arXiv:1902.04795v1 [math.NT] 13 Feb 2019 ntrso ioac-ifrc rm seDefinition (see prime Fibonacci-Wieferich of terms in euto rebr [ Greenberg of result a o h osrcinof construction the for td the study Q ( epltscnetr.Rcnl,R rebr sdcomple used Greenberg R. Recently, conjecture. Leopoldt’s setosaeequivalent: are assertions onee yWl nhsppri h td ftehypothesis the of study the is paper his in Wall by countered equivalent: are assertions charac another give we periods, these of properties and fact 1.1. Theorem sequence prec More by numbers. Denote Fibonacci classical . the of properties to hs eid n[ in periods these rpsto 1.2. Proposition Q h aosgopo h aia pro- maximal the of group Galois the pro- [ oinof notion M-N F ( ( h anrsl fti ae stefloigterm which theorem, following the is paper this of result main The e od n phrases. and words Key 2010 ti nw htfreeypstv positive every for that known is It o h lsia ioac eune namely sequence, Fibonacci classical the For Let n √ √ 1 h field the (1) 1 h field the (1) (2) (2) ) p n IOAC EUNE N ELQUADRATIC REAL AND SEQUENCES FIBONACCI d d ,[ ], gopo rank of -group ntrso h eid fteascae ioac numbers Fibonacci associated the of periods the of terms in ) let and ) K sproi fpro oiieinteger positive a period of periodic is ahmtc ujc Classification. Subject Mathematics Abstract. oac ubr n hi eid ouopstv positive modulo periods their and numbers bonacci k p Mo88 ( eanme edand field number a be sntaFbnciWeeihpiefor prime Fibonacci-Wieferich a not is F p p p ) rtoa ubrfilshsbe nrdcdb oahd and Movahhedi by introduced been has fields number -rational ( rtoaiyo elqartcnme ed.I at egiv we fact, In fields. number quadratic real of -rationality ε 6= d ,[ ], + k ε d ( Q Q N Mo90 ,N p Let Wall 2 esuythe study We ( ( ( ( ) √ √ Let ε . eteaslt om eascaet h field the to associate We norm. absolute the be ) d . r )) d d p 2 ,weeh rvdmn rpriso hs nees n pro One integers. these of properties many proved he where ], ,adi sdfrtecntuto fnnaeinextension non-abelian of construction the for used is and ], ) ) ,where 1, + ( = ≥ G p F p is is p oolr ..]wihrltsthe relates which 4.1.5] Corollary , ≥ ai aosrpeettoswt pniae.I hs pap these In images. open with representations Galois -adic 3 n rtoa ed,Fbncinmes periods, numbers, Fibonacci fields, -rational ε F +2 p p d 3 ea d rm ubrsc that such number prime odd an be -rational, -rational, n and ) eapienme uhthat such number prime a be ( = n p ≥ p rtoaiyo elqartcfilsi em fgeneralize of terms in fields quadratic real of -rationality 0 ε AAIEBOUAZZAOUI ZAKARIAE nodpienme.Tefield The number. prime odd an h d endby defined r d 2 + h udmna ntadtecasnme ftefield the of number class the and unit fundamental the stenme fpiso ope medn of embedding complex of pairs of number the is 1. p ε 13,1M6,20G05. , 11M06 11B39, etninof -extension d ) Introduction F FIELDS n +1 F − 1 0 m k 0, = N ( h euto modulo reduction the , a Q m ( ε ( = [ ) K √ d F ) Wall b F d 1 3.3 hc surmfidoutside unramified is which ) n ,DD ali h rtt study to first the is Wall D.D. 1, = n h euso formula recursion the and 1 = . o n for , (1) p o ioac-ifrc primes). Fibonacci-Wieferich for . hoe .,[ 1.], Theorem , abelian x ∤ sl,let isely, eiaino the of terization p k ( rtoaiyo h field the of -rationality ε ( p d p L ∤ − ) ≥ fntos. -functions ( K 6= ε ecie the describes ε 0 d d . . ) k ssi obe to said is − p > d 2 ( rtoa ubrfields number -rational h p ε d 2 p eeaiainof generalization a e d hntefollowing the Then . .H se whether asked He ). Q -RATIONAL ) eafundamental a be 0 m 2 gynQagDo Quang Nguyen ( h D-R √ d ftesequence the of h following The . p d rtoaiyof -rationality Fibonacci a ) .Uigthis Using ]. p p Fi- d satisfying s -rationality rtoa if -rational p lmen- blem safree a is K Q The . rwe er ( √ 5) 2 ZAKARIAE BOUAZZAOUI the equality k(p) = k(p2) is possible. This question is still open with strong numerical evi- dence [E-J]. By Proposition 1.2, it is equivalent to whether the number field Q(√5) is not p-rational for some p. It is generalized to Fibonacci sequences F (a,b) where for some sequences we have an affirmative answer, for example the Fibonacci sequence F (2,−1) gives that k(13) = k(132) and k(31) = k(312), which means that the field Q(√2) is not p-rational for p = 13, 31. Under the light of the above characterization of the p-rationality, the conjecture of G. Gras [Gr, Conjecture 7.9] on the p-rationality of real quadratic fields, it is suggested that for almost all primes p we have k(p) = k(p2). 6

2. p-rational fields In this section we give a characterization of the p-rationality of real quadratic fields in terms of values of the associated L-functions at odd negative integers. In fact, the p-rationality of totally real abelian number fields K is intimately related to special values of the associated zeta functions ζK. The relation is as follows. For any finite set Σ of primes of K, we denote by GΣ(K) the Galois group of the maximal pro-p-extension of K which is unramified outside Σ. Let S be the finite set of primes Sp S , where S is the set of infinite primes of K ∪ ∞ ∞ and Sp is the primes above p in K. It is known that the group GSp (K) is a free pro-p-group 2 on r2 + 1 generators if and only if the second Galois cohomology group H (GSp (K), Z/pZ) vanishes. This vanishing is related to special values of the zeta function ζK via the conjecture of Lichtenbaum. More precisely, let S be the Galois group of the maximal extension of K which is unramified outside S. TheG main conjecture of Iwasawa theory (now a theorem of 2 Wiles [W90]) relates the order of the group H ( S, Zp(i)), for even integers i, to the p-adic G valuation of ζK(1 i) by the p-adic equivalence: − 2 wi(K)ζK(1 i) p H ( S, Zp(i)) , (1) − ∼ | G | 0 where for any integer i, wi(F ) is the order of the group H (GF , Qp/Zp(i)), and p means 2 ∼ having the same p-adic valuation, see e.g [Kol]. Moreover, the group H ( S, Zp(i)) vanishes 2 G if and only if H ( S, Z/pZ(i)) vanishes. Let µp be the group of p-th unity. The periodicity 2 G of the groups H ( S, Z/pZ(i)) modulo δ = [K(µp) : K] gives that G 2 2 H ( S, Z/pZ(i)) = H ( S, Z/pZ(i + jδ)), G ∼ G 2 for any integer j. In addition, since p is odd, the vanishing of the group H ( S, Z/pZ(i)) 2 G is equivalent to the vanishing of the group H (GSp (K), Z/pZ(i)). Number fields such that 2 H (GSp (K), Z/pZ(i)) = 0 are called (p, i)-regular [A]. In particular, the field K is p-rational if and only if wp−1(K)ζK(2 p) p 1. This leads to the following characterization of the p-rationality of totaly real number− ∼ fields. Proposition 2.1. Let p be an odd prime number which is unramified in an abelian totally real number field K. Then we have the equivalence K is p-rational L(2 p, χ) is a p-adic unit, (2) ⇔ − where χ is ranging over the set of irreducible characters of Gal(K/Q).

Proof. First, the zeta function ζK decomposes in the following way:

ζK(2 p)= ζQ(2 p) Y L(2 p, χ). − − × χ6=1 −

Second, it is known that ζQ(2 p) is of p-adic valuation 1 and that wp (K) has p-adic − − −1 valuation 1, giving that wp (K)ζQ(2 p) p 1. Then from (1) we obtain the formula −1 − ∼ 2 Z Y L(2 p, χ) p H ( S, p(p 1)) . χ6=1 − ∼ | G − | FIBONACCI SEQUENCES AND REAL QUADRATIC p-RATIONALFIELDS 3

Since, for every character χ = 1, the value L(2 p, χ) is a p-integers [Wa, Corollary 5.13], 2 6 − we have H ( S, Zp(p 1)) = 0 if and only if for every χ = 1, L(2 p, χ) is a p-adic unit. G − 2 6 − Furthermore, the vanishing of the group H ( S, Zp(p 1)) is equivalent to the vanishing 2 G − of the group H ( S, Z/pZ(p 1)), which turns out to be equivalent to the vanishing of 2 G − H (GSp (K), Z/pZ) (by the above mentioned periodicity statement). This last vanishing occurs exactly when the field K is p-rational.  In the particular case of a real quadratic field K = Q(√d), we have the decomposition d ζK(2 p)= ζQ(2 p)L(2 p, ( )), − − − . d √ where ( . ) is the quadratic character associated to the field K = Q( d). Corollary 2.2. For every odd prime number p ∤ d, we have the equivalence d Q(√d) is p-rational L(2 p, ( )) 0 (mod p). (3) ⇔ − . 6≡ 

Remark 2.3. The properties of special values of p-adic L-functions tells us that the p- rationality is related to the class number and the p-adic regulator. More precisely, let K be a totally real number field of degree g. Under the Leopoldt conjecture, class field theory gives ab r2+1 ab that GS (K) = Z K , where K is the Zp-torsion of GS (K) . Then the field K is p ∼ p × T T p p-rational precisely when K = 0 [M-N, Théor`eme et Definition 1.2]. Moreover, the order of T K satisfies T −1 Rp(K).hK K p w(K(µp)) Y(1 N(v) ). , (4) |T | ∼ v|p − dk q| | ([Coa, app]), where hK is the class number, Rp(K) is the p-adic regulator, N(v) is the absolute norm of v, w(K(µp)) = µ(K(µp)) the number of roots of unity of K(µp) and dK is the discriminant of the number| field K.| Hence for every odd prime number p such that (p,dKhK )=1, the field K fails to be p-rational if and only if vp(Rp(K)) >g 1. − Under the light of Remark 2.3, for a real quadratic field Q(√d) we have the equivalence

2 Q(√d) is p-rational Rp(Q(√d)) 0 (mod p ). (5) ⇔ 6≡ Recall that Rp(Q(√d)) = logp (εd), where εd is a fundamental unit of K and logp is the p-adic logarithm.

3. Fibonacci number The classical Fibonacci sequence is an interesting linear recurrence sequence, in part be- cause of its applications in several areas of sciences. Here we consider a class of linear recurrence sequences which arise from real quadratic fields and that we use for the study of the p-rationality of these fields. As mentioned in the introduction, Greenberg [G, Corollary 4.1.5.] used classical Fibonacci numbers to give a characterization for the p-rationality of the field Q(√5). In this paper we give a generalization of this result to any real quadratic field. The Fibonacci numbers associated to real quadratic fields are given as follows. Let d> 0 be a fundamental discriminant and let hd, εd be respectively the class number and the funda- mental unit of the field Q(√d) with ring of integers d. We denote by εd the conjugate of O(εd+εd,N(εd)) εd and N(.) the absolute norm. Define the sequence F =(Fn)n such that F0 = 0, F1 = 1 and Fn =(εd + εd)Fn N(εd)Fn. +2 +1 − 4 ZAKARIAE BOUAZZAOUI

The Binet formula [D-R, page 173] gives that n n εd εd Fn = − , n 0. εd εd ∀ ≥ − We establish a relation between Fibonacci numbers and the p-adic regulator which allows us to prove the main result. Definition 3.1. Let a be a non trivial element of the ring of integers of the field Q(√d) such that (a, p)=1. Then the prime p is said to be Wieferich of basis a if the following congruence holds: r ap −1 1 0 (mod p2), − ≡ where r is the residue degree of p in the quadratic field Q(√d). Otherwise, the prime number p is said to be non-Wieferich of basis a. We have the following equality

r r 1 r log ((εp −1 1)+1)=(εp −1 1) (εp −1 1)2 + ... p d − d − − 2 d − where logp is the p-adic logarithm and as before r is the residue degree of p in the quadratic × r field Q(√d). Since Rp = log (εd) and the group ( d/p d) is cyclic of order p 1, where p O O − d is the ring of integers of Q(√d), we obtain the equivalences O r p −1 2 2 εd 1 0 (mod p ) Rp p (mod p ), − 6≡ ⇔ ≡ 2 (6) Rp 0 (mod p ). ⇔ 6≡ Then combining this last equivalence with the equivalence (5) we obtain

Proposition 3.2. Let p be an odd prime number such that p ∤ dhd. Then the field Q(√d) is p-rational if and only if p is a non- of basis εd.  Very little is known about these primes and it is conjectured that the set of Wieferich primes is of density zero [Si]. In the following we are interested with the set of Fibonacci- Wieferich primes defined as follows. Definition 3.3. A prime number p is said to be a Fibonacci-Wieferich prime for the field Q(√d) if 2 Fp−( d ) 0 (mod p ), p ≡ d √ where ( . ) is the Legendre symbol associated to the quadratic field Q( d). We give the main result of this section which describe the p-rationality in terms of Fibonacci- Wieferich primes.

2 Theorem 3.4. Let p 5 be a prime number such that p ∤ (εd εd) hd. Then the following assertions are equivalent:≥ − (1) the field Q(√d) is p-rational, (2) p is not a Fibonacci-Wieferich prime for Q(√d). Proof. Using the equivalence (6), it suffices to prove that: pr−1 2 2 εd 1 0 (mod p ) Fp−( d ) 0 (mod p ). (7) − 6≡ ⇔ p 6≡ Let Qp(εd) be the residue class r εp −1 1 d − (mod p). p FIBONACCI SEQUENCES AND REAL QUADRATIC p-RATIONALFIELDS 5

A prime number p satisfying Qp(εd) 0 (mod p) is non-Wieferich of basis εd. d 6≡ First suppose that ( p ) = 1. Then r = 1 and

p−1 εd 1 Qp(εd) − (mod p). (8) ≡ p

The Binet formula gives that

p−1 p−1 1−p (p−1) (p−1) (εd εd)Fp = ε ε = ε (ε 1)(ε + 1). − −1 d − d d d − d

1−p (p−1) Since εd is a unit and p ∤ (εd εd), we have εd (εd + 1)(εd εd) 0 (mod p). Hence we obtain the equivalence − − 6≡

2 Qp(εd) 0 (mod p) Fp 0 (mod p ). (9) 6≡ ⇔ −1 6≡

Second, suppose that the prime number p is inert in the field Q(√d). Then we have

p2−1 εd 1 Qp(εd) − (mod p). (10) ≡ p

The Galois group of Q(√d)/Q is generated by an element σ of order two such that σ(εd)= εd. × 2 p+1 Since the group ( d/p d) is cyclic of order p 1, we have εd x (mod p) for some x Z. p+1 O O − ≡ ∈ Hence ε x (mod p) and Fp 0 (mod p). Note that since d ≡ +1 ≡

−1 p+1 2(p+1) Fp =(εd εd) ε (ε 1), +1 − d d − we have

ε2(p+1) 1 0 (mod p). d − ≡ Moreover,

− 2 p−3 p 1 1 p −1 1 2(p−1) 1 2(p+1) 2(p+1) 2 Qp(εd)= (ε 1) = ((ε ) 2 1) = (ε 1)(ε + ... + 1). p d − p d − p d − d

Since x2 1 (mod p), we then obtain the congruence ≡

1 p 1 2(p+1) Qp(εd) − (ε 1) (mod p). ≡ p 2 d −

Hence we have the equivalence

2 Qp(εd) 0 (mod p) Fp 0 (mod p ). (11) 6≡ ⇔ +1 6≡

Then in all cases we obtain that the field Q(√d) is p-rational precisely when p is not a Fibonacci-Wieferich prime.  6 ZAKARIAE BOUAZZAOUI

Using this characterization of the p-rationality on pariGP, we obtain some numerical evi- dence for the primes p for which a given real quadratic number field is not p-rational. Discriminant Primes< 109 5 8 13, 31, 1546463 12 103 13 241 17 21 46179311 24 7, 523 28 29 3, 11 33 29, 37, 6713797 37 7, 89, 257, 631 40 191, 643, 134339, 25233137 41 29, 53, 7211 44 53 5 56 6707879, 93140353 57 59, 28927, 1726079, 7480159 60 181, 1039, 2917, 2401457 61 65 1327, 8831, 569831 69 5, 17, 52469057 73 5, 7, 41, 3947, 6079 76 79, 1271731, 13599893, 31352389 77 3, 418270987 85 3, 204520559 88 73, 409, 43, 28477 89 5, 7, 13, 59 92 7, 733 93 13 97 17, 3331

With the help of these results and further computations, we could construct examples of multi-quadratic p-rational fields. The first example is the field K1 = Q(√2, √3, √5, √7, √11, √ 1), which is p-rational for all primes 100

4 n 2t−1 3. Then there exists a continuous homomorphism ≤ ≤ − ρ : GQ GLn(Zp), → with open image. Based on the above computations and Proposition 3.5, we have the following corollary. Corollary 3.6. For any integer 4 n 25 3 and any prime 100

p−1 d − 2 d ( p ) 1 1 hdF d /p 2( )N 2 βp(i) (mod p), (13) p−( p ) X ≡ − p i=1 i √ 1 where hd is the class number of the field Q( d), and i is the inverse of i modulo p. 8 ZAKARIAE BOUAZZAOUI

An interesting problem of combinatorics and additive number theory is the study of sums of reciprocals in finite fields. Here we are concerned with the linear combinations d X βp(i)αp(i) (mod p), i=1 where 1 αp(i)= X . p−1 k 1 k 2 k ≤i (mod≤ d) ≡ We have the following description of the p-rationality of the field Q(√d). 2 Theorem 4.2. If p does not divides (εd εd) , then − d Q √ ( d) is p-rational X βp(i)αp(i) 0 (mod p). (14) ⇔ i=1 6≡

Proof. It is known that Fp−( d ) 0 (mod p) [Wi, page 431 formula (1.2)]. Then by Theorem p ≡ 1.1, the field Q(√d) is p-rational if and only if hdFp−( d )/p 0 (mod p). Using Theorem 4.1, p 6≡ this occurs precisely when p−1 d − 2 d ( p ) 1 1 2 2( )N X βp(i) 0 (mod p). − p i=1 i 6≡

( d )−1 d p 2 √ Since p is an odd prime number and the term 2( p )N equals 1 or 2, the field Q( d) is p-rational if and only if p−1 2 1 X βp(i) 0 (mod p). (15) i=1 i 6≡ Recall that for any integer i, i is the least non-negative residue class of i modulo d. Hence by definition we have i + kd{ }= i for any integer k 0 and the following equality holds for any integer i 1,{ ..., d : } { } ≥ ∈{ } βp(i + kd)= βp(i). 1 1 Then the terms and of (15) have the same coefficient βp(i). For i 1, ..., d regrouping i i+kd ∈{ } the integers 1 such that j lies in the equivalence class of i modulo d and j 1, ..., p−1 , the j ∈{ 2 } sum in (15) can be written p−1 2 1 d X βp(i) = X βp(i)αp(i). i=1 i i=1 d Then the field Q(√d) is p-rational if and only if βp(i)αp(i) 0 (mod p).  Pi=1 6≡ As a consequence we have the following characterization of the p-rationality of the field Q(√5). Corollary 4.3. For every prime p 1 (mod 5), the field Q(√5) is p-rational if and only if ≡ αp(1) + αp(2) αp(4)+2αp(5) 0 (mod p). (16) − 6≡ 5 5 Proof. Let ℓ be a prime number, then ( ℓ ) = 1 if and only if ℓ 1, 4 (mod 5), and ( ℓ )= 1 if and only if ℓ 2, 3 (mod 5). Since p 1 (mod 5), we have≡ for i 1, ..., 5 , − ≡ ≡ ∈{ } i−1 5 βp(i)= X( ), j=1 j such that βp(1) = 1, βp(2) = 1, βp(3) = 0, βp(4) = 1 and βp(5) = 2.  − FIBONACCI SEQUENCES AND REAL QUADRATIC p-RATIONALFIELDS 9

If we fix the prime number p, we obtain a description of the set of fundamental discriminants d for which the field Q(√d) is p-rational. For the particular cases p = 3 and p = 5 we obtain the following proposition.

2 Proposition 4.4. Let d be a fundamental discriminant such that 3, 5 ∤ (εd εd) then we have the equivalence: − (1) Q(√d) is 3-rational β (1) 0 (mod 3), ⇔ 3 6≡ (2) Q(√d) is 5-rational β (1) 2β (2) (mod 5). ⇔ 5 6≡ 5 Proof. By Theorem 4.1, we have for p = 3 the equality

3−1 2 1 X β3(i) = β3(1), i=1 i and for = 5, 5−1 2 1 X β5(i) = β5(1)+3β5(2). i=1 i Then the equivalences in (1) and (2) follow from Theorem 4.2.  In general, given an odd prime number p, it is not known wether there exist infinitely many real quadratic fields which are p-rational. This is known for the cases of p = 3 which is proved by Dongho Byeon in [B, Theorem 1], and the other case is p = 5 (see [A-B]). Both cases are proved using divisibility properties of Fourier coefficients of half-integer weight modular forms.

Acknowledgement. I would like to thank my advisor J.Assim for his guidance and patience during the preparation of this paper. Many thanks goes to H.Cohen and B.Abombert for their help on pariGP computations during the Atelier pariGP in Besan¸con.

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[P] PARI Group. Bordeaux. PARI/GP, version 2.10.0, (2017). [R] Renault, M. The period, rank, and order of the (a,b)-Fibonacci sequence mod m. Mathematics Mag- azine 86.5: 372-380 (2013). [Si] Silverman, J. H. Wieferich’s criterion and the abc-conjecture. Journal of Number Theory, 30(2). 226-237 (1988). [Wall] Wall, D. D. Fibonacci series modulo m, Amer. Math. Monthly 67. 525-532 (1960). [Wa] Washington, L. Introduction to Cyclotomic Fields (2nd ed.), Graduate Texts in Math. 83, Springer- Verlag (1997). [W90] Wiles, A. The Iwasawa conjecture for totally real fields. Annals of mathematics, 131(3), 493-540 (1990). [Wi] Williams H.C. Some formulas concerning the fundamental unit of a real quadratic field. Discrete mathematics. 17;92(1-3):431-40 (1991).

(1) Université Moulay Ismaïl, Département de mathématiques, Faculté des sciences de Meknès, B.P. 11201 Zitoune, Meknès, Maroc. E-mail address: [email protected]