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INTERPOLATION OF NUMBERS OF CATALAN TYPE IN A LOCAL FIELD OF POSITIVE CHARACTERISTIC GREG W ANDERSON Assumed Let k be a locally compact topological eld of positive chaxacr terlstlc Let L be a cocompact discxete addltlve subgroup of 0 Let U be an open compact addltlve subgroup of 0 Let 2 u and a be elements of 1c with a nonzero We study the behavior of the product Homewmmw 2 as a varies uslrg tools fzom local class eld theory and harmonic analysis Typically ratios of such products occur as partial products grouped by degree fox the in nite products representlrg special values of Gammatunctlons fox function elds Our main result provides local con rmation fox a tworvaziable re nement of the Stark conjecture in the function eld case recently proposed by the author CONTENTS I Introduction I 2 Theta and Catalan symbols 6 3 Shadow theta and Catalan symbols 18 4 lnterpolability and related notions 32 5 Concrete examples of strict interpolation 36 6 The interpolation theorern 44 References 50 1 INTRODUCTION 11 Preliminary discussion of the main objects of study 111 The pmduct 0f lattzce paints m a 1m We are going to study products in a locally cornpact topological eld 1c of positive characteristic of the form L c k cocornpact discrete additive subgroup 1 H r for U c h open cornpact additive subgroup 0 ZL auI A u e k and U 7 a e k with the goal of understanding how such numbers depend on the pararneter a using tools from harmonic analysis and local class eld theory We think of z L as a lattice and ofauU as a box which grows as a grows in absolute value our rnain result Theorern 61 below gives detailed information about ratios of numbers of the form 1 and provides local con rmation of a tworvariable re nement of the Stark conjecture in the function eld case recently proposed in 3 Data Match 26 2006 MSO2000 11Sz111580 2 GREGW ANDERSON 112 Theta aaa catalaa symbals The notation used in 1 is inconvenient es pecially when as is usually the case in practice we wish to consider ratios or complicated rnonornials in such products We introduce a rnore strearnlined systern of notation now Following the prohahilists let 15 X a 01 c C denote the indicator function of a suhset s c X i e 15r 7 1 for r e s and 15r 0 otherwise Let 3 denote the cardinality of a set 3 Let AX denote the group of units of a ring A with unit Let p he the characteristic of 1c Let Zlp he the result of adjoining an inverse of p to Z Note that for every r e kx and a e Zlp the expression a We rde nes a nonzero elernent ofthe perfect closure W of h Given ta h a C let f a 1c X h a C he de ned hy f axriy frgy We corne now to the key de nition We say that a function ltlgt 1c X h a Z1p is a mtwnal nggfid atataal latttce if it is a nite Zlprinear cornhination of functions of the forrn 12L laip where t L a and U are as in 1 We warn the reader that the de nition ofrational rigged virtual lattice made here is not quite the sarne as that made in the rnain hody of the paper hut the di erence is inconsequenr tial A similar warning applies to the other de nitions made and applied in the introduction Given a rational rigged virtual lattice ltlgt and a e M we de ne a lt1gt qurair e Zlp wek lt E gt H Ilt1gtaa a 6 atst We call 9 the theta symaal lfthe function ltlgt is supported in a cornpact subset oka we say that ltIgt is prayer For the rnost part our attention is going to he focused on proper rational rigged virtual lattices We call 0 the parttal Catalan symaal Of the Catalan syrnhol itself we speak a hit later For exarnple notation as in 1 we have a 12L law w L m au U and the expression a lt 12L 1aU gt represents the product 1 Having introduced the formalism of theta and Catalan syrnhols we are now ahle to handle ratios and rnonornials in products of the form 1 in an ef cient way 113 Rattaaale far the thaw termtaalaaay aaa caaaecttaa wtth erunctwns The qunction evaluators em guring in Tate s formulation 9 of the Stark cork jecture can in the function eld case he related to Mellin transforrns of functions of the forrn ltIgt hy the methods of Tate s thesis In particular partial zeta values at s 0 can he represented as values of the theta syrnhol in certain situations We do not discuss the connection of the theta syrnhol with qunction evaluators here hut see 3 12 for a discussion of the analogous connection in the glohal setting Theta syrnhols and qunction evaluators a la Tate also have sorne relationship with theta divisors For exarnple see 2 Thrn 411 While we do not discuss qunctions explicitly in the paper the desire to make our results easily applicahle to the study of qunctions has dictated our heavy emphasis on harrnonic analysis INTERPOLATION OF NUMBERS OF CATALAN TYPE 3 114 Ratlaaale fwthe Catalan tm mlnalagy We regard the values ofthepartial Catalan symhol 1 as analogues ofthe classical Catalan numhers The formula 1 2a Heeaaoorai w a 1 a l39leeaaooai w Heeaamani w HmRx Ilwlmaar lwlmarlwlmampww for the usual Catalan numhers suggests how we see the analogy Actually the formula 2quot 7 12 1u 21 2 12 1u 1 l7 71 lt a gt H I welkx for the Catalan numhers of type B shows o the analogy in a slightly hetter way 115 Eaalaatlaa af partlal catalaa spmlals m slmple eaamples aaep FAT Let FAT he the ring of polynomials in a variahle T over the eld T1 of q elements Let FAT he the fraction eld oleqm Let us temporarily just in this paragraph and the next identify the local eld lc considered in 1 with the completion 1Fq1T oleqT at the place T no Let lTFq1T he the open unit hall in JFq1T We have lt T gt 2 a H Mng llFalTl 111TFa1T for all integers N 2 01 The product on the left is the Carlitz analogue of the factorial of we For hackground concerning the latter see 11 4151 We have s TN o lt gt MET1lt n ashramleam 1ilt1Tgtwall1Tll for all s e lTFq1T and integers N 2 0 The product on the left is for s at 0 a reciprocal partial product grouped hy degree for the in nite product 1 s 1 g H 1 l MEAT e W representing the value at s of the geometric Prfunctjon over mm For hachground on Prfunctjons for function elds see 5 Chap 9 or 11 Chap 4 The general theory of Prfunctjons for function elds provides a vast supply of arithmetically signi cant in nite products whose partial products grouped hy degree can he rep resented as values of partial Catalan symhols and thus made accessihle to study hy our methods 116 Eramplas af lntarpalatwn fw mulas By a calculation with Moore deterrml nants it is possihle to rewrite 2 in the form Nl TN1 lt4 Tq 7T lt 1 1 1 gt i FAT lt 11T1Fa1T 7 q 1T11T Fq1TH for all integers N 2 0 There are two things to notice here Firstly dependence of the right side on N is explained hy a varying exponent H on the left Secondly 4 GREG W ANDERSON the rational rigged virtual lattice appearing on the right is proper Similarly by a slightly more involved manipulation of Moore determinants one veri es that qN1 qN171 17alt7Tr v limbo qr lt5 TN lt Mum 11mm 11lt1Tgt1Fq1T11 gt for 0 f s with 15 6 qu and integers N 2 0 Once again we see that dependence on N of the right side is explained by a varying exponent qN1 on the left and that the rational rigged virtual lattice on the right is proper Many more interpolation formulas analogous to 4 and 5 are known which could in principle be expressed in terms of values of the partial Catalan symbol on proper rational rigged virtual latticesi For example there is a generalization of 5 for all 0 f s E quT lTqulT too complicated to repeat hereisee 11 8i475 which plays a key role in the analysis 4 oftranscendence properties of special values of the geometric Ffunction over quTi These examples suggest the possibility of a general interpolation formula relating partial Catalan symbols to variable expo nent expressionslli We prove here in the local setting that such a thing does indeed existi Analogous global phenomena are expected The author s paper 3 explains the still conjectural global picture and makes the link to Starkls conjecture 12 The asymptotic interpolation theorem We formulate a weakened version of our main result preciselyi We then indicate without many details the directions in which this weakened result is strengthened in order to achieve our main result We comment brie y on the global situationi li2ili Apparatus from local class eld theory Let be the canonical absolute value of k i e the modulus for Haar measure on kl Let kabk be the abelian closure of k in a xed algebraic closure of kl Let p kX A Galkabk be the reciprocity law homomorphism of local class eld theory normalized as in 10 Under that normalization we have CM Cllall for all C in the algebraic closure of the prime eld in kab and a E kxi 122 Asymptotic inteipolability Let 0 be the ring of local integers of kl Let L be the cardinality of the residue eld of Oi Let qu be the eld of Teichmiiller represen tatives of 0 Let X and Y be independent variables Let quXYX 1Y 1 be the ring obtained by adjoining inverses of X and Y to the twovariable power series ring quHX We call 5 E kab a basepoint if there is some nite totally rami ed subextension Kk of kabk such that 5 uniformizes the ring of local integers of Ki Given F E quHX YX 1Y 1 a basepoint 5 and a proper rational rigged virtual lattice 39i39 we say that 39i39 is asymptotically yoked to the pair F if 6 lt i gt F pa 1 llall for all a E kX such that Hall gt ll In this situation we also say that 39i39 is asymptotically inteipolablei By a straight forward application of the Weierstrass Division Theorem one veri es that given 39i39 and 5 there exists at most one F such that 39i39 is asymptotically yoked to F We will prove the following result INTERPOLATION OF NUMBERS OF CATALAN TYPE 5 Theorem 123 Every proper rational rigged virtual lattice is a nite Z1plinear combination of asymptotically interpolable rigged virtual lattices This result appears in the main body of the paper as Corollary 611 to our main result7 Theorem 61 124 Problems suggested by the asymptotic interpolability theorem Suppose that 39i39 is asymptotically yoked to F7 We may then consider the following problems 1 Find the prime factorization of F in quHX7 YX 1Y 1 2 Find the multiplicities possibly negative with which X and Y divide F 3 For all a E hX not just for Hall gt 1 a nd F 7 pa 1 llall in case of vanishing7 b nd the order of the zero77 of F at 5 pa 1 llall7 and c nd the leading Taylor coefficient77 of F at 5 pa 1 llall These problems are solved completely by Theorem 61 and the apparatus introduced to prove the theoreml 125 Sketch of solution It turns out that one can read the prime factorization of F directly from the theta symbol 39i39 and that the X and Ymultiplicities ofF can be made explicit in terms of certain integrals involving 39i39 and the rami cation index of 5 over kl Thus problems 1 and 2 are solved Moreover7 problems 1 and 3b turn out to be essentially the same To solve problems 3a and 3c7 we de ne a modi ed version go of the Fourier transform which stabilizes the class of proper rational rigged virtual lattices7 and we de ne a a ail Hall lt gt7lt gtIltgol lgt for all a E hX and proper rational rigged virtual lattices Q We call the Catalan symbol It turns out that whenever we have onesided asymptotic interpolation as in 67 we automatically also have twosided asymptotic interpolation lt i gt F 7 pa 1 llall for a E hX such that maxHaH7 Hall l gtgt 1 Further7 it turns out that there holds a more delicate sort of interpolation which we call strict interpolation7 valid for all a E hx equating the Catalan symbol to a leading Taylor coef cient Thus problems 3a and 3c are solved 126 Relationship to the global picture In the authors paper 3 global adelic versions of the theta and Catalan symbols are defined7 a conjecture relating these objects to twovariable algebraic functions is proposed7 the conjecture is veri ed in the genus zero case7 and it is explained how the conjecture re nes the Stark conjecture The results obtained in this paper7 so we claim7 provide detailed local con rmation of the author s conjecture The claim requires justi cation which we do not provide here We will take up the topic in future publications as part of our ongoing effort to prove the conjecture of This paper is more or less self contained We consistently take here a local point of view complementary to the global point of view of 6 GREG W ANDERSON 13 Organization of the paper In the next second section we x notation and assumptions in force throughout the paper and we develop tools to handle products of the form We de ne rigged virtual lattices the theta symbol the rational Fourier transform go and the Catalan symbol We set up a calculus summarized by a short list of simple rules We emphasize scaling rules and functional equations In the third section of the paper we develop what amounts to a theory of expressions of the form F pa 1 llall and in particular make rigorous sense of leading Taylor coefficients for such expressions We develop a calculus of shadow theta and Catalan symbols with scaling rules and functional equations parallel to those satis ed by the theta and Catalan symbolsi In the fourth section of the paper we de ne asymptotic interpolability strict interpolability and interpolability We work out the formal properties of these notions in some detail In the fth section we explicitly construct examples of strict interpolation In the nal sixth section of the paper we prove the main result by showing that from the examples constructed in the fth section all possible examples of strict interpolation can be built up by natural operations 2 THETA AND CATALAN SYMBOLS We develop local analogues of global notions introduced in We work in the setting of harmonic analysis on local elds The main result of this section is Theorem 25 see equation 52 for a simpli ed version which is analogous to the adelic Stirling formula 3 Thmi 7 7 21 The set up We specify the basic data for our constructionsi We introduce notation and terminology used throughout the paper Zillli General notation Let Agtlt denote the group of units of a ring A with unit Let 15 denote the function equal to l on a set S and 0 elsewherei Let 5 denote the cardinality of a set Si 212 The local eld 16 Fix a local eld k of characteristic p gt 0 with maximal compact subring 0 Let qu z E k l 1 1 z where q is the nite cardinality of the residue eld of 0 Then qu is both a sub eld of k and a set of representatives for the residue eld of 0 Moreover for every uniformizer 7r 6 O we have 0 qu7r and k quHTth ll 213i Chevalley di erentials Let Q be the space of locally constant qulinear func tionals k A qui Elements of Q are called Chevalley di cerentz39alsi For every 6 E 9 let Res6 61 6 qui For every I E k and 6 E 9 let 16 E Q be de ned by the formula 16 6zy for all y E k in this way 9 becomes a vector space one dimensional over kl It is wellknown that there exists a unique qulinear derivation d z k A 9 such that Resz d7r a1 I Z aini ai E El ai 0 for i lt 0 i7oo for all uniformizers 7r 6 O and z E kl Given 577 6 Q with 77 f 0 and z E k such that 5 177 we write I INTERPOLATION OF NUMBERS OF CATALAN TYPE 7 214 The Chevalley di erentz39al w Fix 0 f w E Q and let 9 be the fractional Oideal de ned by the rule 9 1 E k l Res zw 0 for all z E 0 Many constructions below depend on the choice of am But for the most part we suppress reference to w in the notation 215 Valuations and measures Let a be Haar measure on kl Normalize a by requiring that 0 M971 1 Let ax be Haar measure on kxi Normalize ax by requiring that ax OX 1 For each a E k put HaH Mamta orda elogHaHlogq The function is an absolute value of k with respect to which Is is complete The function ord is an additive valuation of kl Note that for every uniformizer 7r 6 O we have 1717 ord7r 1 Note also that fa 11dMI HaH f1dMI for all a E kX and Mintegrable complexvalued functions For every 6 E 9 put W ll5d7rll7 0M5 ilogll5ll1qu7 where 7r 6 O is any uniformizeri Note that 1 2 MO M a 216 The eld lF and character A Fix a sub eld lF C qui Fix a nonconstant homomorphism A from the additive group 0le to the multiplicative group of nonzero complex numbers Many of our constructions depend on lF or A or both7 but for the most part we suppress reference to lF and A in the notation 21 The character e We de ne 7 31 Atrpqp Reszw for all z E kl This is the character we will use to de ne the Fourier transform on kl 218 Bruhat Schwartz space Given a locally compact totally disconnected Haus dorff space X7 eigi7 X k or X kgtlt 16 let 8X denote the space of complexvalued compactly supported locally constant functions on Xi 219 Further notation Let Tc be a xed algebraic closure of kl Let kperf be the closure of k in Tc under extraction of 10 rootsi Let kab be the abelian closure of k in 176 For each positive integer n7 let qun be the unique sub eld of Tc of cardinality qni Put quoo U1 qun C kabi Let the absolute value and additive valuation ord on k be canonically extended to ls and again denoted by and ord7 respectivelyi 8 GREGW ANDERSON 2239 Apparatus from harmonic analysis 221 Frarler treasfrrms Given o e 5a put fWE 35 WIer dl rr thus de ning the Frarrer transfrrm NW 3 6 500 Given o e 80c and a e kx put WM WWII We have a squaring rule 2 1 FM holding for all 4 e 5a Our normalization of additive Haar measure hwas chosen to put the Fourier inversion formula into this simple form Further we have by an evident transformation of integrals a scaling rule HM HaHfMW holding for all or e 5a and a e kx 222 The Palsszm summatlzm frrmala Let L c k be a cocompact discrete sub group The Palsszm summatlzm frrmala states that for every n e 5a we have 1 8 Z W 7 2 36 m MICL 56 where Li is the cocompact discrete subgroup of k de ned by the rule Li gehleaglforallaeL g e k l Resrgo 0 for all r e L and hoeL is the Mameasure of any fundamental domain for L 223 Tularvmlablf Frarrer treasfrrms Given complexrvalued functions f1 and f2 on k set f1 8 f2I1i r2 f1I1f2I2i thus de ning a complexrvalued function f1 f2 on k X la As is We rknown the roperatlon identi es soc X k with the tensor square of 5a over 0 Given o e soc X k put QWKEW wrlyer syrldrfrldrfyli thus de ning the tmroarralle Frarrer traasfrrm gm 6 sfa X a Note that 9 QWUXWD f 1 gt1 w2 E71 gt2 for all on on 6 son The asymmetry of the de nition of g is dictated by our goal of simplifying formula 19 below as much as possible INTERPOLATION OF NUMBERS OF CATALAN TYPE 9 224 Generalized functions of two variables Let 806 X k be the space of gener alized functions on k X k i el7 the Clinear dual of 8k X k and let ltgtS k X k gtltSkgtlt k QC be the canonical pairingl The theory of generalized functions in our context follows the pattern set classically by the theory of distributions We identify 8k X k With a subspace of 806 X k by the rule MW 0 IWWWWWMIWMQ for all 711 E 8kl We have inserted the factor MOVQ HwHil to make the identi cation independent of the choice of Am We extend the Fourier transform Q to 806 X k by the rule mm mew for all T 6 806 X k and go 6 8k X k Given go 6 8k X k and 125 6 16X put 90b cr7y 9445417519 We extend the operation go gt gt 300 to 806 X k by the rule 10 ltFb c780b cgt llbcllmsol for all T 6 806 X k and go 6 8k X k We have squaring and scaling rules 11gt 92m WW 12gt WW Hbclglriltb c 1gt respectively7 holding for all T 6 806 X k and 125 6 16X in evident analogy With the squaring and scaling rules obeyed by F 23 Rigged virtual lattices 231 De nition We say that 39i39 6 806 X k is a primitive rigged virtual lattice if there exist 0 a cocompact discrete subgroup L C k 0 numbers L w E k and o a number r E kX such that 13gt ltlt1gt790gtM0 IZ 90I7ydy 161L HO for all go 6 8k X k We have inserted the factor n0 1 HwHilQ so that the notion of primitive rigged virtual lattice is de ned independently of the choice of the Chevalley differential am We de ne a rigged virtual lattice to be a nite linear combination of primitive rigged virtual lattices With complex coefficients7 and we denote the space of such by RVLUc It is convenient to associate to each 39i39 E RVLUc a complexvalued function 3 on k X k in the usual rather than generalized sense by the rule 14 Ivy hm lt 711a0 1yaOgtllall HallHO 1o GREG W ANDERSON Note that the lower star rule 14 for passing from vae to funetions on k X k is independent of the ehoiee of w To see that lt1 is Wellrde ned note that if lt1gt e RVLUc takes the form 13 then 15 1 12L 110440 as one veri es by a straightforward ea1eu1ation Via 13 and 15 we have us ltltIgtrtgtgt wow 2 ltIgttltnygttgtltzrygtdultygt 616 for a11 lt1gt e vae and o 6 son Thus lt1 unique1y deterrnines lt1gt By 10 and 16 we have the seahng ru1e 17 W 6 731500 bmdray HbHiNUflrrfly ho1ding for an lt1gt e RVLh and he e M The ru1e 17 is a erueia11y important bookkeeping detai1 in our theory The spaee RVLh is stah1e under the aetion of the twowariah1e Fourier transforrn g as the next 1ernrna and its proof show Lemma 232 Let L c k be a oooompaot desorete subgmup and r z e 1e Fee p 6 son If lt1gt e vae takes the fon 1 12L 8 A than 18gt m1 6 7mm amen lmem and mareaver 19 21mm Zgiwof atek 56k The surns on both sides of 19 are Wellrde ned sinee both have only nitely rnany nonzero terms anf We dairn that e 20gt ltltIgtrgmgtlteogt1 Z Memewe ateLi M for a11 to e 5h X k The e1airn granted 18 fo11ows by 16 and the de nitions To prove 20 we may assurne without 1oss of generahty that t gt1 we where Who 6 son P1ugging now into 13 using 9 to calculate gm m the 1eft side of 20 takes the form lt 2 WW morl gt2ltzgtpltygtelt7zygtdeltxgtdeltygtgti w ZL Whereas the right side takes the form 2 mow 2r y8rydttrdtfygtv Li INTERPOLA I ION OF NUMBERS OF CATALAN TYPE 11 Now from the Poisson summation formula 8 by substituting a H or 2 for ya we deduce the slightly more general formula 21 2 WI 2 9 9 w ZL geLi MICL holding for all o e soc Equality of left and right sides of 20 reduces to the special case or rim of 21 The claim 20 is proved Similarly and nally formula 19 reduces to the special case or p of 21 The lemma is proved D 2439 The theta symbol 241 Da mtwn For a e M and ltlgt e vac put 9a lt2 Zlt2lt1 gtgtltm Zimria lrh wek ask where the second equality is justi ed by scaling rule 17 Only nitely many nonzero terms appear in the sum on the extreme right as one can verify by passing without loss of generality to the special case 13 and applying formula 15 Indeed one nds that ltlgty 1242 to 1mm 22 a 9a lt2 Z 14Lnammr z L o aw rm ek Therefore a lt2 is wellede ned We call y the theta symbol The object de ned here is a local version of the theta symbol de ned in 3 242 Baszc fowoal pmpm tzas Fix ltlgt e vac and a e lax Via scaling rule 17 and the de nition of the theta symhol we have the scaling rule 23 9arlt1gtb llbllQWh lt18 holding for all lt1gt e mm and a b o e kquot We claim the following 24 9a lt18 Hallma lrgl 7 eyoo if Hall is suf ciently small 25 MM glt1gt00llall if Hall is suf ciently large 25 The function a H a ltlgt M a Z1p is locally constant To prove all three claims we may assume without loss of generality that ltlgt is of the form 13 to which Lemma 232 applies Functional equation 24 follows from formula 19 scaling rule 23 and the de nitions Further 25 for small Hall and 25 are easy to check using 22 Finally 25 for large Hall follows by 24 from 25 for small Hall The claims are proved We call 24 the functwnal rqaatm satis ed by the theta symhol 12 GREG W ANDERSON 243 Pmpar mzss and a actwanass of nggaa39 omaal lattzcas We say that lt2 6 RVLUc is praparjf ltIgt is cornpactly supported i e if 9alt1gt 0 for maxHaHHaH71gtgt 0 Equivalently lt2 is proper if mayo 0 and QltIgty00 0 The space of proper rigged virtual lattices is stable under the action of the ton variable Fourier transforrn g stable under formation of nite Crljnear combinar tions and for all 27 e M stable under the operation ltlgt gt gt lt20 We say that lt2 6 vac is a actwf if 9a lt2 is a nonnegative real number for all a e lax The space of e ective rigged virtual lattices is stable under the tworvarl able Fourier transforrn g stable under formation of nite nonnegative real linear combinations and for all ha e M stable under the operation lt2 H lt20 Theorem 25 The Stirling formula for rigged virtual lattices F07 all a e lax and lt2 6 vac the fwmula 9alt ordw Z ltIgtyra71aordr Z HaHgltIgt ia Ord aost gem HatHQbPMU 2f M gt 1 em lt1 7 9a 1gtgt 2f M 1 was 2f M lt 1 27 H1 etu M 7lt1gtdorogtordalt1gtvltortgtn1gtdorogt1olt1gtgtdmgt HaH lt9llt1gtlltoiogtorda 1140 n lt1gt0i0lotd xtgt halds The surns on the left side of 27 are absolutely convergent since there appear in each only nitely rnany nonzero terms The integral involving on the right side of 27 is absolutely convergent in view of the forrnal properties of the theta syrnbol that we have just proved The rernaining integrals on the right side of 27 are absolutely convergent since their integrands are cornpactly supported in M We remark that 27 sirnpli es considerably if ltlgt is proper The theorern is closely analogous to 3 Theorern 77 and a generalization of 3 corollary 611 Pmaf Replacernent ofthe pair a lt2 by the pair lltlgtlt1 7gt leaves both sides of 27 unchanged as one veri es with the help of the scaling rules l2 l7 and 23 along with sorne evident transforrnations of integrals We rnay therefore assurne without loss of generality that a l for the rest of the proof Further we rnay assurne without loss of generality that ltlgt is as in Lemma 232 narnely ME 71 1 E A i MICL L MW for sorne p 6 son in which case functional equation 24 written out explicitly witht e kx in place of a e kx takes the form 28gt Z 771 11H1H Z w ZL eLi ltIgt 12L 8 p and 9914571 INTERPOLA I ION OF NUMBERS OF CATALAN TYPE 13 The surn on the left equals t ltlgt and the surn on right equals Htuer1gltlgt Now put H 10 e 1x For every 4 6 son and a e 1 put MUM gtt n 1olttgtaltogtgtdm ix 3 71 ii x 3 WW Hlttgtaltt 1760 w W0 Hltr1gtlltualtxtgt n tlomwmww HlitH By the loeal Stirling formula 3 Thrn as there exists unique to 6 son sueh that 7 e aloroloorolac 1a l fx7 0 W 2 7p0ordwl 1 ilro A lordword impel 9 2 7 0ordwM1 jf 0t Using the fact that the left side of 28 represents 90 ltIgt and using 3 Lemma 69 to justify the exehange of summation and integration processes we have M I o e ZL Ht t P i A 10xt 1 A HlitH where A 1mm Mt 0 Reasoning sirnilarly but this tirne using the fact that the right side of 28 represents HtH t 1gltIgt we have 2 WOUWO ML dMXGL Ht 1HtH t 1 lt1gtDB 10xt 1 lt1gt B HlitH Where NJ B 7 gltlgt 00 MicL lt is now only a rnatter ofbookkeeplng to verify that relation 27 in the ease a l coincides with relation 28 in the ease to p andt 1 these ealeulations are quite similar to those undertalaen to prove 3 cor all and so we ornit further details Thus the theorem is proved D 14 GREG W ANDERSON 26 The rational Fourier transform 261 Tnmal zdmtztzas Put 1 ifa0 29 A0 rH 71 ifal E qa7l0l 0 otherwise Let 25 abbreviate 2C6 We have 30gt 13 Mr you nxltcgtgtxltnoxgti 31gt 22mm wows for all r 6 F 252 Da mtwn Ufgo put e0 a H gtrFqFResru k a7101 Equivalentlv via 7 and 30 we have lt32 eor1F 12 cl n xltcgtgtelt703gti Given o e 5h X k pnt admin metersyngtdtltxgtdtltygti thus de ning a eornplexevalned function on k X k which by 32 belongs to son X h We eall gov the mtwml Framer tmnsfwm of o The operator go de ned here is a loeal version ofthe global rational Fourier transform de ned in 3 By an evident transformation of integrals we have a sealing rule 33gt golwcl llballgolvlltbquot cquotgt holding for all ha 6 kx By 30 31 and the de nitions we have lt34 gold Ilrlz au n 3lt0gtgtglvgtlltcquot cquotgti 35gt 229mm EerWC CW We extend go to an operator on s39n X k by the rule lt oPi gtgt ltPr oMgt for all r e 539h X k and o e 5h X In The operator go so extended satis es the evident generalizations of 33 34 and 35 Lemma 263 F22 Zw 6 k T 6 M and a cacampact ducm a subgmup L C k 02mm ltlgt e van such that ltlgtv 12L 1mm0i we bane grim e mm 35 90 lt43 wwwo olt1gtrry ML 1rlltrgt1 naltygt INTERPOLA I ION OF NUMBERS OF CATALAN TYPE 15 anf By Lemma 232 we have g2 e RVLUc WI 7 WWW MkL By identity 34 in its generalized form we have Qolt1gtF 1E C1 e 40g2ltc c gt e RVLh We get the claimed formula for go2 y now by scaling rule 17 and formula 32 D 1423 1r4r1yerneiy 264 Basic fwmal pmpartzas 47f gr By the scaling rule 17 for the lower star operation and the scaling rule 33 in its generalized forrn we have a scaling rule 37 olt1gtb rry HAHQoWLUm Ly holding for all 2 e vac and 174 6 lax By the squaring rule 11 and scaling rule 12 for g scaling rule 17 for the lower star operation and relation 34 in its generalized form we have a squaring rule 38 JF 1gtIry lt1gtIry22lt1gt0r0y for all 2 e vac after a brief calculation which we omit By the scaling rule 17 for the lower star operation and relation 35 in its generalized forrn we have 39 229M402 Cy E olt1gt0rr0yv By scaling rule 23 and relation 39 the functional equation 24 satis ed by the theta syrnhol can he rewritten in the forrn 40 9a lt18 HaH a 14 olt1gtD holding for all a e M and 2 e vac ln other words we can sirnply replace g by go in 24 without invalidating the latter and this noted it follows that we can test 2 e vac for properness by checking that 2y0 0 0 and go2y0 0 0 lt follows in turn that properness and edectivity are preserved by go lt follows sirnilarly by 39 that Theorern 25 rernains valid with 9 replaced by go 265 Dependence 47f the apamtw go 4m 4 Let us ternporarily write Bo goo in stead of 1 and go respectively in order to keep track of dependence on w Fix u e M We have 1 2 Ban HuH so hence godoM HuH lgoow for all 4 e soc X h as one verifies by an evident transformation of integrals It follows that the analogous relation holds for the canonical extension of go to S Uc X k and in particular 41 goawlt1gtrr y Quwlt1gturruy for all 2 e vac via 37 Hereafter until further notice we revert to the systern of notation in which reference to w is largely suppressed 16 GREG W ANDERSON 266i Dependence of the operator go on lF In this paragraph we temporarily write gm instead of go in order to keep track of dependence on F Consider an interme diate eld lF C G C qul For each c E lF let VcGlF v E G l trgFv c From Wilson s theorem HOE7X fl and its analogue for G we deduce that 42 0 1 H v H v l H v veVcGr veV1Gr o vevoor holds for all c E lFXl We have a trivial identity 0trGF 1 Z 401711 Z 401711 veV1Gr o vevoor holding for all z E G which leads by the de nitions and scaling rule 17 to the identity 90117 PL 17 y Z gO l39i39ldvilIv U719 Z QOG vilzvily v6V1GlF o vevoor 43 holding for all 39i39 E RVLUc Hereafter until further notice we revert to the system of notation in which reference to lF is largely suppressed 2 6 Rationality and separability of rigged virtual lattices We say that 39i39 E RVLUc is rational if 39i39 can be expressed as a nite Zlp linear combination of primitive rigged virtual latticesl By Lemma 263 the class of rational rigged virtual lattices is stable under the operation gel The class of such is stable also for every b c E kx under the operation 39i39 gt gt 45 Given rational 39i39 E RVLUc we say that 39i39 is separable if both 3 and QOK39L are Zvaluedl For example the rational primitive rigged virtual lattice considered in Lemma 263 is separable if nrOnkL E Z By 41 the notion of separability is independent of the choice ofwl By 43 we are not permitted to conclude that the notion of separability is independent of lF but at least we can say that if lF C G C qu then Gseparability implies lF separabilitylll 21 The Catalan symbol 2l7lll De nition Recall that kperf is the closure of k under the extraction of 17 roots in a xed algebraic closure la Note that 3 zy E Zlp for all rational 39i39 E RVLUc and zy E kl Now given a E kX and a rational rigged virtual lattice H Iqgtxa1x6 kx 39i39 put lt a gt perf7 t 16ch 1 H meme H Egollt1gthlt gta gtllall E kpxerf xekx gekx thus de ning the Catalan symbol along with a partial version 24r of it The object de ned here is a local version of the global Catalan symbol de ned in INTERPOLATION OF NUMBERS OF CATALAN TYPE 17 27in Basic formal properties Fix a E kx and rational 39i39 E RVLUc It is clear that 104r and depend Zlp linearly on Q We have a scaling rule a 7 b 110 blt1gt 41gt 00 aCb 44 lt bcgtbuultlt gt lt gtgtlt I gt holding for all 120 6 kx by scaling rule 17 for the lower star operation and the de nitions Directly from the de nitions we deduce a factorization 45 21gt lt 1 M 53 gt and an implication W ekxi a maxlt1uar1gt ltIgt 46 39i39 is separable lt For the Catalan symbol we have a functional equation 4 lt gig gtHaH 719aqgt7lt1gt00 lt 1 gt holding by squaring rule 38 scaling rule 44 Wilson s theorem HOE7X C 71 and 45 Furthermore we have a scaling rule 48 1 7bHacHQolt1gt0gt07H5Hlt1gt00 aCb W NW 7 ltIgt holding for all 125 6 kX by scaling rule 37 functional equation 40 scaling rule 44 and 45 Note that 44454748 all simplify considerably if 39i39 is proper We claim the following 49 The function lta gt gt lt i kX A kgerf is locally constantl To prove the claim we may assume without loss of generality that 39i39 is of the form 13 and it is enough to prove the analogous statement for the partial Catalan symboll Local constancy is then easy to check Indeed we have 11L 1w7 0gtltgt H IA o xezLmawrO The claim is proved We have 50 lt i gt 1 for Hall suf ciently smalll This is also proved by reducing to the case 13 It follows nally that 51 lt II forHaHgtgt1 via factorization 45 and functional equation 47 18 GREG W ANDERSON 273 sampll ed statement af Theanem 25 Fix a e M and proper rational ltlgt e mime In this case the Stirling formula 27 for rigged virtual lattices simpli es a lot especially terms on the right side can be dropped and if we fur ther simplify by writing the left side in terms ofthe Catalan symbol recall that by 39 we may replace 9 by go in 27 without invalidating the latter we arrive at the more comprehensible relation alt1gtordw ordlt g Catltlgtel xtegtaltlgt X 52gt WW 0 ltIgtl0itd xt Hallgokfllth x l In the sequel we always apply Theorem 25 in simpli ed form 52 274 Dependence af the Catalan symbal an al Fix a e lcx and rational ltlgt 6 male Let us temporarily write goo and Cg instead of go and g respectively in order to keep track of dependence on o Fix a e lcx We have 53 lt E gt uguab le00ll ll9 ligtlt gt an a by 41 the de nition of the Catalan symbol and the functional equation 40 for the theta symbol suppose now that ltlgt is proper and let on 6 Q be a xed nonzero Chevalley di erential It follows that the product wwogteltmgtlt g is independent of be As a consistency check note that the right side of formula 52 does not involve ol at all Hereafter we revert to the system of notation in which reference to o is largely suppressed 275 Dependence af the Catalan symbal an F In this paragraph we temporarily write gap and gap instead of go and g respectively in order to keep track of dependence on F By direct substitution into the de nition of on we nd via 42 and 43 that p n for any intermediate eld F c G c L after a straightforward calculation which we omit In other words the Catalan symbol is independent of the choice of sub eld F c Fq Hereafter we revert to the system of notation in which reference to F is largely suppressed 3 SHADOW THETA AND CATALAN SYMBOLS We develop a second theory of symbols parallel to but independent of that de veloped in the preceding section This second theory concerns objects built from the raw materials oflocal class eld theory The main result ofthis section is Theo rem 36 which is parallel to Theorem 25 in its simpli ed form 52 Basic notation introduced in 2l remains in force INTERPOLATION OF NUMBERS OF CATALAN TYPE 19 Bill Apparatus from local Class eld theory Sill The reciprocity law homomorphism p Let p hX A Galhabh be the reciprocity law homomorphism of local class eld theory7 renormalized after the modern fashion so that GM CH0 0 6 Elm a 6 My Our normalization is the same as in 107 but the opposite of that in 312 The Lubih Tate description of cabh From 7 we take the following simple explicit description of the extension habhi Fix a uniformizer 7r E 0 Let X be a variable Put l7rX X E OXl For n gt 07 put l l X Wl nillMX W L IMXW E OleA For general a Za ri E 0 hi 6 qu7 put 10 MAX Zailwnl X 6 Ollel Note that the power series MAX is qulineari Put E7r 0 f 5 E h l 0 for some positive integer Since the polynomials W WX have distinct roots7 every element of E7r is separable over h The set E7r U 0 is a vector space over qu and becomes an Omodule isomorphic to 160 when equipped with the Oaction ai H MAE I 0 X E U 0 A E U 0 This Oaction commutes with the action of the Galois group of the separable alge braic closure of h in h and hence E7r C habi According to the explicit reciprocity law of Lubin Tate 7 we have 54 005 57 M005 MAE for all a 6 OX and 5 E E7P Lest the reader be jarred by the seeming inconsistency of 54 with 7 Formula 2 pl 3807 we remind himher that we renormalized the reciprocity lawi Now for every positive integer n the polynomial l l Xl7r 1lwX satis es the Eisenstein criterion for irreducibility over 0 It follows via 54 that 55 ordE h hxa 6 OX lpa for all 5 6 ET 20 GREG W ANDERSON 32 Path calculus 321i Paths Let X be avariablel Put 73 U XMXHgt 0 The set 73 forms a monoid with unit under power series composition We equip the monoid 73 with an action of Galkabk by declaring that O39V 2aaiXi V ZaiXi ai E quoogt 71 71 for all a E Galkabk and V E 73 We call elements of 73 paths 322i Basepoints Let 73 be the set of 5 E kab satisfying the condition 0rd MXa 0 pltagt55 1pltagt5 a e 0 Clearly Galkabk stabilizes the set 73 Note that E7r C 73 for every uniformizer 7r 6 O by 55 HE E 73 then for every element 1 of the eld generated over k by quoo and 5 there exists a positive integer n such that I can be expanded as a Laurent series in 5 with coef cients in quni Given a pair 5 77 E 73 such that 5 belongs to the eld generated over k by quoo and 77 we write 5 S 77 Note that for each 5 E 73 and uniformizer 7r 6 0 there exists some 77 E E7r such that 5 S 77 Note that any two elements of 73 have a common upper bound We call elements of 73 basepoints 323 The special paths U6quot Given a pair 577 6 73 such that 5 S 77 we note that there exists unique U m US700 E 73 such that U m 77 5 In particular U X X i We nd it helpful to think of Usquot as a path leading from 77 to We have amp Do U nXX E U FqnllelXA n1 Since kabk5 is separable we have d UgmltXgt 314700 y 0 We have UU gtUTI UU m for all a E Galkabk and U nXHaH aU nXHaH for all a E kX such that Hall 2 1 We have Um 0 m4 Us whenever we have basepoints 5 S 77 S Combining the rules noted above we have U anUn b4XHbHHaH Uspabgt4XH bH INTERPOLA I ION OF NUMBERS OF CATALAN TYPE 21 for all 176 M and 571 6 73 such that 5 gr g 4 Finally note that Usvplta5X U pb XHbH 5 HaH W and WM 770 for all at e M such that rninHaH M 2 1 and g e 73 We take these rules for granted in the calculations to follow Proposition 324 F215 6 kx such that 2 F2 57 6 73 such that 5g 7 F21 a set s c oX 0f Taprfzsantatwas far the quatzant Then we haw U5 pbwXHbH U HCWYHCH VltXa Y HltUrporpnltXlWlgt n NC 365 far some VX Y e UflllqrXYX Pmaf Since U b XHbH Umw uc Umcrltmm we Upltc4mYMw we rnay assurne without loss of generality that o 1 Fix a positive integer a such that 55 Umm U Um U Unmw s 6 S C Fq X39 By the Weierstrass Division Theorem we have a unique factorization U pb XHbH UtmY VX YWX Y where VXrY 6 FLAP YHX WX Y e quXY is rnonic in Y of degree e g and oxdm W0 Y Y9 lt suf ces to prove that lt57 Arman HltUrroorltXHbHgt 7 Y 565 Since for each s 6 S we have U VU V p5b U p5b 075U s pb nXHbH UtpltbvX 7 i each factor on the right side of 57 divides the left side of 57 in the ring qu XY Since for s ranging over S the power series Uwygp XHbH are distinct the left side of 57 divides the right side of 57 in the ring 1anXY Finally since e equals the cardinality of S by de nition of 73 equality indeed holds in 57 D 22 GREG W ANDERSON 33 Twovariable extension of path calculus 331 The rings A and A0 Let X and Y be independent variables Put Ao U Fqnuxyn A AolX HY ll n1 Given nonzero F G E A we write F N G if there exists H 6 Ag such that F CH Given any 15 E It such that 0 lt lal lt l and 0 lt lt 1 note that each power series F FX Y E A specializes to a value Fa E It and that if 15 6 kab then Fa E kab Proposition 332 A is a principal ideal domain Proof Given an integer d 2 0 and W WXY E A let us say that W is distinguished of degree d if WX Y 6 A0 WX Y is a monic polynomial in Y of degree d W0Y Yd and WX 0 f 0 Now suppose we are given nonzero F E A There are unique integers u and 1 and unique F0 6 A0 divisible in A0 by neither X nor Y such that F XMYVFO Choose n such that F0 6 qunHX By the Weierstrass Division Theorem there exists unique distinguished F1 6 qun X Y and unique F2 6 qunHXYX such that F0 F1F2 and moreover F1 and F2 are independent of the choice of n We declare the degree in Y of F1 to be the Adegree of F If F is a polynomial in Y then the Adegree of F cannot exceed the degree in Y of F Now let there be given a nonzero ideal I C A Let 0 f F E I be of minimal possible Adegree Choose G E I arbitrarily We claim that F divides C To prove the claim after multiplying both F and G by suitably chosen invertible elements of A we may assume without loss of generality that F is distinguished and that G is a polynomial in Y After dividing G by F according to the usual polynomial division algorithm we may assume that G is of degree in Y strictly less than that of F in which case G must vanish identicallyiotherwise we have a contradiction to minimality of the Adegree of F The claim is proved and with it the proposition D 333 Valuations of A Given nonzero F E A we de ne the X and Yvaluations of F to be the unique integers in and m respectively such that X mY F belongs to A0 but is divisible in A0 by neither X nor Y Given P 6 A0 remaining irreducible in A eg any power series P of the form X 7 or 7 Y with f E 73 and nonzero F E A we de ne the Pvaluation of F to be the multiplicity with which P divides F in A which is wellde ned since A is a principal ideal domain 334 Natural operations on A We equip A with an action of Galkabk stabilizing A0 by declaring that aF Z Emijwyi F Z Zainin aij 6 my for all a E Galkabk and F E A Given F FX Y E A put FlXY FYX Given also 577 6 73 such that 5 S 77 and 125 6 kX such that minHbH 2 l we de ne Flb clg nlX7Y FU gt bTXllbll7U gtPCTIYHCH FU b 1 nXHbH 7 U c 1 mXHCH7 INTERPOLATION OF NUMBERS OF CATALAN TYPE 23 which belongs again to A To abbreviate notation we write Fbc Fbc 7 FUNSW Flbgtb ml7 Flb l Fbb 7 ii er7 we drop doubled letters when possible From the rules noted in 3i2i3 we deduce the following rules obeyed by the square bracket77 and dagger operations We have Flinn mlyr FTcb T We have Flb clg quotln7n F b 15llbll7 0 15llcllA Given also a E GalUcabk7 we have UFbc n 0Fbc060n Given also a E 16X we have Fbc n X7 yHaH Mam bcpa pan xHaH 7 Wall paFabgtacma mX7y Given also d7e E kx such that HdH Hell 2 1 and Q E 73 such that 77 S Q we have Fbc ndem4 Fbdce 4 We take these rules for granted in the calculations to followi 335 The power series Zt Given 5 E 73 and t E 16X put 7 Xll ll Upzgt Y iflltll 217 quot6 l W 7 Upme ithH lt17 which is a power series belonging to the ring Aoi Note that Zl X 7 Y Note that aZt Zt g for all a E Galkabki We claim that 2 N xiiWE WM 21 X7 Y1gt 396 1f g 1 It is enough by the Weierstrass Division Theorem to show that the power series on the right divides the power series on the left In the case gt 17 we have such divisibility because Upt U pt Xll ll Xll lli The remaining cases of the claim are handled similarlyi Thus the claim is proved The symmetry T Zz N Zt l follows Denoting by Zhg the ideal of the principal ideal domain A generated by Zt it is clear that an is prime7 and moreover that Zus Z1894 Zus Zn gt Hill WM and t W35 We take these properties of an for granted in the calculations to followi 24 GREG W ANDERSON Proposition 336 Fix bc E hX such that minHbH7 2 1 Fix 577 E 73 such 11ng p a that 5 S 77 Fix a set S of representatives for the quotient aeox pann We have 506 min In 0 min t 1 58 gig n N H gabCl H H w u u gt 565 for every t E hX We also have 59 thm m N Xllbll37 ylhm m N yllcll For all nonzero F E A we have 60 X 7 Y 7valuation of Flbgtcl gtquotl max I 7 c Z0 1 7valuation ofF 6 Proof Proposition 324 in the present system of notation takes the form M 2 Hell 7 X 7 WWW H X 7 Y sbCJWUHC N 56 Symmetrically we have M HcH 7 X 7 WW H X 7 Y li l bmblb 56 Formula 58 now follows after a slightly tedious case analysis the further details of which we omit Statement 59 is clear We record it here for convenient reference since it is usually applied in conjunction with 58 We turn nally to the proof of 607 which consists of a study of divisibility properties in the principal ideal domain A Let E be the Zcbfvaluation of F and write F meSW7 where ZCb and W are relatively primel It follows that Zibff n and Wlml ml are also relatively prime By 58 we can write Zibff nl X 7 YmaquotllbllgtllcllV7 where X 7 Y and V are relatively primel We conclude that Flbw ml X y1maXHbHgtHcHWlhcfmlvi where X 7 Y and Wlhm mll are relatively prime7 which proves 60 D Proposition 337 Fix bc E hX such that minHbH7 2 1 Fix 577 E 73 such that 5 S 77 Let A be a copy ofA viewed as a Aalgebra via the homomorphism F gt gt 2 Flbgtclggtquotl A is afree Amodule of rank HbcH If 1 then ordn NA is a Galois extension with Galois group isomorphic to the product of two copies of the quotient For all F E A we have F 0 42gt Flbgtcl gtquotl 0 and F E Agtlt 42gt Flbgtcl gtquotl E AX Proof Let A6 be a copy of A0 viewed as Aoalgebra via the ring homomorphism F gt gt Flbgtclgmll In view of formula 597 it is enough to prove that A6 is a free d 2 AomoduleofrankHbcH 311 dent also of X and Y Let N be a positive integer such that Ug n X E quN X7 Y for all a E Galhabhl For any positive integer n divisible by N the quotient Mn 7 FanX Y V WHX 7 Ugiww w w 7 U pcnWHCH is a free qun V7 WHmodule with basis 1 and a free qun X7 YHmodule with basis ord ord 27o le d 17107m7ll0ll 61571 or 77 0F 77 Let V and W be independent variables indepen Vin INTERPOLA I ION OF NUMBERS OF CATALAN TYPE 25 by the Weierstrass Division Theorem It follows that M UN nMyL is a free 2 Ayrnodule ofrank HbaH 333 Since M is a Ayrnodule isomorphic to Ag we are done ii The discriminant of the nite at extension A A of principal ideal dornains factors as a product of a function of X tirnes a function of Y and hence is a unit of A Thus A A is nite etale Call the group in question 0 The group G acts faithfully on the ring extension A A hy the square bracket operation and since 0 has cardinality equal to the rank of the extension A A in fact 0 rnust be the Galois group iii By i we rnay identify A with a subring of A39 whence the result iv By i and the Coheaneidenberg theorern every maximal ideal of A lies below sorne maximal ideal of A39 whence the result a 34 Interpolating gadgets and Coleman units 341 laterpalattaa aaaaets Given a pair F5 where 0 at F e A and g e 73g we call F 5 an taterpalattaa aaaaetif Flti l FM for all a e M such that M 2 1 Proposition 342 Ft a basepamt E 6 73 Ft mmzem F 6 A The falllmlmg caaatttaas are equwalentx 1 F 5 ts aa taterpalattaa aaaaet 2 Flhci mlw ts aa lntanalatzng gadgetfm EUE ry V e 73 and ba e kx such that min ini M 2 1 3 Flhc mlya ts aa taterpalattaa aaaaet far same 7 e 73 and ha e M such that min ini M 2 1 4 47F1l05 5 F far all a e GalUceblc 5 F pa15HtH e In far all a e kx such that HaH gtgt 1 anf The scheme ofproofis 12314151 To prove the implication 12 just note that Flbvci ml ml Flashlhcsml Flbfc ml iaii for all a e kx such that HaH 2 1 The implication 23 is trivial To prove the implication 31 just note that for any a e kx such that Hall 2 1 we have Flti l 7 FHtH hci m Flhci ml ml 2 FIbvc ml tH 0 and hence FM 7 FM 0 by Proposition 337 To prove the equivalence 14 x F e A a e GalUcebk and a e lcx such that Mal l lt76 darlF 7F Then we have Fits Myip iipor sslyltl aF liU f bHaiL Under hypothesis 4 the last terrn equals FM Under hypothesis 1 the rst terrn equals FM Thus the equivalence 14 is proved We turn next to the proof of the implication 15 Fix a e Gallcablc and a e kx such that M 2 1 Let G Flwitl in which case F pa 1 uau C55 Note that since 12 26 GREG W ANDERSON holds Cf is an interpolating gadget and that since l il holds Cf has property 4 We now have altFlt5ltaltagt15gt lgtgt was aaxaaas canvases Gas Fe pa 1 and hence F pa 1 llall E 16 Thus the implication l 5 is proved We turn nally to the proof of the implication 5 l Fix ab E hX with minHaH 2 l and choose Hall large enough so that F pa 1 llall E Is As before put C Fl1gtai l so that C F pa 1 llall We have Flb 57pa715llbll Fbab 57 5 QM 5 5 G b 15llbll7 Pb 1 llbll Pb 1G7b 157 b 15llbll b 10575llbll C575llbll Fe ltpltagt45gtlalgtlbl It follows that Flbigl iFllbll belongs to in nitely many maximal ideals ofA and hence vanishes identically Thus F is an interpolating gadget Thus the implication 5 l is proved D Proposition 343 Given a uniformizer 7r 6 O a basepoint 5 E E7r C 73 and an interpolating gadget F 5 we have F E quHX YX 1Y 1 Proof We have p7rF p7rF1ip7r gt l F by the explicit reciprocity law 54 and the fourth characterization of interpolating gadgets stated in the preceding proposition D 344 Coleman units Given F E A and 5 E 73 we say that F E A is a Coleman unit at 5 if F factors as a product of elements of the set Zgg l t E hx times a unit of A We say that an interpolating gadget F is of Coleman type if F is a Coleman unit at The rationale for the terminology comes ultimately from Coleman7s marvelous paper We have previously discussed our interest in and admiration for Coleman7s paper in our papers 1 and We refer the reader to the latter for more discussion Proposition 345 Fix 5 E 73 and F E A The following statements are equiva lent 1 F is a Coleman unit at 2 For all 77 E 73 such that 5 S 77 and bc E hX such that minHbH 2 l the power series Flbgtcigml is a Coleman unit at 77 3 For some 77 E 73 such that 5 S 77 and bc E hX such that minHbH 2 l the power series Flb cig m is a Coleman unit at 77 Proof 1 This follows immediately from Proposition 336 23 Trivial 3 l In any case we have a factorization F CH where H is a Coleman unit at 5 and C is relatively prime to Zhg for all t E hx It follows that 6 ng is relatively prime to Zli clg nl for all t E hx By Proposition 336 it follows that Claim is relatively prime to Z for all t E hx But Flbgtcigml is by hypothesis a Coleman unit and 641 ng divides Flmi ml Therefore we have 641 ng E AX By Proposition 337 it follows that C E AX Therefore F is a Coleman unit D INTERPOLA I ION OF NUMBERS OF CATALAN TYPE 27 Proposition 346 Fta au tutetpalattua aaaaet an Fog e 73 such thatg g h Assume that alum Gfm alluu e ox such thatpu g aug Thea there msts a uutque intanalatzng gadget F 5 wtth basepatut 5 such that Film G Mateauet tfGty ts 0f Calemau tune theu sa ts F g Pmaf Existence and uniqueness of F e A sueh that Flii m a we get orn part ii of Proposition 337 One veri es that F 5 is an interpolating gadget by Proposition 342 Assuming that an is of Colernan type one veri es that F 5 is of Colernan type by Proposition 345 D 35 De nition and basic formal properties of the shadow symbols Proposition 351 Fta au tutetpalattua aaaaet Hg Fteh 6 7 such thatg g h F2217 e hx such thatmjri bii HAM 2 1 Put 5 X 7 Yrvaluatwn 0f Flhci ml t Thea ZHbH ts tuaepeuaeut Uf r aha aepeuas aulu the Tatw ah tt Futthetmate the erptesstau 61 W XYWgtgtVHEH ae ues a uauaeta elemeut 0f hoof whtch ts tuaepeuaeut aft aha tr aha uhtch aepeuas aulu ah the mtta ah Pmaf For eaeh nonzero H e A in the unique possible way write H E eH X e Y5ltHgt rnod X 7 10501 where 5H 6 ZN OWL 6H 6HY 6 U FanHYmY le 721 in other words 6H is the order of vanishing of H along the ideal X 7 Y and eH is the leading eoe eient of the Taylor expansion ofH in powers of XeY with eoe eients in Uf l qtnlqnyii The a and e notation de ned here will only be used in this proof not elsewhere in the paper it is eonyenient to set G Flbici lv Note that G g is again an interpolating gadget by Proposition 342 i By the de nitions of Z and G and relation 60 of Proposition 336 we have 2 3Flbici ml 3glli ml 3a Thus 2 is independent of t By de nition of interpolating gadget we have 3Flabiaci l 3Glai51 Maw Haiwg for all a e hx sueh that M 2 1 Thus ZHbH depends only on the ratio ah ii Let t denote the number de ned by 51 By the Chain Rule in the form WAX Ut reUtnXU hXt we have 62gt Uhdegfw W7 eltGW1gtltugtUgmmm 28 GREG W ANDERSON The expression U 5g7 de nes an element of hab independent of 7r by a further appheation of the Chain Ru1e Thus r is independent of or By L Hopjtal s Rule in the form 6X 7 WW 1i X 7 WW Ugmm We have as dam sltagtltUshltYgtgt Ugmw w x and henee the right side of 62 is independent of a Thus r is independent of a We therefore have sirnp1y w 5a 64gt Uhdag TM dam By de nition of interpo1ating gadget we have 5Fabyaci 5glai51 6QM mm for 811a e hX such that M 2 139 Thus r depends only on the ratio Lbt It remains only to show that TM 6 kxt In any case we have TM 6 In by 64 Fix o e GalUCabk arhitrari1y It wi11he enough to show that odin TM We have 6G 6UGW 05 5IXE 6UG0 U gg 5a 06G U gg 5a hy the fourth characterization of an interpo1ating gadget given in Proposition 342 followed by an app1ieation of 63 We have dU AED Ugoga U4oeUoss U4e Uogg hy the Chain Rule Finauy by 64 we indeed have opoH TM D 352 De aotooas For every interpo1ating gadget F g and a e M there exist by the preceding proposition unique 95haiFi 6 mm lt Fj gt h e sueh that the fouowing staternent holds for all am e M a 6 7m unjformjzers 7r 6 o and integers z sueh that a wU minltHUHr MD 2 1 5 a Z X 7 Yevaluation of FWMWIL we have 55 HUHQSMREE 1 56gt F2 gt1 lt14 We call endi a the shadow theta symbal and gsh the shadow catalaa symbol The sornewhat eornpheated de nitions of these syrnhon have been contrived to trivialize the veri cation of their formal properties INTERPOLATION OF NUMBERS OF CATALAN TYPE 29 353i Formal properties of the shadow theta symbol Let F and G be inter polating gadgets with a common basepoint and x a E hX arbitrarily We have sha7FG75 9511a7F75 sha7 G75 We have a functional equation 67 9511017125HaHesma EFts This should be compared with functional equation 40 for the theta symboli Given also 120 6 hX such that minHbH 2 l we have a scaling rule 68 9sha7Fb cl quotl 7 llblleshWCMEE This should be compared with scaling rule 23 for the theta symboli We have 69 Zt valuation of F 95h tF l for all t E hxi In principle 69 could have been taken as the de nition of the shadow theta symboli The function a H 9sha7F75 1 V A lePl is compactly supported and factors through the quotient EXa 6 OX l a5 5 This should be compared with properties 25 and 26 of the theta symboli The de ductions of these formal properties of the shadow theta symbol from the de nitions are straightforward We omit the details 354 Formal properties of the shadow Catalan symbol Again let F and Cf be interpolating gadgets with a common basepoint and x a E hX arbitrarily We have a 7 a a my gt51 lt as M as h We have a functional equation ail HaH em a W Muslim all This should be compared with functional equation 47 for the Catalan symboli Given also 125 6 hX such that minHbH 2 1 and 77 E 73 such that 5 S 77 we have a scaling rule W b 71gt lt minim gt511 lt 355 gt511 A This should be compared with scaling rule 44 for the Catalan symboli We have 72 lt Fag gt F57 Pa 15llall for maxltllall7 llallil gtgt 0 7 sh More precisely the equality above holds for a E hX if and only if 95h a F 0 The function a ltagt gt lt F75 gtShgt hX 4161 factors through the quotient hXa 6 OX l pa This should be compared with property 49 ofthe Catalan symbol The deductions ofthese formal properties so GREG W ANDERSON of the shadow Catalan symbol from the de nitions are straightforward We omit the details Theorem 36 Shadow Stirling formula Ft ah thtehphlatthg gadget F g t FM all h e hx we hate shaF ordw ordlt Fag sh Z shatR 10xtshaaF QWXG 73 H1 tH Xevaluation of F ordg th Yevaluation of F ordg tt The thumthg statements the eqhtohleht F5 ts ht chlemhh type 0 Equaltty hhhts th 73 the all h e hx o Equaltty hhhts th 73 the at least the h e hx ht Let 6a aehhte the left stae ht 73 mthhs the ght stae ht 73 Then we hate lirnsup 6a lt oo lirn Hale Hall This theorem is parallel to the simpli ed statement 52 ofthe Stirling formula for rigged virtual lattiees sug 6aHaH lt oo PTl7l7f We tum rst to the proof of part i of the theorem which is most of the work Choose he 6 hX such that rninHl7H Hall 2 1 and a eh Also choose a uniformizer 7r 6 o and h e a sueh that g g h Put 5 X e Yevaluation of Flb i ml F For Y Flatten YWX 7 W CX Y X 7Y W1gt Xay CXaylX YY noting that 97 W UAMY and also that Z HblleshaR E by de nition of the shadow theta symbol By de nition of the shadow Catalan symbol 0rd Fm W 0rd 501 W equals the lefthand side of 73 multiplied by M INTERPOLATION OF NUMBERS OF CATALAN TYPE 31 We turn to the investigation of the righthand side of 73 Choose a set of representatives T6 C kX for the quotient kXa 6 OX 17005 5 By 69 we can write F N W I Xeraluation of F I YYrvaluation of F 74 I H Z hltzFsgtmaxlturu1gt zeT where W 6 A0 is a power series divisible neither by X nor by Y nor by an for any t E T61 Now let n be the smallest positive integer such that 7r r77 0 By the explicit reciprocity law 54 of Lubin Tate a 6 0 13007 n 17r 0 Choose a set Tn of representatives for the quotient EX1 TWO making this choice so that 1 E Tni We have p N Wham I Xeraluation of FHb jn IyYrvaluation of FHCH 75 b o taF max t 1 I H Ztlw h H H gran by 58 59 and 74 after a calculation which we omit Note that Wlbfigml is divisible by neither X nor Y nor by Z for any t E kxi We have also Hblles 1136 G N H Z1 h 1 zeTmOx by 58 and the de nition of the shadow theta symboli For 1 f t E Tn we have via the explicit reciprocity law 54 and the de nition of Zhg that ordzmm Vlt1 W9 maxltltll71 lllitll A straightforward calculation now shows that cram n 7 ord mn e ordWab 15 b 7pc 15 equals the right side of 73 multiplied by Since 0rdWJb 15 b 7 c 15 2 07 the proof of part of the theorem is complete We turn to the proof of part ii of the theoremi Now F is of Coleman type if and only if W N 1 only if equality holds in 73 for all a E kxi Thus the rst of the three given statements implies the second Of course the second statement trivially implies the third Clearly F is not of Coleman type if and only if W is a nonunit of A0 if and only if inequality holds in 73 for all a E kxi Thus the negation of the rst statement implies the negation of the third statement The proof of part ii of the theorem is complete 76 32 GREG W ANDERSON We turn nally to the proof of part iii of the theoremi By the Weierstrass Division Theorem applied in qumHX Y for some m we may assume that W is a monic polynomial in X of degree d and that WX 0 Xdi Then we have llbllil 0rd WPb 15llbll7 P0715llcll dord for Hall gtgt 1 which proves lim supHaHH00 6a lt 00 The niteness of lim supHaHH0 6aHaH is proved similarlyione instead reduces to the case in which W is a monic polynomial in Y of degree 6 such that W0Y Yer The proof of part iii of the theorem is complete The proof of the theorem is complete D 4 lNTERPOLABILITY AND RELATED NOTIONS We now bring the two theories of symbols into alignmenti We de ne the notion of interpolability along with several variants We work out key consequences of these de nitions 4 1 De nitions 4ilili Asymptotic interpolability Given a proper rational rigged virtual lattice 39i39 and a pair Ff with 0 f F E A and E E 73 if lt 1 gt F57pa 1 llall for all a E kx such that Hall gtgt 1 we say that 39i39 is asymptotically yoked to F Note that F is in this situation automatically an interpolating gadget by Propo sition 342 We say that a rigged virtual lattice 39i39 is asymptotically interpolable if 39i39 is rational proper and asymptotically yoked to some interpolating gadgeti 4 12 Remark The notion of asymptotic interpolability de ned above seems to differ from that de ned in the introduction Let us now reconcile the de nitions by proving asymptotic interpolability in the two senses to be equivalent In any case it is clear that given any proper rational virtual rigged lattice 39i39 asymptotic interpolability of 3 in the sense de ned in the introduction implies asymptotic interpolability of 39i39 in the sense de ned immediately above Conversely suppose that 39i39 is asymptotically yoked to F in the sense de ned immediately above The problem here is that we might not have F E quXYX 1Y 1i Fix a uniformizer 7r 6 O and then choose 77 E E7r such that 5 S 77 Put G Flligmli By Proposition 342 both F and Gn are interpolating gadgetsi By Propo sition 343 we have C E quXYX 1Y 1i One veri es easily that Gn is asymptotically yoked to 3 in the sense de ned in the introduction The proof of equivalence is complete 413 Strict interpolability Given a proper rational rigged virtual lattice 39i39 and an interpolating gadget F such that 77 015 eshavF75v 11 lt F75 gt51 for all a E kX we say that 39i39 and F are yoked We say that a rigged virtual lattice 39i39 is strictly interpolable if 39i39 is proper rational and yoked to some interpolating gadgeti A strictly interpolable rigged virtual lattice is necessarily effective and moreover necessarily asymptotically interpolable by 51 and 72 INTERPOLATION OF NUMBERS OF CATALAN TYPE 33 4 14 Dependence of the yoking relation on w and lF The relation 39i39 and F7 are yoked77 is invariant under change of the Chevalley differential w as one sees by comparing the de nition 66 of the shadow Catalan symbol to formula 53 describing the effect of change of w on the Catalan symboll In other words7 w cancels when one forms the ratio ash The yoking relation also remains invariant under change of sub eld lF C qu since neither the Catalan symbol nor its shadow depend on lF Thus the yoking relation is intrinsic Similarly7 but much more trivially7 the asymptotic yoking relation is intrinsic since neither w nor lF have anything to do with that de nitionl 4 13 Interpolability Given 39i39 E RV k7 we say that 39i39 is interpolable if N ltIgt ngw i1 for some integer N 2 07 numbers ai E ZHp7 numbers bi7ci E kx7 and strictly interpolable rigged virtual lattices i Given also for each i the basepoint of some interpolating gadget yoked to i and some common upper bound 5 E 73 for the 51 we say that 39i39 is of conductor S An interpolable rigged virtual lattice is automatically proper and rational By de nition the space of interpolable rigged virtual lattices is a Z plmodule7 and also for every b7c E kgtlt stable under the operation 39i39 gt gt ll Proposition 42 Let F7 and G777 be interpolating gadgets asymptotically yoked to the same proper rational rigged virtual lattice Let Q E 73 be a common upper bound forE and 77 Then we have Flli gt4l Gulfgt4 Proof Put F Flli gt4l and C3 CWTgt4 For every a E kgtlt such that Hall gt 1 we have Flt5ltpltagt15gtlal 0077 a 1n Gltpa 1ltllall Thus the difference F7 C belongs to in nitely many maximal ideals of the principal ideal domain A and hence vanishes identicallyl D Fltltltpltagt1ltgtla gt Proposition 43 Let 39i39 be a strictly interpolable rigged virtual lattice yoked to an interpolating gadget F7 Then QOl39i39 is strictly interpolable and yoked to Fl7 Proof Fix a E kgtlt arbitrarilyl We have Halew gorbb 9017 9140th 5 HalleshW lvFli where the rst and third steps are justi ed by the functional equations 40 and 67 satis ed by the theta symbol and its shadow7 respectively We have lt iii W lt71elta gtlt 1 gt 1 Hall 1les F lt F75 lt its 7 where the rst and third steps are justi ed by the functional equations 47 and 70 satis ed by the Catalan symbol and its shadow7 respectively B 34 GREG W ANDERSON Proposition 44 Let 39i39 be a strictly interpolable rigged virtual lattice yoked to an interpolating gadget F7 Let bc E kgtlt be given such that rninHbH7 2 1 Let 77 E 73 be given such that 5 S 77 Then 45 is strictly interpolable and yoked to Flbw mlm Proof Fix a E kgtlt arbitrarily We have 9a7lt1gtb c llbll9aCb7lt1gt llbllQWCEEEM 9avFlb cl6 m where the rst and third steps are justi ed by the scaling rules 23 and 68 for the theta symbol and its shadow7 respectively We have lt a gt lt aCb gtllbll lt aCb gtllbll lt a gt gtbgt0 i sh F75 sh Flbs mlm sh where the rst and third steps are justi ed by the scaling rules 48 and 71 for the Catalan symbol and its shadow7 respectively B Proposition 45 Let 39i39 be a rational rigged virtual lattice such that 3 07 0 and QO 00 are integers Fix a uniformizer 7r 6 0 Fix 1 E kX Let BX E quHXMX l be the unique Laurent series such that B7r I Put FX Y BX 1gtlt gt gtBYgo1404 e AX Then F7 7r is an interpolating gadget bgtbHbH 739 is strictly interpolable and yoked to F7r Proof For all C E quXYX 1Y 1 and de E kgtlt such that rninHdHHeH 2 1 we have Gldgtei lXY CXlldlhyllell7 and hence consider the case d e in the preceding the pair G77r is an interpolating gadgetl ln particular7 F7r is an interpolating gadgetl ii Fix a E kgtlt arbitrarily By scaling rule 23 we have 9a7gtb bllbll b 0 By scaling rule 48 we have lt yawM id gt bHaHQolt1gtl0gt07lt1gt0gt0 Since F is a unit of A7 GshaF5 0 and lt F gt511 F1a XYW FWWHaH bHaHQolt1gt007lt1gt0gt0 Thus we have strict interpolation as claimed D Proposition 46 Let 39i39 be a strictly interpolable rigged virtual lattice yoked to an interpolating gadget F7 Then F7 is of Coleman type and moreover X Valuation of F ordE 07 tdigtlt t7 78 YValuation of F ordE Qo 07 tdigtlt INTERPOLATION OF NUMBERS OF CATALAN TYPE 35 Proof By our hypotheses combined with Theorem 25 in its simpli ed form 52 and part of Theorem 36 we have 6a lt1gt0tdmt 7 X valuation of F 0rd 5 HaH QO 0 tdigtlt t 7 Yvaluation of F ord 2 0 By part iii of Theorem 36 this is possible only if 6a vanishes identically whence 78 By part ii of Theorem 36 since 6a vanishes identically F must be of Coleman type D Proposition 47 Let 39i39 be a rigged virtual lattice Let r be a power of the char acteristic p of h Assume that r i is strictly interpolable and that max1gtllall 1 w ekthmMWWWW5gtL Then 39i39 is strictly interpolable Recall that if 39i39 is separable then 79 is satis ed by 46 Proof Let r i be yoked to F It is enough to show that FlT E A because once that is shown it is easy to verify that FlT is an interpolating gadget yoked to Q By formula 72 and hypothesis we have F57 a715lalmaxu lalil 6 VV for ma llallvllall l gt1 Choose a0 6 hX such that HH2h MV57PMAFF Then we have 80 Flt5la l25gt7Flt57slaolgt e W for 239 gtgt 0 Write 71 man ZFAXCWXJ39 F109 Y e A j0 in the unique possible way Let K be the eld generated over h by quoo and For some integer i0 2 0 and all integers i 2 i0 we have r71 stzslaol w e W c K j0 by 80 But the powers Ej3 form a basis of K as a vector space over KT and Fj r ll lll E KT for all i 2 0 and j 0Hir 7 1 We therefore have Fj r ll lll 0 for all i 2 i0 and j 1Hiri But then for j 1Hir since FjXY is contained in in nitely many distinct maximal ideals of the principal ideal domain A necessarily Fj X Y vanishes identicallyi Thus we have FX1TY F0X Y e As Symmetrically we have FXY1T e As Finally we have FX1TY1T E A and hence FlT 6 A D 36 GREG W ANDERSON Proposition 48 Fix a basepoint E E 73 Let 39i39 be an interpolable rigged virtual lattice of conductor S Then there exist strictly interpolable rigged virtual gadgets i each of which is yoked to an interpolating gadget with basepoint 5 such that r i 4 7 g for some power r of the characteristic p of 16 Proof By hypothesis N ltIgt Zahgw i1 where ai E ZHp bi7ci E hx 3 is strictly interpolable7 and 3 is yoked to some interpolating gadget such that S For each index i choose di 6 hX such that minHdibiH7 2 1 Then we have N N I ZainiHil wlbnd1c Zai gbciHdiHi1qgtgdbdc i1 i1 By Propositions 44 and 45 we have now exhibited 39i39 as a nite Zlp linear combi nation of strictly interpolable rigged Virtual lattices each yoked to an interpolating gadget with basepoint bounded by After relabeling7 we may simply assume that N r i Z alrig i1 where ai E Z r is a power of characteristic p of 16 3 is strictly interpolable7 and 3 is yoked to an interpolating gadget Fifi such that S By Proposition 44 we may assume that 5 for all i Finally7 after grouping terms according to the signs of their coefl icients7 we may assume that N 2 a1 1 and a2 71 in which case 4 1 and g 2 have the desired properties D Proposition 49 Let a rigged virtual lattice 39i39 and a basepoint E E 73 be given If 39i39 is interpolable of conductor S E e ective and separable then 39i39 is strictly interpolable and yoked to an interpolating gadget with basepoint Proof By Proposition 48 we may assume that for some power r ofthe characteristic p of h and strictly interpolable rigged Virtual lattices i yoked to interpolating gadgets Fi we have r i 4 7 igl Put F FurFm Now F a priori belongs to the fraction eld of the ring A7 but by Proposition 46 the power series Fi are Coleman units with respect to 5 and hence7 since 39i39 is effective7 one can by means of formula 69 verify that F divides F and hence F 6 A It is not dif cult to verify that F7 is an interpolating gadget yoked to r i these details we omit Finally7 FlT7 is an interpolating gadget yoked to 39i39 by Proposition 47 and our hypothesis that 39i39 is separable D 5 CONCRETE EXAMPLES OF STRICT INTERPOLATION Theorem 51 Fix a uniformizer 7r 6 0 Fix a sequence in hab satisfying the relations 1 7 I 7 7517 ifi 1 504 54 HESS 2101 for i gt 0 Put LFq7r icb i1 INTERPOLATION OF NUMBERS OF CATALAN TYPE 37 tht e hx and an thteger M 2 0 such that H7r MH 2 M Let Ta be the uhtque pnmtttoe hyped w rtual latttce such that 9th 1tL 1tOv The Pg 7 ts stnctly t m ETpalabla and can be yakad tt ah l t rp latl g gadgtit wtth basepttht 5M The proof of the theorern whieh is by explicit ealeulation and construction takes up the rest of this seetion The theorern is going to play a vital role in the proof of our rnain result We point out that the sequence 53 is a division tower for the Lubl39rieTate forrnal group diseussed in 3l2 For sirnplieitv we assurne that 81 1F L w d7r As explained in 4l4 these assurnptions entail no loss of generality The calcu lations undertaken here are sirnilar to those in 3 lo but on the whole sirnpler sinee loeal rather than adelie 52 Review of some determinant identities 521 The Mme taehtttp Given r1 In 6 h we de ne the Mme determthaht a a r1 h r Moorer1trn detle 7 e k 7 L V w egg an 2 an Note that Moorer1 V V V an depends E qrh nearly on eaeh of its arguments and van ishes unless its argurnents are E qrh nearly independent Assuming that al V V Wm are E qrh nearly independent we have h 82 Moorer1 V V Wrn H H at M t1ueVe where l4 is the qusparl of MI V V V at This is the Wellrknown Mttre ttlehtttp 522 The ore ttlehtttp Given 4 6 Q and r1 V In 6 h put Resr1 I r1 Ore4r1ra V ekv Resrh r1 an We eall Ore4a1ra the OM determthaht of 4a1ra lt is elear that Ore4 r1 an depends quh nearly on eaeh of its argurnents suppose now that 1 an are quh nearly independent and that Resr4 at 0 for sorne a Let V he the quspan of 1 ra From the Moore identity one easily deduees that 83 Ore r1 trni H 1 u Moorer1ra WV Res114 1 The latter we eall the 0T5 Zda m zty heeause it makes available to us eertain useful features of OrerElklesrPoonen duality 6 4l4 The de nition ofthe Ore determjr nant here is just a slight rnodifaeation of the de nition rnade in 3 2s as GREG W ANDERSON 523 Spacml cases at the Jdctbtedet zdantzty Let r1 aN he distinct e1ements of h Let 2 he a variah1e We have Reszg 2 Egg 7 atz1 71 complete symmetric function of xi t t Wm 84 71er WA er r1 IN INA aer xiv xx 1V 39 39 1 1 1 1 Reszg Zia 110 atz 71m elementary symmetric function of 1 V V V my 85 I1 2 rf1 H rNT1 7 r1 7 1 1 1 1 where the hat indicates omission 1n the rst identity a may take any nomegatjve va1ue whereas in the second we restrict to 0 g a g N The expressions on the left sides are residues of meromorphic dj erentjals on the zehne over k we wi11 make more use of such expressions 1ater in our ca1cu1ations The denominators on the right sides are Vandermonde determinants These identities are specia1 cases ofthe Jdctttedet zdantzty Concerning the 1atter see 8 13 for background and detai1s 5V3 Calculation of symbols Fix a 6 M We will calculate the symbols anthem kiwi expressing the Cata1an symho1 in terms of Moore and Ore determinants Along the way we verify that M 7 T1 is proper 531 Calculatwn 0f the theta symbal Note that k L 69 039 For every n e k we write I m ltrgt m 6 L ltrgt 6 0 in the unique possih1e way We have tLrmtO7 0 a e 1t e L to a e 1tgt e to lt at t e t L at 0 at lttgt 7amp0 5 86 5 5 9 INTERPOLATION OF NUMBERS OF CATALAN TYPE 39 By scaling rule 17 we have unk 1zL 1at0v hence Via 86 we have 87 a 7 Wrote 1Latjlttgtz if lt 1 and a 71t 6 a0 0 if HaH lt 1 and a71t a07 and hence Via scaling rule 23 for the theta symbol and the de nitions we have 1La2jltzgtL aO 11L a0 ifllall 217 ea in 7 m1 7 ea W 7 all 88 1 ifHaH lt1 and a71t 6 a07 0 otherwise Since a7 11 7 111 0 for maxHaH7 Hallil gtgt 07 we have veri ed that 11 7 111 is proper 5 32 Calculation of the rational Fourier transform Under special assumption 81 we have Li L7 9 O7 MUSL Mot and hence 89 9011514957 9 A0Resty 7 Id7r1LI1oy by Lemma 2637 where recall that we have 1 if C 0 A0C 71 ifC17 0 if 0 g 01 for all C E qul 533 Analysis of the rational Fourier transform From 89 Via scaling rule 377 we deduce that GOMWW 7 Maugomwy HaHA0Restay 7 zd7r1Lz1a71Oy and hence go 1gt gttltzzgt 7 HaHlt1LWogtltzgt7 1mm where LtaC Z 6 L ailolResa71tZd7r C for every C E qul Also put Last a U Last C 06117 Note that 90 Lta 0 ltgt a 71t 6 a07 40 GREG W ANDERSON 91 1mm lump 1Llt1atgt luau 7 71717F1L a 10 ijaHltL lt92 LltLa1gte 0 W121 It follows that gum 7 WWW r 0 if M 2 1 93 lt Hauuerxttamla Zea letewweaxx if HaH lt 1 aha a e 1 e ao HaH1L1a1 1L1a 1Lta1 1Ltar if HaH lt 1 aha a e 1 e aov 584 Calculatwn 0f the Catalan symbal We DOW obtain a formula for the Catalan symbol 923 by combining formulae 87793 Wilson s theorerh new 0 71 the de nitions the Moore identity 82 the Ore identity 83 ahd the scaling rule 48 for the Catalan eyrhhoh Write NH V1HV HaH HW H7r a 1 m e r 51 7 up Mooreqatj lttgt7r Moore1 1 V We have if M Z 1 lt94 Moorew1 H q Wt 75gt m if HaH lt 1 and 9a 1t 7 11 1 Oret1 ia rr r 1 Orea7r r if HaH lt 1 and 9a 1t 7 11 0 54 Construction of a candidate We construct the interpolating gadget ultje mately to be yoked to KL 7 EL 0 Z Wu 1399 IO M pXME X m 3 A XW 1M X m D m 0X 3 A XW mp gt1 2U191 9M 0m 401 mZM XW Z zx X 5 x M 0 um XW Z Z x XV 9nd s umaaunf iuumgugg Z V g AWLAAU Z p99 lt0 2 g mama 0z 0 0 7E 07 0 04 m 3 7 4 HWZ 7 qu q x 00 1399 IO 00 k v if 0ZJ jk I 314113 34 qu 0ltz gx q ifz 0z 52X 9qu am sHquyugep sq pu e V9 MEI mammal qysndxe sq Ag 1 2 HM HA I lt 2 4 WK 10 96 Iz HEX 4 H lt z z 4 X I lt w bX XOX emzq 9M A fn x X n X W 0UOJ VD A XHb 3 9nd mqeq ampA Amdmys 09 qnd V g swamp mums T179 IV EXCIle NVWJND d0 saggy 114 d0 NOLLV IOdXEILNI 42 GREG W ANDERSON 543 De mtwn afE Write oo t 2 m tzequ 17M and then put 00 M E may Dewy nZteeaeme 6 Av 10 11 The sum 230 tzfzX Y converges X0 madlcally in lquX Y and hence E is wellede ned We remark that in the case M 0 one simply ignores here and below all functions and formulas connected with g 544 An mphcet Teczpmczty law obeyed by E Given a e 0 we write alZ e quZ for the result of expanding a as a power series in 7r with coelhclents in L and then replacing 7r by 2 Let us agree to extend the square hracket operations from A to AZ coe aclentehyecoe aclent From 96 we deduce that flb 39g 39quotl 1ltXaYZgt failaim albw g m quotl 1ltxaxzgt M glz miwiz where M denotes congruence modulo the ideal 2M c AZ and hence a 97 Fth M 10C FlatbjtXq gt Yqv for all he 6 ox and integers 7 2 0 In particular it is clear that the pair a a FJM 51 g is an interpolating gadget We will yoke FJM 515 to Pg 7 Pl 545 Reductwn 0f the pmaf Note that 98 Pt 900 PWHm Pl for all u e ox via scaling rule 17 On account of 98 and its analogue 97 along with the scaling rules obeyed hy the four types of symbols we have only to prove that enw N Fn W me 99 7rer 7 7r4v Fog sh we 7 021 for all integers N in order to yoke EW39W g to Pg 7 Pl therehy completing the proof of the theorem 5V5 Completion ofthe proof Throughout the calculations rernaining let N V and n be nonnegative integers and let m be an integer in the range 0 g m lt Mt INTERPOLA I ION OF NUMBERS OF CATALAN TYPE 43 551 Spacmlzzatwn of f and y We write E 7 1 E 5X0 UR X 5X for all F e AZ From 95 We deduce the functional and dj ereritjal equations foam lt1 n M fXquiZi fXquiZ liXoZfXiYiZi o ndeiYiz lt1 7X02 gLQXiYi 2 my 2 2 7Y0 so Y2 2 Max 2 M z nXogtaltXiYi 2 5 a 2M 79XiYi z z 7 X0 gown Here as previously EM denotes congruence rnodulo the ideal W C AZt We deduce by iteration and specialization the following list of identities in the power series ring kZ ffofqm iZ Ilium M7252 Hfgoloerrz x qyr rival39l aiwq z 100 aft 67 Z H10 deftz US771 45252 2M n zw Here and below EM denotes congruence rnodulo the ideal W C 1cm 552 Apphcatm of JacaberTudz zdantzty One deduces from rst second fourth and fth identities on the list 100 via the special cases 84 and so of the Jacobl Trudi identity that fafw gt 4 Wsrifea g i 7 7L 7 W V7 M525 7 A ZOoiJaifrl r Mr M may a N may gt t saw it n r aridhence gm MooreQrth lttgt7r 101 M55 gt Moorefrrlma 102 Haida Ore1 in rl dw 7r Ored7r7r 1v by de nition of Fat From the latter formula it follows that 103 M5715 0 es ltlt1 7W1 6 W10 44 GREG W ANDERSON 558 Applccatcoa ofsome rescdae farmulas We have 5 at E nReszeeo dngu err29 ltth E Reszeo 01330 err20 M quotv v39 7 Z 71 7 r dz 7M 14222672 Zem Reszqa W H0 1 7 Z 7 v The rst two formulas follow directly from the third and sixth identities on the list 100 while the last is an application of the Vanderrnonde deterrninant identity By a straightforward if quite tedious calculation exploiting sumrofrresiduesrequalsr Zero77 one Veri es that 1 iwu1 gt6 7rW10 104 BE qr Moore7r 0X0 Moore7r 554 Last detazls By de nition esh7rN1F5 is the X e Yvaluation of EX 19 hence sh7r N 1Fc vanishes by 101 and hence 91 N1E5gt ewwc e 91 by 88 By de nition qvenwiFoe is the X e Yvaluation of FXq Y hence enemas is the X 7 Ya valuation of E hence sh7rVE g 10 according as 1 irrV1tgt does or does not belong to No respectively by 103 and 104 and hence eshw F E 87rv19 e 91 by 88 Recall that since we made our calculations of cata lan syrnhols under the special assurnption 81 and we are obliged to make our calculations of shadow Catalan syrnhols under the sarne special assurnption By de nition gig Fggq and hence gig 311 by 94 and 101 By de nition 53 5512 any 5 according as 9am Fug 9 V1T 7 T1 10 respectively and hence gig 7531 by 88 94 103 and 104 Thus 99 holds and the proof of Theorem 51 is complete a 716 WHH 05gt eua 6 THE INTERFOLATION THEOREM Theorem 61 Every prayer ratwnal ngged mrtual lattzce zs mterpalable This is our rnain result The proof of Theorern 61 takes up the rest of this section Fortunately the proof of Theorern 61 requires none of the details of the exphcit constructions used to prove Theorern 517We will need only the hare staternent of the latter Before commencing the proof we prove two corollaries and make a remark Corollary 611 Every prayer ratzzmal rzgged mrtual lattzce 28 a mte Zlpr lmear cambmatwn 0f asymptatzcally mterpalable rigged wrtual lattzces This is a reiteration of Theorem 123 Prsz By Proposition 48 every interpolahle rigged virtual lattice is a Z1plinear cornhination of strictly interpolahle rigged virtual lattices and as we have already noted strict interpolahility irnplies asyrnptotic interpolahility so the corollary fol lows directly from the theorern o INTERPOLA I ION OF NUMBERS OF CATALAN TYPE 45 Corollary 612 Let T be a prayer mtwnal awed ntrtaal latttce Assume thatlt1gt ts asymptatzcally mterpalable and asymptatzcally poked to aa mterpalatmg gadget E5 Thea T as stnctly mterpalable and poked to F g Pmaf Choose a hasepoint a 6 7 such that g g a and T is of conductor g a By the theorem and Proposition 48 for some power r of the characteristic p of k we can write rT ltIgt e T where Ti are strictly interpolahle rigged virtual lattices yoked to interpolating gadgets GM respectively Put 0 Film Note that an is asymptotically yoked to T lt follows that Glam and Corn are both asymptotically yoked to Tr and hence equal by Proposition 42 lt follows in turn that 071 is yoked to T Finally by the scaling rules for the shadow symbols F g is yoked to T as claimed D 62 Remark Fix a uniformizer 7r 6 0 Let W C k be a cocompact discrete qursubspace such that nosW Mao Consider the rigged virtual lattice T of the form I 1W X 117r 9 lqu7r1 117r0 lt is easy to see that T is rational proper separable and effective By Theorem 61 the rigged virtual lattice T is interpolable and by Proposition 49 it follows that T is strictly interpolahle By means of Proposition 346 and the scaling rules for the Catalan symbol and its shadow it can be veri ed that T is yoked to an interpolating gadget of the form 7W xiea where W many 6 quXYX 1Y 1 The power series W is essentially the same as the one guring in 2 Theorem 241 and so admits interpretation as a Tefunctwn The method used to prove Theorem 61 is actually a re nement of that used to prove 2 Theorem 241 63 Setting for the proof of the main result Fix a uniformizer 7r 6 0 Fix a sequence 5220 in kab satisfying the relations 7 7 753 ifz 1 50 H f ifigt1 for 239 gt 0 For t e k let T he the unique primitive rigged virtual lattice such that 92 1t7r qu7r1 1t0 This is more or less the same setting as that in which we proved Theorem 51 However in the present setting we allow t 0 in the de nition of T we reserve the symbol L for denoting a general cocompact discrete subgroup of ls and we do not make the special assumption 81 64 Ad hue terminology 641 magalaaty and lend Given a rigged virtual lattice T and an integer M 2 0 we say that T is magalar of least g M if it is possible to decompose T as a nite Zlplinear combination of primitive rigged virtual lattices of the form TW for t e k such that M g HrMH and integers 239 and 739 If T is regular of some level we say that T is regular Note that if T is a primitive rigged virtual lattice and T 12 limp where L c h is a cocompact discrete subgroup w e h and r e M a suf cient condition for T to he regular of level g M is that ll rr MH 2 llmrll and L 3 Nqull for some N 46 GREG W ANDERSON 642 Softness We say that a rigged Virtual lattice 39i39 is soft if 39i39 is rational a 0 for all a E hX and 0t 0 QO 0t for all t E 16 Note that softness implies properness and effectiveness If 39i39 is soft strictly interpolable and yoked to an interpolating gadget of the form F EM for some integer M 2 0 then F 6 Ag by Proposition 46 and moreover F E quHX Y X by Proposition 343 Proposition 65 Let 39i39 be a proper rigged virtual lattice If i can be written as a nite Zlplinear combination of primitive rigged virtual lattices of the form 1921 where e 6 01 andi E Z then 39i39 is interpolable of conductor S 7r Proof After replacing 39i39 by M for some integer 2 0 we may assume that 39i39 is a nite Zlp linear combination of rigged Virtual lattices of the form 1921 where e 6 01 and i E Z N 000 Now for every integer i 2 0 and e 6 01 we have 921 471 Z W et EEWI lquM WNO by scaling rule 1 7 and the de nitions Therefore we have N ltIgt Zai litl i1 for some numbers ai E Zlp and ti 6 0 After grouping terms we may assume that the ti are distinct Now in general we have 105 up 0 0 60 900 0 17 by the de nitions in the former case and by Lemma 263 in the latter case Since 3 0 0 0 by hypothesis we may assume by 105 that ti 6 hX for all i Further since go 140 0 0 by hypothesis we have ELI ai 0 by 105 and hence ltIgt Zeal117 L111 i1 Thus 39i39 is interpolable of conductor S 7r by Theorem 51 D Proposition 66 Let 39i39 be a proper rigged virtual lattice If is 7rregular of level S M then 39i39 is interpolable of conductor S 5M Proof By hypothesis after some eVident rearrangement we have N lt1gt Zailtmzwegtlt wgt i1 N Z 010181 7 4 Ijgiv 7riv7r z m i1 N E Z awn i1 where az E lePL i i E Z ti 6 767 6i 5207 lltill S llWTMllA INTERPOLATION OF NUMBERS OF CATALAN TYPE 47 Call the sums on the right 1 2 and 3 respectively The sum 1 is interpolable of conductor S 5M by Theorem Bill The sum 2 is interpolable of conductor S 7r by Proposition 45 Since 39i39 1 and 2 are proper so is 3 Finally 3 is interpolable of conductor S 7r by the preceding propositioni D Proposition 67 Let 39i39 U fil be a family of soft rigged virtual lattices Let M 2 0 be an integer Assume that for every a E hX we have a ikallltMigt4HZO Assume that 3 for everyi is strictly interpolable and yoked to an interpolating gad get with basepoint 5M Then 39i39 is strictly interpolable and yoked to an interpolating gadget with basepoint 5M Proof For each i let FlEM be an interpolating gadget yoked to i noting that E 6 quXYX for all i Assume for the moment that F limFi exists X Yadically in quHX Then necessarily F E quHX Y X and moreover for every a E hX we have convergence lt FifEM gt511 Fi5M7 pa 15MHaH am Flt5M7ltpltagt15Mgt gt FEM l h with respect to By hypothesis we have the corresponding convergence of Catalan symbolsi Therefore F EM must be yoked to Q Thus in order to prove the proposition it is enough just to show that limFi exists X Yadically in quHX Fix a positive integer N arbitrarily For each i use the Weierstrass Division Theorem to write N 71 E1ltX7YgtF1ltX Y Rica Y calx Y H Y 7 X43 F1 where QiXY E quHX Y and RiXY E quXY is of degree lt N in Yr By hypothesis I q if 7r j if Tr j I RZlt5M Mgt7 1EHE75Mlhi 1 17qgtigtmmo for j l i i i Ni By applying the Lagrange lnterpolation Theorem one deduces that Ri MY E h MY tends coef cientbycoe icient to 0 with respect to as n A 00 and hence that RiXY tends X adically to 0 as i A 00 We conclude that Fi1Fi E lmod X YN for all i gtgt 0 Since N was arbitrarily chosen the X Yadic convergence of the sequence is proved and with it the proposition D 68 An approximation scheme We give the set up for the last proposition of the paper which is the core of the proof of Theorem Gill Fix Lw E h and r E hxi Fix a cocompact discrete subgroup L C h Let 39i39 be the unique primitive rigged virtual lattice such that b 11L 1wr0 48 GREG W ANDERSON Note that lt1gt is the general examp1e of a primitive rigged virtua11attice Fix an integer no 2 0 such that FMMTOVMkL 6 Z Fix an integer M 2 0 such that HW MH 2 HwTHr For each a e Z put Ln7 eo 4 0me lt 69 E17471 Ck 1M1 and 1et Ta he the unique primitive rigged virtua11attice such that 1 12La 8 1wmr By construction Ta is a 7r7regular primitive rigged virtua1 1attice of 1eve1 g M for every a The intuition here is that Ta is a good approximation to lt1gt for a gtgt 0 The next resu1t makes precise what we mean by good approximation Proposition 681 Th5 7299521 mrtual lattzca pm 7 in 25 50ft and stnctly m7 tarpalabla far a gtgt 0 The proof is structured as a long series of lemmas Lemma 682pmlt1gt 7 in 23 sapambla far n gt U Pmaf If a is sumcient1y 1arge then nonL nonLa in which case both pm and pm Ta are separah1e by Lemma 263 m Lemma 683 ltIgt 7 by 23 50ft far n gt U Pmaf We have 1 MAO 12L 12L01wm 5 for allt e k and hence ltIgt 7 memo vanishes identica11y int for a gtgt 0 By Lemma 263 we have 1 1 Q ltIgt7ltIgt 0te wt To 77717 it o n o M MICL MICLn r W 0 and hence Qolt1gt 7 140 vanishes identica11y in t for a gtgt 0 It remains on1y to prove that o lt1gt 7 Ta vanishes identica11y in a for a gtgt 0 GivenneZa6kX andrehput an r 12Loamm 12Lanamm IX 7 i l l 7 1Limilrmil r ataltrgt7eoltzltaw zgtgtlt1r 7 n7 WWWmy gt By scaling rule 17 we have 106 ltltltIgt 7 lt1gtagtlt1 gtgtltmgt 7 Faarv By Lemma 263 and scahng ru1e 37 we have 107 olt1gt 7 ltIgtagtlt1 gtvltrirgt 7 Warm INTERPOLATION OF NUMBERS OF CATALAN TYPE 49 Therefore by the de nition of the theta symbol along with the scaling rule 23 and functional equation 40 satis ed by the theta symbol we have 9W7 39i39 39i39n 917 n1 a ZFnaI7 16k eltalt1gt 7 ltIgtngt 90 9M 7 ltIgtn1 20mm 16k Now look closely at the formulas for Fma and Gad We can nd some a1 6 kx and n1 such that Gn vanishes identically for llall 2 Hall and n 2 n1 We can find some n such that Fma vanishes identically for llall lt Hall and n 2 n2 It follows that a 39i39 7 n vanishes identically in a for it gtgt 0 D Lemma 684 For i 2 it gtgt 0 the rigged virtual lattice p 0 Q 7 n is stiictly interpolable and yoked to an interpolating gadget with basepoint 5M Proof It is clear that p 0 i 7 n is in all cases 7r regular of level S M By the preceding two lemmas p 0 Q 7 n is separable and soft for i 2 it gtgt 0 Since softness implies properness p 0 Q 7 n is interpolable of conductor S 5M for i 2 it gtgt 0 by Proposition 66 Since softness implies effectiveness p 0 Q 7 n is strictly interpolable and yoked to an interpolating gadget with basepoint 5M for i 2 it gtgt 0 by Proposition 49 D Lemma 685 Let U C k be an open compact subset of k We have Li n U 7 Li n U for it gtgt 0 Proof Put so filllyl and select N such that go is supported in the set Tr NO By the Poisson summation formula 8 we have U n Li akLn Z W EEW NO Ln For it gtgt 0 the right side does not change if we replace Ln by L D Lemma 686 For each a E kx we have a algal This is the heart of the proof of the proposition Proof By 106 107 the definition of the Catalan symbol and the scaling rule 48 for the Catalan symbol we have lt lt1gtflt1gtn gt lt sigma H NW H szW tekx tekx Hallwho W Now the function Fma vanishes identically for it gtgt 0 and moreover we can find an annulus A I 6 V l0 ltllaollSllIllSHa1Hlt 00 50 GREG W ANDERSON such that Gma is supported in A for n gt 0 It is important here that a is xed no claim of uniformity of convergence is being made in the lemma Then we have simply a k L 108 lt a gt a l H Esme 39i39 7 39i39n 16A for n gt 0 Note that Gma is Z valued for n gt 0 Now select any open compact subgroup U C hX small enough so that the function r gt gt e0 1awii restricted to the compact set A C hX is constant on cosets of Ur Then by the preceding lemma the summation of Gnaz over any coset of U contained in A vanishes for n gt 0 and hence HEB IGW W E U for n gt 0 Since U is arbitrarily small convergence is proved D Proof of the proposition By the preceding lemmas the hypotheses of Proposition 617 are ful lled by the family PM P i 1 U 10 0 4 11 provided that n gt 0 D 69 End of the proof of Theorem 61 By Proposition 681 every proper rational rigged virtual lattice 39i39 can be decomposed as a Z1plinear combination of soft strictly interpolable rigged virtual lattices plus a proper 7rregular rigged virtual lattice Which is interpolable by Proposition 616 D REFERENCES 1 Anderson C W A twoedimensional analogue of Stickelberger s theorem in The Arithmetic of Function Fields ed by D Goss D R Hayes and M I Rosen W de Gruyter Berlin 1992 51777 2 Anderson C W Rank one elliptic Aemodules and Aeharmonic series Duke Math J731994 4917541 A twoe39uariable re nement of the Stark conjecture in the function eld case Come positio Math to appear mathNT0407535 4 Anderson C W BroWnaWell W D Papanikolas M A Determination of the algebraic relations among special Peialues in positive characteristic Ann of Math 1602004 2397 315 5 Coleman R On the Frobenius endomorphisms of Fermat and Artineschreier cur39ues Proc Amer Math Soc 102 1988 461466 6 Goss 13 Basic Structures of Function Field Arithmetic Ergeb der Math und ihrer Grene zgebiete 3 Folge Vol 35 Springer Verlag Berlin Heidelberg 1996 7 Lubin J Tate J Formal complex multiplication in local elds Ann of Math 811965 38387 8 Macdonald I G Symmetric functions and Hall polynomials Second edition Oxford Math ematical Monographs Oxford University Press 1995 9 Tate J Les Conjectures de Stark sur les Fonctions L d7Artin en s 0 Prog in Math 47 Birkhauser Boston 1984 10 Tate J Numberetheoretic background Proc of Symp in Pure Math 331979 part 2 pp 3726 11 Thakur D S Function Field Arithmetic World Scienti c Pub 2004 UNIVERSITY OF MINNESOTA MPLS MN 55455 Eemail address gwandersQumn edu

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