On Class Numbers of Real Quadratic Fields with Certain Fundamental Discriminants

EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 8, No. 4, 2015, 526-529 ISSN 1307-5543 – www.ejpam.com

1, 2 Ayten Pekin ∗, Aydın Carus 1 Department of Mathematics, Faculty of Sciences, Istanbul University, Istanbul, Turkey 2 Department of Computer Engineering, Faculty of Engineering, Trakya University, Edirne, Turkey

Abstract. Let N denote the sets of positive integers and D N be square free, and let χ , h = h(D) ∈ D denote the non-trivial Dirichlet character, the class number of the real quadratic field K = Q(pD), respectively. Ono proved the theorem in [2] by applying Sturm’s Theorem on the congruence of modular form to Cohen’s half integral weight modular forms. Later, Dongho Byeon proved a theorem and corollary in [1] by refining Ono’s methods. In this paper, we will give a theorem for certain real quadratic fields by considering above mentioned

studies. To do this, we shall obtain an upper bound different from current bounds for L(1, χD) and use Dirichlet’s class number formula. 2010 Mathematics Subject Classiﬁcations: 11R29 Key Words and Phrases: Class number, Real quadratic number ﬁeld

1. Introduction

Let Zp, N, Q denote the the ring of p-adic integers, positive integers and rational numbers, respectively. Let R (D) denote the p-adic regulator of K, . denote the usual multiplicative p | |p p-adic valuation normalized p = 1 , and let L(s, χ ) denote the L-function attached to χ . | |p p D D Throughout D N will be assumed square free, K = Q(pD ) will denote the real quadratic ∈ ﬁeld, and the class number of K will be denoted by h = h(D). Let ∆ denote the discriminant,

ǫD the fundamental unit of K.

2. Preliminaries

Theorem 1 ([2]). Let p > 3 be prime. If there is a fundamental discriminant D0 coprime to p for which

∗Corresponding author.

Email addresses: [email protected] (A. Pekin), [email protected] (A. Carus)

A. Pekin, A. Carus / Eur. J. Pure Appl. Math, 8 (2015), 526-529 527

p 1 (i) ( 1) −2 D > 0 − 0 p 1 (ii) B − , χ = 1 | 2 D0 | then

Rp(D) pX # 0 < D < X h(D) 0 (mod p), χD(p)= 0, = 1 p . | 6≡ | pD |p ≫ logX § ª p 1 p 1 Here B( − , χ ) is the − st generalized Bernoulli number with character χ . 2 D0 2 D0 D. Byeon given in [1] the following theorems by reﬁning Ono’s theorem above mentioned for any prime p > 3.

Theorem 2. Let p > 3 be prime.

(a) If p 1 (mod 4), then the fundamental discriminant D > 0 of the real quadratic field ≡ 0 Q( p 2) satisfies the conditions (i) and (ii). − (b) If p 3 (mod 4), then the fundamental discriminant D < 0 of the real quadratic field ≡ 0 Q( (p 4) satisfies the conditions (i) and (ii). − − Theoremp 3. Let p > 3 be prime. Then

1 pX # 0 < D < X h(D) 0 (mod p), χ (p)= δ, R (D) = . | 6≡ D | p |p p ≫p logX § ª 3. Main Theorem

Main Theorem. Let p > 3 be a prime. If p 3 (mod 4), then the fundamental discriminant ≡ D > 0 of the real quadratic ﬁelds K = Q( p2 4) and K = Q( p2 2) satisﬁes the conditions 0 − − (i) and (ii). p p In order to prove the main theorem we need the following lemmas.

Lemma 1. Assume D is a prime.

(i) If D 1 (mod 4), we have ≡ pD 1 if D = n2 4, (n Z), ǫ > k ∓ k ∓ ∈ D p4D 1 otherwise, ¨k ∓ k where ” ” represents great value function of a real number. k∗k (ii) If D 3 (mod 4), we have ≡ 2D 1 if D = n2 2, ǫ > ∓ ∓ D 8D 1 otherwise. ¨ ∓ A. Pekin, A. Carus / Eur. J. Pure Appl. Math, 8 (2015), 526-529 528

t+upD p Proof. (i) Let ǫD = 2 > 1 be the fundamental unit of K = Q( D). Since ǫD is equal to the fundamental solution of the Pell’s equation x2 Dy2 = 4, then we can write − ∓ 2 D 1 t + upD 1 2 Du2 1 ∓ if u = 1, 2 2 p p2 ǫD =( ) = ( Du 4 + u D) > ∓ 4D 1 2 4 ∓ p ≥ ∓ if u > 1. 2 ( p2 p and pD 1 if D = n2 4 ǫ > k ∓ k ∓ D p4D 1 otherwise. ¨k ∓ k p (ii) If D 3 (mod 4), then we have ǫ2 =( t+u D )2 from the least positive integer solution ≡ D 2 (x, y)=(t, u) of Pell’s equation x2 Dy2 = 1 [4]. Similarly, we can write − ∓ 2D 1 if D = n2 2 for some odd integer n, ǫ > ∓ ∓ D 8D 1 in the other cases. ¨ ∓

Lemma 2. Let D N be square free, then h(D) < pD. ∈ In order to prove this we need the following Lemma [3].

Lemma 3. Let γ be Euler’s constant, then

1 4 (log∆+ 2 + γ logπ) if 2 ∆, L(1, χD) − | | |≤ 1 (log∆+ 2 + γ log4π) otherwise. ¨ 2 − Proof. [Lemma 2] By Dirichlet’s class number formula, we have

p∆ h(D)= L(1, χD) 2logǫD | | where ∆ is a fundamental discriminant of a quadratic ﬁeld deﬁned by

4D if D 2, 3 (mod 4), ∆= ≡ D if D 1 (mod 4). ¨ ≡ First, we consider the case D 1 (mod 4) and D = n2 4. Thus, by the upper bound for ≡ ∓ L(1, χD) in Lemma 3, and from Lemma 1 we have that

pD(logD + 1, 478) pD(logD + 1, 478) h(D) < = < pD, (D > 5). 4log pD 1 2log (D 1) k ∓ k k ∓ k pD(logD+1,478) Moreover, we can write h(D) 2log(D 1) where " x " is the greatest integer less ≤ k ∓ k k k than or equal to x. It is also h(D) < pD for D = n2 4. 6 ∓ REFERENCES 529

Now, we consider the case D 3 (mod 4) and D = n2 2 (n Z is odd ). Similarly, by ≡ ∓ ∈ applying Lemmas 1, 3 and using class number formula, we get

pD(log4D + 1, 478) pD(log4D + 1, 478) h(D) < < pD and h(D) 2log(2D 1) ≤k 2log(2D 1) k ∓ ∓ It is also true for D = n2 2. 6 ∓

4. Proof of Main Theorem

Specially, if we write Ds depend on prime p in the forms of D = p2 4, D = p2 2 we can − − immediately prove that h(D) < p for the class numbers of real quadratic ﬁelds K = Q( p2 4), K = Q( p2 2) from Lemma 2. Therefore we have h(D) 0 (mod p) and − R (D) − 6≡ it is clear that p = 1 for above mentioned real quadratic ﬁelds. p | pD |p p

References

[1] D. Byeon. Existence of certain fundamental discriminants and class numbers of real quadratic ﬁelds. Journal of Number Theory, 98(2):432 – 437, 2003.

[2] O. Ken. Indivisibility of class numbers of real quadratic ﬁelds. Compositio Mathematica, 119(1):1–11, 1999.

[3] S. Louboutin. Majorations explicites de l(1, χ) (troisième partie). Comptes Rendus de | | l’Académie des Sciences - Series I - Mathematics, 332(2):95 – 98, 2001.

[4] R.A Mollin. Diophantine equations and class numbers. Journal of Number Theory, 24(1):7 – 19, 1986.