Bessel F-isocrystals for reductive groups

Abstract

We construct the Frobenius structure on a rigid connection ${{\,\mathrm{Be}\,}}_{\check{G}}$ on ${\mathbb {G}}_{m}$ for a split reductive group $\check{G}$ introduced by Frenkel–Gross. These data form a $\check{G}$-valued overconvergent F-isocrystal ${{\,\mathrm{Be}\,}}_{\check{G}}^{\dagger }$ on ${\mathbb {G}}_{m,{\mathbb {F}}_p}$, which is the p-adic companion of the Kloosterman $\check{G}$-local system ${{\,\mathrm{Kl}\,}}_{\check{G}}$ constructed by Heinloth–Ngô–Yun. By studying the structure of the underlying differential equation, we calculate the monodromy group of ${{\,\mathrm{Be}\,}}_{\check{G}}^{\dagger }$ when $\check{G}$ is almost simple (which recovers the calculation of monodromy group of ${{\,\mathrm{Kl}\,}}_{\check{G}}$ due to Katz and Heinloth–Ngô–Yun), and prove a conjecture of Heinloth–Ngô–Yun on the functoriality between different Kloosterman $\check{G}$-local systems. We show that the Frobenius Newton polygons of ${{\,\mathrm{Kl}\,}}_{\check{G}}$ are generically ordinary for every $\check{G}$ and are everywhere ordinary on $|{\mathbb {G}}_{m,{\mathbb {F}}_p}|$ when $\check{G}$ is classical or $G_2$.

Appendix A. A 2-adic proof of Carlitz’s identity and its generalization

As mentioned in introduction, Carlitz [24] proved the following identity between Kloosterman sums:

$$\begin{aligned} {{\,\mathrm{Kl}\,}}(3;a)={{\,\mathrm{Kl}\,}}(2;a)^2-1,\qquad \forall ~ a\in {\mathbb {F}}_{2^s}^{\times }. \end{aligned}$$

In this appendix, we reprove and generalize this identity by establishing an isomorphism between two Bessel F-isocrystals ${{\,\mathrm{Be}\,}}_{2n+1}^{\dagger }$ and ${{\,\mathrm{Be}\,}}_{{{\,\mathrm{SO}\,}}_{2n+1},{{\,\mathrm{Std}\,}}}^{\dagger }$. The following is a restatement of Proposition 5.2.10(ii).

Proposition A1

There exists an isomorphism between following two overcovergent F-isocrystals on ${\mathbb {G}}_{m,{\mathbb {F}}_2}$:

Our strategy is first to show that their maximal slope quotient convergent F-isocrystals are isomorphic. Then we conclude the proposition by a dual version of a minimal slope conjecture (proposed by Kedlaya [56] and recently proved by Tsuzuki [74]) that we briefly recall in the following.

Let X be a smooth k-scheme and $\mathscr {M}^{\dagger }$ an overconvergent F-isocrystal on X/K. We denote the associated convergent F-isocrystal on X/K by $\mathscr {M}$. When the (Frobenius) Newton polygons of $\mathscr {M}$ are constant on X, $\mathscr {M}$ admits a (dual) slope filtration, that is a decreasing filtration

of convergent F-isocrystals on X/K such that

• $\mathscr {M}^{i}/\mathscr {M}^{i+1}$ is isoclinic of slope $s^i$ and

• $s^0>s^1>\cdots >s^{r-1}$.

Theorem A2

(Tsuzuki, [74] theorem 1.3) Let X be a smooth connected curve over k. Let $\mathscr {M}^{\dagger },\mathscr {N}^{\dagger }$ be two irreducible overconvergent F-isocrystals such that the corresponding convergent F-isocrystals $\mathscr {M},\mathscr {N}$ admit the slope filtrations $\{\mathscr {M}^i\}$, $\{\mathscr {N}^i\}$ respectively. Suppose there exists an isomorphism $h:\mathscr {N}/\mathscr {N}^1\xrightarrow {\sim } \mathscr {M}/\mathscr {M}^1$ of convergent F-isocrystals between the maximal slope quotients. Then there exists a unique isomorphism $g^{\dagger }:\mathscr {N}^{\dagger }\xrightarrow {\sim } \mathscr {M}^{\dagger }$ of overconvergent F-isocrystals, which is compatible with h as morphisms of convergent F-isocrystals.

A3 Following Dwork’s strategy [39, § 1-3], we study the maximal slope quotients of ${{\,\mathrm{Be}\,}}_{2n+1}^{\dagger }$ and of ${{\,\mathrm{Be}\,}}_{{{\,\mathrm{SO}\,}}_{2n+1},{{\,\mathrm{Std}\,}}}^{\dagger }$ in terms of their unique solutions at 0.

In the following, we assume $k={\mathbb {F}}_p$. We first recall Dwork’s congruences and show a refinement of his result in the 2-adic case. Consider for every $i\ge 0$, a map $B^{(i)}(-):{\mathbb {Z}}_{\ge 0}\rightarrow K^{\times }$ and the following congruence relation for $0\le a<p$ and $n,m,s \in {\mathbb {Z}}_{\ge 0}$:

• (a) $B^{(i)}(0)$ is a p-adic unit for all $i\ge 0$,

• (b) $\displaystyle \frac{B^{(i)}(a+np)}{B^{(i+1)}(n)} \in R$ for all $i\ge 0$,

• (c) $\displaystyle \frac{B^{(i)}(a+np+mp^{s+1})}{B^{(i+1)}(n+mp^{s})} \equiv \frac{B^{(i)}(a+np)}{B^{(i+1)}(n)} \mod p^{s+1}$ for all $i\ge 0$.

• (c’) When $p=2$, $\displaystyle \frac{B^{(i)}(a+n2+m2^{s+1})}{B^{(i+1)}(n+m2^{s})} \equiv u(i,s,m) \frac{B^{(i)}(a+n2)}{B^{(i+1)}(n)} \mod 2^{s+1}$ for all $i\ge 0$, where $u(i,s,m)=1$ if $s\ne 1$ and $u(i,1,m)=1$ or $-1$ depending on i and m.

If conditions (a–c) (or (a,b,c’)) are satisfied, then $B^{(i)}(n)\in R$ for all $i,n\ge 0$. We set

$$\begin{aligned}&F^{(i)}(x)=\sum _{j=0}^{\infty } B^{(i)}(j) x^j ~ \in K \llbracket x \rrbracket , \\&F^{(i)}_{m,s}(x)=\sum _{j=mp^s}^{(m+1)p^{s}-1} B^{(i)}(j) x^j~ \in K[x],\quad s\ge 0. \end{aligned}$$

We write $F^{(i)}_{0,s}$ by $F^{(i)}_{s}$ for simplicity.

Theorem A4

(i) [39, theorem 2] If conditions (a–c) are satisfied, then

(i$^{\prime }$) If conditions (a,b,c$^{\prime }$) are satisfied (in particular $p=2$), then

(ii) [39, theorem 3] Under the assumption of (i) or (i’) and suppose moreover that

• (d) $B^{(i)}(0)=1$ for $i\ge 0$.

• (e) $B^{(i+r)}=B^{(i)}$ for all $i\ge 0$ and some fixed $r\ge 1$.

Let U be the open subscheme of ${\mathbb {A}}^1_{k}$ defined by $F_{1}^{(i)}(x)\ne 0$, for $i=0,1,\ldots ,r-1$. Then the limit

defines a global function on the formal open subscheme $\mathfrak {U}$ of $\widehat{{\mathbb {A}}}^{1}_R$ associated to U, which takes p-adic unit value at each rigid point of $\mathfrak {U}^{{{\,\mathrm{rig}\,}}}$.

We prove assertion (i’) in the end (A11). We briefly explain Dwork’s result (ii) in the language of formal schemes. The assumption implies that $F_s^{(i)}\ne 0$ on U (cf. [39] 3.4). For $s\ge 1$, the congruences (A41) or ((A42)) imply that

$$\begin{aligned} F_{s+1}^{(0)}(x)/ F_s^{(1)}(x^p)= F_s^{(0)}(x)/F_{s-1}^{(1)}(x^p) \quad \in \Gamma (\mathfrak {U},\mathscr {O}_{\mathfrak {U}}/p^{s-1}\mathscr {O}_{\mathfrak {U}}). \end{aligned}$$

This allows us to use (A43) to define a global function f of $\mathscr {O}_{\mathfrak {U}}$.

A5 Let $F(x)=\sum _{j\ge 0} B(j) x^{j}$ be a formal power series in $R\llbracket x \rrbracket$. We say F satisfies Dwork’s congruences if by setting $B^{(i)}(j)=B(j)$ for every $i\ge 0$, conditions of Theorem A4(ii) are satisfied.

We take such a function F and then we obtain a function $f\in \Gamma (\mathfrak {U},\mathscr {O}_{\mathfrak {U}})$ coinciding with $F(x)/F(x^p)$ in $K\{x\}$ (2.3.1.1) (i.e. the open unit disc). Moreover, by [39, lemma 3.4(ii)], there exists a function $\eta \in \Gamma (\mathfrak {U},\mathscr {O}_{\mathfrak {U}})$ coinciding with $F'(x)/F(x)$ in $K\{x\}$ defined by

$$\begin{aligned} \eta (x)\equiv F'_{s+1}(x)/F_{s+1}(x) \mod p^{s}. \end{aligned}$$

The functions f(x) and $\eta (x)$ satisfy a differential equation:

$$\begin{aligned} \frac{f'(x)}{f(x)}+px^{p-1}\eta (x^{p})=\eta (x). \end{aligned}$$

Note that $f(0)=F(0)/F(0)=1$. Then we deduce that the following corollary.

Corollary A6

The connection $d- \eta$ on the trivial bundle $\mathscr {O}_{\mathfrak {U}^{{{\,\mathrm{rig}\,}}}}$ and the function f form a unit-root convergent F-isocrystal $\mathscr {E}_F$ on U/K, whose Frobenius eigenvalue at 0 is 1.

A7. Let $\mathscr {M}^{\dagger }$ be an overconvergent F-isocrystal on ${\mathbb {G}}_{m,k}$ over K of rank r whose underlying bundle is trivial and the connection is defined by a differential equation:

$$\begin{aligned} P(\delta )=\delta ^r+p_{r}\delta ^{r-1}+\cdots + p_{1}=0, \end{aligned}$$

where $\delta =x\frac{d}{dx}$, $p_i\in \Gamma (\widehat{{\mathbb {A}}}_R^1,\mathscr {O}_{\widehat{{\mathbb {A}}}_R^1})[\frac{1}{p}]$. We assume moreover that $\mathscr {M}^{\dagger }$ is unipotent at 0 with a maximal unipotent local monodromy. Then $\mathscr {M}^{\dagger }$ extends to a log convergent F-isocrystal $\mathscr {M}^{\log }$ on $({\mathbb {A}}^1,0)$ and its Frobenius slopes at 0 are

$$\begin{aligned} s^0>s^1=s^0-1>\cdots > s^{r-1}=s^0-(r-1). \end{aligned}$$

Note that $\mathscr {M}^{\dagger }$ is indecomposable in $F\text {-}{{\,\mathrm{Isoc}\,}}^{\dagger }({\mathbb {G}}_{m,k}/K)$ and so is $\mathscr {M}$ in $F\text {-}{{\,\mathrm{Isoc}\,}}({\mathbb {G}}_{m,k}/K)$. Then by Drinfeld–Kedlaya’s theorem on the generic Frobenius slopes [38], we deduce property (i):

(i) The generic Frobenius slopes (mult-)set is $\{s^0,\ldots ,s^{r-1}\}$ with $s^i=s^0-(i-1)$.

(ii) In view of (2.3.1.2), the differential equation $D=0$ admits a unique solution at 0:

$$\begin{aligned} F(x)=\sum _{n\ge 0} A(n) x^n \quad \in K\{x\},\quad \text { with A(0)=1}. \end{aligned}$$

Proposition A8

Suppose the function F(x) satisfies Dwork’s congruences (A5) and let $\mathscr {E}_F$ be the associated unit-root convergent F-isocrystal on $U \subset {\mathbb {A}}^1_k$. Then

(i) There exists an epimorphism of log convergent isocrystals $\mathscr {M}^{\log }\rightarrow \mathscr {E}_F$ on (U, 0).

(ii) As convergent isocrystals, $\mathscr {E}_F$ coincides with the maximal slope quotient $\mathscr {M}^{\log }/\mathscr {M}^{\log ,1}$ of $\mathscr {M}^{\log }$ (A12).

Proof

(i) We set $A=\Gamma (\mathfrak {U},\mathscr {O}_{\mathfrak {U}})[\frac{1}{p}]$. We claim that there exists a decomposition of differential operators:

Indeed, by the Euclidean algorithm [54, 5.5.2], there exists $r\in A$ such that $P=Q(\delta -x\eta )+r$. By evaluating the above identity at F (in the ring $K\{x\}$ containing A), we obtain

$$\begin{aligned} P(\delta )(F)=0=Q(\delta )(\delta -x\eta )(F)+rF=rF. \end{aligned}$$

Then we deduce $r=0$ and (A81) follows.

Let $e_1,\ldots ,e_r$ be a basis of $\mathscr {M}$ such that $\nabla _{\delta }(e_i)=e_{i+1}, 1\le i\le r-1$ and $\nabla _{\delta }(e_r)=-(p_r e_r+\cdots +p_1 e_1)$. We consider a free $\mathscr {O}_{\mathfrak {U}^{{{\,\mathrm{rig}\,}}}}$-module with a log connection $\mathscr {N}$ with a basis $f_1,\ldots ,f_{r-1}$ and the connection defined by $\nabla _{\delta }(f_i)=f_{i+1}$, $\nabla _{\delta }(f_{r-1})=-(q_{r-1} f_{r-1}+\cdots +q_{1}f_{1})$. By (A81), the morphism $f_1\mapsto e_2- x\eta e_1$ induces a horizontal monomorphism $\mathscr {N}\rightarrow \mathscr {M}^{\log }$ whose cokernel is isomorphic to $\mathscr {E}_F$.

(ii) Note that ${{\,\mathrm{Pic}\,}}(\mathfrak {U}^{{{\,\mathrm{rig}\,}}})\simeq {{\,\mathrm{Pic}\,}}(U)$ [75, 3.7.4] is trivial. Then the rank one convergent isocrystal $\mathscr {M}^{\log }/\mathscr {M}^{\log ,1}$ can be represented as a connection $d-\lambda$ on the trivial bundle $\mathscr {O}_{\mathfrak {U}^{{{\,\mathrm{rig}\,}}}}$.

Since $\mathscr {M}^{\log }$ has a maximal unipotent at 0, the rank one quotient of the restriction $\mathscr {M}^{\log }|_0$ of $\mathscr {M}^{\log }$ at the open unit disc around 0 is unique (2.3.1). In particular, $d-\lambda$ kills the unique solution F of $P(\delta )=0$. By analytic continuation, we have $\lambda =\eta$ and the assertion follows. $\square$

Remark A9

The unique solution F(x) belongs to the ring $K\llbracket x \rrbracket _0 = R \llbracket x \rrbracket \otimes _R K$ of bounded functions on open unit disc, which is a subring of $K\{x\}$. Assertion (ii) can be viewed as an example of Dwork–Chiarellotto–Tsuzuki conjecture on the comparison between the log-growth filtration (of solutions) and Frobenius slope filtration [28]. This conjecture was recently proved by Ohkubo [65].

Proof of Proposition A1

We set $k={\mathbb {F}}_2$ and apply the above discussions to overconvergent F-isocrystals $\mathscr {M}^{\dagger }={{\,\mathrm{Be}\,}}_{2n+1}^{\dagger }$ and $\mathscr {N}^{\dagger }={{\,\mathrm{Be}\,}}_{{{\,\mathrm{SO}\,}}_{2n+1},{{\,\mathrm{Std}\,}}}^{\dagger }$ on ${\mathbb {G}}_{m,{\mathbb {F}}_2}/K$ (A11). Their unique solutions at 0 are:

$$\begin{aligned} F(x)= \sum _{r\ge 0} \frac{(-2)^{(2n+1)r}}{(r!)^{2n+1}} x^r,\qquad G(x)=\sum _{r\ge 0} \frac{2^{(2n+1)r}(2r-1)!!}{(r!)^{2n+1}} x^r. \end{aligned}$$

In the following lemma, we show that F and G satisfy Dwork’s congruences and that the associated maximal slope quotients $\mathscr {E}_F$ and $\mathscr {E}_G$ (A8) are isomorphic. Then Proposition A1 follows from theorem A2 and the following lemma. $\square$

Lemma A10

(i) The functions F(x) and G(x) satisfy Dwork’s congruences (A5) and define unit-root convergent F-isocrystals $\mathscr {E}_F$ and $\mathscr {E}_G$ on ${\mathbb {A}}_k^1$ respectively.

(ii) The function F(x)/G(x) extends to a global function of $\widehat{{\mathbb {A}}}^1_R$ and induces an isomorphism $\mathscr {E}_G\xrightarrow {\sim } \mathscr {E}_F$.

Proof

(i) Conditions (a,b,d,e) are easy to verified. The coefficients of F(x) (resp. G(x)) satisfy condition (c’) (resp. (c)), that is

$$\begin{aligned}&\frac{(-2)^{(2n+1)(a+\ell 2+m2^{s+1})}/((a+\ell 2+m2^{s+1})!)^{2n+1}}{(-2)^{(2n+1)(\ell +m2^{s})}/((\ell +m2^{s})!)^{2n+1}} \\&\quad \equiv u(s,m) \frac{ (-2)^{(2n+1)(a+\ell 2)}/ ((a+\ell 2)!)^{2n+1}}{(-2)^{(2n+1)\ell }/ (\ell !)^{2n+1}} \mod 2^{s+1}, \end{aligned}$$

where $u(1,m)=(-1)^m$ and $u(s,m)=1$ if $s\ne 1$, and

$$\begin{aligned}&\frac{(2(a+\ell 2+m2^{s+1})-1)!! 2^{(2n+1)(a+\ell 2+m2^{s+1})}/((a+\ell 2+m2^{s+1})!)^{2n+1}}{(2(\ell +m2^{s})-1)!!2^{(2n+1)(\ell +m2^{s})}/((\ell +m2^{s})!)^{2n+1}} \\&\quad \equiv \frac{(2(a+\ell 2)-1)!!2^{(2n+1)(a+\ell 2)}/ ((a+\ell 2)!)^{2n+1}}{(2\ell -1)!!2^{(2n+1)\ell }/ (\ell !)^{2n+1}} \mod 2^{s+1}. \end{aligned}$$

Since $F_1(x)\equiv G_1(x)\equiv 1\mod 2$, the F-isocrystals $\mathscr {E}_{F},\mathscr {E}_G$ are defined over ${\mathbb {A}}_k^1$.

(ii) We set $B^{(0)}(r)=\frac{(-2)^{(2n+1)r}}{(r!)^{2n+1}}$ and $B^{(1)}(r)=\frac{2^{(2n+1)r}(2r-1)!!}{(r!)^{2n+1}}$ and $B^{(i+2)}=B^{(i)}$. Then these sequences satisfy conditions (a,b,c’,d,e). For condition (c’), the constants u(i, 1, m) are given by

$$\begin{aligned} u(0,1,m)=1, \quad u(1,1,m)=(-1)^m,\quad u(i+2,1,m)=u(i,1,m). \end{aligned}$$

Since $F_1(x)\equiv G_1(x)\equiv 1\mod 2$, $F(x)/G(x^2)$ extends to a global function of $\mathscr {O}_{\widehat{{\mathbb {A}}}_R^1}$ by Theorem A4 and so is F(x)/G(x). Then the assertion follows. $\square$

A11 Proof of Theorem A4(i’). We prove assertion (i’) by modifying the argument of [39, theorem 2]. Note that condition (c’) implies the following congruence relation:

When $n<0$, we set $B^{(i)}(n)=0$. We set $A=B^{(0)}$, $B=B^{(1)}$ and for $a\in \{0,1\}$, $j,N\in {\mathbb {Z}}$, we set

$$\begin{aligned}&U_{a}(j, N) =A(a+2(N-j)) B(j)-B(N-j) A(a+2j), \\&H_{a}(m, s, N) = \sum _{j=m2^s}^{(m+1)2^s-1} U_{a}(j, N). \end{aligned}$$

Then the assertion is equivalent to

By condition (b), we have $A(a+2m)/B(m)\in R$ and hence

$$\begin{aligned} U_a(m,N) \equiv 0 \mod B(m). \end{aligned}$$

Then equation (A112) for $s=0$ follows from the fact that $H_a(m,0,N)=U_a(m,N)$.

We now prove by induction on s. We write the induction hypothesis

$$\begin{aligned} \alpha _s: H_a(m,u,N)\equiv 0 \mod 2^u B^{(u+1)}(m),\quad \text {for } u\in [0,s), m,N\ge 0. \end{aligned}$$

We may assume $\alpha _{s}$ for fixed $s\ge 1$. The main step is to show for $0\le t\le s$ that

$$\begin{aligned}&\beta _{t,s}: v(s,t,m) H_a(m,s,N+m2^s)\\&\quad \equiv \sum _{j=0}^{2^{s-t}-1} B^{(t+1)}(j+m2^{s-t})H_a(j,t,N)/B^{(t+1)}(j)\mod 2^s B^{(s+1)}(m), \end{aligned}$$

where $v(s,t,m)=1$ or $-1$ depending on s, t, m.

We list some elementary facts (cf. [39, 2.5–2.7])

We first prove $\beta _{0,s}$. We have

By (A111), we have

$$\begin{aligned} A\left( a+2 j+m 2^{s+1}\right) = A(a+2j)B(j+m2^s)/B(j)+X_j 2^s B(j+m 2^s), \end{aligned}$$

where $X_j\in R$. Then the right hand side of (A116) is

$$\begin{aligned} B\left( j+m 2^{s}\right) \biggl (U_{a}(j, N) / B(j)-2^{s} X_{j} B(N-j)\biggr ). \end{aligned}$$

Since $U_{a}(j,N)=H_a(j,0,N)$, we obtain

$$\begin{aligned} H_{a}\left( m, s, N+m 2^{s}\right)= & {} \sum _{j=0}^{2^{s}-1} B\left( j+m 2^{s}\right) H_{a}(j, 0, N) / B(j)\\&-2^{s} \sum _{j=0}^{2^{s}-1} X_{j} B\left( j+m 2^{s}\right) B(N-j). \end{aligned}$$

Since $X_jB(N-j)\in R$, it follows from (A115) ($B=B^{(1)}$) that the second sum is congruent to zero modulo $2^s B^{(s+1)}(m)$. This proves $\beta _{0,s}$ with $v(s,0,m)=1$.

With s fixed, $s\ge 1$, t fixed, $0\le t \le s-1$, we show that $\beta _{t,s}$ together with $\alpha _s$ imply $\beta _{t+1,s}$. To do this we put $j=\mu +2i$ in the right side of $\beta _{t,s}$ and write it in the form

$$\begin{aligned} \sum _{\mu =0}^{1} \sum _{i=0}^{2^{s-t-1}} B^{(t+1)}\left( \mu + 2i+m 2^{s-i}\right) H_{a}(\mu +2 i, t, N) / B^{(t+1)}(\mu +2i). \end{aligned}$$

By condition (c’), we have,

$$\begin{aligned}&B^{(t+1)}\left( \mu +2i+m2^{s-t}\right) \\&\quad = u(t+1,s-t-1,m)\left( B^{(t+1)}(\mu +2 i) B^{(t+2)}\left( i+m2^{s-t-1}\right) / B^{(t+2)}(i)\right) \\&\qquad +X_{i, \mu } 2^{s-t} B^{(t+2)}\left( i+m 2^{s-t-1}\right) , \end{aligned}$$

where $X_{i,\mu }\in R$. Thus the general term in the above double sum is

$$\begin{aligned} u(t+1,s-t-1,m)\biggl (B^{(t+2)}(i+m2^{s-t-1})H_a(\mu +2i,t,N)/B^{(t+2)}(i)\biggr )+Y_{i,\mu }, \end{aligned}$$

where the error term:

$$\begin{aligned} Y_{i, \mu }=X_{i, \mu } 2^{s-t} B^{(t+2)}\left( i+m 2^{s-t-1}\right) H_{a}(\mu +2 i, t, N) / B^{(t+1)}(\mu +2i). \end{aligned}$$

For this error term, since $t<s$, we can apply $\alpha _s$ to conclude that

$$\begin{aligned} Y_{i,\mu }\equiv 0 \mod B^{(t+2)}(i+m2^{s-t-1})2^s. \end{aligned}$$

Then we can use (A115) to conclude that

$$\begin{aligned} Y_{i,\mu }\equiv 0 \mod 2^s B^{(s+1)}(m). \end{aligned}$$

After modulo $2^s B^{(s+1)}(m)$, the right side of $\beta _{t,s}$ is equal to

$$\begin{aligned}&u(t+1,s-t-1,m)\sum _{\mu =0}^{1} \sum _{i=0}^{2^{s-t-1}-1} B^{(t+2)}\left( i+m2^{s-t-1}\right) \\&\quad \times H_{a}(\mu +2 i, t, N) / B^{(t+2)}(i).\end{aligned}$$

By reversing the order of summation and using (A114), the above sum is the same as

$$\begin{aligned}&u(t+1,s-t-1,m)\sum _{i=0}^{2^{s-t-1}-1} B^{(t+2)}\left( i+m2^{s-t-1}\right) \\&\quad \times H_{a}(i, t+1, N) / B^{(t+2)}(i),\end{aligned}$$

which proves $\beta _{t+1,s}$. In particular, we obtain $\beta _{s,s}$, which states

We now consider the statement (with s fixed before)

$$\begin{aligned} \gamma _{N}:H_a(0,s,N)\equiv 0 \mod 2^s. \end{aligned}$$

We know that $\gamma _N$ is true for $N<0$. Let $N'$ (if it exists) be the minimal value of N for which $\gamma _{N'}$ fails. For $m\ge 1$, since $B^{(s+1)}(0)$ is a unit, we have by (A117)

$$\begin{aligned} H_a(m,s,N')\equiv & {} v(s,s,m) B^{(s+1)}(m)H_a(0,s,N'-m2^s)/B^{(s+1)}(0)\\\equiv & {} 0 \mod 2^s. \end{aligned}$$

Applying this to (A113), we obtain that

$$\begin{aligned} H_a(0,s,N')\equiv 0 \mod 2^s. \end{aligned}$$

Thus $\gamma _N$ is valid for all N and Eq. (A117) implies $\alpha _{s+1}$. This proves assertion (i$^{\prime }$). $\square$