Journal of Ramanujan Mathematical Society, Vol. 24, No. 2 (2009)

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1 Joural of Ramaua Mathematical Society, Vol. 4, No. (009) IWASAWA λ-invariants AND Γ-TRANSFORMS Aupam Saikia 1 ad Rupam Barma Abstract. I this paper we study a relatio betwee the λ-ivariats of a p-adic measure ad its Γ-trasform exploitig certai combiatorial idetities. Alog the way we also determie p-adic properties of certai Mahler coefficiets. Key Words: p-adic measure, Γ-trasform, Iwasawa ivariats, Mahler coefficiets. 000 Mathematics Classificatio Numbers: Primary 11F85, 11S80 1. Itroductio Fix a odd prime p. Let O be the rig of itegers i a fiite extesio of Q p with a local parameter π. We write Z p = V U where V is the group of (p 1)st roots of uity i ad U = 1 + p. Let u be a topological geerator of U. The proectios from Z p oto V ad U are deoted by ω ad <> respectively. We have a isomorphism φ : U give by φ(y) = u y. Let Λ deote the O-valued measures o. It is well-kow, (see e.g. [1]), that Λ is a rig uder covolutio, ad is isomorphic to the formal power series rig O[[T 1]]. Explicitly, for x, let ( ) x T x = (T 1) O[[T 1]]. =0 The power series associated to a measure α Λ is the defied by ˆα(T ) = T x dα(x) = b (α)(t 1) where b (α) = =0 ( ) x dα(x). A classical theorem of Mahler states that ay cotiuous fuctio f : Q p may be writte uiquely i the form ( ) x f(x) = a (f), =0 1 Departmet of Mathematics, Idia Istitute of Techology, Guwahati , Assam, INDIA Departmet of Mathematical Scieces, Tezpur Uiversity, Napaam-78408, Assam, INDIA. 1

2 Saikia ad Barma where a (f) Q p, a (f) 0 as. I fact ( ) a (f) = ( 1) f(). (1.1) =0 This theorem may be geeralized to cotiuous fuctios f : K, where K is ay fiite extesio of Q p. Usig this geeralizatio, we obtai the followig ( ) x f(x)dα(x) = a (f) dα(x) = a (f)b (α). =0 Note that if O is the rig of itegers of K ad f : O, the a (f) O. For a Z p, deote by α a the measure o give by α a(a) = α(aa) for all compact ope subsets A of. Also, for a compact ope subset A, we let α A deote the measure obtaied by restrictig α to A ad extedig by 0. The Γ-trasform of a measure α is defied as a fuctio of the p-adic variable s give by Γ α (s) = < x > s dα(x). Z p Splittig up the itegral, ad puttig dα(ax) for dα a(x), we ca also write Γ α (s) = < ηx > s dα(ηx) = x s dβ(x), η V U U where β = η V (α η) U, a measure o U. Now the measure β may be viewed as a measure o via the isomorphism φ: β(a) = β(φ(a)). It is customary to write dβ(u y ) for d β(y). Let G(T ) be the power series associated to β, that is, G(T ) = T y dβ(u y ). The Γ α (s) = G(u s ), so that Γ α (s) is a Iwasawa fuctio over O. =0. Iwasawa λ-ivariats ad Γ- trasforms The Iwasawa µ ad λ- ivariats of a power series F (T ) = a (T 1) O[[T 1]] are defied by =0 µ(f (T )) = mi{ord(a ) : 0} λ(f (T )) = mi{ : ord(a ) = µ(f (T ))}

3 Iwasawa λ-ivariats ad Γ-trasforms 3 For a measure α, we uderstad µ(α) ad λ(α) to mea µ(ˆα(t )) ad λ(ˆα(t )). Let α Λ be a O-valued measures o. Let u be a fixed topological geerator of U = 1 + p, ad let G(T ) satisfy G(u s ) = Γ α (s), so that G(T ) = T y dβ(u y ), where β = (α η) U. (.1) η V Note that β is a measure o U. We exted β to by 0 ad the we get a power series ˆβ(T ) = =0 b (T 1). Suppose that G(T ) = =0 g (T 1). Siott i his paper [4] proved that µ(g(t )) = µ(α + α ( 1)), if ˆα(T ) is a ratioal fuctio of T. Here α = α Z p. It was Kida who first obtaied a relatio betwee the λ-ivariat of a measure ad its Gamma-Trasform with a fixed topological geerator []. Later, Nacy Childress proved the followig results i her paper [1]: Result.1. µ(g(t )) = µ(β). Result.. Suppose λ(g(t )) p, the λ(β) = pλ(g(t )). She remarked that it would be iterestig to kow whether her methods ca be exteded for larger λ(g(t )). Satoh obtaied the same result without ay coditio o λ(g(t )), but his approach was based o certai properties of Stirlig umbers [3]. I this paper we prove the followig mai result i the spirit of Childress. Theorem.3. Suppose λ(g(t )) p, the λ(β) = pλ(g(t )). We will prove this theorem exploitig certai combiatorial idetities, which we shall prove i the ext sectio. Through our approach we also derive certai p-adic properties of Mahler coefficiets. Note that the relatio betwee b m ad g m is give by the followig result i Childress [1]. Result.4. If ord p (m!), the b m r=0 g ra r (f m ) (mod p). Here, a m (f )s are the Mahler coefficiets of f (x) = ( ) u x = m=0 a m(f ) ( x m). We will ivestigate p-adic properties of the Mahler coefficiets a m (f ). I order to study the Mahler coefficiets a m (f ) we will require certai idetities ivolvig biomial coefficiets, which will be established i a combiatorial fashio i the ext sectio. 3. Certai Combiatorial Idetities The followig result was a crucial igrediet i the work of Childress [1]. Result 3.1. ti ( 1) i = t. i i=1 Here we will prove a more geeral result. Lemma 3.. For o-egative itegers, t, k, we have t(i + k) ( 1) i = t. (3.1) i

4 4 Saikia ad Barma Proof. The result is obvious for t = 0 or = 0. So we assume, t 1 ad k 0. Let N, N, T be sets such that N N, N =, N = + k, ad T = t. Let R be the set of all -subsets of N T. Clearly R = ( ) t(+k). Also, for a N, let Ra be the set of all -subsets A of N T such that (a, b) / A for ay b T. Obviously R a is the set of all -subsets of (N {a}) T ad hece R a = ( ) t(+k 1). For I N, let R I be the set of all -subsets A of N T such that (a, b) / A for ay a I ad for ay b T. Clearly R I is the set of all -subsets of (N I) T ad hece ( ) t( + k i) R I =, where I = i. (3.) If I = {a 1,, a i }, the clearly R I = R a1 R ai. Thus R a1 R ai = ( ) t(+k i). By iclusio-exclusio priciple, we get a N R a = a N = Therefore, R a {a 1,a } N + + ( 1) +1 a N R a R a1 R a + + ( 1) i+1 {a 1,,a i } N R a1 R ai ( + k i)t ( 1) i+1. (3.3) i i=1 R R a = R ( ) t( + k) ( + k i)t R a = ( 1) i+1 i a N a N i=1 t( + k i) = ( 1) i i t(i + k) = ( 1) i. (3.4) i A fuctio f : N T may be viewed as a -subset of N T. Coversely, a -subset A N T defies a fuctio f : N T if ad oly if the cardiality of the set {a N : (a, b) A for some b T } is equal to. Therefore, it is ot difficult to see that there is a oe-to-oe correspodece betwee R a N R a ad the set of all fuctios from N to T. Thus R a N R a = t, which proves the result because of (3.4). Remark 3.3. The result (3.1) of Childress is othig but lemma (3.) with k = 0. Lemma 3.4. For o-egative itegers, t with > 1, we have ti ( 1) i = 0. i 1

5 Iwasawa λ-ivariats ad Γ-trasforms 5 Proof: Sice > 1, we have ( ) ( i = 1 ) ( i + 1 i 1). Usig this ad Lemma (3.) for k = 1, we get ti ( 1) i i 1 { } { } 1 ti 1 ti = ( 1) i + ( 1) i i 1 i 1 1 { 1 } { 1 ti 1 } 1 t(i + 1) = ( 1) 1 i + ( 1) 1 i i 1 i 1 = t 1 + t 1 = p-adic properties of Mahler coefficiets a m (f ) Let us fix a topological geerator u = 1 + t 1 p + t p + of 1 + p. Hece t 1 is a uit. It is ot difficult to see that (1 + T ) up+ (1 + T )(1 + T p ) t 1 (1 + T p ) t 1+ ( 1) t 1 +t + higher order terms (mod p). (4.1) (1+T ) u (1+T )(1+T p ) t 1 (1+T p ) ( 1) t 1 +t + higher order terms (mod p). (4.) Usig these biomial expasios, we prove the followig lemmas about the Mahler coefficiets a m (f ) for differet m ad. Lemma 4.1. Suppose that 1 k < p ad p + (k 1)p m < p + kp. The Proof: From (1.1), we have a p+k (f m ) 0 (mod p). p+k p + k u a p+k (f m ) = ( 1) p+k. (4.3) m =0 But, ( u m) is the co-efficiet of T m i the expasio of (1+T ) u. Clearly, if p +(k 1)p m < p + kp ad m p + (k 1)p, p + (k 1)p + 1, the from (4.1) ad (4.) we fid that the co-efficiet of T m i (1 + T ) u is zero modulo p. Also, co-efficiets of T m modulo p i (1 + T ) u are equal for m = p + (k 1)p, p + (k 1)p + 1. Thus, to prove that a p+k (f m ) is zero modulo p whe m = p + (k 1)p, p + (k 1)p + 1, we eed to prove for m = p + (k 1)p oly. If k = 1, the ( ) ( ) ( ) u u p+1 a p+1 (f p ) + t t 1 + (t 1 + t ) 0 (mod p). (4.4) p p p

6 6 Saikia ad Barma Therefore, we assume that k > 1. From (4.1) ad (4.), we have ( ) u = co-efficiet of T m i the expasio of (1 + T ) u m ( ) { } t1 ( 1) t 1 + t (mod p) if < p (4.5) k 1 ad (u ) m Now, ( ) { it1 t 1 + k 1 p+k ( p + k a p+k (f m ) = ( 1) p+k =0 k =0 =p ( 1) p+k ( p + k } i(i 1) t 1 + it )( ) u m p+k p + k u + ( 1) p+k m + k =0 ( 1) k ( p + k (mod p) if = p + i, 0 i < p. (4.6) )( ) { } t1 ( 1) t 1 + t k 1 )( ) { } t1 ( 1) t 1 + t k 1 k { p + k ( 1) k t1 t 1 + k k 1 ) (mod p) ad hece (4.7) implies that =0 Agai, ( ) ( p+k k a p+k (f m ) =0 } ( 1) t 1 + t (mod p). (4.7) k k ( 1) k t1 t 1 (mod p). (4.8) k 1 Usig Lemma (3.4), we complete the proof of a p+k (f m ) 0 (mod p) whe m = p + (k 1)p ad this completes the proof of the lemma. Lemma 4.. Suppose that 1 k < p. The a p+k (f p +kp) t k+1 1 (mod p) ad a p+k+1 (f p +kp) 0 (mod p). Proof: Proceedig as Lemma (4.1), we fid that { k ( k a p+k (f p +kp) t 1 ( 1) k ad a p+k+1 (f p +kp) t 1 =0 { k+1 =0 ( 1) k+1 ( k + 1 )( ) } t1 (mod p) k )( ) } t1 (mod p). Usig result (3.1) ad lemma (3.4), we complete the proof of the lemma. k

7 Iwasawa λ-ivariats ad Γ-trasforms 7 Lemma 4.3. Suppose that p p m < p. The a p (f m ) 0 (mod p). Also, a p (f p ) t 1 (mod p), a p+1 (f p ) 0 (mod p), ad a p+ (f p ) 0 (mod p). Proof: Suppose that p p m < p. From (1.1), we have p a p (f m ) = ( 1) p m =0 ( ) p + p m m { ( ) } p co-efficiet of T m i (1 + T ) up + (1 + T ) up p 0 (mod p). (4.9) We obtai (4.9) usig the biomial expasio (4.1). Agai, ( ) p a p (f p ) + p p p { ( ) } p co-efficiet of T p i (1 + T ) up + (1 + T ) up p ( ) ( ) t1 t1 + Also, modulo p t 1 (mod p). (4.10) ( ) ( ) {( ) ( )} ( ) ( ) u p + 1 u p+1 u p+1 a p+1 (f p ) + + p p p p p p ( ) p + 1 { co-efficiet of T p i (1 + T ) u + (1 + T ) up (1 + T ) up+1} p ( ) t + (1 + T ) up + (1 + T ) up+1 ( ) {( ) ( )} ( ) ( p + 1 t1 t1 + t t1 t1 + t + p But, ( ) p+1 p (mod p). Usig this i (4.11), we fid that Fially, we prove that a p+ (f p ) 0 (mod p). ). (4.11) a p+1 (f p ) 0(mod p). (4.1)

8 8 Saikia ad Barma Usig ( ) ( p+ p (mod p) ad p+ ) p+1 4 (mod p), we fid that ( ) ( ) ( ) ( ) u u u p+1 a p+ (f p ) p p p p ( ) ( ) ( ) ( ) u p+ u p+1 u p+ + + p p p p co-efficiet of T p i (1 + T ) u + (1 + T ) u (1 + T ) up + 4(1 + T ) up+1 (1 + T ) up+ + (1 + T ) up (1 + T ) up+1 + (1 + T ) up+ ( ) ( ) ( ) ( ) ( ) t t t t1 t1 + t t t 1 + t ( ) ( ) ( ) t1 t1 + t t t 1 + t 0(mod p). (4.13) This completes the proof of the lemma. 5. Proof of Mai Result Now we have all the igrediets for the proof of the mai result. We may assume that µ(g(t )) = 0, because µ(g(t )) = µ(β) by result (.1), ad for ay power series F (T ) O[[T 1]], if π F (T ) the λ(π 1 F (T )) = λ(f (T )). Childress i her paper [1] proved that if λ(g(t )) p, the λ(β) = pλ(g(t )). Hece it is eough to prove the Theorem (.3) for p < λ(g(t )) p. Case (i): Suppose that λ(g) = p + k where 0 < k < p. The g i 0 (mod π) for i = 0,, p + k 1 ad g p+k is a uit. Clearly, ord p ((p + kp)!) = p + k + 1 ad if m < p + kp, the ord p (m!) p + k. Also, if m < p + (k 1)p, the ord p (m!) < p + k. Usig result (.4) ad g i 0 (mod π) for i = 0,, p + k 1, we have b m 0(mod π) if m < p + (k 1)p (5.1) ad b m g p+k a p+k (f m )(mod π) if p + (k 1)p m < p + kp. (5.) From lemma (4.1) ad (5.), we get b m 0 (mod π) ad hece b m 0(mod π) if m < p + kp. (5.3) Sice ord p ((p + kp)!) = p + k + 1, usig Lemma (4.), we have b p +kp p+k+1 r=0 g r a r (f p +kp) (mod p) g p+k a p+k (f p +kp) + g p+k+1 a p+k+1 (f p +kp) (mod π) g p+k t k+1 1 (mod π), (5.4) which is a uit i O. This proves that λ(β) = p + kp = pλ(g(t )). Case (ii): Now suppose that λ(g(t )) = p. The g i 0 (mod π) for i = 0,, p 1 ad g p is a uit i O. If m < p p, the ord p (m!) < p ad hece from result (.4),

9 Iwasawa λ-ivariats ad Γ-trasforms 9 we have b m 0 (mod π). If p p m < p, the ord p (m!) p ad hece from result (.4) ad lemma (4.3), we have b m p r=0 g r a r (f m ) (mod p) g p a p (f m ) 0 (mod π). (5.5) Thus, if m < p, the b m 0 (mod π). Agai, ord p ((p )!) = p + ad hece b p p+ r=0 g r a r (f m ) (mod p) g p a p (f p ) + g p+1 a p+1 (f p ) + g p+ a p+ (f p ) (mod π). (5.6) From (4.10), (4.1), (4.13), ad (5.6), we have b p g p t 1 (mod π). Therefore, b p is a uit i O ad hece λ(β) = p = pλ(g(t )). This completes the proof of the mai theorem. 6. Ackowledgmet We are very grateful to R. Suatha for her advice, helpful discussios ad ecouragemet. The secod author gratefully ackowledges the fiacial support ad the accommodatio of the School of Mathematics, Tata Istitute of Fudametal Research Idia, durig July-August 008. Refereces [1] N. Childress, λ-ivariats ad Γ-trasforms, Mauscripta math. 64, (1989). [] Y. Kida, The λ-ivraits of p-adic measures o ad 1 + q, Sci. Rep. Kaazawa Uiv. 30, (1986). [3] J. Satoh, Iwasawa λ-ivariats of Γ-Trasforms, Joural of Number Theory, 41, (199). [4] W. Siott, O the µ-ivariat of the Γ-trasform of a ratioal fuctio, Ivet. Math. 75, 73-8 (1984). Departmet of Mathematics, Idia Istitute of Techology, Guwahati , Assam, INDIA address: a.saikia@iitg.eret.i Departmet of Mathematical Scieces, Tezpur Uiversity, Napaam-78408, Soitpur, Assam, INDIA address: rupamb@tezu.eret.i