THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 6, Number 1/2005, pp

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROANIAN ACADEY, Sere A, OF THE ROANIAN ACADEY Volume 6, Number /005, ON THE TRANSCENDENCE OF THE TRACE FUNCTION Vctor ALEXANDRU Faculty o athematc, Uverty o Bucharet, Str. Academe, r. 4, Bucharet, Romaa E-mal:vcale@al.math.ubuc.ro ; vrale@.ro The am o th aer to how that the reult rom [APZ] o the trace ucto ( amely the tracedece o th ucto over C ( Z ) rove alo that, ome codto, the dervatve o the trace ucto ( over C ( Z ). F,,, are vald more geeral ramewor. We F, alo tracedetal Deote by Let be a rme umber, Q a ed algebrac cloure o. INTRODUCTION Reccomeded by Ncolae POPESCU, member o the Romaa Academy Q the eld o -adc umber ad the atural -adc module o Q ad alo by the uque eteo o to comleto o Q wth reect to (ee [APZ]). Deote G Gal( Q Q ) toology. The grou G caocaly deted to G Gal cat ( C Q ) automorhm o C. Uualy we hall aume G = ( C Q ). Q ad Q C the = edowed wth o called Krull = the grou o all Q cotou Gal cat For a elemet C, deote H = { σ G σ() = } ad O = { σ() σ G}. Accordg to [APZ] the ma σ σ() dee a atural homeomorhm to G H edowed to quotet toology, o to O edowed to toology duced by C. For ay real umber ε > 0 let u deote H ( = σ G σ ε [ G : H(, ], ad H (, = H. It clear that ε ε, the H (, ε ) H (,! { },. Oe ha. For ay ε> 0 σ G deote B( σ(), = { y O uch that σ() y ε} the oe ball wth radu ε O. The N (, ε ) = [ G : H(, ] jut the umber o all dtct oe ball O whch cover O. Further, deote H (, = { σ G σ ε}, I (, ε ) = [ G : H (, ] ad B ( σ(), = { y σ σ y ε}, the cloed ball o radu ε O. I (, jut the umber o dtct cloed boll o radu ε whch cover O. Accordg to (APZ) or ay ε > 0 deote µ ( B(, ) = the Haar meaure duced by N(, π the -adc meaure (whch uual Haar meaure µ o G (ormalzed uch that µ ( G ) = ). Alo deote

2 Vctor ALEXANDRU geeraly ot bouded) deed o ball B (,, (, B jut a µ,.e. vewed a a -adc N( umber. Accordg to (APZ) a elemet C called to be Lchtza lm ε 0 ε 0 N(, =. For α + α eamle a elemet = lm a, a Q uch that (*) 0 where d = deg α ma( d,d + ) Lchtza. Alo [APZ] t roved that Lchtza ucto : O C, where a Lchtza elemet are tegrable wth reect to -adc meaure π. Partculary, or z C {} \ O(), the ucto, are Lchtza, ad oe ca ea that the ucto z z F(, ) = d π () t, F (, = d () t zt π are aalytcal o z C {} \ O( ) zt O( ) O( ) (reectvely o z C {} \ O() ). oreover the elemet o O ( ) (reectvely O ) are the z z F, z called trace gular ot o F (, ) (reectvely o F (, )). I [APZ] the ucto ucto ad the act that the elemet o O ( ) are jut the gular ot or ( F, roved oly the cae where the elemet very the troger codto (*). I act, [APZ] roved that or a elemet wth codto (*) there et a equece ( z ), z uch that F (, z ). By th remar (whch roved [APZ], but ot o eay) t ollow that F (, tracedetal over C () z tracedetal over z F, z are related by the ormula Q. Sce the ucto F (, ) ad (, = zf(, ), z C {} O() F z there rezult that eough to tudy the the behavor o oe o there ucto aroud o t gular ot.,. THE TRANSCENDENCE OF THE TRACE FUNCTION The am o th aer to how that the reult rom [APZ] wth reect to the ucto F (, whe a (*) elemet, are alo vald more geeral ramewor, amely, or the elemet deed the trace. Tr = t d π() t deed (ee (PVZ)) ad get oe addtoal aumto. Partculary th O() vald or all Lchtza ot. I what ollow we hould deote by F (, = dπ () t. z t O( ) Accordg to [APZ] let { α } be a dtguhed equece o (.e. = lm α, α Q ) ad deote D = deg α, ad ε = α, 0. Oe ha ε 0. Accordg to [APZ] ad [PVZ], ( ε ) a decreag equece o otve real umber wth lmt zero ad S a yteme o rereetatve o let coet o G modulo H(, ε ) or modulo H (, ε ), the oe ha: () F, z = lm (where d = S = [ G : H (, ε )]), the lmt beg d σ σ S z

3 3 O the tracedece o the trace ucto tae or ay z C {} \ O(). Alo t clear that the covergece o the above lmt uorm o the comlemet o ay eghbourhood o O. I what ollow we hall ue the ollowg reult roved [APP]. I a Lchtza elemet, α there et a equece ( α ), α Q [] Q uch that lm α =, ad 0 whe d d = deg Q ( α ) = [ G : H(, ε )], ad Q ( α ) = FH(, ε ) (o coure, ε α ad ε = 0 ). By th reult, there reult that F, z = lm = lm, where d z d S σ α σ = IrrQ ( α ). O coure, oe ha d d +, or all. It clear that ad o ( α ) Q ( α ) olyomal o H Q ad thu (, ε ) H ( α ) = { σ G σ( α ) = ( α )}, d dvde D, D = d q. Let u deote ( X ) the mmal α over Q. Alo deote S a ytem o rereetatve o let coet o G wth reect to H α. Oe ca aume S S. oreover oe ca aume that the elemet o S are o the orm { } g g ru a ytem o let coet o H ( ε ) q ). It clear that B(, ε ) O cota jut g α where g belog to the chooed ytem o let rereetatve o the H ( ε ) σ, σ S, ad j umber o thee coet jut orm H( α )., wth reect to By the above coderato oe ow that: F( {} O() z C \. Now oe have: = D σ α σ S z z C \ O H α (remd that the q cojugate o Remar. Oe ha: F(, lm, z C {} \ O() α o the, wth reect to, = lm, d σ α σ S z. Proo. Ideed, or ay ed {} () ad a utable ball ( z, δ) O, ad ug the orm o uorm covergece o B ( z, δ) oe ha: ( g ) B whch do ot terect σ α σ α ε = 0 d σ S z σ( α ) d S z d σ σ α σ S ( z σ α )( z σ( α) ) d δ whe. Furthermore, oe ca wrte: ( ) F, z = lm = lm = lm q g H, ε S H α d σ D z σ g α σ S z σ α D z (

4 Vctor ALEXANDRU 4 g By a reeated ue o the algorthm o dvo wth remader there reult that ay olyomal Q [] ca be uquely wrtte a: = a e e =... q q, oreover oe ha g u a Prooto. Oe ha: g (te um) where deg =, ad 0 0 e D, =,..., q., 0 e D, =,..., q. =, ee [APZ.] 0 u = Proo. Aume, that the Prooto ot true. The there et a real umber > 0, uch that, or all or equvaletly,. But the oe mut have : or all. Ideed, σ S α the reult σ ( α ), where S α the et o all cojugate o α over,. Now let D Q [] Q [] σ S σα ( ) = α : the ma deed by: D P() a um o term o orm Q. Now ce: = P, P Q [], whe P mea the ormal dervatve. Oe ow that D a lear ma ad oe ha: ( P()() Q ) = P DQ Q DP(). Sce D + or all the by revou Remar, there rezult that D cotuou, ad o t ca be eteded by cotuty to a ma D : Q [] Q [], uch that D( PH ) = PD( H ) + HD( P) or all P, H Q []. Deote L = Q [] Q. Sce L de Q [], the rezult that D ot detcal, zero o L. Hece let α uch that D ( α) 0, ad q = Irr ( α) L X. Q 4

5 5 O the tracedece o the trace ucto = D q D There oe ha: 0 ( α) = q ( α) ( α) ad o q ( α) = 0 Hece oe mut have u =, a clamed., a cotradcto. Now ug the ame argumet a ([APZ], ag 44) there rezult that z ear by O, the value o the ucto F(, = lm, ca be a bg a we wat abolute value. But the d (ee alo [APZ]) rom th there rezult that the ucto F (, tracedetal over [ Z] Theorem. Wth the revou otato aume that 0 d. The F (, C. ot bouded where z ear by O. The F (, a tracedetal over C [ Z ]. oreover by tracedece o F (, ollow the tracedecy o ( obta: Proo. Oe ha: Now ce F,. ( ) z z z F(, = lm = lm = d d ( ( ) ( ( = lm d ( d ( d F (, t clear that d 0 ad o oe O aother art oe ha: F (, = lm d () = () ()

6 Vctor ALEXANDRU 6 ad o would be a real umber > 0 uch that ad, the oe would obta:, or all. Now ce th ot oble accordg to above codto, oe mut have: But there rezult that F (, u () = or u =. ot bouded aroud o the argumet o [APZ]. For the dervato o hgher order oe ca obta mlar reult. () O, ad o t tracedetal over [ Z] C ee REFERENCES. [APZ] V. Aleadru, N. Poecu, ad A. Zaharecu, O cloed ubeld o C, J. Number Th. 68 (998), [APZ] V. Aleadru, N. Poecu, ad A. Zaharecu, Trace o C, J. Number Th. 88 (00), [APP] V. Aleadru, E. L. Poecu, ad N. Poecu, O the cotuty o the trace, Proceedg o the Romaa Academy, 004 (to aear) 4. [Ar] E. Art, Algebrac Number ad Algebrac Fucto, Gordo & Breach, New Yor, [B] D. Bary, Traormato de Cauchy -adque et algébre d Iwaawa, ath. A. 3 (978) [F.V.] J. Freel, ad. Va der Put, Géométre Aaltque Rgde et Alcato, Brhäuer, Bael, [Sch] W.H.Schho, Ultrametrc Calculu, A Itroducto to -adc Aaly, Cambrdge Uv. Pre, Cambrdge, UK, 984 Receved December, 004 6