The Rubin–stark Conjecture for Imaginary Abelian Fields of Odd Prime Power Conductor

We build upon ideas developed in [9], as well as results of Greither on a strong form of Brumer’s Conjecture ([2]–[4]), and prove Rubin’s integral version of Stark’s Conjecture for imaginary abelian extensions of Q of odd prime power conductor. INTRODUCTION In the present paper, we prove Rubin’s integral version (Conjecture B, [10], §2.1) of Stark’s general conjecture (see Conjecture 5.1 in [12], Chap. I), for abelian extensions of type K/Q, where K is an abelian, imaginary number field of odd prime power conductor l. These extensions belong to the class of “nice” CM extensions of totally real number fields, introduced by Greither in [4]. In [9], we used results obtained by Greither in [4] on the “odd part” of a strong form of Brumer’s Conjecture, and settled Rubin’s Conjecture up to a power of 2, for general “nice” extensions. In this paper, we restrict ourselves to the subclass of “nice” extensions K/Q described above and, by using ideas developed in [9], as well as results of Greither ([2] and [3]) on the 2–part of the strong Brumer conjecture, we completely settle Rubin’s Conjecture in this case. Unlike in [9], where we attack the “odd” part of Rubin’s Conjecture “one prime at a time”, all the arguments in this paper are global in nature. This is made possible by the crucial observation that, for general abelian CM extensions of totally real number fields K/k, one can state a conjecture equivalent to Rubin’s, over the ring Z[Gal(K/k)]/(1 + j), rather than Z[Gal(K/k)] itself (see Conjecture B− in §2 below), where j is the complex conjugation automorphism of K. As it turns out, for K/Q as above, many arithmetically meaningful Z[Gal(K/Q)] –modules which, in general, are not of finite projective dimension over Z[Gal(K/Q)], have projective dimension at most 1 over Z[Gal(K/Q)]/(1 + j), after tensoring with Z[Gal(K/Q)]/(1+ j). We take full advantage of this fact and prove Conjecture B− in the context described above, and conclude that the equivalent Conjecture B of Rubin is also true. The paper is organized as follows. In §1, we introduce the notations and main definitions, state Rubin’s Conjecture B and list some of its functoriality properties. In §2, we state the equivalent Conjecture B− for general abelian CM extensions of totally real number fields. In §3, we describe results of Greither on the strong Brumer Conjecture which are of relevance in this context, and draw several conclusions which are used in the subsequent sections. In §4, we prove Conjecture B− for 1991 Mathematics Subject Classification. 11R42, 11R58, 11R27. Research on this project was partially supported by NSF grants DMS–9801267 and DMS–0196340 1 2 CRISTIAN D. POPESCU extensions K/Q as described above, under certain additional hypotheses. In §5, we use the results of §4 and some functoriality properties stated in §1 to prove Rubin’s Conjecture B for the class of extensions K/Q under consideration. Acknowledgements. We would like to thank Cornelius Greither for helpful conversations and especially for sharing his results with us prior to publication. 1. PRELIMINARY CONSIDERATIONS 1.1 Notation. In what follows, Z(p) will denote the localization of Z at its prime ideal pZ, for all prime numbers p. Let us assume that M is an arbitrary Z– module. Then M̃ , will denote the quotient of M by its submodule of Z–torsion points. If A is an arbitrary (commutative) ring, AM := A ⊗Z M . In particular, for any prime number p, M(p) := Z(p)M . If G is a finite, abelian group, then Ĝ denotes the set of its irreducible, complex–valued characters. For any χ ∈ Ĝ, eχ := 1/|G| ∑ σ∈G χ(σ ) · σ is the idempotent associated to χ in the group–ring C [G]. Also, if M is a Z [G]–module, ∧GM := ∧ r Z[G]M . Let us assume that K/k is a finite, abelian extension of number fields, of Galois group G := Gal(K/k). Let S be a finite set of primes in k, which contains the infinite primes as well as the ones which ramify in K/k, and let SK be the set of primes in K, sitting above primes in S. Then, US and AS will denote respectively the group of units and the ideal–class group of the ring of SK–integers OS of K. If T is an additional finite, non–empty set of primes in k, disjoint from S, then US,T denotes the subgroup of US consisting of S–units congruent to 1 modulo every prime in TK , and AS,T denotes the (S, T )–ideal-class group of K, as defined in [10],§1.1. For a finite prime w in K, we denote by K(w) its residue field. Then, as pointed out in [10], §1.1, we have an exact–sequence of Z [G]–modules (0) 0 −→ US,T −→ US resT −−→ ∆T −→ AS,T −→ AS −→ 0 , where ∆T := ⊕w∈TKK(w) , and resT (x) := (xmodw; w ∈ TK), for all x ∈ US . As in [10], to any S, T and χ as above, one can associate a meromorphic L–function LS(s, χ) of complex variable s, holomorphic away from s = 1, and an everywhere holomorphic L–function LS,T (s, χ) := ∏ v∈T (1 − χ(σ −1 v )Nv ) · LS(s, χ), where Nv = |k(v)|, and σv is the Frobenius morphism associated to v in G, for all primes v in T . As shown in [12], Chapitre I, §3, the orders of vanishing at s = 0 of these L–functions, rχ,S := ords=0LS(s, χ) = ords=0LS,T (s, χ), are given by (1) rχ,S := { card{v ∈ S |χ|Gv = 1Gv}, if χ 6= 1G card(S)− 1, if χ = 1G , where Gv is the decomposition group of v relative to K/k. Now, we combine the L–functions above, to obtain the associated Stickelberger functions