, N^{(1)}, N^{(2)}, …, N^{(k)} ≥ 0, we have

B_{N}(m)(H_{∑_{i=1}^{d} L_{i}m_{i}p^{s}} − H_{∑_{i=1}^{d} L_{i}⌊\frac{m_{i}}{p} ⌋p^{s+1}}) ∈ \frac{1}{p^{s}} ℤ_{p}.  (2.6)

Consequently, putting the congruences (2.5) and (2.6) together, it follows from (2.4) that C(a + pk) is congruent mod pℤ_{p} to a multiple sum (over s and m) whose terms are all in pℤ_{p}. Hence, we have established that

C(a + pk) ∈ pℤ_{p}.

This concludes our outline of the proof of Theorem 2.

3. Proof of Lemma 1

Proof of the “only if” part. We have to show that if S(z) ∈ 1 + ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]], then

\frac{S(z^{p})}{S(z)^{p}} ∈ 1 + p ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]].

To do this, we set S(z) = ∑_{i≥0} a_{i}z^{i}. The congruence (u + v)^{p} ≡ u^{p} + v^{p} mod pℤ_{p} and Fermat’s Little Theorem imply that

S(z)^{p} = ( ∑_{i≥0} a_{i}z^{i})^{p} ≡ ∑_{i≥0} a_{i}^{p}z^{pi} mod p ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]]

≡ ∑_{i≥0} a_{i}z^{pi} mod p ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]].

This means that S(z)^{p} = S(z^{p}) + pH(z) with H(z) ∈ ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]]. Hence,

\frac{S(z^{p})}{S(z)^{p}} = 1 − p \frac{H(z)}{S(z)^{p}} ∈ 1 + p ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]],

because the formal series S(z) ∈ 1 + ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]] is invertible in ℤ_{p}[[z]].

Proof of the “if” part. Suppose that S(z^{p}) = S(z)^{p}R(z) with R(z) = 1 + p ∑_{|i|≥1} b_{i}z^{i} ∈ 1 + p ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]] and S(0) = 1. Set S(z) = ∑_{i≥0} a_{i}z^{i}. We have a_{0} = 1, and we proceed by induction on |i| to show that a_{i} ∈ ℤ_{p}.

So, let us assume that a_{i} ∈ ℤ_{p} for all vectors i ∈ ℤ^{d} with |i| ≤ r − 1. Let n ∈ ℤ^{d} be a vector with |n| = r. The Taylor coefficient C_{n} of z^{n} in S(z^{p}) is

\begin{cases}a_{\frac{1}{p} n} if p | n_{1}, p | n_{2}, …, p | n_{d}; \\ 0 otherwise. \end{cases}

The Taylor coefficient C_{n} is at the same time also equal to the coefficient of z^{n} in the expansion of the series

( ∑_{i≥0} a_{i}z^{i})^{p}(1 + p ∑_{i≥0, |i|≥1} b_{i}z^{i}).

The coefficient of z^{n} in this series is thus C_{n} = B_{n} + pD_{n}, where

B_{n} = ∑_{i^{(1)}+···+i^{(p)}=n} a_{i^{(1)}} ⋯ a_{i^{(p)}}  (3.1)

and

D_{n} = _{∑_{i^{(1)}+···+i^{(p+1)}=n} ; |i^{(p+1)}|>0}a_{i^{(1)}} ⋯ a_{i^{(p)}}b_{i^{(p+1)}}.  (3.2)

Case 1. If p | n_{1}, …, p | n_{d}, in the multiple sum B_{n} a term ∏_{ℓ=1}^{m} a_{i_{ℓ}}^{e_{ℓ}} with a_{i_{ℓ_{1}}} ≠ a_{i_{ℓ_{2}}} occurs

\frac{(e_{1} + ⋯ + e_{m})!}{e_{1}! ⋯ e_{m}!} = \frac{p!}{e_{1}! ⋯ e_{m}!}  (3.3)

times. The multinomial coefficient (3.3) is an integer divisible by p, except if m = 1 and e_{1} = p; that is, if we are looking at the term a_{\frac{1}{p} n}^{p}, which occurs with coefficient 1 in B_{n}. The term a_{n} appears in the form pa_{n}a_{0}^{p−1} = pa_{n} in the expression (3.1) for B_{n}. For all other terms in the sum on the right-hand side of (3.1), we have |i^{(ℓ)}| < |n| for ℓ = 1, 2, …, p. Hence, the induction hypothesis applies to all the factors in the corresponding terms a_{i^{(1)}} ⋯ a_{i^{(p)}}, whence B_{n} = pa_{n} + a_{\frac{1}{p} n}^{p} mod pℤ_{p}.

In the multiple sum (3.2) for D_{n}, the condition |i^{(p+1)}| > 0 guarantees that |i^{(ℓ)}| < |n| for ℓ = 1, …, p, and therefore we can apply the induction hypothesis to each factor a_{i^{(ℓ)}}. This shows that D_{n} ∈ ℤ_{p}.

We therefore have

a_{\frac{1}{p} n} = C_{n} ≡ pa_{n} + a_{\frac{1}{p} n}^{p} mod pℤ_{p},

whence,

pa_{n} ≡ a_{\frac{1}{p} n} − a_{\frac{1}{p} n}^{p} mod pℤ_{p}.

This shows that a_{n} ∈ ℤ_{p} since a_{\frac{1}{p} n} − a_{\frac{1}{p} n}^{p} ∈ pℤ_{p} by Fermat’s Little Theorem.

Case 2. If p ∤ n_{i} for some i between 1 and d, the only change compared to the preceding case is that the term a_{\frac{1}{p} n}^{p} does not occur. Therefore, in this case we have

0 = C_{n} ≡ pa_{n} mod pℤ_{p}.

Hence,

pa_{n} ≡ 0 mod pℤ_{p},

which shows again that a_{n} ∈ ℤ_{p}.

This completes the proof of the lemma. □

4. Proof of Lemma 2

We begin by showing that, if T (z) ∈ ∑_{i=1}^{d} z_{i}ℚ_{p}[[z]], then

exp(T (z)) ∈ 1 + ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]] if and only if T (z^{p}) − pT (z) ∈ p ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]].

The formal power series exp(X) and log(1 + X) are defined by their usual expansions.

Proof of the “if” part. By Lemma 1 with T (z) replaced by exp(T (z)), we have

exp (T (z^{p}) − pT (z)) ∈ 1 + p ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]].

Therefore, we have T (z^{p})−pT (z) = log(1+pH(z)) with H(z) ∈ ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]]. This yields

T (z^{p}) − pT (z) = − ∑_{n=1}^{∞} \frac{p^{n}}{n} (−H(z))^{n} ∈ p ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]]

since v_{p}(p^{n}/n) ≥ 1 for all integers n ≥ 1.

Proof of the “only if” part. We have T (z^{p}) − pT (z) = pJ(z) with J(z) ∈ ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]]. Therefore, we have

exp (T (z^{p}) − pT (z)) = 1 + ∑_{n=1}^{∞} \frac{p^{n}}{n!} J(z)^{n} ∈ 1 + p ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]],

since

v_{p} (\frac{p^{n}}{n!} ) = n − ∑_{k=1}^{∞} ⌊\frac{n}{p^{k}} ⌋ > n − ∑_{k=1}^{∞} \frac{n}{p^{k}} = \frac{p − 2}{p − 1} n ≥ 0.

By Lemma 1 with S(z) replaced by exp (T (z)), it follows that

exp (T (z)) ∈ 1 + ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]].

In order to finish the proof of the lemma, we observe that for T = G/F with F (z) ∈ 1 + ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]], we have the equivalence

T (z^{p}) − pT (z) ∈ p ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]] if and only if F (z)G(z^{p}) − pF (z^{p})G(z) ∈ p ∑_{i=1}^{d} z_{i}ℤ_{p}[[z]],

since F (z) is invertible in ℤ_{p}[[z]]. □

5. Proof of Lemma 3

The following proof was kindly provided by an anonymous referee. It differs from the proof in Section 6 of [14] even in the case d = 1, and thus provides an alternative argument.

For convenience, we shall drop the upper index in N_{i}^{(1)} in this section, that is, we write

B(N^{(1)}, m) = \frac{(N^{(1)} · m)!}{∏_{i=1}^{d} m_{i}!^{N_{i}^{(1)}}} = \frac{(N · m)!}{∏_{i=1}^{d} m_{i}!^{N_{i}}} .

We claim that one may restrict oneself to the case L = N. Indeed, we have the factorisation

\frac{(N · m)!}{∏_{i=1}^{d} m_{i}!^{N_{i}}} = \frac{(L · m)!}{∏_{i=1}^{d} m_{i}!^{L_{i}}} × \frac{(N · m)!}{(L · m)! ∏_{i=1}^{d} m_{i}!^{N_{i}−L_{i}}} ,

where the second factor is obviously an integer. If this is inserted in (2.2), with m_{i} = a_{i}+pk_{i} for all i, then the claim becomes evident.

We continue to use the notation m_{i} = a_{i} + pk_{i} for all i, so that we have to prove

\frac{(N · m)!}{∏_{i=1}^{d} m_{i}!^{N_{i}}} (H_{⌊\frac{1}{p} N·m⌋} − H_{N·⌊\frac{1}{p} m⌋}) ∈ pℤ_{p},  (5.1)

where ⌊\frac{1}{p}m⌋ = (⌊\frac{1}{p}m_{1}⌋, ⌊\frac{1}{p}m_{2}⌋, …, ⌊\frac{1}{p}m_{d}⌋). If ⌊\frac{1}{p}N · m⌋ = N · ⌊\frac{1}{p}m⌋, then the difference of harmonic numbers in (5.1) is zero, whence Lemma 3 is trivially true in this case. On the other hand, if ⌊\frac{1}{p}N · m⌋ > N · ⌊\frac{1}{p}m⌋, then we claim that the first factor in the expression in (5.1) is divisible by

p(N · ⌊\frac{1}{p}m⌋ + 1)(N · ⌊\frac{1}{p}m⌋ + 2) ⋯ (⌊N · \frac{1}{p}m⌋).

Clearly, this would immediately imply Lemma 3 in this case also.

In order to establish the claim, we observe that

v_{p}(n!) = ⌊n/p⌋ + v_{p}(⌊n/p⌋!),

and hence

v_{p} (\frac{(N · m)!}{∏_{i=1}^{d} m_{i}!^{N_{i}}} ) = ⌊\frac{1}{p}N · m⌋ − N · ⌊\frac{1}{p}m⌋ + v_{p} (\frac{⌊\frac{1}{p}N · m⌋!}{∏_{i=1}^{d} ⌊\frac{1}{p}m_{i}⌋!^{N_{i}}} ) .  (5.2)

We have the factorisation

\frac{⌊\frac{1}{p}N · m⌋!}{∏_{i=1}^{d} ⌊\frac{1}{p}m_{i}⌋!^{N_{i}}} = (⌊N · \frac{1}{p}m⌋) ⋯ (N · ⌊\frac{1}{p}m⌋ + 2)(N · ⌊\frac{1}{p}m⌋ + 1)\frac{(N · ⌊\frac{1}{p}m⌋)!}{∏_{i=1}^{d} ⌊\frac{1}{p}m_{i}⌋!^{N_{i}}} ,

where the last factor on the right-hand side is an integer. If this is inserted in (5.2), then together with our assumption that ⌊\frac{1}{p}N · m⌋ > N · ⌊\frac{1}{p}m⌋ and the fact that (5.2) holds for all primes p, the claim follows. This finishes the proof of the lemma. □

6. A combinatorial lemma

In this section, we generalise a combinatorial lemma due to Dwork (see [11, Lemma 4.2]) to several variables.

Lemma 5. Let r be a non-negative integer, let Z and W be maps from ℤ^{d} to a ring R, and let

\overline{W}_{r}(m) = ∑_{p^{r}m≤k≤p^{r}(m+1)−1} W (k).

Then

∑_{0≤k≤(p^{r}−1)1} Z(k)W (k) = Z(0)\overline{W}_{r}(0)

+ ∑_{s=0}^{r−1} ( ∑_{0≤m≤(p^{r−s}−1)1} (Z(m_{1}p^{s}, …, m_{d}p^{s}) − Z (⌊\frac{m_{1}}{p} ⌋ p^{s+1}, …, ⌊\frac{m_{d}}{p} ⌋ p^{s+1})) \overline{W}_{s}(m)). (6.1)

We give two proofs, both of which have their merits. The first one shows that behind the formula there is a combinatorial decomposition of the summation range, see (6.2). The second one, which was kindly supplied to us by an anonymous referee, provides a recursive “construction” of the summation formula.

First proof of Lemma 5. Let

X_{s} = ∑_{0≤m≤(p^{r−s}−1)1} Z(m_{1}p^{s}, …, m_{d}p^{s})\overline{W}_{s}(m)

and

Y_{s} = ∑_{0≤m≤(p^{r−s}−1)1} Z (⌊\frac{m_{1}}{p} ⌋ p^{s+1}, …, ⌊\frac{m_{d}}{p} ⌋ p^{s+1}) \overline{W}_{s}(m).

By definition, we have

X_{s} = ∑_{0≤m≤(p^{r−s}−1)1} ( ∑_{p^{s}m≤k≤p^{s}(m+1)−1} Z(m_{1}p^{s}, …, m_{d}p^{s})W (k_{1}, …, k_{d})).

For k_{j} ∈ {m_{j}p^{s}, …, (m_{j} + 1)p^{s} − 1}, we have m_{j} = ⌊k_{j}/p^{s}⌋, j = 1, …, d, and furthermore we have the partition

{0, …, p^{r} − 1}^{d} = ⋃_{0≤m≤(p^{r−s}−1)1} ∏_{j=1}^{d} {m_{j}p^{s}, …, (m_{j} + 1)p^{s} − 1}.  (6.2)

Hence, it follows that

X_{s} = ∑_{0≤k≤(p^{r}−1)1} Z (⌊\frac{k_{1}}{p^{s}} ⌋ p^{s}, …, ⌊\frac{k_{d}}{p^{s}} ⌋ p^{s}) W (k_{1}, …, k_{d}).

Similarly, we have

Y_{s} = ∑_{0≤k≤(p^{r}−1)1} Z (⌊\frac{k_{1}}{p^{s+1}} ⌋ p^{s+1}, …, ⌊\frac{k_{d}}{p^{s+1}} ⌋ p^{s+1}) W (k_{1}, …, k_{d}),

where we used that ⌊\frac{1}{p} ⌊\frac{k}{p^{s}} ⌋⌋ = ⌊\frac{k}{p^{s+1}} ⌋. We therefore have

∑_{s=0}^{r−1}(X_{s} − Y_{s}) = ∑_{0≤k≤(p^{r}−1)1} W (k_{1}, …, k_{d})

× ∑_{s=0}^{r−1} (Z (⌊\frac{k_{1}}{p^{s}} ⌋ p^{s}, …, ⌊\frac{k_{d}}{p^{s}} ⌋ p^{s}) − Z (⌊\frac{k_{1}}{p^{s+1}} ⌋ p^{s+1}, …, ⌊\frac{k_{d}}{p^{s+1}} ⌋ p^{s+1}))

= ∑_{0≤k≤(p^{r}−1)1} W (k)(Z(k) − Z(0)),

because the sum over s is a telescoping sum. Since

∑_{0≤k≤(p^{r}−1)1} W (k) = \overline{W}_{r}(0),

this completes the proof of the lemma. □

Second proof of Lemma 5. We observe that

∑_{0≤k≤(p^{r}−1)1} Z(k)W (k) = ∑_{0≤k≤(p^{r}−1)1} (Z(k) − Z (⌊\frac{k_{1}}{p} ⌋ p, …, ⌊\frac{k_{d}}{p} ⌋ p)) W (k)

+ ∑_{0≤k′≤(p^{r−1}−1)1} Z(pk′)\overline{W}_{1}(k′),

where k′ = (⌊\frac{k_{1}}{p} ⌋, ⌊\frac{k_{2}}{p} ⌋, …, ⌊\frac{k_{d}}{p} ⌋). This construction is now iterated. □

7. Proof of Theorem 1

We adapt Dwork’s proof [11, Theorem 1.1] of the special case d = 1, that is, the case in which there is just one variable.

For integer vectors k, K, v ∈ ℤ with k ≥ 0 and 0 ≤ v_{i} < p for i = 1, 2, …, d, set

U(k, K) = A(v + p(K − k))A(k) − A(v + pk)A(K − k),

being 0 if k_{i} > K_{i} for some i, which is the case in particular if K_{i} < 0 for some i. Furthermore, for a vector m ∈ ℤ^{d} with m ≥ 0, set

H(m, K; s) = ∑_{p^{s}m≤k≤p^{s}(m+1)−1} U(k, K),  (7.1)

being 0 if K_{i} < 0 for some i. (The reader should recall that, by definition, 1 is the all 1 vector.) We omit to indicate the dependence on p and v in order to not overload notation.

Lemma 6. Let k, K, v ∈ ℤ with k ≥ 0 and 0 ≤ v_{i} < p for i = 1, 2, …, d. Then there hold the following three facts:

(i) We have U(K − k, K) = −U(k, K).

(ii) For all integer vectors M with p^{s+1}(M + 1) > K, we have

∑_{0≤m≤M} H(m, K; s) = 0.

(iii) We have

H(k, K; s + 1) = ∑_{0≤i≤(p−1)1} H(i + pk, K; s).

Proof. The assertion (i) is obvious.

(ii) We have

∑_{0≤m≤M} H(m, K; s) = ∑_{0≤m≤M} ( ∑_{p^{s}m≤k≤p^{s}(m+1)−1} U(k, K))

= ∑_{0≤k≤p^{s}(M+1)−1} U(k, K)

= ∑_{0≤k≤K} U(k, K)

= 0.

Here, in order to pass from the second to the third line, we used the fact that U(k, K) = 0 if k_{i} > K_{i} for some i between 1 and d. To obtain the last line, we used the functional equation given in (i).

(iii) We have

∑_{0≤i≤(p−1)1} H(i + pk, K; s) = ∑_{0≤i≤(p−1)1} ( ∑_{p^{s}(i+pm)≤k≤p^{s}(i+pm+1)−1} U(k, K)),

and it is rather straightforward to see that this sum simply equals H(m, K; s + 1). □

Proof of Theorem 1. We define two assertions, denoted by α_{s} and β_{t,s}, in the following way: for all s ≥ 0, α_{s} is the assertion that the congruence

H(m, K; s) ≡ 0 mod p^{s+1}g(m)ℤ_{p}

holds for all vectors m, K ∈ ℤ^{d} with m ≥ 0.

For all integers s and t with 0 ≤ t ≤ s, β_{t,s} is the assertion that the congruence

H(m, K + p^{s}m; s) ≡ ∑_{0≤k≤(p^{s−t}−1)1} \frac{A(k + p^{s−t}m)}{A(k)} H(k, K; t) mod p^{s+1}g(m)ℤ_{p} (7.2)

holds for all vectors m, K ∈ ℤ^{d} with m ≥ 0.

Moreover, we define three further assertions A1, A2, A3:

A1: for all vectors k, K ∈ ℤ^{d} with k ≥ 0, we have U(k, K) ∈ pg(k)ℤ_{p}.

A2: for all vectors m, k, K ∈ ℤ^{d} and integers s ≥ 0 with m ≥ 0 and 0 ≤ k_{i} < p^{s} for i = 1, 2, …, d, we have

U(k + p^{s}m, K + p^{s}m) ≡ \frac{A(k + p^{s}m)}{A(k)}U(k, K) mod p^{s+1}g(m)ℤ_{p}.

A3: for all integers s and t with 0 ≤ t < s, we have

“α_{s−1} and β_{t,s} together imply β_{t+1,s}.”

In the following, we shall first show that Assertions A1, A2, A3 altogether imply Theorem 1, see the “first step” below. Subsequently, in the “second step,” we show that Assertions A1, A2, A3 hold indeed.

First step. We claim that Theorem 1 follows from A1, A2 and A3. So, from now on we shall assume that A1, A2 and A3 are true. Our goal is to show that α_{s} holds for all s ≥ 0. We shall accomplish this by induction on s ≥ 0.

We begin by establishing α_{0}. To do so, we observe that

H(m, K; 0) = U(m, K),  (7.3)

that is, that Assertion α_{0} is equivalent to A1. Hence, Assertion α_{0} is true.

We now suppose that α_{s−1} is true. We shall show by induction on t ≥ 0 that β_{t,s} is true for all t ≤ s. Because of A3, it suffices to prove that β_{0,s} is true. To do so, we see that

∑_{0≤k≤(p^{s}−1)1} \frac{A(k + p^{s}m)}{A(k)} H(k, K; 0)

= ∑_{0≤k≤(p^{s}−1)1} \frac{A(k + p^{s}m)}{A(k)} U(k, K)

≡ ∑_{0≤k≤(p^{s}−1)1} U(k + p^{s}m, K + p^{s}m) mod p^{s+1}g(m)ℤ_{p}

≡ H(m, K + p^{s}m; s) mod p^{s+1}g(m)ℤ_{p}.  (7.4)

Here, the first equality results from (7.3), the subsequent congruence results from A2, and the last line is obtained by remembering the definition (7.1) of H (there holds in fact equality between the last two lines). The congruence (7.4) is nothing else but Assertion β_{0,s}, which is therefore proved under our assumptions.

The above argument shows in particular that Assertion β_{s,s} is true, which means that we have the congruence

H(m, K + p^{s}m; s) ≡ \frac{A(m)}{A(0)} H(0, K; s) mod p^{s+1}g(m)ℤ_{p}.  (7.5)

Let us now consider the property γ_{K} defined by

γ_{K}: H(0, K; s) ≡ 0 mod p^{s+1}ℤ_{p}.

This property holds certainly if K_{i} < 0 for some i because in that case each term of the multiple sum that defines H vanishes. We want to show that the assertion also holds when K ≥ 0. Let K′ be one of the vectors of non-negative integers (if there is at all) such that |K′| = K′_{1} + K′_{2} + ⋯ + K′_{d} is minimal and γ_{K′} does not hold. Let m ∈ ℤ^{d} be a vector with m ≥ 0 and |m| > 0, and set K = K′ − p^{s}m. Since |K| < |K|′, we have

H(0, K; s) ≡ 0 mod p^{s+1}ℤ_{p}

because γ_{K} holds by minimality of K′. Since A(m)/A(0) ∈ ℤ_{p} by Properties (i) and (ii) in the statement of Theorem 1, it follows from (7.5) that

H(m, K′; s) ≡ 0 mod p^{s+1}ℤ_{p}  (7.6)

provided m ≥ 0 et |m| > 0.

However, by Lemma 6, (ii), we know that

∑_{0≤m≤M} H(m, K′; s) = 0

if one chooses M sufficiently large. Isolating the term H(0, K′; s), this equation can be rewritten as

H(0, K′; s) = −_{∑_{0≤m≤M} ; |m|>0}H(m, K′; s).

The sum on the right-hand side is congruent to 0 mod p^{s+1} by (7.6), whence

H(0, K′; s) ≡ 0 mod p^{s+1}.

This means that γ_{K′} is true, which is absurd. Assertion γ_{K} is therefore true for all K ∈ ℤ^{d}.

Let us now return to Assertion β_{s,s}, which is displayed explicitly in (7.5). We have just shown that H(0, K; s) ≡ 0 mod p^{s+1}, while A(m)/A(0) ∈ g(m)ℤ_{p} by Properties (i) and (ii) in the statement of Theorem 1. Hence, we have also

H(m, K + p^{s}m; s) ≡ 0 mod p^{s+1}g(m)ℤ_{p}.

By replacing K by K − p^{s}m (which is possible because K can be chosen freely from ℤ^{d}), we see that this is nothing else but Assertion α_{s}. Thus, Theorem 1 follows indeed from the truth of A1, A2 and A3.

Second step. It remains to prove Assertions A1, A2 and A3 themselves, which we shall do in this order.

Proof of A1. The assertion holds if k_{i} > K_{i} or if K_{i} < 0 for some i. If K ≥ k ≥ 0, we have

U(k, K) = A(K − k)A(v) (\frac{A(v + pk)}{A(v)} − \frac{A(k)}{A(0)})

+ A(k)A(v) (\frac{A(K − k)}{A(0)} − \frac{A(v + p(K − k))}{A(v)}) .

Property (iii) in the statement of Theorem 1 with u = 0, n = k, s = 0 says that

\frac{A(v + pk)}{A(v)} − \frac{A(k)}{A(0)} ∈ p \frac{g(k)}{g(v)} ℤ_{p}

while its special case in which u = 0, n = K − k, s = 0 reads

\frac{A(K − k)}{A(0)} − \frac{A(v + p(K − k))}{A(v)} ∈ p \frac{g(K − k)}{g(v)} ℤ_{p}.

Hence,

A(K − k)A(v) (\frac{A(v + pk)}{A(v)} − \frac{A(k)}{A(0)}) ∈ p g(k)A(K − k)\frac{A(v)}{g(v)} ℤ_{p} ⊆ pg(k) ℤ_{p}

and

A(k)A(v) (\frac{A(K − k)}{A(0)} − \frac{A(v + p(K − k))}{A(v)}) ∈ pg(k)g(K − k)\frac{A(k)}{g(k)} \frac{A(v)}{g(v)} ℤ_{p} ⊆ pg(k) ℤ_{p},

where the inclusion relations result from Property (ii) in the statement of Theorem 1. It therefore follows that

U(k, K) ∈ pg(k)ℤ_{p},

which proves Assertion A1.

Proof of A2. By a straightforward calculation, we have

U(k + p^{s}m, K + p^{s}m) − \frac{A(k + p^{s}m)}{A(k)}U(k, K)

= −A(K − k)A(v + pk) (\frac{A(v + pk + p^{s+1}m)}{A(v + pk)} − \frac{A(k + p^{s}m)}{A(k)}) .

If K_{i} < 0 for some i, the right-hand side is zero since A(K − k) = 0, whence Assertion A2 is trivially true. If K ≥ 0, by Properties (iii) and (ii) in the statement of Theorem 1, the right-hand side is an element of

A(K − k)A(v + pk)\frac{g(m)}{g(v + pk)} p^{s+1}ℤ_{p} ⊆ g(m)p^{s+1}ℤ_{p},

which proves Assertion A2 in this case as well.

Proof of A3. Let 0 ≤ t < s, and assume that α_{s−1} and β_{t,s} are true. Under these assumptions, we must deduce the truth of Assertion β_{t+1,s}.

In the assertion β_{t,s}, we replace the summation index k in the sum on the right-hand side of (7.2) by i + pu, where 0 ≤ i_{ℓ} < p − 1 and 0 ≤ u_{ℓ} < p^{s−t−1} for ℓ = 1, 2, …, d. Thus, we obtain that

H(m, K + p^{s}m; s) ≡ ∑_{0≤i≤(p−1)1} ( ∑_{0≤u≤(p^{s−t−1}−1)1} \frac{A(i + pu + p^{s−t}m)}{A(i + pu)} H(i + pu, K; t))

mod p^{s+1}g(m)ℤ_{p}. (7.7)

Define

X := H(m, K + p^{s}m; s) − ∑_{0≤u≤(p^{s−t−1}−1)1} \frac{A(u + p^{s−t−1}m)}{A(u)} ∑_{0≤i≤(p−1)1} H(i + pu, K; t).

Since β_{t,s} (in the form (7.7)) is true, we have

X ≡ ∑_{0≤i≤(p−1)1} ( ∑_{0≤u≤(p^{s−t−1}−1)1} H(i + pu, K; t)

× (\frac{A(i + pu + p^{s−t}m)}{A(i + pu)} − \frac{A(u + p^{s−t−1}m)}{A(u)}) ) mod p^{s+1}g(m) ℤ_{p}.

Since u_{i} < p^{s−t−1} for all i, Property (iii) in the statement of Theorem 1 implies that

\frac{A(i + pu + p^{s−t}m)}{A(i + pu)} − \frac{A(u + p^{s−t−1}m)}{A(u)} ∈ p^{s+1} \frac{g(m)}{g(i + pu)} ℤ_{p}.  (7.8)

Moreover, since t < s, Assertion α_{s−1} implies that

H(i + pu, K; t) ∈ p^{t+1}g(i + pu)ℤ_{p}.  (7.9)

It now follows from (7.8) and (7.9) that X ≡ 0 mod p^{s+1}g(m)ℤ_{p}.

However, by Lemma 6, (iii), we know that

∑_{0≤i≤(p−1)1} H(i + pu, K; t) = H(u, K; t + 1),

which can be used to simplify X to

X = H(m, K + p^{s}m; s) − ∑_{0≤u≤(p^{s−t−1}−1)1} \frac{A(u + p^{s−t−1}m)}{A(u)}H(u, K; t + 1).

Since X ≡ 0 mod p^{s+1}g(m)ℤ_{p}, the preceding identity shows that

H(m, K + p^{s}m; s) ≡ ∑_{0≤u≤(p^{s−t−1}−1)1} \frac{A(u + p^{s−t−1}m)}{A(u)}H(u, K; t + 1) mod p^{s+1}g(m)ℤ_{p}.

This is nothing else but Assertion β_{t+1,s}. Hence, Assertion A3 is established.

This completes the proof of Theorem 1. □

8. Theorem 1 implies Theorem 2

We want to prove that Theorem 1 can be applied for A = g = B_{N}. In order to see this, we first establish some intermediary lemmas, extending corresponding auxiliary results in Section 7 of [14] to higher dimensions.

Lemma 7. Under the assumptions of Theorem 1, we have

\frac{B_{N}(v + pu + p^{s+1}n)}{B_{N}(pu + p^{s+1}n)} = \frac{B_{N}(v + pu)}{B_{N}(pu)} + O(p^{s+1}),

where O(R) denotes an element of Rℤ_{p}.

Proof. Recalling the definition of B(N^{(j)}, m) in (2.1), we have

\frac{B(N^{(j)}, v + pu + p^{s+1}n)}{B(N^{(j)}, pu + p^{s+1}n)}

= \frac{(∑_{i=1}^{d} N_{i}^{(j)}(pu_{i} + p^{s+1}n_{i}) + ∑_{i=1}^{d} N_{i}^{(j)}v_{i}) ⋯ (∑_{i=1}^{d} N_{i}^{(j)}(pu_{i} + p^{s+1}n_{i}) + 1)}{∏_{i=1}^{d} ((v_{i} + pu_{i} + p^{s+1}n_{i}) ⋯ (1 + pu_{i} + p^{s+1}n_{i}))^{N_{i}^{(j)}}}

= \frac{(∑_{i=1}^{d} N_{i}^{(j)}(pu_{i}) + ∑_{i=1}^{d} N_{i}^{(j)}v_{i}) ⋯ (∑_{i=1}^{d} N_{i}^{(j)}(pu_{i}) + 1) + O(p^{s+1})}{∏_{i=1}^{d} ((v_{i} + pu_{i}) ⋯ (1 + pu_{i}))^{N_{i}^{(j)}} + O(p^{s+1})}.

We claim that this implies

\frac{B(N^{(j)}, v + pu + p^{s+1}n)}{B(N^{(j)}, pu + p^{s+1}n)}

= \frac{(∑_{i=1}^{d} N_{i}^{(j)}(pu_{i}) + ∑_{i=1}^{d} N_{i}^{(j)}v_{i}) ⋯ (∑_{i=1}^{d} N_{i}^{(j)}(pu_{i}) + 1)}{∏_{i=1}^{d} ((v_{i} + pu_{i}) ⋯ (1 + pu_{i}))^{N_{i}^{(j)}}} + O(p^{s+1})

= \frac{B(N^{(j)}, v + pu)}{B(N^{(j)}, pu)} + O(p^{s+1}).

Indeed, if v = 0, then this holds trivially. If v > 0, then, together with the hypothesis v_{i} < p, we infer that (v_{i} + pu_{i})(v_{i} + pu_{i} − 1) ⋯ (1 + pu_{i}) is not divisible by p, which implies in particular that B(N^{(j)}, v + pu)/B(N^{(j)}, pu) ∈ ℤ_{p}. This allows us to arrive at the above conclusion in the same style as in Section 7.1 in [14].

By taking products, we deduce

∏_{j=1}^{k} \frac{B(N^{(j)}, v + pu + p^{s+1}n)}{B(N^{(j)}, pu + p^{s+1}n)} = ∏_{j=1}^{k} (\frac{B(N^{(j)}, v + pu)}{B(N^{(j)}, pu)} + O(p^{s+1}).) .

By expanding the product on the right-hand side and using that \frac{B(N^{(j)}, v + pu)}{B(N^{(j)}, pu)} ∈ ℤ_{p},

we obtain the assertion of the lemma. □

For the proof of Lemma 9 below, we will use the p-adic gamma function, which is defined on integers n ≥ 1 by

Γ_{p}(n) = (−1)^{n} ∏_{(k,p)=1^{k=1}}^{n−1} k.

In the following lemma, we collect some facts about Γ_{p}.

Lemma 8. (i) For all integers n ≥ 1, we have

\frac{(np)!}{n!} = (−1)^{np+1}p^{n}Γ_{p}(1 + np).

(ii) For all integers k ≥ 1, n ≥ 1, s ≥ 0, we have

Γ_{p}(k + np^{s}) ≡ Γ_{p}(k) mod p^{s}.

The above two properties of the p-adic gamma function are now used in the proof of the following result.

Lemma 9. We have

\frac{B_{N}(pu + p^{s+1}n)}{B_{N}(u + p^{s}n)} = \frac{B_{N}(pu)}{B_{N}(u)}(1 + O(p^{s+1})).

Proof. We have

\frac{B(N^{(j)}, pu + p^{s+1}n)}{B(N^{(j)}, u + p^{s}n)} = (−1)^{1+|N^{(j)}|} \frac{Γ_{p}(1 + N^{(j)} · (pu + p^{s+1}n))}{∏_{i=1}^{d} Γ_{p}(1 + pu_{i} + p^{s+1}n_{i})^{N_{i}^{(j)}}}  (8.1)

= (−1)^{1+|N^{(j)}|} \frac{Γ_{p}(1 + pN^{(j)} · u) + O(p^{s+1})}{∏_{i=1}^{d} Γ_{p}(1 + pu_{i})^{N_{i}^{(j)}} + O(p^{s+1})}  (8.2)

= (−1)^{1+|N^{(j)}|} \frac{Γ_{p}(1 + pN^{(j)} · u)}{∏_{i=1}^{d} Γ_{p}(1 + pu_{i})^{N_{i}^{(j)}}} (1 + O(p^{s+1}))  (8.3)

= \frac{B(N^{(j)}, pu)}{B(N^{(j)}, u)}(1 + O(p^{s+1})).  (8.4)

where (i) of Lemma 8 is used to see (8.1) and (8.4), and (ii) is used for (8.2). Equation (8.3) holds because Γ_{p}(1 + pu_{i}) and Γ_{p}(1 + pN^{(j)} · u) are both not divisible by p. Taking the product over j = 1, 2, …, k, we obtain the assertion of the lemma. □

Before proceeding, we remark that v_{p}(B(N^{(j)}, p^{s}u)/B(N^{(j)}, u)) = 0 for any integer s ≥ 0, which can be proved in the same way as Lemma 13 in [14]. This property will be used twice below.

We now multiply both sides of the congruences obtained in Lemmas 7 and 9. Thus, we obtain

\frac{B_{N}(v + pu + np^{s+1})}{B_{N}(u + np^{s})} = \frac{B_{N}(v + pu)}{B_{N}(u)}(1 + O(p^{s+1})) + \frac{B_{N}(pu)}{B_{N}(u)}O(p^{s+1})

= \frac{B_{N}(v + pu)}{B_{N}(u)}(1 + O(p^{s+1})) + O(p^{s+1})

(since v_{p}(B_{N}(pu)/B_{N}(u)) = 0 by the remark above), which, in its turn, can be rewritten as

\frac{B_{N}(v + pu + np^{s+1})}{B_{N}(v + pu)} = \frac{B_{N}(u + np^{s})}{B_{N}(u)} + \frac{B_{N}(u + np^{s})}{B_{N}(u)}O(p^{s+1}) + \frac{B_{N}(u + np^{s})}{B_{N}(v + pu)}O(p^{s+1}).

It remains to show that

\frac{B_{N}(u + np^{s})}{B_{N}(u)} ∈ \frac{B_{N}(n)}{B_{N}(v + pu)} ℤ_{p}  (8.5)

and

\frac{B_{N}(u + np^{s})}{B_{N}(v + pu)} ∈ \frac{B_{N}(n)}{B_{N}(v + pu)} ℤ_{p}.  (8.6)

These two facts will follow from the next lemma.

Lemma 10. For all non-negative integers s, all integer vectors n ∈ ℤ^{d} with n ≥ 0, and all integer vectors u ∈ ℤ^{d} with 0 ≤ u_{i} < p^{s}, i = 1, 2, …, d, we have

\frac{B_{N}(u + np^{s})}{B_{N}(u)} ∈ B_{N}(n)ℤ_{p}.

Proof. We have

\frac{B(N^{(j)}, u + np^{s})}{B(N^{(j)}, u)} = ^{\begin{matrix}( & ∑_{i=1}^{d} N_{i}^{(j)}(u_{i}+n_{i}p^{s}) ; ∑_{i=1}^{d} N_{i}^{(j)}u_{i} & ) \\ ∏ & {d \atop i=1} & \binom{u_{i}+n_{i}p^{s}}{u_{i}} & N_{i}^{(j)} \end{matrix}} · \frac{B(N^{(j)}, np^{s})}{B(N^{(j)}, n)} · B(N^{(j)}, n).

On the right-hand side, the term B(N^{(j)}, np^{s})/B(N^{(j)}, n) and the binomial coefficients \binom{u_{i}+n_{i}p^{s}}{u_{i}} have vanishing p-adic valuation (this has already been observed in the paragraph after the end of the proof of Lemma 9). Thus we have

\frac{B(N^{(j)}, u + np^{s})}{B(N^{(j)}, u)} ∈ B(N^{(j)}, n)ℤ_{p}.  (8.7)

The lemma follows by taking the product over j ∈ {1, …, k} of both sides of (8.7). □

The preceding lemma implies

\frac{B_{N}(u + np^{s})}{B_{N}(u)} ∈ B_{N}(n)ℤ_{p} ⊆ \frac{B_{N}(n)}{B_{N}(v + pu)} ℤ_{p},

which proves (8.5). Moreover, still due to Lemma 10, we have

\frac{B_{N}(u + np^{s})}{B_{N}(v + pu)} = \frac{B_{N}(u + np^{s})}{B_{N}(u)} · B_{N}(u) · \frac{1}{B_{N}(v + pu)}

∈ B_{N}(u) · \frac{B_{N}(n)}{B_{N}(v + pu)} ℤ_{p} ⊆ \frac{B_{N}(n)}{B_{N}(v + pu)} ℤ_{p},

which proves (8.6). Therefore,

\frac{B_{N}(v + pu + np^{s+1})}{B_{N}(v + pu)} − \frac{B_{N}(u + np^{s})}{B_{N}(u)} ∈ p^{s+1} \frac{B_{N}(n)}{B_{N}(v + pu)} ℤ_{p},

which shows that Property (iii) of Theorem 1 is satisfied. Since Properties (i) and (ii) are trivially true, we can hence apply the latter theorem.

9. Proof of Lemma 4

The claim is trivially true if p divides m_{i} for all i. We may therefore assume that p does not divide m_{i} for some i between 1 and d for the rest of the proof. Let us write m = a+pj, with 0 ≤ a_{i} < p for all i (but at least one a_{i} is positive). We are apparently in a similar situation as in Lemma 3. Indeed, we may derive Lemma 4 from Lemma 3. In order to see this, we observe that

H_{∑_{i=1}^{d} L_{i}m_{i}p^{s}} − H_{∑_{i=1}^{d} L_{i}⌊\frac{m_{i}}{p} ⌋p^{s+1}} = ∑_{ε=1}^{p^{s}L·a} \frac{1}{p^{s+1}L · j + ε}

= ∑_{ε=1}^{⌊L·a/p⌋} \frac{1}{p^{s+1}L · j + p^{s+1}ε} +_{∑_{ε=1}^{p^{s}L·a} ; p^{s+1}∤ε} \frac{1}{p^{s+1}L · j + ε}

= \frac{1}{p^{s+1}} (H_{L·j+⌊L·a/p⌋} − H_{L·j}) +_{∑_{ε=1}^{p^{s}L·a} ; p^{s+1}∤ε} \frac{1}{p^{s+1}L · j + ε}.

Because of v_{p}(x + y) ≥ min{v_{p}(x), v_{p}(y)}, this implies

v_{p}(H_{∑_{i=1}^{d} L_{i}m_{i}p^{s}} − H_{∑_{i=1}^{d} L_{i}⌊\frac{m_{i}}{p} ⌋p^{s+1}}) ≥ min\{−1 − s + v_{p}(H_{L·j+⌊L·a/p⌋} − H_{L·j}), −s\}.

It follows that

v_{p}(B_{N}(m)(H_{∑_{i=1}^{d} L_{i}m_{i}p^{s}} − H_{∑_{i=1}^{d} L_{i}⌊\frac{m_{i}}{p} ⌋p^{s+1}}))

≥ −1 − s + min {v_{p}(B_{N}(a + pj)(H_{L·j+⌊L·a/p⌋} − H_{L·j})), 1 + v_{p}(B_{N}(a + pj))} .

Use of Lemma 3 then completes the proof. □

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C. Krattenthaler, Fakultät für Mathematik, Universität Wien, Nordbergstraße 15,

A-1090 Vienna, Austria. WWW: http://www.mat.univie.ac.at/~kratt.

T. Rivoal, Institut Fourier, CNRS UMR 5582, Université Grenoble 1, 100 rue des Maths, BP 74, 38402 Saint-Martin d’Hères cedex, France.

WWW: http://www-fourier.ujf-grenoble.fr/~rivoal.