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L-FUNCTIONS and CYCLOTOMIC UNITS 1. Introduction

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L-FUNCTIONS and CYCLOTOMIC UNITS 1. Introduction L-FUNCTIONS AND CYCLOTOMIC UNITS TOM WESTON, UNIVERSITY OF MICHIGAN 1. Introduction Let K be a number field with r1 real embeddings and r2 pairs of complex con- jugate embeddings. If ζK (s) is the Dedekind zeta function of K, then ζK (s) has a zero of order r1 + r2 1 at s = 0, and the value of the first non-zero derivative is given by the Dirichlet− class number formula: (r1+r2 1) hK RK ζK − (0) = : − wK Here RK is the regulator of K, wK is the number of roots of unity in K and hK is the class number of K. This formula is a striking connection between arithmetic and analysis, and there have been many attempts to generalize it to other L-functions: one expects that the value of a \motivic" L-function at an integer point should involve a transcendental factor, a boring rational factor, and an interesting rational factor. In the case of the Dedekind zeta function, these roles are played by RK , wK , and hK , respectively. In the general case one expects that the interesting rational factor is the order of a certain Selmer group. 2. Selmer groups and Kolyvagin systems Let us recall the definition of the Selmer group of a p-adic Galois representation. Let T be a free Zp-module with a continuous action of the absolute Galois group ¯ GK = Gal(K=K). We define the Cartier dual of T by T ∗ = HomZp (T; µp1 ) with the adjoint Galois action. Let be a Selmer structure on T . Recall that consists of choices of local Selmer structuresF F 1 1 H (Kv;T ) H (Kv;T ) F ⊆ for each place v of K; these are assumed to coincide with the unramified choice at almost all places v. Tate local duality yields a Cartier dual Selmer structure ∗ on 1 F T ∗, and the Selmer group S(K; T ∗) is defined to be the subgroup of H (K; T ∗) of 1 cocycles whose local restrictions lie in H (Kv;T ) for every place v. ∗ ∗ Kolyvagin has shown that in some situationsF one can use duality theorems in Galois cohomology to bound the order of such a Selmer group. In this section we will explain Kolyvagin's general strategy; the key step is the production of what we will call a Kolyvagin system. Date: May 4, 2001. Notes from Arizona Winter School, March 10{15, 2000. 1 2 TOM WESTON, UNIVERSITY OF MICHIGAN Fix a Selmer structure on T . If a is a fractional ideal of K, we define modified F Selmer structures on T and T ∗ by H1 (K ;T ) v - a; ( v a = F1 F H (Kv;T ) v a; j and 1 H (Kv;T ∗) v - a; ∗ = ( F ∗ Fa 0 v a: j (Here for simplicity of notation we say that v divides the fractional ideal a if v occurs with non-zero exponent in the prime factorization of a.) We will just write the Selmer groups S (K; T ) (resp. S (K; T ), resp. S (K; T ), resp. S (K; T )) ∗ ∗ a a∗ ∗ F F a F F as S(K; T ) (resp. S(K; T ∗), resp. Sa(K; T ), resp. S (K; T ∗)). For any place v we 1 1 1 will write Hs (Kv;T ) for the singular quotient H (Kv;T )=H (Kv;T ). F If M is a Zp-module, we write M _ for its dual Hom(M; Qp=Zp). Unwinding the discussion of local and global duality in Mazur's lecture, one finds that for any and a as above there is an exact sequence F 1 a (1) 0 S(K; T ) Sa(K; T ) Hs (Kv;T ) S(K; T ∗)_ S (K; T ∗)_ 0: ! ! !v ⊕a ! ! ! j This exact sequence suggests the following approaching to bounding the order of a the Selmer group S(K; T ∗): first, one bounds the order of S (K; T ∗) for “sufficiently divisible" ideals a. In practice it is often not difficult to show that one can choose a an ideal a so that S (K; T ∗) vanishes. For such an a, (1) yields an isomorphism 1 S(K; T ∗)_ = Hs (Kv;T ) = img Sa(K; T ): ∼ v⊕a j 1 In particular, if one can exhibit elements of Sa(K; T ) which fill up v aHs (Kv;T ) ⊕ j with finite index η, then one concludes that #S(K; T ∗) η. (More generally, if a j a S (K; T ∗) is non-zero but computable, one has #S(K; T ∗) η #S (K; T ∗).) An j · 1 explicitly given submodule of Sa(K; T ) which has finite index in v aHs (Kv;T ) is called a Kolyvagin system. ⊕ j a In practice, one can bound S (K; T ∗) (for sufficiently divisible a) for a very general class of Galois representations T ; it involves only the coarse structure of T . In contrast, constructing a Kolyvagin system for T is quite difficult and has only been done for certain special Galois representations. In the remainder of this paper we will study the situation in the simplest case of the Galois representation T = Zp(1). 3. Example: Ideal class groups We now set T = Zp(1), so that T ∗ = Qp=Zp. We will define the Selmer structure on T momentarily. F To exhibit a Kolyvagin system, we need some method of constructing elements of H1(K; T ) with controlled ramification. We will use the Kummer map 1 κ : K× H (K; µn) ! σ(α1=n) α σ : 7! 7! α1=n L-FUNCTIONS AND CYCLOTOMIC UNITS 3 Here α1=n is a fixed choice of nth root of α in K¯ ; changing the choice of α1=n changes κ by a coboundary. The Kummer map is compatible with changing n; in particular, 1 if we take the inverse limit over powers of p we obtain a map κ : K× H (K; T ). The same construction yields local Kummer maps ! 1 κv : K× H (Kv;T ) v ! for each v. In fact, κv induces an isomorphism 1 κv : Kv× H (Kv;T ) d ! where Kv× is the p-adic completion of Kv×. Considerd the exact sequence ordv 0 × K× Z 0 !OKv ! v −! ! with p-adic completion ordv (2) 0 × Kv× Zp 0: ! OKv ! −! ! d d1 We define the local Selmer structure H (Kv;T ) by requiring that the exact se- quence F 1 1 1 0 H (Kv;T ) H (Kv;T ) Hs (Kv;T ) 0 ! F ! ! ! corresponds to (2) under the isomorphism κv of the middle terms. (It is straight- forward to check that this choice agrees with the natural local Selmer structure if v - p.) As always we take the Cartier dual Selmer structure on T ∗. We see from the definition that for each v there is a canonical isomorphism 1 ∼= H (Kv;T ) Zp s −! fitting into a commutative diagram κ 1 1 (3) K× / H (K; T ) / Hs (Kv;T ) ord v ∼= Z / Zp 1 In particular, it follows from (3) that for any α K×, κ(α) H (K; T ) satisfies the local condition given by at every place which2 is relatively2 prime to α. In the notation introduced earlier, thisF means that κ(α) S (K; T ): 2 (α) We now use κ to produce a Kolyvagin system for T . To motivate our construc- tion, we recall that class field theory yields a canonical isomorphism (4) S(K; T ∗) ∼= A_ where A is the p-part of the ideal class group of K and A_ = Hom(A; Qp=Zp). 1 (To see this, note that H (K; T ∗) = Hom(GK ; Qp=Zp). The local conditions ∗ correspond to allowing only those homomorphisms which factor through Gal(L=KF ) for L=K unramified. Since the Galois group of the maximal unramified abelian extension of K is canonically isomorphic to the ideal class group of A, the iso- morphism (4) follows.) Furthermore, for any ideal a, the restricted Selmer group a S (K; T ∗) corresponds to the dual of the quotient of A by the subgroup generated a by the prime factors of a. In particular, S (K; T ∗) will vanish if the prime factors 4 TOM WESTON, UNIVERSITY OF MICHIGAN of a generate A. (One can also easily prove this vanishing by using the Tcheba- torev density theorem instead of (4); this approach has the advantage of making the results below somewhat less vacuous.) Write A = Z=pn1 Z Z=pnr Z ⊕ · · · ⊕ with generators l1;:::; lr (which we can take to be prime). Set a = l1 lr; as we a ··· observed above, we have S (K; T ∗) = 0. In particular, by (1) we have 1 (5) S(K; T ∗)_ = Hs (Kv;T )= img Sa(K; T ): ∼ v⊕a j We will now compute this directly, without any reference to (4). pni By definition, li is principal; choose a generator αi and set ci = κ(αi) ni 2 Sli (K; T ). Since ordli (αi) = p , by (3) we have 1 ni (6) H (Kl ;T )=(Zp img ci) = Z=p Z: s i · ∼ These elements c1; : : : ; cr form a Kolyvagin system. Indeed, let C be the Zp-span of c1; : : : ; cr in Sa(K; T ). By (6) we have 1 n1 nr Hs (Kv;T )= img C = Z=p Z Z=p Z = A: v⊕a ∼ ⊕ · · · ⊕ ∼ j Combining this with (5) yields a surjection (7) A S(K; T ∗)_: In fact, it is not difficult to see that img C = img Sa(K; T ), so that the surjection (7) is an isomorphism. Of course, all we have done is reproduced the result (4).
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