Bull. Kyushu Inst. Tech.
(Math. Natur. Sci.) No. 36, 1989, pp. 11-20
ON A CONVOLUTION ON THE SPACE
OF p-ADIC FUNCTIONS
By
Masaya NoHMi
It is the purpose of this note to give some generalization of C. F. Woodc{ ck's result [7] that gives a valuation of an expression involving the Bernoulli numbers.
1. Preliminaries
Let p be an odd prime number, and Z., e., C. be the ring of p-adic integers, the field of rational p-adic numbers and the completion of the algebraic closure of e.
respectively. Let v. be the p-adic normalized additive valuation of C. such that v.(p) =1.
First, we recall some requisite definitions from Woodcock [4]. Put Cpn= {coE Cp l co P" = 1},
Tp= U Cpn, n20
U, = {coE C. 1 v. (co - 1 )År O} .
All C,-valued continuous functions on Z. form a p-adic Banach algebra over C.
under the norm
v (f)= inf v. (f (x )) XEZp
and the pointwise operations. This Banach algebra is denoted by `{27(Z.).
All C.-valued uniformly differentiable functions on Z. form a p-adic Banach algebra over Cp under the norm
with V(f)== min {v (f ), R(f )}
R(f)==.,l,e.lq,vp(f(X.)--f,('))
and the pointwise operations. This Banach algebra is denoted by `ZrS2)(Z.).
Let ip. be a uniformly differentiable function on Z. belonging to (D ET. defined by
12 Masaya NoHMi
ip.(x)=cox. It is known that {gb.1ctEU} has a dense linear span in (if(Z,) and
`Z(9(Z,), where U is any non-empty open subset of U,.
For any fECZ(9(Z.), we have a p-adic integral as follows:
1 pn-1 Io (f )- lim :i 2 f(i )•
n. oo P i =1
From this definition, the fundamental property holds as follows:
Io( f,) =- I,(f) +f'(o)
where f. (x ) ==f (x + z ).
Now, we define the p-adic convolution due to Woodcock [5, 6, 7] as follows:
f*g - Z Io (fÅë. )Io (g ip. )ip. - i, coE Tp '
for any f, gECtzO(Z,). Then we have f*gedtz9(Z,) and Io (f*g)ip. - Io (fip. )Io (g ip. ),
for f, gEav9(Z,) and coET..
It is known that a mapping
a(g(z,)-v(z,)
wu f-fr
induces an isomorphism
cZ(9 (Z. )/Ctz9o (Z. ) = V(Z, ),
where av9, (Z, ) = {fE `Z(9 (Z. ) l f' = O}. And p-adic convolution f oog is induced by this isomorphism as follows:
f' opg' =- - (f*g)',
for any f, gEav9(Z,). We havef(abgE`tz9(Z.) and v (f oo g) ). v (f )+v (g ),
for any fl gEav9(Z,).
From these definitions, it follows that fundamental formulae hold:
(1) f*g (z )=I6X' (f (x )g (z -x)) --f oog' (z ),
onaconvolution on the space of p-adic functions 13
(2) z(f*g)= zf*g +f*zg +fpa g, (3) (fQg)ip. -f ip.Qg ip..
2. The case of a root of unity
We shall try to compute the p-adic valuation of Io(zMip.(z)Qz"ipB(z)), ior ct, fiET,. The expression obtained by integrating the above equality involves the Euler numbers. The Euler number is defined as follows:
Hi (" )= .lii, (1 -1u )m ,i;, (- 1 )k(ii(l)k"
We define the real numbers a.,i by am•i = k]Ilio (- i )k(T)ki
belonging to non-negative integers m, l, And we shall try to compute the valuation of
am,l•
First, we give a few properties with respect to a.,i. For non-negative integers m and l, we have
(4) a.,i=1ma.,i9i-ma.-i,i-i iMi;:i-ii'mO.si,
t (-1)Mm! ,m=l
which is easily seen by the induction on m. Moreover, for all integers m, l with 2S.m S.p and l= 1, we have
(5) vp(a.,i) lll l,
where 1$lS.p-1 and l=l(modp-1). Indeed, for all integers m, l with 1S.mS.p and l
=p-1, we have a.,i!-1 (mod p). Hence (5) is trivial from (4).
For 1:.{mS.l, we define b.,i by
(6) bm,i :[Mp--l i 1] - [Mp-(pl-P il)1]
where [•] is the Gauss symbol. Later, we show that v.(a.,i))-b.,i•
LEMMA 1. For two integers m,lwith 1;SmS.l, we have
(7) b.,, =[M-i- l;l+i-i]
14 Masaya NoHMi
Proof We denote the right hand of (7) by b'.,i. Ifm=1, then the both sides of (7) equal to O. Now, assume that b.,i=b'.,i. In the case of mll(modp-1), it is easily seen that b.,i=b.+i,b b'.,i=b'.+i,i, and therefore b.+i,i=b'.+i,i. In the case of m!i!l
(modp-1), we can choose two non-negative integers s, t(OS.tSp-1) such that m-l=sp (p-1)+t(p-1),
from which follows that b.,i and bin,i are equal to
sp-s+t+[-i;+l]
From Lemma 1, when we keep the value of m-l constant and increase the values of m and l, we see that the value of b.,i increases one by one with the interval of p. We show the chart of b.,i below.
5lp-i)
EZ Z2ik/EY•
4lp-1)
[illZi 1
3lp-1)
2lp-1)
p-1
1
1 p 2p 3p 4p
LEMMA 2. For all integers m,lwith 1;.Sm:.fl, we have vp (am,i ) l-ll bm,i'
Proof I'11 give the proof only for the integers m, l which satisfy 1S.mSIS.m+p -2. The proof for the remainder part is almost the same. I'11 prove that
Vp (a (n -1)p+i,np ) lll bnp,np = n, i -- 2, ''', p.
We show this by the induction on n. It is trivial in the case of n=1.
Consider the process of computing {ai,,-,1i=1,•••, p-1} from {ai,ili=1,•••, p-1}
by (4). We get
Onaconvolution on the space of p-adic functions 15 p-1
(8) aj,.-i=2cjiaii, 1'=1,•••,p-1
i=1 with some integers cij.
Similarly, we have p-1
(9) anp+j,(n+op-i=: 2 Cnp+j,ianp+i,np+i,j-- 1,''', P-1•
i=1
Because (8) and (9) are derived from the formula (4), we have
(10) c...j,i!cj,i (modp), j=1,•••,p-1.
For 1S.iS.p-1, we have ai,i=-1, and for 1S.m:.{p-1, we have a.,.-iii-1 (modp). Then we have from (8)
p-1
1=- Z cji (modp), 1'=1,•••, p-1.
i=1
Therefore from (10), we have p-1
(1 1) Z c.,.j,,i1 (m odp ), J'= 1, •••, p- 1.
i=1
Assume that v, (a(. -i)p+i,., )l b.p,.p =n (i -- 2, •••, p ). From (4), we have Vp (anp+i,np+i)lln+1, i=2,•••, p-1.
So in the case of vp(a.p+i,.p+i)ln+1, our result is trivial. Assume that vp(a.p+i,.p+i) =n. In this case, we have
anp+iinp+iEt (modp) p
for some IStS.p-1. Then we have by (4) anp+i,np+iit (modp) pn
which, by (7) and (11), implies p-1
a"p'j•(".'i)P-ii 2 c.p.j,it=-t (modp), j--1,''', P-1•
P i=1
, Therefore a.p+j,(.+i)p+i are equal to each other modulo p"+i. Hence, vp (a.p+j;(.+i)p) lb(n+i)p,(.+i).==n+1 O'=2,•••, p), from (4).
LEMMA 3. ifuEU, satisfies
16 Masaya NoHMi
Tl
v. (u-1) l;S T+ 1 p-1' then we have
v. (H'(u))S -lv. (u-1).
Proof From the definition of a.,b we have i
Hi(u)=.\.,(ua-Mi').,
and therefore
v.(H'(u))l inf (-mv,(u-1)+v,(a.,i)) Osmsl
l.lr inf (-mv.(u-1)+b.,i).
Osmsl
It is trivial that if mS.l, then b.,t=O. Therefore, it is sufficient to prove that
(12) -lv. (u-1);;$ -mv. (u-1)+b.,i
for IS m.
Assume that m)-p(p-1)-l. In this case, we can choose two integers s, m'(s). 1, O
$m' S.p(p-1)) so that
m= sp (p-1)+ m' +l.
From this equation and the definition of b.,i, we have
-mv.(u-1)+b.,i
= - (m'+l )v. (u-1)+ b.•.i,i- sp (p-1 )v, (u-1)+sp -s
ll; - (m' + l )v. (u - 1 )+ b.• +7,i•
Therefo re it is suMcient to prove (1 2) for T S. m S. p (p - 1 )+ l,
In the case ofl5mS.p(p-1)+l and mfl (modp-1), we can choose s(O5s$p-1)
so that
m=s(p-1)+l.
From the definition of b.,b we have
b.,l,.lS' Ol:ls$l .
ks-1, 7+ISsS.p-1
onaconvolution on the space of p-adic functions 17
And (12) will be proved in the same manner.
In the case of l5mS.p(p-1)+l and mll(modp-1), we can choose s, t(1 S.sS.p, 1 StS.p-2) so that
m=:s(p-1)+l-t,
and therefore
- mv. (u - 1 )+ b.,i
== - (s (p-1)+7)v. (u-1)+[S (P-1)+lit+t+1-1].
We see that t+1=t+1 forIS.t;$p-2. Then we have
- mv, (u - 1 )+ b.,i il: - (s (P - 1 )+l )v. (u - 1 )+ bs(p- i) +zi•
Because we proved (12) for m=s(p-1)+l above, we have proved all cases.
THEoREM 1. For two integers m, n and ct, 6, ct6-iETp-Cpi, we have
vp(Io(zM ip.(z) oo z" ipfi(z)))
i(IPpV,;1ct.r,i6):iP)V-'(p6,IIi.)Li), ,V.pf.ctIlj=#,V.p((66:ii;.
Proof Subordinating f=zMip. and g=z"'iip6 into (4), and integrating the both hand with respect to z, we get
Io (zMip. (z )Q z" ip6 (z ) )
=.ii[jZ=", (-i)j(n t• i).-M, 6+-j i Hm+j-i (ct-i6)"6+-i iJ' Hn-j (6-i)
= -.!1Hm-i (ct-i )Z +- 11 Hn (6-1).
Then, from the formula
.t1("; 1) (n+1-j)=(J".),
we obtain the result.
3. The case of a principal unit
In this section, we discuss the case where tueU.. First, we compute the p-adic
18 Masaya NoHMi
valuation of the logarithm of tu.
LEMMA 4. if coEU. satisLfies 1 v. (tu-1)$
2(p-1)' then v. (log to)ÅqO.
Proof We have 1
v. (log co)).
Our case is the latter one. In the case wheres=1,
and in the case wheresÅr1,
v. (to-1 ), p-1$Vp (tu-1)
11
pSv. (tu -1)- s, p, ip .1) SVp (tu -1)$ ps-i (p -1)
1
v, (log tu)=pv, (to-1)-1S2ip-1)N1ÅqO,
p v, (log tu)== pSv, (tu-1)-sS
-sÅqO, p-1
as required.
LEMMA 5. if coEU. satisLfies 1 vp (tu-1)$2(p-1)'
then we have
vp (Io (zMÅë.))il -pvp (co-1)+vp (log co), for all mlO.
Proof In the case of tuET,, the result is trivial from Lemmas 3 and 4, So, assume that coEU,-T,. Ifm=O, then
log tu Io( ipca) = to - 1 '
and the result is trivial. Making use of (1) with f=etZip.(z), we obtain co-1
(co - 1 )Io (e'Z ip. (z ))= ,,,, - 1 (t +10g CO )•
onaconvolution on the space of p-adic functions 19
On the other hand, from the definition of the Euler number
co 1
(`O - 1 ).ll], Io (zM iptu (z ) )ifiT/ tM = .\l, HM (.CC'! -i ) tm (t + iog co ).
Compairing the coeMcients of the two equations, we get mHM-i(co-i)+mHM(coMi) log co
Io (zMiptu)= co-1 '
And the conclusion follows immediately from Lemma 3.
THEoREM 2. If m, n are two non-negative integers, and ct, 6EU. satisfy 1
v, (ct '1 ), v, (6-1 ), v. (ct-'6-1)$
2(p-1)' then we have
vp (Io (zM ip. (z )oo z"ipfi (z )) )
-1
;i!r]
