aa r X i v : . [ m a t h . N T ] J un RATIONAL -FUNCTIONS ARE ABELIAN L. FELIPE M ¨ULLER
Abstract.
We show that the coefficients of rational 2-functions are containedin an abelian number field. More precisely, we show that the poles of suchfunctions are poles of order one and given by roots of unity and residue whichis integral outside the discriminant of the underlying number field.
Contents
1. Introduction 1Acknowledgment 2Notation 22. Preliminaries 33. Proof of Theorem 1.2 7References 111.
Introduction
Fermat’s and Euler’s congruences are well-known in number theory and are richof remarkable consequences. In the following we will give a short survey of thesecongruences. We start with the famous
Theorem 1.1 (Euler) . The congruence a p r ” a p r ´ mod p r (1.1) holds for all integers a P Z , all primes p , and all natural numbers r P N . A sequence p a k q k P N of rational numbers is called an Euler sequence (or
Gausssequence as in [4]) for the prime p , if a k is a p -adic integer for all k P N and a mp r ” a mp r ´ mod p r (1.2)for all integers r ě m ě
1. A survey of these congruences has been given in[11] and [15].Beukers coined the term supercongruence : A supercongruence (with respect toa prime p ) refers to a sequence p a n q n P N P Z N p that satisfies congruences of the type a mp r ” a mp r ´ mod p sr , (1.3)for all m, r P N and a fixed s P N , s ą Date : June 12, 2020. to name a few. Note that all the above mentioned supercongruences are valid for s “ p ě K be an algebraic number field and D its discriminant. In Section 2 weconsider a generalization of supercongruences to sequences p a n q n P N P K N of alge-braic numbers, as has been done in [14]. We will refer to such sequences whichsatisfy eq. (1.3) as s -sequences. Their generating functions then integrate to whatis referred to as s -functions in [14]. More precisely, for a natural number s P N , an s -sequence a P K N corresponds to the s -function (see Proposition 2.2 in Section 2)given by the (formal) power series V p z q “ ÿ n “ a n n s z n P zK J z K . By Theorem 22 in [14], 2-functions appear as the non-singular part of the su-perpotential function (without the constant term) with algebraic coefficients, i.e.algebraic cycles on Calabi-Yau three-folds provide a source of 2-functions that areanalytic and furthermore satisfy a differential equation with algebraic coefficients.As has been discussed in [14], it is interesting to find a “basis of s -functions”. Asa first attempt, the author characterized rational 2-functions with algebraic coeffi-cients. This is the main result of the present paper: Theorem 1.2.
Let p a n q n P N P K N be a -sequence such that its generating function ř n “ a n z n is the Maclaurin expansion of a rational function F . Then there is aninteger N P N and coefficients A i P Z “ D ‰ for i “ , ..., N such that F p z q “ N ÿ i “ A i ζ i z ´ ζ i z , where ζ is a suitable N -th root of unity. Note that all such functions are rational s -functions for all s ě
2. The firstreduction in the proof of Theorem 1.2 is given by Theorem 2.4, a statement due toMinton (cf. [11]). It states, that the generating functions of Euler sequences aregiven by sums of logarithmic derivatives of polynomials with integral coefficients.Theorem 1.2 is restated in Section 3 as a corollary of Theorem 3.5.In [7] Dold classified all rational 1-functions with rational coefficients: he showedthat the generating function of an Euler sequence is rational if and only if thissequence can be realized as the sequence of fixed point indices t I p f n qu n P N of theiterates f n of a map f : V Ñ Y , where Y is an euclidean neighborhood retract and V Ă Y is an open subset. Acknowledgment.
The author is grateful to Johannes Walcher for providing theinitial motivation for this work.
Notation.
Throughout this paper, the natural numbers will be meant to be theset of all positive integers, N “ t , , ... u , while N “ N Y t u . If X is a set, then X N denotes the set of all sequences indexed by the natural numbers, p x n q n P N P X N .For a ring R let R J z K denote the ring of formal power series in the variable z withcoefficients in R . We identify R N with zR J z K in terms of generating functions,that is, we denote this identification by G : R N Ñ R J z K , where G p r q “ ř n “ r n z n .For k P N , let r´s k : zR J z K Ñ R be given by “ř n “ r n z n ‰ k “ r k . Then the ATIONAL 2-FUNCTIONS ARE ABELIAN 3 inverse of G is given by pr´s k q k P N : zR J z K Ñ R N , i.e. G ˝ pr´s k q k “ id zR J z K and pr´s k q k ˝ G “ id R N . 2. Preliminaries
In this section, we introduce the definitions and notational conventions that willbe used throughout the paper. We mainly follow the conventions given in [14].Let K be a fixed algebraic number field. Denote by O the ring of integers of K .Let D be the discriminant of K { Q . Note that an unramified prime p is characterizedby the property that p ∤ D . For any prime ideal p , O p denotes the ring of p -adicintegers. The O p are integral domains and their field of fractions K p “ Quot p O p q is the p -adic completion of K . For an unramified prime p , we set O p to be given by O p “ ź p |p p q O p . Analogously, K p “ ź p |p p q K p . Multiplication is realized by component-wise multiplication, that is, for p x p q p |p p q and p y p q p |p p q P O p (resp. K p ) we have p x p q p |p p q ¨ p y p q p |p p q “ p x p ¨ y p q p |p p q P O p ( K p resp.). Therefore, K p is a K -algebra. Let ι p : K ã Ñ K p be the canonical embeddingsof K into its p -adic completion, then K is embedded in K p by the map ι p : K ã Ñ K p , x ÞÑ p ι p p x qq p |p p q . Nonetheless, if it is clear from the context, we will use the samesymbol x for ι p p x q or ι p p x q , whenever x P K . We say that x P K is a p -adic integer ( p -adic unit resp.), if x P O p ( x P O ˆ p resp.) with respect to all prime ideals p suchthat p | p p q . ι p ( ι p resp.) can be extended to the ring K p J z K ( K p J z K resp.) by setting ι p p z q “ z and ι p p z q “ z , i.e. linearity induces the maps ι p : K J z K ã Ñ K p J z K and ι p : K J z K ã Ñ K p J z K . Again, for V P K J z K , we will use the same symbol V to refer to the powerseries ι p p V q P K p J z K and ι p p V q P K p J z K .For p | p p q , the Frobenius element Fr p at p is the unique element satisfy-ing the following two conditions: Fr p is an element in the decomposition group D p p q Ă Gal p K { Q q of p and for all x P O , Fr p p x q ” x p mod p . By Hensel’s Lemma,Fr p can be lifted to O p and then extended to an automorphism Frob p : K p Ñ K p .We obtain the isomorphism Frob p : K p Ñ K p , by setting Frob p “ p Frob p q p |p p q .By declaring Frob p p z q “ z , Frob p can be (linearly) extended to an endomorphismFrob p : K p J z K Ñ K p J z K .Let R be a Q -algebra. The logarithmic derivative δ R is an operator δ R : R J z K Ñ zR J z K , where δ R is given by z dd z , i.e. δ R « ÿ n “ r n z n ff “ ÿ n “ nr n z n . L.F. M¨ULLER
Its (partial) inverse of δ R is the logarithmic integration I R : zR J z K Ñ zR J z K givenby I R « ÿ n “ r n z n ff “ ÿ n “ r n n z n . For a prime p P Z let C R,p be the operator C R,p : R J z K Ñ R J z K , called the Cartieroperator , given by C R,p « ÿ n “ r n z n ff “ ÿ n “ r pn z n . Hereafter, we will omit R from the notation of δ R , I R and C R,p .In [14], an s -function with coefficients in K (for s P N ) is defined to be a formalpower series V P zK J z K such that for every unramified prime p P Z in K { Q we have1 p s Frob p V p z p q ´ V p z q P z O p J z K . In the following, we will identify s -functions with s -sequences. Definition 2.1.
A sequence p a n q n P N P K N is said to satisfy the local s -functionproperty for p , if p P Z is an unramified prime in K { Q , and a n P O p is a p -adicinteger for all n P N , andFrob p ` a mp r ´ ˘ ” a mp r mod p sr O p , for all m, r P N . p a n q n P N is called an s -sequence if it satisfies the local s -functionproperty for all unramified primes p in K { Q .From the above definition of an s -sequence a P K N it is evident that a P O “ D ‰ N .The above two definitions are equivalent in the following sense: Proposition 2.2 (Lemma 4 in [14]) . Let s P N . Then:(i) If a P O “ D ‰ N is an s -sequence then I s r G p a qs P zK J z K is an s -function.(ii) If V P zK J z K is an s -function then pr δ s V s n q n P N P K N is an s -sequence. Note that a sequence p a n q n P N P K N is an s -sequence if and only if, for all un-ramified primes p and all r ě C r ´ p p Frob p G p a n q ´ C p G p a n qq ” p sr O p J z K . Let us rephrase
Dwork’s Integrality Lemma in the setting of 1-functions as hasbeen done in [14]:
Lemma 2.3.
Let V P zK J z K and Y P ` zK J z K be related by V “ log Y , Y “ exp p V q . Then the following are equivalent(i) V is a -function.(ii) Y P ` z O “ D ‰ J z K . The next Theorem 2.4 is a modified version of Theorem 7.1 in [4], which onthe other hand is a re-proven statement from [11]. The crucial point is, that arational 1-function only admits poles of order 1. We give a proof for the sake ofcompleteness.
ATIONAL 2-FUNCTIONS ARE ABELIAN 5
Theorem 2.4 (Thm. 7.1 in [4]) . Let p a n q n P N P K N be a -sequence with generatingfunction G p a n q P zK J z K representing a rational function F P K p z q as a Taylorexpansion in . Then there are r P N and distinct α i P Q ˆ , A i P Q ˆ , for i “ , ..., r ,such that F has the form F p z q “ r ÿ i “ A i α i z ´ α i z . Proof.
Let F be given by the fraction of P, Q P K r z s , Q ı
0, i.e. F “ PQ . Wemay assume that Q p q ‰ P p q “
0. By Proposition 3.5 in [4], we havedeg p P q ď deg p Q q . By adding a constant C P K to F it does not affect the 1-function condition but we may assume deg p P q ă deg p Q q . Then, by the PartialFraction Decomposition ˜ F “ PQ ` C has the form˜ F “ r ÿ i “ m i ÿ j “ A i,j p ´ α i z q j , where the α i P Q ˆ , i P t , ..., r u are distinct algebraic numbers, m i P N and A i,j P Q for all p i, j q P t , ..., r u ˆ t , ..., m i u . Now, let p be a sufficiently large unramifiedprime, such that following conditions are simultaneously satisfied:(i) α i are p -adic units,(ii) α i ´ α j is a p -adic unit for i ‰ j , and(iii) p ą m i .What we need to show is m i “ i P t , ..., r u . We have1 p ´ α i z q j “ ÿ k “ ˆ k ` j ´ j ´ ˙ α ki z k . Therefore, if f is the Maclaurin series expansion of ˜ F , we have C p p f q “ ÿ k “ « r ÿ i “ m i ÿ j “ A i,j ˆ pk ` j ´ j ´ ˙ α pki ff z k . Since p ą m i , we find ` pk ` νν ˘ ” p for all 0 ď ν ă m i . Consequently, C p p f q ” ÿ k “ « r ÿ i “ m i ÿ j “ A i,j α pk ff z k mod p “ r ÿ i “ m i ÿ j “ A i,j ´ α pi z “ r ÿ i “ A i ´ α pi z , where A i “ ř m i j “ A i,j . Hence, C p p ˜ F q is equal to a rational function with simplepoles modulo p . The 1-function property ensures that ˜ F has simple poles as well.Therefore, we write from now on˜ F “ r ÿ i “ A i ´ α i z mod p, L.F. M¨ULLER where A i , α i P Q ˆ and α i ‰ α j for i ‰ j . Evaluating ˜ F at z “ C “ ř ri “ A i . Therefore, F “ ˜ F ´ C “ r ÿ i “ A i ´ α i z ´ r ÿ i “ A i “ r ÿ i “ A i α i z ´ α i z . In particular, we have a n “ r ÿ i “ A i α ni for all n P N . The 1-function property then gives0 ” Frob p p a m q ´ a mp “ r ÿ i “ p Frob p p A i q Frob p p α mi q ´ A i α mpi q mod p, for all m P N . Since Fr p is given by taking component-wise the p -th power mod p ,we conclude 0 ” r ÿ i “ p A pi ´ A i q α mpi mod p, for all m P N . The Vandermonde type matrix MM “ ¨˚˚˚˝ α p α p . . . α rp α p α p . . . α rp ... ... . . . ... α pr α pr . . . α rpr ˛‹‹‹‚ . is invertible mod p . Indeed, its determinant is given bydet p M q “ ` r ź i “ α pi ˘ ˆ ź ď i ă j ď r ` α pj ´ α pi ˘ . Since α i ´ α j is a p -adic unit for i ‰ j by assumption (ii) we obtain det p M q P O p isinvertible. Hence, A pi ” A i mod p for all i P t , ..., r u . From Frobenius’s DensitiyTheorem , see for instance [10], it follows that A i P Q for all i P N . (cid:3) Corollary 2.5.
Let V P zK J z K be an s -function and Y “ exp p´ δ s ´ V q . Then δ s V is the series expansion of a rational function if Y is the series expansion of arational function. Conversely, if δ s V represents a rational function, then there isan M P N such that Y M is the series expansion of a rational function.Proof. Let Y be the series expansion of a rational function, then so is δY . Hence, δYY is the series expansion of a rational function. Consequently, δ s V “ ´ δYY representsa rational function. Conversely, let δ s V represent a rational function at zero. ByTheorem 2.4 there exists a natural number r P N , and distinct α i P Q ˆ , A i P Q ˆ ,for i “ , ..., r , such that δ s V “ r ÿ i “ A i α i z ´ α i z “ r ÿ i “ A i ÿ n “ α ni z n “ r ÿ i “ A i δ ÿ n “ α ni n z n “ ´ r ÿ i “ A i δ log p ´ α i z q . Therefore, Y “ exp p´ δ s ´ V q “ exp ˜ r ÿ i “ A i log p ´ α i z q ¸ “ r ź i “ p ´ α i z q A i . ATIONAL 2-FUNCTIONS ARE ABELIAN 7
Taking M P N to be the least common multiple of the denominators of A i we findthat Y M is a rational function. (cid:3) Proof of Theorem 1.2
Let p a n q n P N P O “ D ‰ N be a 2-sequence, such that its generating function G p a n q is rational. Then p a n q n P N is in particular an 1-sequence and, by the above The-orem 2.4, there is a N P N , A i P Q and distinct α i P Q ˆ for i P t , ..., r u suchthat a n “ r ÿ i “ A i α ni for all n P N . In the following, let us assume α i P K , since we might otherwise substitute K by anormal closure of K p α , ..., α r q . As pointed out by Minton in [11] the Chebotar¨evdensity theorem implies
Theorem 3.1 (Thm. 3.3. in [11]) . Let K be a Galois number field. For any σ P Gal p K { Q q , there exists infinitely many primes p of K such that Fr p “ σ . Let p P Z be an unramified prime in K , splitting completely in K , i.e. Fr p “ id K for all p | p p q . By the density theorem of Chebotar¨ev there are infinitely many suchprimes p . Let m, n P N then the local 2-function property reads a p n m ´ Frob p p a p n ´ m q “ a p n m ´ a p n ´ m “ r ÿ i “ A i ´ α p n mi ´ α p n ´ mi ¯ ” p n . At this point, one might guess that those numbers α , . . . , α r are (somehow relatedto) roots of unity (note that for α i being roots of unity, these congruences are indeedequations). In fact, since the above congruence is valid for an infinite number ofprimes p it should be true that these congruences already hold for each summand.But in the case r “ α is already a root of unity as shown in the followingLemma 3.2 for α “ x . Lemma 3.2.
Let x P K ˆ and p P Z a prime, which splits completely in K { Q , suchthat x is a p -adic unit. Suppose that x p n ´ x p n ´ ” p n , for all n P N . Then x is a root of unity in K .Proof. We have x p n ´ p p ´ q ” p n . Recall that the Iwasawa logarithm preserves the p -adic order, therefore p n ´ log p p x p ´ q ” p n . Hence, log p p x p ´ q ” p n ` for all n P N , i.e. x is a root of unity. (cid:3) As Lemma 3.2 suggests it should be sufficient to investigate the 2-function prop-erty for only one suitably chosen prime p . This justifies the restriction of theproblem to the local case as we will see in the proof of Corollary 3.6. L.F. M¨ULLER
Let x P Z ˆ p and m P N . By Euler’s Theorem 1.1 there is a sequence p ρ n p m qq n P N P Z N p such that x mp n ´ x mp n ´ “ p n ρ n p m q . We also write κ n p m q : “ ord p p ρ n p m qq P N Y t8u . As we will see in Proposition 3.3and Proposition 3.4, κ n p m q is independent of n, m P N for p p, m q “ Proposition 3.3.
Let p ą and x P Z ˆ p . Then the sequence κ n p m q P N Y t8u isa constant in n . If κ p m q “ κ p m q ‰ 8 , then ρ n ` p m q ” ρ n p m q mod p n ` κ p m q for all n P N .Proof. To simplify the notation, let ρ n : “ ρ n p m q , x “ x m and κ n “ κ n p m q . Sup-pose ρ n “ n P N . But then, the equation x p n ´ p p ´ q “ x is a root of unity in Z p and therefore x p n ´ p p ´ q “ n P N , i.e. ρ n “ n P N . Suppose therefore, that ρ n ‰ n P N , i.e. x is not a root of unityin Z p . Then the statement follows by using the Binomial Theorem . We have x p n “ ´ x p n ´ ¯ p “ ´ x p n ´ p n ρ n ¯ p “ p ÿ k “ ˆ pk ˙ x kp n p´ q p ´ k p n p p ´ k q ρ p ´ kn “ x p n ` ` p ´ ÿ k “ ˆ pk ˙ x kp n p´ q p ´ k p n p p ´ k q ρ p ´ kn ` p´ q p p np ρ pn . “ x p n ` ` p ´ ÿ k “ ˆ p ´ k ˙ x kp n p´ q p ´ k p ´ k ρ p ´ kn p n p p ´ k q` ` p´ q p ρ pn p np . Therefore, p n ` ρ n ` “ p ´ ÿ k “ ˆ p ´ k ˙ x kp n p´ q p ´ k ` p ´ k ρ p ´ kn p n p p ´ k q` ´ p´ q p ρ pn p np , ô ρ n ` “ p ´ ÿ k “ ˆ p ´ k ˙ x kp n p´ q p ´ k ` p ´ k ρ p ´ kn p n p p ´ k ´ q ´ p´ q p ρ pn p n p p ´ q´ . If p ą
2, we find modulo p n ` κ n , that is, we take the p p ´ q -th summand ρ n ` ” x p p ´ q p n ρ n mod p n ` κ n . From this congruence it is evident, that κ n ` “ κ n . Hence (i) follows recursively.We therefore write κ for κ n . Furthermore, ρ n ` ” x p p ´ q p n ρ n mod p n ` κ “ ´ ` p n ` x ´ p n ρ n ` ¯ ρ n ” ρ n mod p n ` κ , which finishes the proof. (cid:3) ATIONAL 2-FUNCTIONS ARE ABELIAN 9
Proposition 3.4.
Let x P Z ˆ p be a p -adic unity and n, m P N be integers with m coprime to p . Then κ : “ κ p m q P N Y t8u does not depend on m . Furthermore, if x is not a root of unity in Z p , then ρ n p m q ” mx p m ´ q p n ´ ρ n p q mod p n ` κ . Proof.
Fix n P N . If x is a root of unity in Z p , then κ p m q “ 8 for all m P N .Conversely, if ρ n p m q vanishes for some m P N , then x is a root of unity in Z p .Therefore, let x be not a root of unity in Z p . Since ρ n p q ‰ x ´p m ´ q p n ´ ρ n p m q ρ n p q “ ´ x mp n ´ p p ´ q ´ x p n ´ p p ´ q “ m ´ ÿ k “ x kp n ´ p p ´ q “ m ` p n m ´ ÿ k “ x ´ kp n ´ ρ n p k q . The above computation shows that κ p m q “ ord p p ρ n p m qq “ ord p p ρ n p qq “ : κ , since m is coprime to p . In particular, the p -order of the sum ř m ´ k “ x ´ kp n ´ ρ n p k q is atleast κ (since every single summand has p -order greater or equal to κ ). Therefore, ρ n p m q ” mx p m ´ q p n ´ ρ n p q mod p n ` κ , as stated. (cid:3) Theorem 3.5.
Let r P N and let (for i “ , ..., r ) x i , B i P Z ˆ p with x k ‰ x ℓ for k ‰ ℓ . Suppose that for all n, m P N we have r ÿ i “ B i ´ x mp n i ´ x mp n ´ i ¯ ” p n . (3.1) Then x i is a root of unity in Z p for i “ , ..., r .Proof. In the following, we will write ρ i,n p m q : “ p n ´ x mp n i ´ x mp n ´ i ¯ , and σ n p m q : “ p n r ÿ i “ B i ρ i,n p m q for suitable ρ i,n p m q , σ n p m q P Z p . Note that σ n p m q is indeed in Z p by (3.1). Itsuffices to show that one of the x i is a root of unity, because then one can substitutethe sum on the left of (3.1) recursively by a sum of the same type with r ´ α i is a root of unity.In particular, ρ i,n p m q ‰ i, n , and m . By Proposition 3.3 and Proposition 3.4we have for every i “ , ..., r a κ i P N such that κ i “ ord p p ρ i,n p m qq for all n, m P N with m coprime to p . Define κ : “ min t κ i | i “ , ..., r u .Within this scenario, we prove the following statement: For all n, m P N with p m, p q “ we have r ÿ i “ B i x p m ´ q p n ´ i ρ i,n p q ” p n ` κ . (3.2)By applying Proposition 3.3 to each ρ i,n ` separately, we obtain p n ` σ n ` p m q “ r ÿ i “ B i ρ i,n ` p m q ” r ÿ i “ B i ρ i,n p m q “ p n σ n mod p n ` κ . Dividing the above equation by p n , we obtain pσ n ` p m q ” σ n p m q mod p κ , (3.3)for all n P N . Inductively, σ n p m q ” p κ σ n ` κ p m q ” p κ , for all n P N . Therefore, r ÿ i “ B i ρ i,n p m q ” p n ` κ for all n P N . (3.4)From (3.4) the assertion (3.2) for m “ m P N bearbitrary again. Using Proposition 3.4 gives r ÿ i “ B i x p m ´ q p n ´ i ρ i,n p q ” m r ÿ i “ B i ρ i,n p m q mod p n ` κ . (3.5)Since p m, p q “
1, applying (3.4) on the right-hand side of (3.5) yields the formula(3.2).Inserting n “ m “ , ..., r into (3.2) yields the following system of linearequations, since r ă p , ¨˚˚˚˝ B B ¨ ¨ ¨ B r B x B x ¨ ¨ ¨ B r x r ... ... . . . ... B x r ´ B x r ´ ¨ ¨ ¨ B r x r ´ r ˛‹‹‹‚¨˚˚˚˝ ρ , p q ρ , p q ... ρ r, p q ˛‹‹‹‚ ” p ` κ . The determinant of the above Vandermonde matrix is given bydet ¨˚˚˚˝ B B ¨ ¨ ¨ B r B x B x ¨ ¨ ¨ B r x r ... ... . . . ... B x r ´ B x r ´ ¨ ¨ ¨ B r x r ´ r ˛‹‹‹‚ “ r ź i “ B i ˆ ź ď k ă ℓ ď r p x k ´ x ℓ q . Hence, the determinant does not vanish mod p , since x k ´ x ℓ is a p -adic unit for k ‰ ℓ . In other words, κ ě ` κ , which is a contradiction. Therefore, the set t x , ..., x r u contains at least one root of unity. Inductively, all x , ..., x r are roots ofunity. (cid:3) Corollary 3.6.
Let a P K N be a -sequence such that its generating function ř n “ a n z n is the Maclaurin expansion of a rational function F . Then there isan integer N P N and coefficients A i P Z “ D ‰ for i “ , ..., N such that F p z q “ N ÿ i “ A i ζ i z ´ ζ i z , where ζ is a suitable N -th root of unity.Proof. As mentioned at the beginning of this section, we may assume α i P K for all i . By Theorem 2.4 the 2-sequence a is given by a power sum a n “ ř ri “ A i α ni , forfixed r P N , where A i P Q ˆ and where the α i P Q ˆ are distinct algebraic numbers.Now, choose a prime p P Z such that(1) p splits completely in K { Q ,(2) α i , A i and α k ´ α ℓ are p -adic units for all i “ , ..., r and k ‰ ℓ , ATIONAL 2-FUNCTIONS ARE ABELIAN 11 (3) max t r, u ă p .This choice of p is possible by Theorem 3.1. Therefore, we have K p – Q p and O p – Z p for all prime ideals p Ă O dividing p p q . Hence, O p may be identified with ś p |p p q Z p . Since Fr p “ id K , the local 2-function condition for p then reads r ÿ i “ A i ´ α mp n i ´ α mp n ´ i ¯ ” p n O p , for all m, n P N . For B i “ ι p p A i q and x i “ ι p p α i q for p | p p q , Theorem 3.5 statesthat α i are all roots of unity. Therefore, choose a primitive root of unity ζ of order,say, N such that α i “ ζ ν i for a suitable ν i P N . Without loss of generality, we mayassume A i P Q (zeros are allowed) and a n “ N ÿ i “ A i ζ in . We observe that the sequence a n P O “ D ‰ X Q p ζ q . Therefore, without loss ofgenerality we may also assume K “ Q p ζ q . Furthermore, the set t , ζ, ...ζ N ´ u is anintegral basis of K . Since a P O “ D ‰ N , we conclude A i P Z “ D ‰ . (cid:3) References
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