The exact multiplicity of the Wilson supercongruence in F_q[t]

Introduction

Let $q=p^s$ with $p$ an odd prime, $A=\mathbb{F}_q[t]$, and $[i]=t^{q^i}-t$. For a monic irreducible $\wp\in A$ of degree $d$ let $F_d$ be the product of all nonzero polynomials of degree less than $d$. Then $F_d\equiv-1\pmod{\wp}$, the analogue of Wilson’s theorem, and

where $D_d=\prod_{i=1}^{d-1}[i]^{q^{d-i}}$ and $L_d=\prod_{i=1}^{d-1}[i]$ [2 p. 1842]. Following Thakur, $\wp$ is a Wilson prime if $F_d\equiv-1\pmod{\wp^2}$. We write

for the exact multiplicity of the Wilson congruence. For $d=1$ one has $F_1=-1$, so we assume $d\geq2$ throughout.

Thakur proved that $\wp$ is a Wilson prime if and only if $d^2\wp/dt^2=0$, that is, all exponents of $\wp$ are $\equiv0$ or $1\pmod p$, and that in this case the congruence holds modulo $\wp^{p-1}$ [2 Thm. 2.9]; see also the remarks after [3 Thm. 5.1]. For the primes $t^p-t-a$ over $\mathbb{F}_p$ the exponent $p-1$ was already known to be exact [1]; see also [2 Rem. 2.12]. Thakur also reported numerical data of G. Todd suggesting that $p-1$ is exact even when $q$ is not prime, and noted that it was unclear whether the power can be increased [2 Rem. 2.11]. In Question (1) of the closing list of questions of [3], he asked whether, for $d>1$ and $p>2$, the congruence ever (or infinitely often) holds modulo a higher power, and what the exact multiplicities are.

Definition. The depth $r(\wp)$ is the largest integer $r\geq0$ such that every exponent $n$ with a nonzero coefficient in $\wp$ satisfies $n\equiv0$ or $1\pmod{p^r}$. Since $d\geq2$ is an exponent of $\wp$, the depth is finite. Equivalently, $r(\wp)\geq\rho$ if and only if $\wp=U(t^{p^\rho})+tV(t^{p^\rho})$ for some polynomials $U,V$.

Thus the only possible exact multiplicities are $1$, $p-1$, $p^2-1$, $\ldots$, $q-1$ and $q+1$, and the Wilson congruence never holds modulo $\wp^{q+2}$. Over $\mathbb{F}_p$ the only possible values are $1$, $p-1$ and $p+1$. Since depth $r$ forces $d \ge p^r$, the depth is at most 1 when $d < p^2$, and then the value $p-1$ of [2] is exact, in agreement with the data of [2 Rem. 2.11]. The case $r=1$ of Lemma 4 below is [2 Thm. 2.9(iii)]. We assume $p$ odd, as in the question; for $p=2$ see [2 Thm. 2.9(iv)]. The answer to the first part of the question is positive: the examples of Section 4 have multiplicity $q+1$ for $q=3,5,9$, and $p^2-1=8$ for $q=9$. We do not address the infinitude of such primes.

Thakur also proved [3 Thm. 7.1] that the primes of degree $d$ dividing the numerator of $H_d = 1/[1] + \cdots+ 1/[d-1]$ are exactly the Wilson primes, with multiplicity at least $p-2$. Theorem 1 makes this exact.

Proof. Since $[i]'=-1$ and $q^{d-i}-1\equiv-1\pmod p$, logarithmic differentiation of (1) gives $F'_d=F_dH_d$. As $\wp\nmid F_d$, $v_{\wp}(H_d)=v_{\wp}(F'_d)=v_{\wp}((F_d+1)')$. Write $F_d+1=\wp^mG$ with $\wp\nmid G$; then $(F_d+1)'=\wp^{m-1}(m\wp'G+\wp G')$, and $\wp\nmid\wp'$ because $\wp$ is separable. Since none of the values $p^r-1$, $q+1$ is divisible by $p$, the valuation is $m-1$. The primes $\wp$ of degree $d$ do not divide $L_{d-1}=[1]\cdots[d-1]$, so $v_{\wp}(H_d)$ is also the valuation of the numerator of $H_d$. □

Local expansion at a root

Fix $\wp$ as in Theorem 1 and a root $\theta$ of $\wp$, so that $\mathbb{F}_q(\theta)=\mathbb{F}_{q^d}$ and $\theta^{q^d}=\theta$. Put

these are nonzero, and $\beta_1,\ldots,\beta_{d-1}$ together with 0 are the roots of $\wp(X+\theta)$. For a polynomial $G\in\mathbb{F}_q[t]$, the Hasse derivatives $G^{[k]}$ are defined by $G(t+u)=\sum_k G^{[k]}(t)u^k$. Since $\wp$ is separable, $v_{\wp}(G)$ is the order of vanishing of $G$ at $\theta$, that is

Proof. In characteristic $p$, $[i](\theta+u)=\theta^{q^i}+u^{q^i}-\theta-u=\beta_i-u+u^{q^i}$; apply (1). □

Proof. We have $\wp(u+\theta)=\sum_j\wp^{[j]}(\theta)u^j=u\prod_{i=1}^{d-1}(u-\beta_i)$, so $Q(u):=\sum_{j\ge1}\wp^{[j]}(\theta)u^{j-1}=\prod_i(u-\beta_i)$ and $Q(0)=\wp'(\theta)$. Hence $\prod_i(1-x_iu)=Q(u)/Q(0)$, which is the first identity.

Write $\wp=\sum_n c_nt^n$, so that $\wp^{[j]}=\sum_n\binom{n}{j}c_nt^{n-j}$. For $j\ge1$ this polynomial has degree less than $d$, so it vanishes at $\theta$ if and only if it is zero, that is, if and only if $p\mid\binom{n}{j}$ for every $n$ with $c_n\ne0$. Let $2\le j\le p^r-1$ and $c_n\ne0$. The lowest $r$ base-$p$ digits of $n$ are all 0, or are $1,0,\ldots,0$; since $j<p^r$ has $j\ge2$, some digit of $j$ exceeds the corresponding digit of $n$, and Lucas’ theorem gives $p\mid\binom{n}{j}$. If $r\ge1$, by maximality of $r$ some exponent $n$ has $n\bmod p^{r+1}\notin\{0,1\}$; as $n\equiv0$ or $1\pmod{p^r}$, its digit $n_r$ is nonzero, and Lucas’ theorem gives $\binom{n}{p^r}\equiv n_r\ne0$. Distinct $n$ give distinct monomials $t^{n-p^r}$, so $\wp^{[p^r]}\ne0$. Finally $\wp'(\theta)\ne0$ by separability, and if $r=0$ some exponent has $n\not\equiv0,1\pmod p$, so $\binom{n}{2}=n(n-1)/2\ne0$ and $\wp^{[2]}(\theta)\ne0$.

If $r \ge1$, the denominator in the first identity is $\wp'(\theta)(1+cu^{p^r-1}+O(u^{p^r}))$ with $c=\wp^{[p^r]}(\theta)/\wp'(\theta)\ne0$, so the quotient is $1-cu^{p^r-1}+O(u^{p^r})$. If $r=0$, the same computation gives $h_1=-\wp^{[2]}(\theta)/\wp'(\theta)\ne0$.

Proof of Theorem 1

Let $f_k=F_d^{[k]}(\theta)$, the coefficient of $u^k$ in $F_d(\theta+u)$. By (2), $m=\min\{k\ge1:f_k\ne0\}$ and $f_0=F_d(\theta)=-1$.

Step 1: $f_k=-h_k$ for $0\le k\le q-1$. Modulo $u^q$ all terms $u^{q^i}$ in Lemma 3 vanish. Moreover

and $q^{d-i}\ge q$. Hence, modulo $u^q$,

Step 2: the case $r\le s$. If $r=0$, Step 1 and Lemma 4 give $f_1=-h_1\ne0$, so $m=1$. If $1\le r\le s$, then $p^r-1\le q-1$, and Step 1 with Lemma 4 gives $f_1=\cdots=f_{p^r-2}=0$ and $f_{p^r-1}\ne0$, so $m=p^r-1$.

Step 3: the case $r>s$. Then $p^r-1\ge pq-1\ge q+2$, so Lemma 4 gives $h_1=\cdots=h_{q+1}=0$, and by Step 1 $f_1=\cdots=f_{q-1}=0$. We compute modulo $u^{q+2}$. For $i\ge2$ the term $u^{q^i}$ vanishes, since $q^2\ge q+2$. As in Step 1, $(\beta_i-u+\delta_{i,1}u^q)^{e_i}=N_i/D_i$ with $D_i=\beta_i-u+\delta_{i,1}u^q$ and $N_i=D_i^{q^{d-i}}$. Modulo $u^{q+2}$, $N_i=\beta_i^{q^{d-i}}$ for $i\le d-2$, since $q^{d-i}\ge q^2$, while $N_{d-1}=\beta_{d-1}^q-u^q$; here $d\ge p^r\ge pq\ge9$, so the factors $i=1$ and $i=d-1$ are distinct. Hence, with $y=\beta_{d-1}^{-q}$,

Now $(1-x_1u+x_1u^q)^{-1}=(1-x_1u)^{-1}-x_1u^q(1-x_1u)^{-2}+O(u^{2q})$, so

Reading off coefficients,

Since $\theta^{q^d}=\theta$, we have $\beta_{d-1}^q=\theta^{q^d}-\theta^q=-\beta_1$, so $y=-x_1$. With $h_q=h_{q+1}=h_1=0$ we get $f_q=0$ and $f_{q+1}=x_1^2=\beta_1^{-2}\ne0$. Hence $m=q+1$.

Remark 5. In Step 3 the vanishing of $f_q$ depends on the identity $\beta_{d-1}^q=-\beta_1$, which links the factors $i=1$ and $i=d-1$ of Lemma 3: the term $-x_1$ coming from $i=1$ cancels the term $-y$ coming from $i=d-1$.

Examples and verification

We write $\mathbb{F}_9=\mathbb{F}_3[a]/(a^2-a-1)$. The following polynomials are irreducible, with the stated depth and multiplicity:

$q$	$\wp$	$r$	regime	$m$
3	$t^3-t-1$	1	$r\le s$	2
3	$t^{18}-t^{10}+1$	2	$r>s$	4
5	$t^5-t-1$	1	$r\le s$	4
5	$t^{51}-2t^{50}-t^{26}+1$	2	$r>s$	6
9	$t^3-(a+1)t-(a+1)$	1	$r\le s$	2
9	$t^{10}+(a+1)t^9+t+1$	2	$r=s$	8
9	$t^{54}-at^{28}+at^{27}+at+1$	3	$r>s$	10

The ancillary file verify_wilson_multiplicity.py (Python 3, no dependency, exact arithmetic in $\mathbb{F}_q[t]$, about fifteen seconds) checks:

• the product formula (1) against the definition of $F_d$ over $\mathbb{F}_3$ for $d=2,3$;

• for every monic irreducible $\wp$ of degree $2 \le d \le7$ over $\mathbb{F}_3$, $2 \le d \le4$ over $\mathbb{F}_5$ and $2 \le d \le3$ over $\mathbb{F}_9$ (981 primes in all): the valuation $v_\wp(F_d+1)$, computed in $\mathbb{F}_q[t]$ modulo a power of $\wp$ from (1), equals the value of Theorem 1, and $v_\wp(H_d)$ equals $m-1$ (Corollary 2);

• for the examples above: irreducibility (Rabin’s test), the depth, and the same two valuations computed directly in $\mathbb{F}_q[t]$ for the five examples of degree at most 18;

• for all the examples, the coefficients $f_k$ of $F_d(\theta+u)$ computed in the residue field $\mathbb{F}_q[t]/\wp$ from Lemma 3: the first nonzero one is $f_m$; $f_k=-h_k$ for $k<q$, with $h_k$ computed from the first identity of Lemma 4; the vanishing of $\wp^{[j]}(\theta)$ for $2 \le j \le\min(p^r-1,q+3)$, and $\wp^{[p^r]} \ne0$; $\beta_{d-1}^q=-\beta_1$; and, when $r>s$, $f_q=0$ and $f_{q+1}=\beta_1^{-2}$.

In small degree only the depths 0 and 1 occur, and the multiplicity $p-1$ of [2] is then exact. These computations check the statements; the proofs do not depend on them.

Acknowledgement

Acknowledgement. Technical assistance: Claude (Anthropic).

ORCID 0009-0009-1526-1178