Integral structures on p -adic Fourier theory
aa r X i v : . [ m a t h . N T ] M a r INTEGRAL STRUCTURES ON p -ADIC FOURIER THEORY KENICHI BANNAI AND SHINICHI KOBAYASHI
Abstract.
In this article, we give an explicit construction of the p -adic Fourier transform by Schneider and Teitelbaum, which allows forthe investigation of the integral property. As an application, we givea certain integral basis of the space of K -locally analytic functions onthe ring of integers O K for any finite extension K of Q p , generalizingthe basis constructed by Amice for locally analytic functions on Z p . Wealso use our result to prove congruences of Bernoulli-Hurwitz numbersat non-ordinary (i.e. supersingular) primes originally investigated byKatz and Chellali. Introduction
One important method in studying the congruences and p -adic proper-ties of important invariants in number theory is the use of p -adic measuresinterpolating such values. Such theory was applied to obtain the Kummercongruence between special values of Riemann zeta function as well as theconstruction of the p -adic L -functions for elliptic curves with ordinary re-duction at p . When dealing with the non-ordinary case, it is necessary to usethe theory of p -adic analytic distributions, which is a generalization of thetheory of p -adic measures. For such p -adic distributions on Z p , the Amicetransform gives a one-to-one correspondence between C p -valued distribu-tions on Z p and rigid analytic functions on the open unit disc. The generalidea is to study the congruences and p -adic properties of the interpolatedinvariants through the p -adic property of the rigid analytic function corre-sponding to the p -adic distribution. However, contrary to the case of p -adicmeasures, the Amice transform is not well-behaved integrally for general p -adic distributions, hence it is necessary to investigate in detail the pre-cise integral structure of this transform. Amice [Am, §
10] investigated theprecise integral structure of the Amice transform.Let O K be the ring of integers of a finite extension K of Q p . In [ST, § p -adic Fourier transform, whichis a one-to-one correspondence between C p -valued distributions on O K andrigid analytic functions on an open unit disc. The purpose of this article is to Date : March 27, 2015.2010
Mathematics Subject Classification. give an explicit and elementary construction of the p -adic Fourier transformof Schneider-Teitelbaum, which allows investigation of the precise integralstructure of this correspondence. We then determine an integral structureon the ring of locally analytic functions on O K . The integrality of the p -adicFourier transform for general K is even less well behaved than for the case of Q p ; even if the rigid analytic function corresponding to a p -adic distributionhas bounded coefficients, the p -adic distribution may not necessarily be a p -adic measure. As an application of our result, we obtain the congruencesoriginally proved by Katz [Ka2, Theorem 3.11] and Chellali [Ch, Th´eor`em1.1] of Bernoulli-Hurwitz numbers, which are essentially special values of p -adic L -functions of CM elliptic curves at non-ordinary primes.We now give the exact statements of our theorems. Let p be a rationalprime and let | · | be the absolute value of C p such that | p | = p − . Let π be an uniformizer of O K , and let F q be the residue field of O K . We define LA N ( O K , C p ) to be the space of locally analytic functions on O K of order N which take values in C p . Namely, f ( x ) ∈ LA N ( O K , C p ) if and only if f ( x ) is defined as a power series P ∞ n =0 a n ( x − a ) n on a + π N O K for any a ∈ O K . We let k f k a,N := max n {| a n π nN |} . The space LA N ( O K , C p ) is a p -adic Banach space induced by the norm max a ∈O K {k f k a,N } and we denoteby LA N ( O K , C p ) the submodule of elements whose absolute values are lessthan or equal to 1. We let G be a Lubin-Tate group of K corresponding to π , and let ̟ p ∈ C p be a p -adic period of G . We let ρ ( k ) = max k ≤ m {| m ! /̟ mp |} , ρ ( k ) = min ≤ m ≤ k {| m ! /̟ mp |} . Let ϕ ( t ) be a rigid analytic function on the open unit disc. In other words, ϕ ( t ) is a power series of the form ϕ ( t ) = P ∞ n =0 c n t n such that | c n | r n → < r <
1. Let µ ϕ be the distribution on O K corresponding to ϕ ( t )given by Schneider-Teitelbaum’s p -adic Fourier theory [ST, Theorem 2.3].Then we have the following Theorem 1.1.
Let f be a K -locally analytic function in LA N ( O K , C p ) .Then we have (cid:12)(cid:12)(cid:12)(cid:12)Z a + π N O K f ( x ) dµ ϕ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ρ (0) (cid:12)(cid:12)(cid:12)(cid:12) πq (cid:12)(cid:12)(cid:12)(cid:12) N k f k a,N k ϕ k N (1) where (2) k ϕ k N := max k (cid:26) | c k | ρ (cid:18)(cid:20) kq N (cid:21)(cid:19) (cid:27) and [ x ] is the integral part of x . The crucial difference from the case when K = Q p is the fact that | π/q | > K = Q p . A finer version of the above is given as Theorem 4.3. Since ρ (cid:16)h kq N i(cid:17) ∼ p − kr where r = 1 /eq N ( q − k ϕ k N is approximated -ADIC FOURIER THEORY 3 by k ϕ k B ( p − r ) = max x ∈ B ( p − r ) { | ϕ ( x ) | } where B ( p − r ) ⊂ C p is the closed disc of radius p − r centered at the origin.As an application of our main theorem, we obtain an estimate of theFourier coefficients of Mahler like expansion of functions in LA N ( O K , C p ).Let λ ( t ) be the formal logarithm of G , and following [ST], we define thepolynomial P n ( x ) by exp( xλ ( t )) = ∞ X n =0 P n ( x ) t n . Note that when G is the multiplicative formal group G = b G m , then λ ( t ) =1 + t and the above expansion is simply(1 + t ) x = ∞ X n =0 (cid:18) xn (cid:19) t n . Hence the polynomial P n ( t ) is the generalization of the binomial polynomial (cid:18) xn (cid:19) = x ( x − · · · ( x − n + 1) n ! . Then we have the following.
Theorem 1.2. (=Theorem 4.7.) The series P ∞ n =0 a n P n ( x̟ p ) converges toan element of LA N ( O K , C p ) for a n satisfying | a n | ≤ ρ (cid:18)(cid:20) nq N (cid:21)(cid:19) , lim n → | a n | /ρ (cid:18)(cid:20) nq N (cid:21)(cid:19) = 0 . Conversely, if f ( x ) ∈ LA N ( O K , C p ) , then it has an expansion f ( x ) = ∞ X n =0 a n P n ( x̟ p ) of the form | a n | ≤ c (cid:12)(cid:12)(cid:12)(cid:12) πq (cid:12)(cid:12)(cid:12)(cid:12) N ρ (cid:18)(cid:20) nq N (cid:21)(cid:19) , lim n → | a n | /ρ (cid:18)(cid:20) nq N (cid:21)(cid:19) = 0 , where c = 1 if e ≤ p − , and c = ρ (0) , otherwise. Corollary 1.3. (=Corollary 4.8.) Suppose e N,n := γ (cid:18)(cid:20) nq N (cid:21)(cid:19) P n ( x̟ p ) , ( n = 0 , , · · · ) , where γ ( u ) is an element in C p such that ρ ( u ) = | γ ( u ) | . We denote by L N the O C p -module topologically generated by e N,n , then ρ (0) − (cid:12)(cid:12)(cid:12) qπ (cid:12)(cid:12)(cid:12) N LA N ( O K , C p ) ⊂ L N ⊂ LA N ( O K , C p ) . BANNAI AND KOBAYASHI
In particular, L N ⊗ Q p = LA N ( O K , C p ) , namely, the functions e N,n form aBanach basis of LA N ( O K , C p ) . Moreover, if e ≤ p − , then (cid:12)(cid:12)(cid:12) qπ (cid:12)(cid:12)(cid:12) N +1 LA N ( O K , C p ) ⊂ L N ⊂ LA N ( O K , C p ) . In particular, If O K = Z p , we recover Amice’s result [Am, Th´eor`em 3] ,namely, (cid:20) np N (cid:21) ! (cid:18) xn (cid:19) , ( n = 0 , , · · · ) form a topological basis of LA N ( Z p , C p ) . (Actually, we can show that it isa basis of LA N ( Z p , Q p ) .) As another application, in Theorem 5.8, we derive from our estimate ofthe integral the congruence of Bernoulli-Hurwitz numbers BH ( n ) at super-singular primes established by Katz [Ka2, Theorem 3.11] and Chellali [Ch,Th´eor`em 1.1].1.1. Acknowledgment.
Part of this research was conducted while the firstauthor was visiting the Ecole Normale Sup´erieure in Paris, and the secondauthor Institut de Math´ematiques de Jussieu. The authors would like tothank their hosts Yves Andr´e and Pierre Colmez for hospitality. The authorswould also like to thank Seidai Yasuda for reading an earlier version of themanuscript and pointing out mistakes.2.
Schneider-Teitelbaum’s p -adic Fourier theory. Let K be a finite extension of Q p and k = F q the residue field. Let e be the absolute ramification index of K . We fix a uniformaizer π of K andlet G be the Lubin-Tate formal group of K associated to π . For a naturalnumber N and an element a of O K , we define the space A ( a + π N O K , C p )of K -analytic functions on a + π N O K as follows. { f : a + π N O K → C p | f ( x ) = ∞ X n =0 a n ( x − a ) n , a n ∈ C p , π nN a n → } . We equip the space A ( a + π N O K , C p ) with the norm k f k a,N := max n {| π nN a n |} = max x ∈ a + π N O C p {| f ( x ) |} . We also define the space LA N ( O K , C p ) of locally K -analytic functions on O K of order N by { f : O K → C p | f | a + π N O K ∈ A ( a + π N O K , C p ) for any a ∈ O K } which is a Banach space by the norm max a {k f k a,N } . We denote by LA N ( O K , C p ) the submodule of elements whose absolute values are lessthan or equal to 1. We put LA ( O K , C p ) = [ N LA N ( O K , C p ) -ADIC FOURIER THEORY 5 and equip it with the inductive limit topology. A continuous C p -linear func-tion LA ( O K , C p ) → C p is called an C p -valued distribution on O K . Wedenote the space of C p -valued distributions on O K by D ( O K , C p ), namely, D ( O K , C p ) = lim ←− N Hom cont C p ( LA N ( O K , C p ) , C p ) . We write an element of D ( O K , C p ) symbolically as Z dµ : LA ( O K , C p ) → C p , f Z f dµ = Z O K f ( x ) dµ ( x ) . The space D ( O K , C p ) has a product structure given by the convolutionproduct. For a compact open set U of O K , we let Z U f ( x ) dµ ( x ) := Z O K f ( x ) · U ( x ) dµ ( x )where 1 U is the characteristic function of U .The structure of D ( O K , C p ) is well-known for the case K = Q p anddescribed through the so called Amice transform. We denote by R rig thering of rigid analytic functions on the open disc of radius 1, namely, the ringof power series of the form ϕ ( T ) = P ∞ n =0 c n T n such that | c n | r n → < r <
1. Then there exists an isomorphism of topological C p -algebras(3) D ( Z p , C p ) ∼ = R rig , µ ϕ that is characterized by the equation c n = Z Z p (cid:18) xn (cid:19) dµ ( x )or equivalently ϕ ( T ) = Z Z p (1 + T ) x dµ ( x ) . For the Mahler expansion f ( x ) = ∞ X n =0 a n (cid:18) xn (cid:19) of f ∈ LA ( Z p , C p ), Amice showed that | a n | r n → r > Z Z p f ( x ) dµ = ∞ X n =0 a n c n . Schneider-Teitelbaum [ST, Theorem 2.3] constructed an isomorphism anal-ogous with (3) for a general local field K .Let ̟ p be a p -adic period of G . Namely, by Tate’s theory of p -divisiblegroups and the Lubin-Tate theory we haveHom O C p ( G , b G m ) ∼ = Hom Z p ( T p G , T p b G m ) ∼ = O K . BANNAI AND KOBAYASHI (The last isomorphism is non-canonical.) Hence there exists a generator ofthe O K -module Hom O C p ( G , b G m ), which is written in the form of the integralpower series exp( ̟ p λ ( t )) ∈ O C p [[ t ]] where λ ( t ) is the logarithm of G . Theelement ̟ p ∈ O C p is determined uniquely up to an element of O × K . We fixsuch a ̟ p and call it the p -adic period of G . (If the height of G is equal to 1,the inverse of ̟ p is often called a p -adic period of G , for example, see [dS]).It is known that | ̟ p | = p − s , where s = p − − e ( q − (see Appendix of [ST]or an elementary proof in [Box1] when K/ Q p is unramified). We define thepolynomials P n ( X ) ∈ K [ X ] by the formal expansionexp( Xλ ( t )) = ∞ X n =0 P n ( X ) t n . Note that in the case G = b G m , π = p and λ ( t ) = log(1 + t ), the polynomial P n ( X ) is no other than the binomial polynomial (cid:0) Xn (cid:1) . By construction, P n ( x̟ p ) is in O C p if x ∈ O K . Theorem 2.1. ( Schneider-Teitelbaum [ST, § : i) The series ∞ X n =0 a n P n ( x̟ p ) converges to an element of LA ( O K , C p ) if lim n | a n | n < . Conversely, anylocally K -analytic function f ( x ) on O K has a unique expansion f ( x ) = ∞ X n =0 a n P n ( x̟ p ) for some sequence ( a n ) n in C p such that lim n | a n | n < .ii) There exists an isomorphism of topological C p -algebras (5) D ( O K , C p ) ∼ = R rig . having the following characterization property: if ϕ ( T ) = P ∞ n =0 c n T n corre-sponds to a distribution µ , then c n = Z O K P n ( x̟ p ) dµ ( x ) or equivalently ϕ ( t ) = Z O K exp( x̟ p λ ( t )) dµ ( x ) . Schneider and Teitelbaum called the power series ϕ ( t ) corresponding to µ the Fourier transform of µ and denoted it by F µ ( t ) . -ADIC FOURIER THEORY 7 Power sums
In this section, we give an estimate of the absolute value of the powersum S N,n,k := ∂ n G X t N ∈G [ π N ] ( t ⊕ t N ) k | t =0 , where x ⊕ y = G ( x, y ), ∂ G is the differential operator λ ′ ( t ) − ( d/dt ), and G [ π N ] is the kernel of the multiplication [ π N ] of G . This estimate is crucialfor everything in this paper. We use Newton’s method to compute thisvalue.We define ρ [ l, n ] and ρ [ l, n ] by ρ [ l, n ] = max l ≤ m ≤ n {| m ! /̟ mp |} , ρ [ l, n ] = min l ≤ m ≤ n {| m ! /̟ mp |} for l ≤ n . For l > n , we put ρ [ l, n ] = 0 and ρ [ l, n ] = ∞ . Then ρ ( k ) = ρ [ k, ∞ ]and ρ ( k ) = ρ [0 , k ] are as in the introduction. Proposition 3.1. i) The values ρ ( k ) and ρ ( k ) are decreasing with k .ii) We have ρ ( k ) ≤ ρ ( k ) , ρ ( k ) ≤ ρ (0) ρ ( k ) . iii) We have ρ ( k + · · · + k n ) ≤ ρ ( k ) · · · ρ ( k n ) . iv) We have p p − − ke ( q − ≤ ρ ( k ) ≤ . Proof. i) is clear. For ii), first we have ρ ( k ) ≥ | k ! /̟ kp | ≥ ρ ( k ). Suppose ρ ( k ) = | k ! /̟ k p | and ρ ( k ) = | k ! /̟ k p | . Then k ≥ k ≥ k and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ! ̟ k p / k ! ̟ k p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) k k (cid:19) ( k − k )! ̟ k − k p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ρ (0) . For iii), suppose that ρ ( k i ) = | l i ! /̟ l i p | for l i ≤ k i . Then the assertion for ρ follows from ρ ( k + · · · + k n ) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( l + · · · + l n )! ̟ l + ··· + l n p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ( l + · · · + l n )! l ! · · · l n ! (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l ! ̟ l p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l n ! ̟ l n p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . For iv), suppose that ρ ( k ) = | l ! /̟ lp | for l ≤ k . Then p p − − ke ( q − ≤ p p − − le ( q − ≤ (cid:12)(cid:12)(cid:12)(cid:12) l ! ̟ lp (cid:12)(cid:12)(cid:12)(cid:12) = ρ ( k ) . (cid:3) If e ≤ p −
1, then we can determine ρ ( k ) and ρ ( k ) explicitly. Lemma 3.2.
Let k be a non-negative integer and let q be a power of p .i) For any integer ≤ r < q , we have (cid:0) kq + rr (cid:1) ≡ p. ii) We have (cid:0) kq (cid:1) ∈ [ k/q ] Z p . BANNAI AND KOBAYASHI
Proof. i) is clear. For ii), we write k = aq + r with 0 ≤ r < q . We put(1 + x ) q = 1 + x q + pf ( x ) for some integral polynomial f ( x ). Then(1 + x ) k = (1 + x q + pf ( x )) a (1 + x ) r ≡ (1 + x q ) a (1 + x ) r mod ap Z p [ x ] . Hence the coefficient of x q in the above is in a Z p . (cid:3) Proposition 3.3.
Let i , e and h be natural numbers. We put q = p h . Thenwe have v p ( i !) ≥ ip − − ie ( q − − h + 1 e + (cid:20) iq (cid:21) (cid:18) e − p − e ( q − (cid:19) + v p (cid:18)(cid:20) iq (cid:21) ! (cid:19) . In the above, the equality holds if and only if i ≡ − q . In particular,if e ≤ p − or i < q , we have v p ( i !) ≥ ip − − ie ( q − − h + 1 e and the equality holds if and only if i = q − , and we have ρ (0) = | π/q | .Proof. First, we assume that i < q . We prove the inequality by inductionon h . If h = 1, then i < p . Hence the left hand side is equal to zero, namely v p ( i !) = 0, and the right hand side take the maximum value when i = p − h . Since the right hand side is strictly increasing for i ,and v p ( i !) strictly increase only when p divides i , we may assume that i isof the form i = kp − k ≤ p h − . We have v p ( i !) = v p (( kp )!) − v p ( kp ) = k − v p (( k − . On the other hand, we have ip − − ie ( q − − h + 1 e = ( k −
1) + k − p − − k − e ( p h − − − ( h −
1) + 1 e + k − e ( p h − − − kp − e ( q − ≤ k − v p (( k − . In the last inequality, we used the inductive hypothesis and k ≤ p h − . Hencewe have the desire inequality and the equality holds only when k = p h − ,namely, i = q −
1. For i ≥ q , by Lemma 3.2 ii) and induction, we have v p ( i !) ≥ v p (( i − q )!) + v p ( q !) + v p (cid:18)(cid:20) iq (cid:21)(cid:19) ≥ ip − − ie ( q − − h + 1 e + (cid:20) iq (cid:21) (cid:18) e − p − e ( q − (cid:19) + v p (cid:18)(cid:20) iq (cid:21) ! (cid:19) . From the above argument and the induction, to have the equality, i mustbe congruent to −
1. On the other hand, if i ≡ − q , then directcalculations give the equality. (cid:3) -ADIC FOURIER THEORY 9 Proposition 3.4.
Suppose that e ≤ p − , and that e > or h > .i) We have | n ! /̟ np | > for < n < q .ii) For any non-negative integer n , ρ ( n ) = | n ! /̟ n p | where n = [ n/q ] q .iii) For n ≡ − q and a natural number i = q , we have (cid:12)(cid:12)(cid:12)(cid:12) n ! ̟ np (cid:12)(cid:12)(cid:12)(cid:12) > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( n + q )! ̟ n + qp (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > (cid:12)(cid:12)(cid:12)(cid:12) ( n + i )! ̟ n + ip (cid:12)(cid:12)(cid:12)(cid:12) In particular, for any non-negative integer n , we have ρ ( n ) = | n ! /̟ n p | where n = [ n/q ] q + q − .Proof. We prove i) by induction for h of q = p h . If h = 1, then n ! is a p -adicunit and the assertion is clear. Assume that h >
1. We write as n = kp + r with 0 ≤ r < p . Then n ! ̟ np = (cid:18) nr (cid:19) ( kp )! ̟ kpp r ! ̟ rp . Hence by Lemma 3.2 i) and the induction for n , we may assume that r = 0and k ≥
1. Then v p ( kp )! ̟ kpp ! = v p (( kp )!) − kpp − kpe ( q − < v p ( k !) − kp − ke ( p h − − . By the inductive hypothesis for h , the right hand side is negative or 0.Next we prove ii). Suppose that m < n . Then (cid:12)(cid:12)(cid:12)(cid:12) n ! ̟ n / m ! ̟ mp (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) n ̟ p (cid:18) n − m (cid:19) ( n − m − ̟ n − m − p (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) n ̟ p (cid:12)(cid:12)(cid:12)(cid:12) ρ (0) = (cid:12)(cid:12)(cid:12)(cid:12) n πq̟ p (cid:12)(cid:12)(cid:12)(cid:12) < . Suppose that n ≥ m > n . We write as m = [ n/q ] q + r with 0 < r < q .Then i) and Lemma 3.2 i) show that (cid:12)(cid:12)(cid:12)(cid:12) n ! ̟ n p / m ! ̟ mp (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) mr (cid:19) − ̟ rp r ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < . Finally, we show iii). Let n be such that n ≡ − q . We have( n + i )! ̟ n + ip / ( n + q )! ̟ n + qp = ( n + i )!( n + q )! ̟ q − ip = u qπ ( i − ̟ i − p π̟ q − p q !where u = (cid:0) n + qq − (cid:1) − (cid:0) n + ii − (cid:1) is a p -adic integer by Lemma 3.2 i). By Proposition3.3, the p -adic (additive) valuation of the right hand side is positive. Since v p ( π/̟ p ) >
0, the p -adic (additive) valuation of( n + q )! ̟ n + qp / n ! ̟ np = (cid:18) n + qq (cid:19) q ! π̟ q − p π̟ p is positive. (cid:3) Next we investigate the absolute values of the coefficients of a power ofthe logarithm and the exponential map of the Lubin-Tate group. The case k = 1 in the proposition below is obtained in [IS]. Proposition 3.5.
We put ∂ = d/dt . Then we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ̟ kp ∂ n λ ( t ) k k ! n ! | t =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ρ [ k, n ] − , (cid:12)(cid:12)(cid:12) ∂ n exp k G ( t ) | t =0 (cid:12)(cid:12)(cid:12) ≤ | ̟ np | ρ [ k, n ] . Proof.
The case for n < k or k = 0 is trivial. Suppose that n ≥ k ≥
1. Wefirst assume that the formal logarithm of G is given by λ ( t ) = ∞ X m =0 t q m π m . Then it suffices to show inequalities (cid:12)(cid:12)(cid:12) ∂ n λ ( t ) k | t =0 (cid:12)(cid:12)(cid:12) ≤ | k ! ̟ n − kp | , (cid:12)(cid:12)(cid:12) ∂ n exp k G ( t ) | t =0 (cid:12)(cid:12)(cid:12) ≤ | k ! ̟ n − kp | . When k = 1, the inequality for λ ( t ) is proven by direct calculations. Weprove the general case by induction on k . We have ∂ n λ ( t ) k | t =0 = k∂ n − ( λ ( t ) k − λ ′ ( t )) | t =0 = k∂ n − ∞ X m =0 λ ( t ) k − q m t q m − π m | t =0 = ∞ X m =0 (cid:18) n − q m − (cid:19) q m ! kπ m ∂ n − q m λ ( t ) k − | t =0 . Hence we have | ∂ n λ ( t ) k | t =0 | ≤ | k ! ̟ n − kp | . We put exp k G ( t ) = P ∞ n = k a n t n . We prove that | n ! a n | ≤ | k ! ̟ n − kp | by induc-tion for n . If n = k , this is true since a k = 1. We assume that the assertionis true for integers less than n . Since exp k G ( λ ( t )) = t k , we have t k = a k λ ( t ) k + a k +1 λ ( t ) k +1 + · · · + a n λ ( t ) n + · · · . By i) and the inductive hypothesis, we have | a m ∂ n λ ( t ) m | t =0 | ≤ | k ! ̟ n − kp | for m < n . Since ∂ n λ ( t ) n | t =0 = n ! and ∂ n λ ( t ) m | t =0 = 0 for n < m , theassertion is also true for n .Now we consider a general parameter s . Then the logarithm and the ex-ponential for G with parameter s are of the form λ ( φ ( s )) and ψ (exp G ( s )) forsome φ ( s ) , ψ ( s ) ∈ s O K [[ s ]] × . We put λ ( t ) k = P ∞ n = k c ( k ) n t n and λ ( φ ( s )) k = P d ( k ) n s n . Then we have shown | c ( k ) n | ≤ | k ! ̟ n − kp /n ! | . Since d ( k ) n is a linearsum of c ( k ) l ( k ≤ l ≤ n ) with integral coefficients, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ̟ kp d ( k ) n k ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max k ≤ l ≤ n ((cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c ( k ) l ̟ kp k ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)) ≤ max k ≤ l ≤ n ((cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ̟ lp l ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)) = ρ [ k, n ] − . Hence we have the inequality for the logarithm. The inequality for theexponential is straightforward. (cid:3) -ADIC FOURIER THEORY 11
Lemma 3.6. i) Suppose that f ( t ) ∈ O K [[ t ]] satisfies f ( t ⊕ t N ) = f ( t ) forall t N ∈ G [ π N ] . Then there exists a power series g ( t ) ∈ O K [[ t ]] such that f ( t ) = g ([ π N ] t ) . ii) There exists an integral power series g k ( t ) ∈ O K [[ t ]] such that π − N X t N ∈G [ π N ] ( t ⊕ t N ) k = g k ([ π N ] t ) . Proof.
See [Col], Chapter III. (cid:3)
We put F ( t, X ) = Y t N ∈G [ π N ] (1 − ( t ⊕ t N ) X ) = 1 + α ( t ) X + · · · + α q N ( t ) X q N . For ∂ X = ∂/∂X , we consider the power series(6) π − N ∂ X F ( t, X ) F ( t, X ) = − ∞ X k =0 π − N X t N ∈G [ π N ] ( t ⊕ t N ) k +1 X k . By Lemma 3.6 and the above formula, we have π − N ∂ X F ( t, X ) ∈ O K [[ t ]][ X ]. Proposition 3.7.
Let k, n be non-negative integers and N a natural number.Then we have (7) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π − N X t N ∈G [ π N ] ∂ n G ( t ⊕ t N ) k | t =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) π Nn + k (1 − q − ) ̟ np (cid:12)(cid:12)(cid:12) ρ (cid:18)(cid:20) kq N (cid:21)(cid:19) ρ (0) where k = max { [ k/q N ] − n, } . We also have (8) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π − N X t N ∈G [ π N ] ∂ n G ( t ⊕ t N ) k | t =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12) π Nn ̟ np (cid:12)(cid:12) ρ [0 , n ] . Moreover, if e ≤ p − , we have (9) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π − N X t N ∈G [ π N ] ∂ n G ( t ⊕ t N ) k | t =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12) π Nn ̟ np (cid:12)(cid:12) ρ (cid:18)(cid:20) kq N (cid:21)(cid:19) . Proof.
We put G ( t, X ) = F (0 , X ) − F ( t, X ), then G (0 , X ) = G ( t,
0) = 0.We have 1 F ( t, X ) = 1 F (0 , X ) − G ( t, X ) = ∞ X l =0 G ( t, X ) l F (0 , X ) l +1 ∈ O K [[ t, X ]] . Since G (0 , X ) = 0 and G ( t, X ) is invariant for the translation t t N , it isof the form(10) G ( t, X ) = ([ π N ] t ) H ([ π N ] t, X ) for some element H in O K [[ t ]][ X ]. Since F (0 , X ) ≡ π , the powerseries F (0 , X ) − l − is equal to ∞ X m =0 (cid:18) − l − m (cid:19) ( F (0 , X ) − m = ∞ X m =0 (cid:18) l + mm (cid:19) π m (cid:18) − F (0 , X ) π (cid:19) m . Hence we have π − N ∂ X F ( t, X ) F ( t, X ) = ∞ X l =0 π − N ∂ X F ( t, X ) · G ( t, X ) l · F (0 , X ) − l − (11) = ∞ X l =0 ∞ X m =0 (cid:18) l + mm (cid:19) π m ( π − N ∂ X F ( t, X )) G ( t, X ) l (cid:18) − F (0 , X ) π (cid:19) m . (12)To show the assertion for k + 1, we look the coefficient of X k of (12). Weconsider the coefficients of the terms X a , X b and X c with a + b + c = k of π − N ∂ X F ( t, X ), G ( t, X ) l and (1 − F (0 , X )) m π − m respectively. Sincedeg ∂ X F ( t, X ) = q N −
1, deg G ( t, X ) = q N and deg (1 − F (0 , X )) = q N − X , we have a ≤ q N − , b ≤ lq N and c ≤ m ( q N − (cid:18) l + mm (cid:19) π m G l ([ π N ] t )where G l ( t ) is a power series in t l O K [[ t ]] and l , m satisfies(13) a + lq N + m ( q N − ≥ a + b + c = k. We estimate the absolute value of(14) (cid:18) l + mm (cid:19) π m ∂ n G G l ([ π N ] t ) | t =0 . By Proposition 3.5, we have (cid:12)(cid:12)(cid:12) ∂ n G ([ π N ] t ) d | t =0 (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) π Nn d n dz n exp d G ( z ) | z =0 (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12) π Nn ̟ np (cid:12)(cid:12) ρ [ d, n ] . Therefore, we have (cid:12)(cid:12) ∂ n G G l ([ π N ] t ) | t =0 (cid:12)(cid:12) ≤ (cid:12)(cid:12) π Nn ̟ np (cid:12)(cid:12) ρ [ l, n ] . Hence we have (8). If n < l , then (14) is zero and there is nothing to prove.We assume that n ≥ l . We let l ′ ≥ l be such that ρ ( l ) = | l ′ ! /̟ l ′ p | . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) l + mm (cid:19) π m ∂ n G G l ([ π ] N t ) | t =0 (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) l + mm (cid:19) π m + Nn ̟ np (cid:12)(cid:12)(cid:12)(cid:12) ρ [ l, n ](15) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π Nn ̟ np ( l + m )! ̟ l + mp ( l ′ − l )! ̟ l ′ − lp (cid:18) l ′ l (cid:19) ̟ mp π m m ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (16)First we consider the case a ≤ q N − m = 0. Then by (13) we have l + m ≥ (cid:20) k + 1 q N (cid:21) . -ADIC FOURIER THEORY 13 In particular, m ≥ [( k + 1) /q N ] − n and the value (16) is less than or equalto | π Nn + k (1 − q − ) ̟ np | ρ (cid:18)(cid:20) k + 1 q N (cid:21)(cid:19) ρ (0)where k = max { [( k +1) /q N ] − n, } . Hence in this case we have (7). Supposethat e ≤ p −
1. If l ′ < l + m , then (cid:12)(cid:12) ̟ mp (cid:12)(cid:12) < (cid:12)(cid:12)(cid:12) ̟ l ′ − lp (cid:12)(cid:12)(cid:12) and hence the value (16)is less than | π Nn ̟ np | ρ (cid:16)h k +1 q N i(cid:17) . If l ′ ≥ l + m , then ρ ( l ) = (cid:12)(cid:12)(cid:12)(cid:12) l ′ ! ̟ l ′ p (cid:12)(cid:12)(cid:12)(cid:12) ≤ ρ ( l + m ) ≤ ρ (cid:18)(cid:20) k + 1 q N (cid:21)(cid:19) . Hence the value (15) is also less than or equal to | π m + Nn ̟ np | ρ (cid:16)h k +1 q N i(cid:17) .Hence in this case we have (9).Finally we consider the case when a = q N − m = 0. Then thecoefficient of π − N ∂ X F ( t, X ) of degree a is ( q/π ) N α q N ( t ), which is divisi-ble by [ π N ] t . Hence in this case the product of the coefficient of X a in π − N ∂ X F ( t, X ), the coefficient of X b in G ( t, X ) l and the coefficient of X c in (1 − F (0 , X )) m π − m is an integral linear combination of terms in theform G l +1 ([ π N ] t ) for some G l +1 ( t ) ∈ t l +1 O K [[ t ]]. In this case l satisfies l + 1 ≥ (cid:2) ( k + 1) /q N (cid:3) . Therefore (cid:12)(cid:12) ∂ n G G l +1 ([ π ] N t ) | t =0 (cid:12)(cid:12) ≤ (cid:12)(cid:12) π Nn ̟ np (cid:12)(cid:12) ρ [ l + 1 , n ] ≤ (cid:12)(cid:12) π Nn ̟ np (cid:12)(cid:12) ρ (cid:18)(cid:20) k + 1 q N (cid:21)(cid:19) . If n < l + 1, then (14) is zero and there is nothing to prove. We assumethat n ≥ l + 1. In particular, by (13) we have n ≥ [( k + 1) /q N ], and hence k = max { [( k + 1) /q N ] − n, } = 0. Therefore we have (7) and (9). (cid:3) Integral structures on p -adic Fourier theory In this section, we give an explicit construction of Schneider-Teitelbaum’s p -adic distribution associated to a rigid analytic function on the open unitdisc.Let ϕ ( t ) be a rigid analytic function on the open unit disc. We willconstruct a distribution µ ϕ on O K such that Z O K exp( x̟ p λ ( t )) dµ ϕ = ϕ ( t ) . If we first had the Mahler like expansion for K -analytic functions, then itis easy to define the integral like as (4), but as in [ST], we first define theintegral and then the Mahler like expansion for K -analytic function is shownby using this integral. We fix a Lubin-Tate formal group G with a parameter π and denote itsaddition by ⊕ . For a ∈ O K and a natural number N , we let(17) Z a + π N O K ( x − a ) n dµ ϕ := 1 q N ̟ np ∂ n G X t N ∈G [ π N ] ϕ a ( t ⊕ t N ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 where ϕ a ( t ) := exp( − a̟ p λ ( t )) ϕ ( t ) . We put ϕ ( t ) = P ∞ k =0 c k t k and ϕ a ( t ) = P ∞ k =0 c ( a ) k t k . Then by Proposition3.7, we have (cid:12)(cid:12)(cid:12)(cid:12)Z a + π N O K ( x − a ) n dµ ϕ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ρ (0) (cid:12)(cid:12)(cid:12)(cid:12) πq (cid:12)(cid:12)(cid:12)(cid:12) N | π | Nn sup k { | c ( a ) k | ρ (cid:18)(cid:20) kq N (cid:21)(cid:19) } (18) ≤ ρ (0) (cid:12)(cid:12)(cid:12)(cid:12) πq (cid:12)(cid:12)(cid:12)(cid:12) N | π | Nn sup k { | c k | ρ (cid:18)(cid:20) kq N (cid:21)(cid:19) } . (19)Here for the last estimate, we used the facts that c ( a ) k is a integral linercombination of c , . . . , c k and the function ρ ( m ) for m is decreasing.We define µ ϕ on LA N ( O K , C p ) as follows. For an element f of LA N ( O K , C p ),suppose f is of the form P ∞ n =0 a n ( x − a ) n such that a n π nN → n → ∞ on a + π N O K . Then we define the integral of f on a + π N O K by(20) Z a + π N O K f ( x ) dµ ϕ := ∞ X n =0 a n Z a + π N O K ( x − a ) n dµ ϕ . We define(21) Z O K f ( x ) dµ ϕ = X a mod π N Z a + π N O K f ( x ) dµ ϕ . We have to show the well-definedness of the integral.
Proposition 4.1. i) The integral (20) converges and does not depend on thechoice of the representative of a mod π N . The integral (21) does not dependon the choice of N . Hence µ ϕ gives a well-defined element of D ( O K , C p ) .ii) For a polynomial f ( x ) , we have Z O K f ( x ) dµ ϕ = f ( ̟ − p ∂ G ) ϕ ( t ) | t =0 . Proof.
Since ρ ([ k/q N ]) ≤ Ckp − keqN ( q − for some constant C which dependsonly on e, q and N , the value sup k { | c k | ρ (cid:16)h kq N i(cid:17) } is finite. Hence the con-vergence follows from (19). We show that the integral (20) depends only onthe class of a modulo π N . Since the integral is convergent, we may assume -ADIC FOURIER THEORY 15 that f is a monomial ( x − a ) n . For a ′ such that a ′ ≡ a mod π N , we put b = a ′ − a . Since( x − a ) n | a ′ + π N O K = n X l =0 (cid:18) nl (cid:19) b n − l ( x − a ′ ) l | a ′ + π N O K , it suffices to show that(22) Z a + π N O K ( x − a ) n dµ ϕ = n X l =0 (cid:18) nl (cid:19) b n − l Z a ′ + π N O K ( x − a ′ ) l dµ ϕ . However, we have ̟ − np ∂ n G ϕ a ( t ⊕ t m ) = ̟ − np ∂ n G (exp( b̟ p λ ( t )) ϕ a ′ ( t ⊕ t N ))= exp( b̟ p λ ( t )) n X l =0 (cid:18) nl (cid:19) b n − l ̟ − lp ∂ l G ( ϕ a ′ ( t ⊕ t m )) . Hence (22) follows.Now we show that the integral (21) does not depend on N . It is sufficientto show the distribution relation(23) Z a + π N O K f ( x ) dµ ϕ = X b ≡ a mod π N Z b + π N +1 O K f ( x ) dµ ϕ where the sum runs over a representative b of O K /π N +1 such that b ≡ a mod π N . To show this, replacing ϕ by ϕ a , we may assume that a = 0 and f ( x ) = x n . Then q N +1 ̟ np X b ≡ π N Z b + π N +1 O K x n dµ ϕ = X b ≡ π N k X i =0 (cid:18) nk (cid:19) b n − k ̟ n − kp ∂ k G X t N +1 ∈G [ π N +1 ] ϕ b ( t ⊕ t N +1 ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = X b ≡ π N ∂ n G X t N +1 ∈G [ π N +1 ] exp( b̟ p λ ( t )) ϕ b ( t ⊕ t N +1 ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = X t N +1 ∈G [ π N +1 ] X b ≡ π N exp( − b̟ p λ ( t )) | t = t N +1 ! ∂ n G ϕ ( t ⊕ t N +1 ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = q ∂ n G X t N ∈G [ π N ] ϕ ( t ⊕ t N ) | t =0 = q N +1 ̟ np Z π N O K x n dµ ϕ . The above calculation is also true when a = N = 0, and hence we have ̟ np X b ∈O K /π Z b + π O K x n dµ ϕ = ∂ n G ϕ ( t ) | t =0 . From this the assertion ii) follows. (cid:3)
For ϕ ( t ) = P ∞ k =0 c k t k ∈ R rig , we define k ϕ k N by(24) k ϕ k N := max k (cid:26) | c k | ρ (cid:18)(cid:20) kq N (cid:21)(cid:19) (cid:27) . Since ρ (cid:16)h kq N i(cid:17) ∼ p − kr where r = 1 /eq N ( q − k ϕ k N is approxi-mately, k ϕ k B ( p − r ) = max x ∈ B ( p − r ) { | ϕ ( x ) | } where B ( p − r ) ⊂ C p is the closed disc with radius p − r at origin. Lemma 4.2.
For an element a ∈ O K , we put ϕ a ( t ) = exp( − a̟ p λ ( t )) ϕ ( t ) .as before. Then k ϕ a k N = k ϕ k N .Proof. It suffices to show k ϕ a k N ≤ k ϕ k N . This follows from the same argu-ment showing (19). (cid:3) Then Proposition 3.7 may rewritten as follows, which is a precise versionof Theorem 1.1 of the introduction.
Theorem 4.3. i) Suppose that for a ∈ O K , the function f ∈ LA N ( O K , C p ) is given by a polynomial of degree d on a + π N O K . For ϕ k ( t ) = t k , we have (cid:12)(cid:12)(cid:12)(cid:12)Z a + π N O K f ( x ) dµ ϕ k (cid:12)(cid:12)(cid:12)(cid:12) ≤ ρ [0 , d ] (cid:12)(cid:12)(cid:12)(cid:12) πq (cid:12)(cid:12)(cid:12)(cid:12) N k f k a,N . (25) We also have (cid:12)(cid:12)(cid:12)(cid:12)Z a + π N O K f ( x ) dµ ϕ k (cid:12)(cid:12)(cid:12)(cid:12) ≤ ρ (0) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π k (1 − q − )+ N q N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k f k a,N ρ (cid:18)(cid:20) kq N (cid:21)(cid:19) (26) where k = max { [ k/q N ] − d, } . Moreover, if e ≤ p − , then we have (cid:12)(cid:12)(cid:12)(cid:12)Z a + π N O K f ( x ) dµ ϕ k (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) πq (cid:12)(cid:12)(cid:12)(cid:12) N k f k a,N ρ (cid:18)(cid:20) kq N (cid:21)(cid:19) . (27) ii) We have (cid:12)(cid:12)(cid:12)(cid:12)Z a + π N O K f ( x ) dµ ϕ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ρ (0) (cid:12)(cid:12)(cid:12)(cid:12) πq (cid:12)(cid:12)(cid:12)(cid:12) N k f k a,N k ϕ k N . (28) Moreover, if e ≤ p − , then (cid:12)(cid:12)(cid:12)(cid:12)Z a + π N O K f ( x ) dµ ϕ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) πq (cid:12)(cid:12)(cid:12)(cid:12) N k f k a,N k ϕ k N . (29) Corollary 4.4.
We have (cid:12)(cid:12)(cid:12)(cid:12)Z a + π N O K f ( x ) dµ ϕ (cid:12)(cid:12)(cid:12)(cid:12) ≤ p pp − + e ( q − ρ (0) | π | N k f k a,N k ϕ k B ′ ( p − r ) (30) where r = 1 /eq N ( q − and (31) k ϕ k B ′ ( p − r ) := max k n | c k | kp − kr o . -ADIC FOURIER THEORY 17 Moreover, if e ≤ p − , then (cid:12)(cid:12)(cid:12)(cid:12)Z a + π N O K f ( x ) dµ ϕ (cid:12)(cid:12)(cid:12)(cid:12) ≤ p pp − + e ( q − | π | N k f k a,N k ϕ k B ′ ( p − r ) . (32) Proof.
The formula follows from ρ (cid:18)(cid:20) kq N (cid:21)(cid:19) ≤ kq − N p pp − + e ( q − − keqN ( q − . (cid:3) As before, we define polynomials P n byexp( xλ ( T )) = ∞ X n =0 P n ( x ) T n . Then by formal computation, we have P k ( ∂ G ) ϕ ( t ) | t =0 = 1 k ! ∂ k ϕ ( t ) | t =0 where ∂ = d/dt (for example, formula 6 of Lemma 4.2 of [ST]). We let ϕ n ( t ) = t n and µ ϕ n the distribution associated to ϕ n ( t ). Then by Proposi-tion 4.1 ii) we have Z O K P k ( x̟ p ) dµ ϕ n = ∞ X n =0 P k ( ∂ G ) ϕ n ( t ) | t =0 = ( k = n )0 ( k = n ) . Hence if ϕ ( t ) = P ∞ k =0 c k t k , then Z O K P k ( x̟ p ) dµ ϕ = c k . Equivalently, ϕ ( t ) = Z O K exp( x̟ p λ ( t )) dµ ϕ . Proposition 4.5.
For N ≥ , we have (cid:12)(cid:12)(cid:12) qπ (cid:12)(cid:12)(cid:12) N ρ (cid:18)(cid:20) nq N (cid:21)(cid:19) − c − ≤ || P n ( x̟ p ) || N ≤ ρ (cid:18)(cid:20) nq N (cid:21)(cid:19) − where c = 1 if e ≤ p − and c = ρ (0) , otherwise.Proof. We have1 = (cid:12)(cid:12)(cid:12)(cid:12)Z O K P n ( x̟ p ) dµ ϕ n (cid:12)(cid:12)(cid:12)(cid:12) ≤ max a { (cid:12)(cid:12)(cid:12)(cid:12)Z a + π N O K P n ( x̟ p ) dµ ϕ n (cid:12)(cid:12)(cid:12)(cid:12) }≤ (cid:12)(cid:12)(cid:12)(cid:12) πq (cid:12)(cid:12)(cid:12)(cid:12) N || P n ( x̟ p ) k N ρ (cid:18)(cid:20) nq N (cid:21)(cid:19) ρ (0) . Similarly, if e ≤ p −
1, then by using (27), we get the lower estimate.We show the upper estimate. We put P n ( xπ N ̟ p ) = P nk =1 a ( n ) k x k for n ≥
1. By definition of P n , the value a ( n ) k is the coefficient of t n of ̟ kp λ ([ π N ] t ) k /k !. Since ρ ( k ) is decreasing with k , we may assume that λ ( t ) = P ∞ l =0 t q l /π l .Since [ π N ] t ≡ t q N mod π , we have λ ([ π N ] t ) ≡ λ ( t q N ) + πtf ( t )for some f ( t ) ∈ O C p [[ t ]]. (cf. [Ho, Lemma 4].) Hence we have ̟ kp λ ([ π N ] t ) k k ! = k X i =0 t i f ( t ) i ̟ ip π i i ! ̟ k − ip λ ( t q N ) k − i ( k − i )! . Therefore by Proposition 3.5 we have | a ( n ) k | ≤ ρ (cid:18)(cid:20) nq N (cid:21)(cid:19) − . Hence we have || P n ( x̟ p ) || ,N ≤ ρ (cid:16)h nq N i(cid:17) − . Then by the formula beforeLemma 4.4 of [ST], for a ∈ O K , we have || P n ( x̟ p ) || a,N ≤ max ≤ i ≤ n || P i ( x̟ p ) || ,N ≤ ρ (cid:18)(cid:20) nq N (cid:21)(cid:19) − . (cid:3) Now we prove that our definition of the distribution coincides with thatof Schneider-Teitelbaum. Namely, it has the characterization property (5).
Theorem 4.6.
Let µ ϕ be the distribution associated to a rigid analytic func-tion ϕ ( t ) on the open unit disc. Then ϕ ( t ) = Z O K exp( x̟ p λ ( t )) dµ ϕ . Conversely, for every distribution µ , there exists a unique rigid analyticfunction ϕ such that µ = µ ϕ . Then ϕ is the Fourier transform of µ , and wehave F µ ϕ = ϕ . In particular, we have an isomorphism of algebras, D ( O K , C p ) ∼ = R rig . Proof.
We have already shown the first assertion. For a given µ , we put c k := Z O K P k ( x̟ p ) dµ. Since the distribution is a continuous linear operator on the Banach space LA N ( O K , C p ) for every natural number N , there exists a positive constant C depending only on µ and N such that | c k | = (cid:12)(cid:12)(cid:12)(cid:12)Z O K P k ( x̟ p ) dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k P k ( x̟ p ) k N ≤ Cp − p − + keqN ( q − where for the last inequality, we used Proposition 3.1 and Proposition 4.5.Hence for any 0 ≤ r <
1, if we choose sufficiently large N , we have | c k | r k → -ADIC FOURIER THEORY 19 when k → ∞ . Hence ϕ ( t ) = P ∞ k =0 c k t k is a rigid analytic function on theopen unit disc. Then by construction ϕ ( t ) = Z O K exp( x̟ p λ ( t )) dµ. Since the function ( x − a ) | a + π N O K is given by1 q N ̟ np ∂ n G X t N ∈G [ π N ] exp(( x − a ) ̟ p λ ( t )) | t = t ⊕ t N | t =0 , we have Z a + π N O K ( x − a ) n dµ = 1 q N ̟ np ∂ n G X t N ∈G [ π N ] ϕ a ( t ⊕ t N ) | t =0 = Z a + π N O K ( x − a ) n dµ ϕ . Since π − nN ( x − a ) n | a + π N O K for a ∈ O K and n = 0 , , · · · are topologicalgenerators of LA n ( O K , C p ), we have Z O K f ( x ) dµ = Z O K f ( x ) dµ ϕ for all f ∈ LA N ( O K , C p ). Hence µ = µ ϕ . (cid:3) Now we prove Theorem 1.2.
Theorem 4.7. i) The series P ∞ n =0 a n P n ( x̟ p ) converges to an element of LA N ( O K , C p ) for a n satisfying | a n | ≤ ρ (cid:18)(cid:20) nq N (cid:21)(cid:19) , lim n → | a n | /ρ (cid:18)(cid:20) nq N (cid:21)(cid:19) = 0 . ii) If f ( x ) ∈ LA N ( O K , C p ) , then it has an expansion f ( x ) = ∞ X n =0 a n P n ( x̟ p ) of the form | a n | ≤ c (cid:12)(cid:12)(cid:12)(cid:12) πq (cid:12)(cid:12)(cid:12)(cid:12) N ρ (cid:18)(cid:20) nq N (cid:21)(cid:19) , lim n → | a n | /ρ (cid:18)(cid:20) nq N (cid:21)(cid:19) = 0 , where c = 1 if e ≤ p − , and c = ρ (0) , otherwise.Proof. i) follows from Proposition 4.5. For ii), we proceed as in the proof ofTheorem 4.7 of [ST] except the estimate of the Mahler coefficients. We put a n := Z O K f ( x ) dµ ϕ n . Then by Theorem 4.3, we have | a n | = (cid:12)(cid:12)(cid:12)(cid:12)Z O K f ( x ) dµ ϕ n (cid:12)(cid:12)(cid:12)(cid:12) ≤ c (cid:12)(cid:12)(cid:12)(cid:12) πq (cid:12)(cid:12)(cid:12)(cid:12) N ρ (cid:18)(cid:20) nq N (cid:21)(cid:19) . We next prove the limit in ii). We may assume that f ( x ) = P ∞ i =0 c i ( x − a ) i on a + π N O K and f ( x ) = 0 outside of a + π N O K . For a given ǫ >
0, we cantake N so that k ∞ X i = N c i ( x − a ) i k a,N < ǫ. Hence by (26), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z a + π N O K ∞ X i = N c i ( x − a ) i dµ ϕ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ C ρ (cid:18)(cid:20) nq N (cid:21)(cid:19) (33)where C is a positive constant independent of n . On the other hand, alsoby (26), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z a + π N O K N X i =0 c i ( x − a ) i dµ ϕ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C p − n e (cid:16) − q − (cid:17) ρ (cid:18)(cid:20) nq N (cid:21)(cid:19) (34)where n = max { [ n/q N ] − N , } and C is a positive constant independentof n . Hence we have (cid:12)(cid:12)(cid:12)(cid:12)Z a + π N O K f ( x ) dµ ϕ n (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ C ρ (cid:18)(cid:20) nq N (cid:21)(cid:19) (35)for sufficiently large n . Hence we have | a n | /ρ (cid:16)h nq N i(cid:17) → n →∞ . Then by i), the series P ∞ k =0 a n P n ( x̟ p ) converges to a function in LA N ( O K , C p ). We put g ( x ) = f ( x ) − ∞ X k =0 a n P n ( x̟ p ) . Then we have R O K g ( x ) dµ ϕ n = 0 for all n , and hence R O K g ( x ) dµ = 0 for alldistribution µ . Considering the Dirac distribution δ a : h h ( a ), we have g ( a ) = 0 for any a . Hence f ( x ) = P ∞ n =0 a n P n ( x̟ p ). (cid:3) Corollary 4.8.
Suppose e N,n = γ (cid:18)(cid:20) nq N (cid:21)(cid:19) P n ( x̟ p ) , ( n = 0 , , · · · ) , where γ ( u ) is an element in C p satisfying ρ ( u ) = | γ ( u ) | . If L N is the O C p -module topologically generated by e N,n , then ρ (0) − (cid:12)(cid:12)(cid:12) qπ (cid:12)(cid:12)(cid:12) N LA N ( O K , C p ) ⊂ L N ⊂ LA N ( O K , C p ) . In particular, the functions e n form a topological basis of the Banachspace LA N ( O K , C p ). Moreover, if e ≤ p −
1, then (cid:12)(cid:12)(cid:12) qπ (cid:12)(cid:12)(cid:12) N +1 LA N ( O K , C p ) ⊂ L N ⊂ LA N ( O K , C p ) . -ADIC FOURIER THEORY 21 In addition, if O K = Z p , we recover Amice’s result, namely, (cid:20) np N (cid:21) ! (cid:18) xn (cid:19) for n = 0 , , · · · form a topological basis of LA N ( Z p , C p ) .5. Relations to Katz’s and Chellali’s results.
As an application, we reprove Katz’s and Chellali’s results ([Ch], [Ka2])by using the p -adic Fourier theory.First we recall results of Katz [Ka2] and Chellali [Ch]. Let E be an ellipticcurve with complex multiplication by the ring of integer O K of an imaginaryquadratic field K . For simplicity, we assume that E is defined over K andfix a Weierstrass model y = 4 x − g x − g , g , g ∈ O K of E/ K . Let p be an odd prime. We assume that p is inert in K and does notdivide the discriminant of the above Weierstrass model, or equivalently, E has good supersingular reduction at p . Then the Bernoulli-Hurwitz number BH ( n ) is defined by ℘ ( z ) = 1 z + X n ≥ BH ( n + 2) n + 2 z n n ! , where ℘ ( z ) is the Weierstrass ℘ -function for the model. Let ǫ be a rootof unity in O K such that the multiplication by − ǫp gives the Frobenius( x, y ) ( x p , y p ) of E mod p . Let γ be a unit in the Witt ring W ( F p )such that γ p − = − ǫ − p ! p p +1 ( p − . For a fixed b ∈ O K prime to p , we put L ( n ) = (1 − b n +2 )(1 − p n ) γ n p [ np/ ( p − BH ( n + 2) n + 2 . Theorem 5.1 (Katz [Ka2]) . The number L ( n ) is integral. Let l and n benon-negative integers. Then L ( n + p l ( p − ≡ L ( n ) mod p l . Later, Chellali [Ch] refined the congruences as follows.
Theorem 5.2 (Chellali [Ch]) . Let l and n be non-negative integers. If n p − , we have L ( n + p l ( p − ≡ L ( n ) mod p l +1 . If n ≡ p − and n = 0 , put L ′ ( n ) = L ( n ) /n , then L ′ ( n + p l ( p − ≡ L ′ ( n ) mod p l +1 . In the following, let K be the unramified quadratic extension of Q p andlet G be the Lubin-Tate group of height h = 2 associated to the uniformizer π = − ǫp . We assume that [ π ] T = πT + T q for q = p is an endomorphismof G . It is known that the formal group of E at p is isomorphic to G . Proposition 5.3.
Let ϕ be an integral power series and let µ ϕ be the cor-responding distribution associated to ϕ .i) We have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z O × K x n dµ ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ p. ii) If m ≡ n mod p l ( q − , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z O × K ( x m − x n ) dµ ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ p − l + pq − . iii) If ( q − | n and m ≡ n mod p l ( q − , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z O × K (cid:18) x m − m − x n − n (cid:19) dµ ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ p − l − pq − . Proof.
We have Z a + π O K x n dµ ϕ = a n Z a + π O K dµ ϕ + n X k =1 Z a + π O K (cid:18) nk (cid:19) ( x − a ) k a n − k dµ ϕ . Then by the estimate (25) the absolute value of the first integral is lessthan or equal to p . By the estimate (27), the absolute value of the secondintegral is also less than or equal to p since || ( x − a ) || a, ρ (0) = 1. We put m − n = k ( q − x m − x n = x n k X i =1 (cid:18) ki (cid:19) ( x q − − i = kx n ( x q − −
1) + x n k X i =2 k (cid:18) k − i − (cid:19) ( x q − − i i = k (cid:18) c + c ( x − a ) + c ( x − a ) c ( x − a ) · · · (cid:19) where c i are integers satisfying p | c . Since k ( x − a ) i /i k a, ≤ p − for i ≥ s , we have( x q − ) s − s = ∞ X i =1 (log p x q − ) i i ! s i − = ∞ X i =1 ∞ X n = i c i,n ( x q − − n n != ∞ X i =1 ∞ X j + k ≥ i c i,j,k π k k ! ( x − a ) j j ! s i − -ADIC FOURIER THEORY 23 for some integers c i,n and c i,j,k . If we write m = s ( q −
1) and n = s ( q − x q − ) s − s − ( x q − ) s − s = ∞ X i ≥ ,j + k ≥ i c i,j,k π k k ! ( x − a ) j j ! ( s i − − s i − )By the estimate (25), the integral of π k k ! ( x − a ) j j ! is divisible by p − pq − . Theassertion iii) follows from this fact. (cid:3) For b ∈ O K prime to p , we put ℘ b ( z ) = (1 − b [ b ] ∗ ) ℘ ( z )and φ ( t ) = ℘ b ( z ) | z = λ ( t ) . Then ℘ b ( z ) has no pole at z = 0 and ℘ b ( z ) = X n ≥ (1 − b n +2 ) BH ( n + 2) n + 2 z n n !It is known that φ ( t ) is an integral power series. Similarly, for c ∈ O K primeto p , we put ζ c ( z ) = ( c − [ c ] ∗ ) ζ ( z ) , ζ b,c ( z ) = (1 − b [ b ] ∗ ) ζ c ( z )where ζ ( z ) is the Weierstrass zeta function and ψ ( t ) = ζ b,c ( z ) | z = λ ( t ) . Notethat ζ c ( z ) is double periodic and ζ b,c ( z ) has no pole at z = 0. Then ζ b,c ( z ) = X n ≥ ( c − c n )(1 − b n +1 ) BH ( n + 1) n + 1 z n n !and ψ ( t ) is an integral power series. Lemma 5.4. X z ∈ p Γ / Γ ℘ b ( z + z ) = p ℘ b ( pz ) , X z ∈ p Γ / Γ ζ c ( z + z ) = pζ c ( pz ) . Proof.
It is known that X z ∈ p Γ / Γ ℘ ( z + z ) = p ℘ ( pz ) . The first formula follows from this. The above formula also show that for aset S of representatives of p Γ / Γ, there exists a constant A ( S ) such that X z ∈ S ζ ( z + z ) = pζ ( pz ) + A ( S ) . We take S so that S = − S . Then since ζ ( z ) is an odd function, A ( S ) shouldbe zero. Therefore, X z ∈ S ζ c ( z + z ) = pζ c ( pz ) . Since ζ c ( z ) is an elliptic function, the left hand side does not depend on thechoice of S . (cid:3) Proposition 5.5.
We put B ( n ) = BH ( n + 2) / ( n + 2) if n ≥ and if n = − , , . For n ≥ , we have ̟ np Z O × K x n dµ φ = (1 − p n )(1 − b n +2 ) B ( n ) ,̟ np Z O × K x n dµ ψ = (1 − p n − )( c − c n )(1 − b n +1 ) B ( n − . Proof.
Since ℘ b ( z ) and ζ b,c ( z ) are double periodic, for t ∈ G [ p ] we have ψ ( t ⊕ t ) = ζ b,c ( z + z ) | z = λ ( t ) and φ ( t ⊕ t ) = ℘ b ( z + z ) | z = λ ( t ) where z isan image of t by G [ p ] → E [ p ] → p Γ / Γ. (See for example, [BK1], Lemma2.18.) From this fact and the previous lemma, we have φ ( t ) − q X t ∈G [ p ] φ ( t ⊕ t ) = ( ℘ b ( z ) − ℘ b ( pz )) | z = λ ( t ) ,ψ ( t ) − q X t ∈G [ p ] ψ ( t ⊕ t ) = ( ζ b,c ( z ) − p − ζ b,c ( pz )) | z = λ ( t ) . Hence ̟ np Z O × K x n dµ φ = ∂ n G φ ( t ) − q X t ∈G [ p ] φ ( t ⊕ t ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = ∂ z ( ℘ b ( z ) − ℘ b ( pz )) | z =0 = (1 − p n )(1 − b n +2 ) B ( n ) . The other equality is also shown similarly. (cid:3)
We put c ( n ) = (1 − p n )(1 − b n +2 ) BH ( n + 2) n + 2 . Corollary 5.6. i) We have (cid:12)(cid:12)(cid:12)(cid:12) c ( n ) ̟ np (cid:12)(cid:12)(cid:12)(cid:12) ≤ p. Furthermore, if n ≡ q − , then (cid:12)(cid:12)(cid:12)(cid:12) c ( n ) ̟ np (cid:12)(cid:12)(cid:12)(cid:12) ≤ p pq − . ii) Suppose that m ≡ n mod p l ( q − . Then c ( m ) ̟ mp ≡ c ( n ) ̟ np mod p l − pq − O C p . Furthermore, if n q − , then c ( m ) ̟ mp ≡ c ( n ) ̟ np mod p l O C p . -ADIC FOURIER THEORY 25 If n ≡ q − , then c ( m ) m̟ mp ≡ c ( n ) n̟ np mod p l +1 − pq − . Proof.
For i), the first inequality follows from Proposition 5.3 i) for µ φ .The second inequality follows from Proposition 5.3 ii) for l = 0. Note that R O × K dµ φ = 0. For ii), the first and third congruences follow from Proposition5.3 for φ , and the second inequality for ψ . (cid:3) Next, we compare c ( n ) with L ( n ). Lemma 5.7.
We choose u ∈ C p so that ̟ q − p = p p u q − . Then u is a unitof O C p and (cid:18) uγ (cid:19) q − ≡ p. Proof.
Simple calculation shows the valuation of u is zero. We have λ ( t ) = t + θt q + · · · with θ = 1 /ǫ ( p q − p ). The q -th coefficient of the integral powerseries exp( ̟ p λ ( t )) is ̟ qp q ! + ̟ p θ = ̟ p θ ̟ q − p θq ! + 1 ! . Since ̟ p θ is not integral, the valuation v p (( ̟ q − p /θq !) + 1) ≥
1. Thus ̟ q − p θq ! + 1 ≡ (cid:18) uγ (cid:19) q − (1 − p q − )( q −
1) + 1 ≡ − (cid:18) uγ (cid:19) q − + 1 mod p. must be congruent to zero. (cid:3) We write n = n ′ ( q − r with 0 ≤ r < q − c r = u − r p − [ pr/ ( q − ̟ rp .Then ̟ np = c r p [ pn/ ( q − u n . Hence we have L ( n ) = c r (cid:18) uγ (cid:19) n c ( n ) ̟ np . Therefore by Corollary 5.6 i), we have | L ( n ) | < p . (Note that if n q −
1, then | c r | < L ( n ) is contained in the unramified field K ,we have L ( n ) ∈ O K . Similary, for m ≡ n mod p l ( q − L ( n ) ∈ O K ,Lemma 5.7 and Corollary 5.6 ii) imply the congruence L ( m ) ≡ L ( n ) (cid:18) uγ (cid:19) m − n ≡ L ( n ) mod p l − pq − . Since this is a congruence between elements of O K , we have L ( m ) ≡ L ( n ) mod p l . Similarly, from Corollary 5.6 we obtain the congruences originally proved byKatz [Ka2, Theorem 3.1] and Chellali [Ch, Th´eor`em 1.1].
Theorem 5.8. i) We have L ( n ) ∈ O K .ii) Suppose that m ≡ n mod p l ( q − . Then L ( m ) ≡ L ( n ) mod p l . Furthermore, if n q − , then L ( m ) ≡ L ( n ) mod p l +1 . If n ≡ q − , then L ′ ( m ) ≡ L ′ ( n ) mod p l +1 . References [Am] Y. Amice, Interpolation p -adiques, Bull. Soc. Math. France t. (1964), 117-180.[BK1] K. Bannai and S. Kobayashi, Algebraic theta functions and p -adic interpolation ofEisenstein-Kronecker numbers, Duke Math. J. Volume , Number 2 (2010), 229-295.[BK2] K. Bannai and S. Kobayashi, Algebraic theta functions and Eisenstein-Kroneckernumbers, RIMS Kˆokyˆuroku Bessatsu B4 : Proceedings of the Symposium on AlgebraicNumber theory and Related Topics, eds. K. Hashimoto, Y.Nakajima and H. Tsunogai,December (2007), 63–78.[Box1] J. Boxall, p -adic interpolation of logarithmic derivatives associated to certainLubin-Tate formal groups. Ann. Inst. Fourier (1986), no. 3, 1–27.[Box2] J. Boxall, A new construction of p -adic L -functions attached to certain ellipticcurves with complex multiplication. Ann. Inst. Fourier (1986), no. 4, 31–68.[Ch] M. Chellali, Congruences entre nombres de Bernoulli-Hurwitz dans le cas supersin-gulier, J. Number Theory, (1990), 157-179.[Col] R. Coleman, Division values in local fields, Invent. Math (1979), 91–116.[dS] E. de Shalit, Iwasawa theory of elliptic curves with complex multiplication , AcademicPress (1987).[Ho] T. Honda, Formal groups and zeta-functions, Osaka J. Math. Volume , Number 2(1968), 199–213.[IS] T. Imada, K. Shiratani, The exponential series of the Lubin-Tate groups and p -adicinterpolation. Mem. Fac. Sci. Kyushu Univ. Ser. A 46 (1992), no. 2, 351–365.[Ka1] N. Katz, Formal groups and p -adic integration, Ast´erisque (1977), 55-65.[Ka2] N. Katz, Divisibilities, Congruences and Cartier Duality, J. Fac. Sci. Univ. Tokyo,Ser. 1A, 28 (1982), 667-678.[ST] P. Schneider, J. Teitelbaum, p -adic Fourier Theory, Documenta Math. (2001),447-481.[T] J. Tate, p -divisible groups, Proc. Conf. On Local fields, ed. T. Springer, Springer-Verlag (1967), 153-183. Department of Mathematics, Keio University, 3-14-1 Hiyoshi, Kouhoku-ku,Yokohama 223-8522, Japan
E-mail address : [email protected] Mathematical Institute, Tohoku University, 6-3 Aramaki-aza-Aoba, Aoba-ku, Sendai 980-8578, Japan
E-mail address ::