THE LAZARD FORMAL GROUP, UNIVERSAL CONGRUENCES AND SPECIAL .

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THE LAZARD FORMAL GROUP, UNIVERSALCONGRUENCES AND SPECIAL VALUES OF ZETAFUNCTIONSPIERGIULIO TEMPESTAAbstract. A connection between the theory of formal groups andarithmetic number theory is established. In particular, it is shown howto construct general Almkvist–Meurman–type congruences for the universal Bernoulli polynomials, that are related with the Lazard universalformal group. Their role in the theory of L–genera for multiplicativesequences is illustrated. As an application, classes of sequences of integer numbers are constructed. Some congruences are also obtained forcomputing special values of a new class of Riemann–Hurwitz–type zetafunctions.Contents1. Introduction: formal groups2. The Lazard universal formal group, the universal Bernoullipolynomials and numbers3. Generalized Almkvist–Meurman congruences3.1. Proof of Theorem 13.2. Related results4. The Todd genus, the L and A genuses and related universalpolynomials5. Universal congruences and integer sequences5.1. Sequences of integer numbers5.2. Applications5.3. Polynomial sequences with integer coefficients6. A class of Riemann–Hurwitz zeta functions and their values atnegative integers6.1. Hurwitz zeta functions and formal groups6.2. χ–universal numbers from Dirichlet characters and congruencesAppendixReferencesDate: June 3, 2012.1235688101011111212131414

2PIERGIULIO TEMPESTA1. Introduction: formal groupsThe theory of formal groups [7], [13] has been intensively investigatedin the last decades, due to its relevance in many branches of mathematics,especially algebraic topology [8], [5], [20], [12], the theory of elliptic curves[24], and arithmetic number theory [1], [2], [26], [28].Given a commutative ring R with identity, and the ring R {x1 , x2 , .}of formal power series in the variables x1 , x2 , . with coefficients in R, acommutative one–dimensional formal group over R is a formal power seriesΦ (x, y) R {x, y} such that [7]1)2)Φ (x, 0) Φ (0, x) xΦ (Φ (x, y) , z) Φ (x, Φ (y, z)) .When Φ (x, y) Φ (y, x), the formal group is said to be commutative (theexistence of an inverse formal series ϕ (x) R {x} such that Φ (x, ϕ (x)) 0follows from the previous definition).As is well known, over a field of characteristic zero, there exists an equivalence of categories between Lie algebras and formal groups. Let V be thevaluation ring of a complete ultrametric field K. Then a commutative diagram of functors holds [23]:Analytic Groups /K Formal Groups /K-%Formal Groups /VGiven a formal group law F (X, Y ), let F2 (X, Y ) denote its quadraticpart. The Lie algebra (over the same valuation ring V ) associated to F isdefined via the identification [x, y] F2 (X, Y ) F2 (Y, X). This equivalenceof categories is no longer true in a field of characteristic p 6 0.A crucial point is the relation between the Lazard’s universal formal groupand cobordism theory [8], [20], [21]. Indeed, the coefficients cn of the universal group can be identified with the cobordism classes of CP n . Let E bea 2–periodic generalized cohomology theory with a complex orientation indegree zero. Then, for each stably almost complex manifold M , we have afundamental class [M ] E 0 . We also haveformal group law FPa canonicalkk 10over E , and it turns out that logF (x) k 0 [CP ]x /k 1.In [8] and [18], the relevance of the Lazard group in the discussion of theclassical and modern theory of unitary cobordisms has been clarified. A niceconnection with combinatorial Hopf algebras and Rota’s umbral calculus [22]was found in [5]. A combinatorial approach has been proposed in [4].In the papers [26], [28] a relation between the universal formal group andthe theory of L–series has been established. A new class of Bernoulli–typepolynomials, called the Universal Bernoulli polynomials, were introduced,and preliminarily studied in [27]. They are Appell polynomials defined viathe universal formal group exponential law.

FORMAL GROUPS, AM CONGRUENCES AND ZETA FUNCTIONS3In [16], [17], the universal Bernoulli polynomials have been related to thetheory of hyperfunctions of one variable by means of an extension of theclassical Lipschitz summation formula to negative powers.Due to the generality of their definition, the universal polynomials includemany (if not all) of the generalizations of the classical Bernoulli polynomials known in the literature. In [1], [2], Kummer–type congruences for theuniversal Bernoulli numbers have been derived.Another interesting aspect is represented by the relation between Bernoullinumbers, Hirzebruch’s theory of genera, and the computation of Todd classes[14]. Indeed, in the context of complex cobordism theory, let ϕ be a multiplicative genus in the sense of Hirzebruch, and cn is the value of ϕ on thecomplex projective n–space. Then the generating function t/G(t), whereG(t) is the universal formal group exponential (see Definition 1), is thecorresponding characteristic power series in Hirzebruch’s formalism. For instance, the power series associated to the L–genus, the Todd genus and theA genus are special cases of the proposed construction. Consequently, theG (x) can be interpreted asgenerating function defining the polynomials Bk,asome special value of the genus.In this paper, we address the following general problem: Construct congruences for the Universal Bernoulli polynomials and apply them to computespecial values of the related zeta functions.Main result. A universal congruence, that generalize the well–knownAlmkwist–Meurman [3] and Bartz–Rutkowski [6] congruences to the case ofthe universal Bernoulli polynomials, is proposed. We introduce two familiesof polynomials related to the characteristic power series of important m–sequences of the genus theory in algebraic topology, that represent nontrivialrepresentations of the universal polynomials. Their arithmetic properties arestudied.A second result, coming from the theory of universal congruences, is aprocedure enabling to construct infinitely many sequences of integers andpolynomials with integral coefficients.Finally, we deduce congruences for the values at negative integers of a newclass of Hurwitz–type zeta functions, that have been recently constructed in[28]. Work is in progress on a p–adic generalization of the previous theory.2. The Lazard universal formal group, the universalBernoulli polynomials and numbersWe recall some basic definitions, necessary in the subsequent discussion.Definition 1. Let us consider the formal group logarithm, over the polynomial ring Q[c1 , c2 , .](1)LogG (s) Xi 0cisi 1,i 1

4PIERGIULIO TEMPESTAwith c0 1. Let G (t) be its compositional inverse (the formal group exponential):(2)G (t) Xγii 0ti 1i 1so that F (G (t)) t. We have γ0 1, γ1 c1 , γ2 32 c21 c2 , . . .The Lazard universal formal group law [13] is defined by the formal powerseries(3)Φ (s1 , s2 ) G (F (s1 ) F (s2 )) .The Lazard ring L is the subring of Q [c1 , c2 , .] generated by the coefficients of the power series G (F (s1 ) F (s2 )).In [26], the following family of polynomials has been introduced.G (x, c , ., c ) Definition 2. The universal higher–order Bernoulli polynomials Bk,a1nG (x) are defined byBk,aµ(4)tG (t)¶aext XGBk,a(x)k 0tk,k!x, a R.Notice that a 1, ci ( 1)i , then F (s) log (1 s) , G (t) et 1,and the universal Bernoulli polynomials and numbers reduce to the standardG (x) B G (x) and B G (0) B G .ones. For sake of simplicity, we will put Bk,1kkkG (0) Q [c , c , .] coincide with Clarke’s universal BernoulliThe numbers Bk,11 2numbers [9].The universal Bernoulli numbers satisfy some very interesting congruences. In particular, we recall the universal Von Staudt’s congruence [9],that generalize the classical Clausen–von Staudt congruence and several variations of it known in the literature, and the universal Kummer congruence[1]. Both congruences have a distinguished role in algebraic geometry [5],[15].The universal Bernoulli polynomials (4) possess many remarkable properties. By construction, for any choice of the sequence {cn }n N they representG (x) a class of Appell polynomials. It means that, for all k N, DBk,aGkBk 1,a (x), where D denotes the standard derivative operator. The binomial property holds:n µ ¶XnGB G (x)y n m .(5)Bn (x y) m mm 0We also mention that in the paper ([27]), several families of polynomialobtained by specializing eq. (4), i.e. the Bernoulli–type polynomials of firstand second kind, as well as related Euler polynomial sequences were studied.

FORMAL GROUPS, AM CONGRUENCES AND ZETA FUNCTIONS53. Generalized Almkvist–Meurman congruencesG (0) : Bcn correspond to the universal Bernoulli numbers.The values Bn,1Here we mention only two of their most relevant properties.i) The universal Von Staudt’s congruence [9].If n is even,(6)cn BX cn/(p 1)p 1p 1 npmod Z [c1 , c2 , .] ;pprimeIf n is odd and greater than 1,(7)n 3ncn c1 c1 c3B2mod Z [c1 , c2 , .] .When cn ( 1)n , the celebrated Clausen–Von Staudt congruence forBernoulli numbers is obtained.ii) The universal Kummer congruences [1], [2].The numerators of the classical Bernoulli numbers play a special rôle, dueto the Kummer congruences and to the notion of regular prime numbers,introduced in connection with the Last Fermat Theorem. The relevance ofKummer’s congruences in algebraic geometry has been enlightened in [5].bn satisfy an universal congruence.As shown by Adelberg, the numbers BSuppose that n 6 0, 1 (mod p 1). Then(8)bn p 1bnBB cp 1n p 1nmodpZp [c1 , c2 , .] .Almkvist and Meurman in [3] discovered a remarkable congruence for theBernoulli polynomials:µ ¶h(9)k n Bn Z,kwhere k, h, n N. Bartz and Rutkowski [6] have proved a Theorem whichcombines the results of Almkvist and Meurman [3] and Clausen–von Staudt.Simpler proofs of the original one proposed in [3] have also been providedin [10] and [25].The main result of this section is a generalization of the Almkvist–Meurmanand Bartz–Rutkowski Theorems for the universal polynomials (4).Theorem 1. Let h 0, k 0, n be integers. Consider the polynomialsdefined bytkt xt X Ge Bk (x) ,G (t)k!k 0

6PIERGIULIO TEMPESTAwhere G (t) is a formal group exponential, such that ci Z for all i 1, 2, . . .Assume that cp 1 1 mod p for all primes p 2. Thenµ ¶hngG(10)k Bn Z,kGgGbwhere Bn (x) Bn (x) Bn .The following lemmas are useful in the proof of Theorem 1.Lemma 1. If s is a positive integer, then, under the same hypotheses ofTheorem 1, we haveX n/(p 1)(11)(k n 1)cp 1/p Z.p 1 np-kProof. The result is a consequence of Fermat’s little Theorem: if s Z, andp - s, then sp 1 1 mod p. Therefore, if cp 1 1 mod p for all primesp 2, we deduce the congruence (11). Lemma 2. Let n be a positive integer and p 2 a prime number. Denote¶· K µXnn,K Sp (n) ,k(p 1)p 1k 1where [α] is the unique integer such that [α] α [α] 1. Then Sp (n) 0 mod p if p 1 - n and Sp (n) 1 mod p if p 1 n.This Lemma is a particular case of an old result of Jenkins (see [11], vol.I, pag 271).3.1. Proof of Theorem 1.Proof. As in [6], we base the proof on the use of the induction principle onh, with s fixed. First observe that, by virtue of (5) for x hk and y k1 wegetµ¶ Xµ ¶n µ ¶h 1n m G hk n BnG k Bm.kmkm 0For h 0 , the Theorem is obviously true. Assume that it holds for h.We obtain:µ ¶µ¶ Xn µ ¶n m G hh 1ngGbn k Bmk Bn kn Bmkkm 0(12)µ ¶ n 1n µ ¶XX µn¶n mghGbm .k Bmkm B mmkm 0m 0Due to the induction hypothesis, we have only to prove that the congruence (10) holds for the second sum in (12).

FORMAL GROUPS, AM CONGRUENCES AND ZETA FUNCTIONS7Notice that, thanks to (7), for every odd m 1 we have thatn 1X µn¶bm Z.km Bmm 1oddConsider first n even. We shall distinguish two subcases.i) Let k be an even integer. Taking into account Lemma 1, we getµ ¶ Xn 1X 1 n 1XX m/(p 1) µ n ¶nm/(p 1)cp 1/p cmmp m 0 p 1m 0 H X1p-kpp-kp 1 mm 1 or m evenp-kn 1Xm 0p 1 mp 1 m µ ¶µ¶nX1 X n n Z. H α mp m 0 mp-kHere H Z, and(p 1-mp 1 m.in. We conclude that the TheoremHere Lemma 2 has been used for K p 1is true for h 1.ii) A similar reasoning applies when k is odd. Now we have to considerseparately the contributions p 2 and p 2.µ ¶ Xn 1Xnm/(p 1)cp 1/p mm 0(13)α 01p 1 mififhm 1 or m even n 1XX1p-kpp 1 mm 0m 1 or m evenp-km/(p 1)cp 1µ ¶n mp 1 m(14) X 1 n 1Xp-kp 2pm 0p 1 mm/(p 1)cp 1µ ¶n 1X cm µ n ¶nn1 c1 Z.m22 mm6 0m evenThe first sum (i.e. the contribution when p 2) can be analyzed asin the previous case. By using the two Lemmas, we are led again to theconclusion that this contribution is an integer. The last two addenda alsogive an integer.Finally, the use of mathematical induction completes the proof for n even.The case n odd is very similar and is left to the reader.

8PIERGIULIO TEMPESTA3.2. Related results. The next theorem is a direct generalization of theBartz–Rutkowski theorem, proved in [6].Theorem 2. Under the hypotheses of Theorem 1, we have that if n is evenor n 1,µ ¶ X n/p 1cp 1n G h(15)k Bn Z;kpp 1 np-kif n 3 is odd(16)gGkn Bnµ ¶h Z.kProof. In order to obtain the result, it is sufficient to combine Theorem 1,the universal von Staudt congruence (6), (7) and Lemma 1. Strictly related to the AM theorem is an interesting problem proposed tothe author by A. Granville [19].Problem: To classify the sequences of polynomials {Pn (x)}n N satisfyingthe Almkvist–Meurman propertyµ ¶hn(17)k Pn Z, h, k, n N.kWe shall call this class of sequences the Granville class.We observe that Theorem 1 provides a tool for generating a large class ofpolynomial sequences satisfying the property (17).4. The Todd genus, the L and A genuses and related universalpolynomialsIn this section, we will show that there exists a close connection betweenthe m–sequences of Hirzebruch’s theory of genera and the Lazard formalgroup, via the universal congruences previously discussed. To this aim, wewill introduce two classes of polynomials related to specific cases of the K–genus of an almost complex manifold, that can be interpreted as particularinstances of the construction of universal Bernoulli polynomials.To fix the notation, let R be a commutative ring with identity, and R R [p1 , p2 , . . .] be the ring of polynomials in the indeterminates pi with coefficients in R. WeP assume that the product pk1 . . . pkr has weight k1 . . . kr ,so that R n 0 Rn , where Rn contains only polynomials of degree n.Consider a multiplicative sequence (or m–sequence) of polynomials {Kn } inthe indeterminates pi with K0 1 and Kn Rn , j N. As is well known, am–sequence is completely determined by the specification of the associatedcharacteristic power sequence Q(z) (see [14] for details).The polynomials {Tk (c1 , . . . , ck } of the m–sequence associated with thepower series Q(x) x/ (1 e x ) are called the Todd polynomials (here we

FORMAL GROUPS, AM CONGRUENCES AND ZETA FUNCTIONS9adopt the conventional notation x z 2 and Xi 0pi ( z)i Xcj ( x)j ·j 0 Xci xii 0for the relation among the pi ’s and the ci ’s).Now, let Mn is a compact, differentiable of class C almost complexmanifold, and let ci H 2i (Mn , Z) denote the Chern classes of the tangentGL(n, C)–bundle of Mn . The K–genus of Mn , denoted by Kn [Mn ], is a ringhomomorphism with respect to the cartesian product of two almost complexmanifolds.The genus associated with the sequence of Todd polynomials can be introduced in general for an admissible space A , i.e. a locally compact, finitedimensional space, with A j N Sj , where Sj are compact sets. Let usdenote by b a continuous GL(q, C)–bundle over A, withPChern classes ci H 2i (A, Z). The total Todd class is given by T d(b) k 0¡Tk (c1 , . . . , ck ).2When q 1 and c1 (b) t H (A, Z), we have T d(b) t/ 1 e t . Foran almost complex manifold Mn , the Todd genus is defined to be the rational number Tn [Mn ] T d(b)(2n) , where u(2n) denotes the 2n–dimensionalcomponent of u H (Mn ).A similar construction applies in relation with the theory of L–genus,which is the genus associated with the m–sequence with characteristic powerseries Q( z) x/ tanh z, and denoted by Lk (p1 , . . . , pk ). The A–genusis associated with the m–sequence with characteristic power series Q(z) 2 z/ sinh 2 z.The aim of this section is to study the algebro–geometric properties of them–sequences discussed above. We introduce now two families of polynomialsrelated to the genus of these sequences.Definition 3. We shall call λ–polynomials the family of polynomials generated by the characteristic power series associated with the L–genus:(18)Xt2k2teytλk (y) .sinh 2tk!k 0The λ–polynomials are particular cases of the Universal Bernoulli polynomials (4); also the Todd polynomials are directly related to the NörlundBernoulli polynomials. The next result relates these polynomials with theuniversal AM congruences.Proposition 1. Let λfm (x) denote the polynomials generated by(19)2t(eyt 1) X ft2k λk (y).sinh 2t4k!k 0The sequence {λfm (x)}n N is in the Granville class.

10PIERGIULIO TEMPESTATo prove it is sufficient to observe that the sequence {λfm (x)}n N satisfiesthe hypotheses of Theorem 1.The Cauchy formula relates Q(z) with sn , which is defined to be thecoefficient of pn in the polynomial Kn : (20)1 zXdlog Q(z) ( 1)j sj z j .dzj 0A similar construction can be proposed in relation with the A–genus.Definition 4. We shall define the α–polynomials to be the family of polynomials generated by the characteristic power series associated with the A–genus:X2teytt2k(21) αk (y).cosh 2t4k!k 0A completely analogous result also holds for the associated polynomials{αfm (x)}n N .Proposition 2. Let αfm (x) denote the polynomials generated by(22)t2k2t(eyt 1) Xαk (y) .cosh 2t4k!k 0The sequence {αfm (x)}n N is in the Granville class.5. Universal congruences and integer sequences5.1. Sequences of integer numbers. In this Section, both sequences ofinteger numbers and Appell sequences of polynomials with integer coefficients are constructed as a byproduct of the previous theory.Lemma 3. Consider a sequence of the form (23)X Nktt tk ,G1 (t) G2 (t)2k!k 0where G1 (t) and G2 (t) are formal group exponentials, defined as in formula(2). Assume that cn Z. Then {Nk }k N is a sequence of integers.ck 1 and Bck 2 associated with the formalProof. The Bernoulli–type numbers Bgroup exponentials G1 (t) and G2 (t), under the previous assumptions mustsatisfy the classical Clausen–von Staudt congruence for k even, as a consequence of Clarke’s universal congruence (6). For k odd these numbers areck 1 also integers or half–integers, due to (7). It follows that the difference Bck 2 for any k even is an integer and for k odd is a half–integer or an integer.BThe thesis follows from the definition of {Nk }k N in (23).

FORMAL GROUPS, AM CONGRUENCES AND ZETA FUNCTIONS115.2. Applications. a) The characteristic power series obtained as the difference between the power series associated with the L–genus and that oneassociated with the Todd genus, i.e. Q(t) t/ tanh t t/(1 e t ) is thegenerating function of the sequence(24)1, 1, 0, 1, 0, 3, 0, 17, 0, 155, . . .b) In [28], realizations of the universal Bernoulli polynomials were constructed by using the finite operator theory [22]. Here we quote two generating functions of these classes of polynomials: XBkV (x) ktext,t 3tk!e 2e2t 2et 2e t e 2tk 0(25) XB V II (x)kk 0k!tk text.e4t e3t e2t 2et e t e 2t e 3tThe sequences associated with the generating functions (25) satisfy theClausen–von Staudt congruence in the strong sense and can be used toconstruct integer sequences of numbers and polynomials. Here are reportedsome representative of generating functions of integer sequences which canbe obtained from the previous considerations.b.1)t(1 et ).2(1 et e2t )The sequence generated is 2, 6, 39, 324, 3365, 41958, . . .b.2)t (1 2 cosh t 4 cosh 2t 6 sinh t).( 6 8 cosh t)(2 cosh t cosh 2t sinh t sinh 2t 2 sinh 3t)The sequence generated is 7, 6

formal group. Their role in the theory of L{genera for multiplicative sequences is illustrated. As an application, classes of sequences of inte-ger numbers are constructed. Some congruences are also obtained for computing special values of a new class of Riemann{Hurwitz{type zeta functions. Contents 1. Introduction: formal groups 2 2. The .

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