1.1 Classical results of Lucas and Kummer

It is a well-known result of Lucas [Reference LucasLuc78] that, for all nonnegative integers $m,n$ and all primes $p$ , we have

where $m=m_{0}+m_{1}p+\cdots +m_{k}p^{k}$ and $n=n_{0}+n_{1}p+\cdots +n_{k}p^{k}$ are the base- $p$ expansions of $m$ and $n$ .

In particular, a prime $p$ divides the binomial $\binom{m}{n}$ if and only if there is $0\leqslant i\leqslant k$ such that $m_{i}<n_{i}$ . Precisely, Kummer proved in [Reference KummerKum52] that, for all nonnegative integers $m\geqslant n$ , the $p$ -adic valuationFootnote ¹ of the binomial $\binom{m}{n}$ is the number of carries which occur when $n$ is added to $m-n$ in base $p$ . As a consequence, we have

In this article, we show that many sequences $(A(n))_{n\geqslant 0}$ of Apéry-like numbers satisfy congruences similar to (1.1), that is, for all nonnegative integers $n$ and all primes $p$ , we have

where $n=n_{0}+n_{1}p+\cdots +n_{k}p^{k}$ is the base- $p$ expansion of $n$ . Furthermore, we prove that an analogue of (1.2) holds for those numbers, that is,

which proves Beukers’ conjectures on the $p$ -adic valuation of Apéry numbers.

1.2 Beukers’ conjectures on Apéry numbers

For all nonnegative integers $n$ , we set

These sequences were used in $1979$ by Apéry in his proofs of the irrationality of $\unicode[STIX]{x1D701}(3)$ and $\unicode[STIX]{x1D701}(2)$ (see [Reference ApéryApé79]). In the $1980$ s, several congruences satisfied by these sequences were proved (see, for example, [Reference BeukersBeu85, Reference BeukersBeu87, Reference Chowla, Cowles and CowlesCCC80, Reference GesselGes82, Reference MimuraMim83]). In particular, Gessel proved in [Reference GesselGes82] that $A_{1}$ satisfies the $p$ -Lucas property for all prime numbers $p$ , that is, for any prime $p$ , all $v$ in $\{0,\ldots ,p-1\}$ and all nonnegative integers $n$ , we have

Thereby, if $n=n_{0}+n_{1}p+\cdots +n_{N}p^{N}$ is the base- $p$ expansion of $n$ , then we obtain

In particular, $p$ divides $A_{1}(n)$ if and only if there exists $k$ in $\{0,\ldots ,N\}$ such that $p$ divides $A_{1}(n_{k})$ . Beukers stated in [Reference BeukersBeu86] two conjectures, when $p=5$ or $11$ , which generalize this property.Footnote ² Before stating these conjectures, we observe that the set of all $v$ in $\{0,\ldots ,4\}$ (respectively $v$ in $\{0,\ldots ,10\}$ ) satisfying $A_{1}(v)\equiv 0\;\text{mod}\;5$ (respectively $A_{1}(v)\equiv 0\;\text{mod}\;11$ ) is $\{1,3\}$ (respectively $\{5\}$ ).

Similarly, Sequence $A_{2}$ satisfies the $p$ -Lucas property for all primes $p$ . Furthermore, Beukers and Stienstra proved in [Reference Beukers and StienstraBS85] that, if $p\equiv 3\;\text{mod}\;4$ , then $A_{2}((p-1)/2)\equiv 0\;\text{mod}\;p$ , and Beukers stated in [Reference BeukersBeu86] the following conjecture.

Conjectures A–C have been extended to generalized Apéry numbers and any prime $p$ by Deutsch and Sagan in [Reference Deutsch and SaganDS06, Conjecture 5.13] but this extension is false for at least one generalization of Apéry numbers. Indeed, a counterexample is given by

since $A(1)=9\equiv 0\;\text{mod}\;3$ but $A(4)=A(1+3)=1152501$ is not divisible by $3^{2}$ .

The main aim of this article is to prove Theorem 1, stated in § 1.4, which confirms and generalizes Conjectures A–C. First, we introduce some notations which we use throughout this article.

1.3 Notations

In order to study arithmetic properties of sums of products of binomial coefficients, such as Apéry numbers, we first study families, indexed by $\mathbb{N}^{d}$ , of ratios of factorials of linear forms with integer coefficients. For example, we will obtain congruences for $A_{1}(n)$ by studying the factorial ratios

as we have the useful formula

Let $d$ be a positive integer. Given tuples of vectors in $\mathbb{N}^{d}$ , $e=(\mathbf{e}_{1},\ldots ,\mathbf{e}_{u})$ and $f=(\mathbf{f}_{1},\ldots ,\mathbf{f}_{v})$ , we shall prove congruences for the factorial ratios

to deduce arithmetic properties of the numbersFootnote ³

Here $\cdot$ denotes the standard scalar product on $\mathbb{R}^{d}$ and $|\mathbf{n}|=n_{1}+\cdots +n_{d}$ if $\mathbf{n}=(n_{1},\ldots ,n_{d})$ . For example, we obtain that $\mathfrak{S}_{e,f}(n)=A_{1}(n)$ with the tuples

Because of the summation in (1.4), it is usually difficult to study arithmetic properties of $\mathfrak{S}_{e,f}(n)$ ; however, we will show that, in many interesting cases, we can transfer the $p$ -Lucas property from ${\mathcal{Q}}_{e,f}(\mathbf{n})$ to $\mathfrak{S}_{e,f}(n)$ . To that purpose, we define the $p$ -Lucas property for families of $p$ -adic integers indexed by $\mathbb{N}^{d}$ . For all primes $p$ , we write $\mathbb{Z}_{p}$ for the ring of $p$ -adic integers.

If $A=(A(\mathbf{n}))_{\mathbf{n}\in \mathbb{N}^{d}}$ is a $\mathbb{Z}_{p}$ -valued family, then we say that $A$ satisfies the $p$ -Lucas property if, for all vectors $\mathbf{v}$ in $\{0,\ldots ,p-1\}^{d}$ and $\mathbf{n}$ in $\mathbb{N}^{d}$ , we have

If $\mathbf{n}$ is nonzero, then we say that $\mathbf{n}=\mathbf{n}_{0}+\mathbf{n}_{1}p+\cdots +\mathbf{n}_{N}p^{N}$ is the base- $p$ expansion of $\mathbf{n}$ if, for all $i$ in $\{0,\ldots ,N\}$ , we have $\mathbf{n}_{i}\in \{0,\ldots ,p-1\}^{d}$ and $\mathbf{n}_{N}\neq \mathbf{0}$ , where $\mathbf{0}:=(0,\ldots ,0)$ . Hence, if $A$ satisfies the $p$ -Lucas property, then we have

We write ${\mathcal{Z}}_{p}(A)$ for the set of all vectors $\mathbf{v}$ in $\{0,\ldots ,p-1\}^{d}$ such that $A(\mathbf{v})$ belongs to $p\mathbb{Z}_{p}$ . Hence, $A(\mathbf{n})$ is in $p\mathbb{Z}_{p}$ if and only if at least one $\mathbf{n}_{i}$ , $0\leqslant i\leqslant N$ , belongs to ${\mathcal{Z}}_{p}(A)$ . To state our generalization of Conjectures A–C, we define the following counting function. For every nonzero vector $\mathbf{n}$ in $\mathbb{N}^{d}$ whose base- $p$ expansion is $\mathbf{n}=\mathbf{n}_{0}+\mathbf{n}_{1}p+\cdots +\mathbf{n}_{N}p^{N}$ , we write $\unicode[STIX]{x1D6FC}_{p}(A,\mathbf{n})$ for the number of $i$ in $\{0,\ldots ,N\}$ such that $\mathbf{n}_{i}\in {\mathcal{Z}}_{p}(A)$ , and we set $\unicode[STIX]{x1D6FC}_{p}(A,\mathbf{0})=0$ . Thereby, to prove Conjectures A–C, it is enough to show that $A_{i}(n)\in p^{\unicode[STIX]{x1D6FC}_{p}(A_{i},n)}\mathbb{Z}$ with $i=1,p=5$ or $11$ and $i=2$ , $p\equiv 3\;\text{mod}\;4$ .

Our generalization of Beukers’ conjectures will apply to sequences $\mathfrak{S}_{e,f}$ restricted to the following two conditions.

The first condition (the $r$ -admissibility) ensures that we can transfer the $p$ -Lucas property from ${\mathcal{Q}}_{e,f}(\mathbf{n})$ to $\mathfrak{S}_{e,f}(n)$ . If $\mathbf{m}=(m_{1},\ldots ,m_{d})$ and $\mathbf{n}=(n_{1},\ldots ,n_{d})$ belong to $\mathbb{R}^{d}$ , then we write $\mathbf{m}\geqslant \mathbf{n}$ if, for all $i$ in $\{1,\ldots ,d\}$ , we have $m_{i}\geqslant n_{i}$ . Furthermore, we set $\mathbf{1}:=(1,\ldots ,1)\in \mathbb{N}^{d}$ and we write $\mathbf{1}_{k}$ for the vector in $\mathbb{N}^{d}$ , all of whose coordinates equal zero except the $k$ th, which is $1$ . Let ${\mathcal{S}}:=\{1\leqslant i\leqslant u:\mathbf{e}_{i}\geqslant \mathbf{1}\}$ . For every positive integer $r$ , we say that $e$ is $r$ -admissible if

We will use this definition with $r=1$ or $2$ . In the case of the Apéry numbers $A_{1}(n)$ , we study the family ${\mathcal{Q}}_{e,f}$ with the tuple $e=((2,1),(2,1))$ , so that $\#{\mathcal{S}}=2$ and $e$ is $2$ -admissible. As another example, we will also prove a result similar to Beukers’ conjectures for the sequence

We can write

so that $A_{6}(n)=\mathfrak{S}_{e,f}(n)$ with $e=((1,1),(2,0),(0,2))$ . In this case, we have $d=2$ , $\#{\mathcal{S}}=1$ but $e$ is also $2$ -admissible because for $k=1$ or $2$ we have $\#\{2\leqslant i\leqslant 3:\mathbf{e}_{i}\geqslant 2\mathbf{1}_{k}\}=1$ .

The second condition is of differential type. To apply our main result, we need the generating series of $(\mathfrak{S}_{e,f}(n))_{n\geqslant 0}$ to be annihilated by a differential operator of a special form that we describe below. We set $\unicode[STIX]{x1D703}:=z(d/dz)$ and we say that a differential operator ${\mathcal{L}}$ in $\mathbb{Z}_{p}[z,\unicode[STIX]{x1D703}]$ is of type I if there is a nonnegative integer $q$ such that:

1. – ${\mathcal{L}}=P_{0}(\unicode[STIX]{x1D703})+zP_{1}(\unicode[STIX]{x1D703})+\cdots +z^{q}P_{q}(\unicode[STIX]{x1D703})$ with $P_{k}(X)\in \mathbb{Z}_{p}[X]$ for $0\leqslant k\leqslant q$ ;

2. – $P_{0}(\mathbb{Z}_{p}^{\times })\subset \mathbb{Z}_{p}^{\times }$ ;

3. – for all $k$ in $\{2,\ldots ,q\}$ , we have $P_{k}(X)\in \prod _{i=1}^{k-1}(X+i)^{2}\mathbb{Z}_{p}[X]$ .

We say that a differential operator ${\mathcal{L}}$ in $\mathbb{Z}_{p}[z,\unicode[STIX]{x1D703}]$ is of type II if:

1. – ${\mathcal{L}}=P_{0}(\unicode[STIX]{x1D703})+zP_{1}(\unicode[STIX]{x1D703})+z^{2}P_{2}(\unicode[STIX]{x1D703})$ with $P_{k}(X)\in \mathbb{Z}_{p}[X]$ for $0\leqslant k\leqslant 2$ ;

2. – $P_{0}(\mathbb{Z}_{p}^{\times })\subset \mathbb{Z}_{p}^{\times }$ ;

3. – $P_{2}(X)\in (X+1)\mathbb{Z}_{p}[X]$ .

For example, the generating series of $(A_{1}(n))_{n\geqslant 0}$ is annihilated by the differential operator

which is of type I for every prime $p$ . We will also prove a result similar to Beukers’ conjectures for the numbers

The generating series of $A_{5}$ is annihilated by the differential operator

which is of type II for every prime $p$ .

Our main result confirms Conjectures A–C, and also provides surprising similar properties for some deformations of Apéry-like numbers. For example, while proving that, for every prime $p$ and all nonnegative integers $n$ , we have

we will also show that, for every nonnegative integer $a$ , we have

More generally, we will obtain congruences for deformations $\mathfrak{S}_{e,f}^{g}$ of the sequences $\mathfrak{S}_{e,f}$ defined as follows. For any prime $p$ , we write $\mathfrak{F}_{p}^{d}$ for the set of all functions $g:\mathbb{N}^{d}\rightarrow \mathbb{Z}_{p}$ such that, for all nonnegative integers $K$ , there exists a sequence $(P_{K,k})_{k\geqslant 0}$ of polynomial functions with coefficients in $\mathbb{Z}_{p}$ which converges pointwise to $g$ on $\{0,\ldots ,K\}^{d}$ . For all tuples $e$ and $f$ of vectors in $\mathbb{N}^{d}$ , all $g\in \mathfrak{F}_{p}^{d}$ and all nonnegative integers $m$ , we set

1.4 Main results

In the rest of the article, if $e=(\mathbf{e}_{1},\ldots ,\mathbf{e}_{u})$ is a tuple of vectors in $\mathbb{N}^{d}$ , then we set $|e|:=\mathbf{e}_{1}+\cdots +\mathbf{e}_{u}$ . The main result of this article is the following.

In § 1.6, we show that Theorem 1 applies to many classical sequences. In particular, Theorem 1 implies Conjectures A–C. Indeed, we have $A_{1}=\mathfrak{S}_{e_{1},f_{1}}$ and $A_{2}=\mathfrak{S}_{e_{2},f_{2}}$ with $d=2$ ,

and

Furthermore, it is well known that $f_{A_{1}}$ , respectively $f_{A_{2}}$ , is annihilated by the differential operator ${\mathcal{L}}_{1}$ , respectively ${\mathcal{L}}_{2}$ , defined by

and

Since ${\mathcal{L}}_{1}$ and ${\mathcal{L}}_{2}$ are of type I for all primes $p$ , the conditions of Theorem 1 are satisfied by $A_{1}$ and $A_{2}$ , and Conjectures A–C hold. In addition, for all primes $p$ and all nonnegative integers $n$ and $a$ , we obtain that

We provide a similar result which applies to the constant terms of powers of certain Laurent polynomials. Consider a Laurent polynomial

where $\mathbf{a}_{i}\in \mathbb{Z}^{d}$ and $\unicode[STIX]{x1D6FC}_{i}\neq 0$ for $i$ in $\{1,\ldots ,k\}$ . Recall that the Newton polyhedron of $\unicode[STIX]{x1D6EC}$ is the convex hull of $\{\mathbf{a}_{1},\ldots ,\mathbf{a}_{k}\}$ in $\mathbb{R}^{d}$ . Hence, we have the following result.

For example, Theorem 2 applies to Apéry numbers $A_{1}$ thanks to the following formula of Lairez (personal communication, 2013):

By a result of Samol and van Straten [Reference Samol and van StratenSvS15], if $\unicode[STIX]{x1D6EC}(\mathbf{x})\in \mathbb{Z}_{p}[x_{1}^{\pm },\ldots ,x_{d}^{\pm }]$ contains the origin as its only interior integral point, then $([\unicode[STIX]{x1D6EC}(\mathbf{x})^{n}]_{\mathbf{0}})_{n\geqslant 0}$ satisfies the $p$ -Lucas property, which is essential for the proof of Theorem 2. Likewise, the proof of Theorem 1 rests on the fact that $\mathfrak{S}_{e,f}$ satisfies the $p$ -Lucas property when $|e|=|f|$ , $e$ is $2$ -admissible and $f=(\mathbf{1}_{k_{1}},\ldots ,\mathbf{1}_{k_{v}})$ . Since those results deal with multisums of factorial ratios, it seems natural to study similar arithmetic properties for simpler numbers such as families of factorial ratios. To that purpose, we prove Theorem 3 below, which gives an effective criterion for ${\mathcal{Q}}_{e,f}$ to satisfy the $p$ -Lucas property for almost all primes $p$ .Footnote ⁴ Furthermore, Theorem 3 shows that if $A:={\mathcal{Q}}_{e,f}$ satisfies the $p$ -Lucas property for almost all primes $p$ , then, for all nonnegative integers $n$ and all primes $p$ , we have $A(n)\in p^{\unicode[STIX]{x1D6FC}_{p}(A,n)}\mathbb{Z}$ .

To state this result, we introduce some additional notations. For all tuples $e$ and $f$ of vectors in $\mathbb{N}^{d}$ , we write $\unicode[STIX]{x1D6E5}_{e,f}$ for Landau’s function defined, for all $\mathbf{x}$ in $\mathbb{R}^{d}$ , by

where $\lfloor \cdot \rfloor$ denotes the floor function. Therefore, according to Landau’s criterion [Reference LandauLan00] and a result of the author [Reference DelaygueDel13], we have the following dichotomy.

1. – If, for all $\mathbf{x}$ in $[0,1]^{d}$ , we have $\unicode[STIX]{x1D6E5}_{e,f}(\mathbf{x})\geqslant 0$ , then ${\mathcal{Q}}_{e,f}$ is a family of integers.

2. – If there exists $\mathbf{x}$ in $[0,1]^{d}$ such that $\unicode[STIX]{x1D6E5}_{e,f}(\mathbf{x})\leqslant -1$ , then there are only finitely many primes $p$ such that ${\mathcal{Q}}_{e,f}$ is a family of $p$ -adic integers.

In the rest of the article, we write ${\mathcal{D}}_{e,f}$ for the semi-algebraic set of all $\mathbf{x}$ in $[\!0,1\!)^{d}$ such that there exists a component $\mathbf{d}$ of $e$ or $f$ satisfying $\mathbf{d}\cdot \mathbf{x}\geqslant 1$ . Observe that $\unicode[STIX]{x1D6E5}_{e,f}$ vanishes on the nonempty set $[\!0,1\!)^{d}\setminus {\mathcal{D}}_{e,f}$ .

Coster proved in [Reference CosterCos88] results similar to Theorems 1–3 for the coefficients of certain algebraic power series. Namely, given a prime $p\geqslant 3$ , integers $a_{1},\ldots ,a_{p-1}$ and a sequence $A$ such that

Coster proved that, for all nonnegative integers $n$ , we have

1.5 Auxiliary results

The proof of Theorem 1 rests on three important results. The first one is stated rather formally but we believe that it may be useful to study results similar to Beukers’ conjectures for other sequences. Throughout this article, if $(A(n))_{n\geqslant 0}$ is a sequence taking its values in $\mathbb{Z}$ or $\mathbb{Z}_{p}$ , then, for all negative integers $n$ , we set $A(n):=0$ .

We will apply Proposition 1 with $A=\mathfrak{S}_{e,f}$ for some tuples $e$ and $f$ satisfying the conditions of Theorem 1 for a fixed prime $p$ . Then we will choose the set $\mathfrak{B}$ to be the set of the deformations $\mathfrak{S}_{e,f}^{g}$ for $g$ in $\mathfrak{F}_{p}^{d}$ . Taking $g$ to be a constant in $\mathbb{Z}_{p}$ shows that the set $\mathfrak{B}$ contains the $\mathbb{Z}_{p}$ -module $\mathfrak{A}$ spanned by $A$ . The main difficulty in this article is to show, by $p$ -adic techniques, that Assertion (a) in Proposition 1 holds with these choices. In particular, we shall prove and use several times the following result.

Our proof of Theorem 2 does not use Proposition 1 but rests on the beautiful result of Mellit and Vlasenko [Reference Mellit and VlasenkoMV16, Lemma 1], which gives useful congruences modulo powers of $p$ for some constant terms of powers of Laurent polynomials. In this case, the $p$ -adic difficulties are hidden in the result of Mellit and Vlasenko.

Finally, we give a general result to prove the $p$ -Lucas property for many sums of products of binomial coefficients. We recall that a tuple $e=(\mathbf{e}_{1},\ldots ,\mathbf{e}_{u})$ of vectors in $\mathbb{N}^{d}$ is $1$ -admissible if either $\mathbf{e}_{i}\geqslant \mathbf{1}$ for some $i$ or if, for every $k$ in $\{1,\ldots ,d\}$ , we have $\mathbf{e}_{i}\geqslant d\mathbf{1}_{k}$ for some $i$ .

1.6 Application of Theorem 1

By applying Theorem 1, we obtain results similar to Conjectures A–C for numbers satisfying Apéry-like recurrence relations, which we list below. Characters in brackets in the last column of the following table form the sequence number in the On-line Encyclopedia of Integer Sequences [OEIS13].

All differential operators listed in the above table are of type I for all primes $p$ , except the one associated with $A_{5}(n):=\sum _{k=0}^{n}\binom{n}{k}^{4}$ , which reads

Hence, ${\mathcal{L}}_{5}$ is of type II for all primes $p$ . By a result of Calkin [Reference CalkinCal98, Proposition 3], for all primes $p$ , we have $A_{5}(p-1)\equiv 0\;\text{mod}\;p$ , i.e.  $p-1$ is in ${\mathcal{Z}}_{p}(A_{5})$ . Thus, we can apply Theorem 1 to $A_{5}$ .

Observe that the generating function of the central binomial coefficients is annihilated by the differential operator ${\mathcal{L}}=\unicode[STIX]{x1D703}-z(4\unicode[STIX]{x1D703}+2)$ , which is of type I for all primes $p$ .

We set $A_{6}(n):=\sum _{k=0}^{n}\binom{n}{k}\binom{2k}{k}\binom{2(n-k)}{n-k}$ . In 1885, Catalan gave in [Reference CatalanCat85] a recurrence relation for the Catalan–Larcombe–French sequence $2^{n}A_{6}(n)$ from which we deduce a recurrence relation for $A_{6}(n)$ (see also Case (d) in [Reference Almkvist and ZudilinAZ06]). According to this relation, $A_{6}(n)$ is also Sequence E in Zagier’s list [Reference ZagierZag09], that is,

Furthermore, according to [Reference Richmond and ShallitRS09], Domb numbers $A_{8}(n)=\sum _{k=0}^{n}\binom{n}{k}^{2}\binom{2k}{k}\binom{2(n-k)}{n-k}$ are also the numbers of abelian squares of length $2n$ over an alphabet with four letters.

Now we consider the numbers $C_{i}(n)$ of abelian squares of length $2n$ over an alphabet with $i$ letters, which, for all positive integers $i\geqslant 2$ , satisfy (see [Reference Richmond and ShallitRS09])

According to [Reference Borwein, Nuyens, Straub and WanBNSW11], $C_{i}(n)$ is also the $(2n)$ th moment of the distance to the origin after $i$ steps travelled by a walk in the plane with unit steps in random directions.

To apply Theorem 1 to $C_{i}$ , it suffices to show that its generating series $f_{C_{i}}$ is annihilated by a differential operator of type I for all primes $p$ . Indeed, by [Reference Borwein, Nuyens, Straub and WanBNSW11, Proposition 1 and Theorem 2], for all $j\geqslant 2$ , $C_{j}(n)$ satisfies the recurrence relation of order $\lceil j/2\rceil$ with polynomial coefficients of degree $j-1$ :

where the sum is over all sequences of positive integers $\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{i}$ satisfying $\unicode[STIX]{x1D6FC}_{k}\leqslant j$ and $\unicode[STIX]{x1D6FC}_{k+1}\leqslant \unicode[STIX]{x1D6FC}_{k}-2$ . We consider $i\geqslant 2$ and $i$ positive integers $\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{i}\leqslant j$ satisfying $\unicode[STIX]{x1D6FC}_{k+1}\leqslant \unicode[STIX]{x1D6FC}_{k}-2$ . We have

with $j-\unicode[STIX]{x1D6FC}_{1}\geqslant 0$ , $\unicode[STIX]{x1D6FC}_{k}-\unicode[STIX]{x1D6FC}_{k+1}\geqslant 2$ and $\unicode[STIX]{x1D6FC}_{i}-1\geqslant 0$ . Then $f_{C_{j}}(z)$ is annihilated by a differential operator ${\mathcal{L}}=P_{0}(\unicode[STIX]{x1D703})+zP_{1}(\unicode[STIX]{x1D703})+\cdots +z^{q}P_{q}(\unicode[STIX]{x1D703})$ with $P_{0}(\unicode[STIX]{x1D703})=\unicode[STIX]{x1D703}^{j-1}$ and, for all $i\geqslant 2$ ,

so that ${\mathcal{L}}$ is of type I for all primes $p$ , as expected.

1.7 Structure of the article

In § 2, we use several results of [Reference DelaygueDel13] to prove Theorem 3. Section 3 is devoted to the proofs of Theorem 2 and Proposition 1. In particular, we prove Lemma 1, which points out the role played by differential operators in our proofs. In § 4, we prove Theorem 1 by applying Proposition 1 to $\mathfrak{S}_{e,f}$ . It is the most technical part of this article.