arxiv:1003.4813v3 [math.gm] 16 Jan 2017 Dedekind zeta function and BDS conjecture Abstract. Keywords: Dang Vu Giang Hanoi Institute of Mathematics Vietnam Academy of Science and Technology 18 Hoang Quoc Viet, 10307 Hanoi, Vietnam e-mail: dangvugiang@yahoo.com November 13, 2018 AMS subject classification: 14R15 1 Introduction Let K be an algebraic number field with degree n = [K : Q]. Dedekind zeta function of K is first defined for complex numbers s with R(s) > 1 by convergent Dirichlet series ζ K (s) = a a s where a ranges through the non-zero ideals of the algebraic integers ring O K of K and a denotes the absolute norm of a which is the cardinality of quotient ring O K /a. It is well known that ab = a b. for every ideals a and 1
b of O K. Clearly, if K = Q, the Dedekind zeta function ζ Q of Q becomes the classical Riemann zeta function ζ. Moreover, for K = Q(e 2πi/m ), ζ Q(exp 2πi m )(s) = ζ(s) χ L(s,χ) p m(1 p s ) 1. Here, χ denotes a non-principal Dirichlet character modulo m. The Euler product of Dedekind zeta function of K is a product over all the prime ideals p of O K 1 ζ K (s) = p (1 p s ) = a µ K (a), for Re(s) > 1. a s This is the expression in analytic terms of uniqueness of the prime factorization of an ideal a in O K (Dedekind domain). The Möbius function µ K in the field K is defined by 1 if a = O K µ K (a) = ( 1) r if a = p 1 p 2 p r. 0 if p 2 a For example, µ K (5) = 1 for K = Q(i), but µ Q (5) = µ(5) = 1. For R(s) > 1, ζ K (s) is given by a convergent product of infinite non-zero numbers, so ζ K (s) is non-zero in this region. Erich Hecke first proved that ζ K (s) has an analytic continuation to the complex plane as a meromorphic function, having a simple pole only at s = 1. Let K denote the discriminant of K, r 1 and r 2 denote the number of real and complex places of K, and Γ R (s) = π s/2 Γ(s/2), Γ C (s) = 2(2π) s Γ(s). Λ K (s) = K s/2 Γ R (s) r 1 Γ C (s) r 2 ζ K (s) We have the functional equation Λ K (s) = Λ K (1 s) Ξ K (s) = 1 2 (s2 + 1 4 )Λ K( 1 2 +is). Ξ K ( s) = Ξ K (s). Moreover, lim (s 1)ζ π K(s) = 2r1+r2 r2 h(k)r(k) s 1 w(k). K In the virtue of this functional equation, it is also conjectured that the complex roots of ζ K (s) are on the critical line Rs = 1/2. This is called the 2
extended Riemann hypothesis. If this hypothesis is true then every complex root of Riemann zeta function ζ and Dirichlet L function L(χ) is on the line Rs = 1/2. The values of the Dedekind zeta function at integers encode important arithmetic data of the field K. For example, the class number h(k) of K relates the residue at s = 1, the regulator R(K) of K, the number w(k) of unity roots in K, the discriminant of K, and the number of real and complex places of K. From the functional equation and the fact that Γ is infinite at all integers less than or equal to zero, it follows that ζ K (s) vanishes at all negative even integers. It even vanishes at all negative odd integers unless K is totally real number field. In the totally real case, Carl Ludwig Siegel showed that ζ K (s) is a non-zero rational number at negative odd integers. Stephen Lichtenbaum conjectured specific values for these rational numbers in terms of the algebraic K theory of K. Another example is at s = 0 where ζ K (s) has a zero with order r equal to the rank the unit group of O K and the leading term given by lim s 0 s r ζ K (s) = h(k)r(k). w(k) Two fields are called arithmetically equivalent if they have the same Dedekind zeta function. There are examples of non-isomorphic fields that are arithmetically equivalent. In particular some of these fields have different class numbers, so the Dedekind zeta function of a number field does not determine its class number. Now note that lnζ K (s) = 1 n p ns, n=1 lnζ(s)+lnζ(s 1) = For an elliptic curve E over Q we define p n=1 p p n +1 n p ns. lnl(e,s) = n=1 p 2N #E(F p n) n p ns We have #E(F p n) = p n +1 if E denotes the elliptic curves y 2 = x(x N)(x+ N) and p = 4k +3. 3
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