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Home , Cyclotomic character, Euler system, Selmer group

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 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− :

(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 .

2. Selmer groups and Kolyvagin systems Let us recall the definition of the of a p-adic Galois representation. Let T be a free Zp-module with a continuous action of the absolute ¯ GK = Gal(K/K). We define the Cartier dual of T by T ∗ = HomZp (T, µp∞ ) 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 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 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; | and 1 H (Kv,T ∗) v - a; ∗ = ( F ∗ Fa 0 v a. | (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 → → → | 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  | 1 In particular, if one can exhibit elements of Sa(K,T ) which fill up v aHs (Kv,T ) ⊕ | with finite index η, then one concludes that #S(K,T ∗) η. (More generally, if a | a S (K,T ∗) is non-zero but computable, one has #S(K,T ∗) η #S (K,T ∗).) An | · 1 explicitly given submodule of Sa(K,T ) which has finite index in v aHs (Kv,T ) is called a Kolyvagin system. ⊕ | 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 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 which∈ is relatively∈ prime to α. In the notation introduced earlier, thisF means that κ(α) S (K,T ). ∈ (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 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 | 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 ∈ 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 ∼ ⊕ · · · ⊕ ∼ | 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). Nevertheless, the point of this was to illustrate the basic idea of a Kolyvagin system: one uses some map (such as the Kummer map) to transform “motivic objects” (such as elements of K×) into cohomology classes. If one can produce enough motivic objects with controlled ramification, then one should be able to form a Kolyvagin system and thus obtain a bound on the relevant Selmer group. We note in passing that the order of the Selmer group S(K,T ∗) is equal to (r1+r2 1) the p-part of the normalized L-value ζK − (0)wK /RK . This fits in with the general conjectural framework. However, this proof is very unsatisfying, as one can not hope to generalize the proof of the Dirichlet class number formula to any other situations. In the remainder of this lecture we will redo the above example in a less naive way. In particular, we will exhibit a Kolyvagin system which is directly connected to special values of the Dedekind zeta function. Applying this Kolyvagin system will yield a bound on S(K,T ∗) in terms of a special value of an L-function without any reference to the Dirichlet class number formula. This is the sort of argument which one hopes to be able to apply in more general situations.

4. An : Cyclotomic units

We continue to let T = Zp(1), and we now restrict our attention to the field + th K = Q(ζp) , the maximal real subfield of the p cyclotomic field. (One can modify the constructions below for the case where K is an arbitrary abelian extension of + Q, but we will stick to the case K = Q(ζp) for simplicity.) We also assume that p is odd for the remainder of the lecture. L-FUNCTIONS AND CYCLOTOMIC UNITS 5

For any integer m > 1, we define the cyclotomic cm = ζm 1, regarded as − an element of Q(ζm)×. (In fact, cm is only actually a unit in the of integers of Q(ζm) when m is composite, but this is irrelevant.) For any prime l we have the fundamental distribution relation c l m; ( m (8) NQ(ζml)/Q(ζm) cml = 1 | (1 Frob− )cm l - m; − l here N is the norm and Frobl Gal(Q(ζm)/Q) is the Frobenius at l. Q(ζml)/Q(ζm) ∈ Note that Gal(Q(ζm)/Q) acts multiplicatively, so that

1 cm (1 Frobl− )cm = 1 Q(ζm)×. − Frobl− cm ∈ Recall that we have fixed a prime p of interest. It is useful to view the cyclotomic fields as living in p-power towers:

Q(ζmlp∞ ) q qqq qqq qqq n Q(ζmp∞ ) Q(ζmlp ) s q sss qqq sss qqq sss qqq n 2 Q(ζp∞ ) Q(ζmp ) Q(ζmlp ) ss qq sss qqq ss qqq sss qq Q(ζpn ) Q(ζmp2 ) Q(ζmlp) s qq sss qq sss qqq sss qqq Q(ζp2 ) Q(ζmp) s sss sss sss Q(ζp) Here m is an integer not divisible by p and l = p is a prime not dividing m. For fixed m, the distribution relation (8) shows that6 the cyclotomic units

(9) cmpn n 1 { } ≥ form a norm compatible system in the middle cyclotomic tower above. Let m U denote the inverse limit of the Q(ζmpn )× with respect to the norm:

m = lim Q(ζmpn )×. U n 1 ←−≥ Let cm m be the element (9). ∞ ∈ U Note that the norm operators from Q(ζmlpn )× to Q(ζmpn )× are compatible as n varies, so that we obtain a norm operator Nml/m : ml m of inverse systems. The distribution relation (8) in this context yields theU relation→ U 1 (10) Nml/mcml = (1 Frobl− )cm ∞ − ∞ where Frobl is a Frobenius element at l in Gal(Q(ζmp∞ )/Q). 6 TOM WESTON, UNIVERSITY OF MICHIGAN

The relation (10) means that the cyclotomic units form an Euler system in the sense of Kolyvagin and Rubin. This means that for each m (divisible by p) one can apply an appropriate Galois operator to cm Q(ζm)× to obtain an element ∈ dm K×. In fact, there is some integer k, depending on m, such that the image ∈ 1 k k of dm in H (K, T/p T ) under the Kummer map lies in S(m)(K, T/p T ). Fixing an integer k, one can choose primes l1, . . . , lr so that the images of the + Gal(Q(ζp )/Q)-conjugates of

dl1 , dl1l2 , dl1l2l3 , . . . , dl1 lr ··· k under the Kummer map form a Kolyvagin system in S(l1 lr )(K, T/p T ). The Euler system formalism allows one to compute the index of the··· module generated by this Kolyvagin system in 1 k Hs (Kv, T/p T ). v l1⊕ lr | ··· Taking k sufficiently large and combining all of this, one obtains the following result.

Theorem. #S(K,T ∗) [ K× : K ] where K is the group of cyclotomic units of K. | O C C

See [4, Chapter 8] for the precise definition of K and see [4, Chapter 15, Section 3] for the details of the argument (although WashingtonC avoids the explicit use of cohomology). Since S(K,T ∗) identifies with the p-ideal class group A, this result is also a consequence of the p-adic class number formula; see [4, Theorem 8.2]. More interesting is that one can apply the same ideas to study the twist of T by a character of finite order; this allows one to prove that the divisibility of the theorem + holds on the level of χ-components for each character χ : Gal(Q(ζp) /Q) Zp×. This result is known as the converse to Herbrand’s theorem; the original proof,→ due to Ribet [2], used subtle facts in the arithmetic of modular curves.

5. Relationship to p-adic L-functions The connection between the Euler system of cyclotomic units and special values of L-functions is through p-adic L-functions. We first set up some notation. Set ¯ GQ = Gal(Q/Q) and G = Gal(Q(ζp∞ )/Q). There is a canonical isomorphism G ∼= Zp× such that the composition ε : GQ  G ∼= Zp× is the cyclotomic character. Set ∆ = Gal(Q(ζp)/Q); there is a canonical isomor- phism ∆ ∼= (Z/pZ)×, and the composition ω : GQ  ∆ ∼= (Z/pZ)× is the Teichm¨ullercharacter. There is a canonical injection (Z/pZ)× , Z×, which yields a canonical injection → p ∆ , G. We let Q be the fixed field of Q(ζp ) by ∆: → ∞ ∞

Q(ζp∞ ) w ∆ ww ww Γ ww ww Q Q(ζp) ∞ ∆ vv Γ vv vv vv Q v L-FUNCTIONS AND CYCLOTOMIC UNITS 7

Define a character ε by h i 1 ε = ω− ε : GQ Γ = 1 + pZp Z×. h i  ∼ ⊂ p The element 1 + p 1 + pZp is a multiplicative generator, and we let γ Γ be the corresponding generator∈ of Γ. ∈ Let Cp be the p-adic completion of Q¯ p. There is a p-adic meromorphic function ζp(s) on the disk

1 1 p 1 (11) s Cp : s < p − − , { ∈ | | } analytic except for a pole at s = 1, such that for any integer k 0, ≤ p k+1 ζ (k) = Lp(k, ω− ).

(Here for any integer m and Dirichlet character χ, we write Lm(s, χ) for the Dirich- let L-function for χ with Euler factors at primes dividing m removed.) See [4, Theorem 5.11] for details. (Of course, the p-adic zeta function ζp(s) is an example of a p-adic L-function.) More generally for any integer m relatively prime to p, we can remove the Euler factors at primes dividing m from ζp(s) to define a p-adic meromorphic function p ζm(s) on the disk (11) with p k+1 (12) ζm(k) = Lpm(k, ω− ) for every integer k 0. In particular, if l is a prime not dividing mp, then the p≤ p relations (12) for ζm(k) and ζml(k) imply that p k 1 k p (13) ζ (k) = (1 ω − (l)l− )ζ (k). ml − m p (The factor appearing here is just the extra Euler factor removed from ζml(s).) p In fact, ζm(s) is not merely a p-adic meromorphic function: there is a power series gm(T ) Zp[[T ]] such that the function fm(T ) defined by ∈ 1 fm(T ) = gm(T ) T p · − satisfies p s ζ (s) = fm (1 + p) 1 m −  for all s in the disk (11); see [4, Theorem 7.10]. Let Λ = Zp[[Γ]] be the Iwasawa algebra. Recall that the correspondence γ 1 − ↔ T sets up an isomorphism Λ ∼= Zp[[T ]]. We wish to use this correspondence to interpret the p-adic zeta functions as elements of the Iwasawa algebra. However, p since ζm(s) has a pole at s = 1 we will need to work with a slightly extended Iwasawa algebra: define Λ˜ to be the localization of Λ at γ p 1. We can now − − define an element Zm Λ˜ by ∈ 1 Zm = fm(γ 1) = gm(γ 1). − γ p 1 · − − − p We claim that we can recover the function ζm(k) from Zm. To do this, first extend ε :Γ Zp× by linearity to a map Λ Zp which we still write as ε . h i → → k h i More generally, for any integer k we can extend the character ε :Γ Z× to a h i → p 8 TOM WESTON, UNIVERSITY OF MICHIGAN

k map ε :Λ Zp. If k 0, we use the fact that ε (γ) = 1 + p to compute h i → ≤ h i k k ε (Zm) = ε fm(γ 1) h i h i −  k = fm ε (γ) 1 h i −  k = fm (1 + p) 1 p −  = ζm(k). We can use the arithmetic of Λ˜ to rewrite the relation (13) in a simpler form. This relation says that k k 1 k k (14) ε Zml = (1 ω − (l)l− ) ε Zm. h i − h i Consider Frobl Γ = Gal(Q /Q); by definition, this element satisfies ω(Frobl) = ∈ ∞ 1 1 ω(l) and ε(Frobl) = l. Thus ε (Frob− ) = ω(l)l− and therefore h i l k 1 k k 1 1 (15) 1 ω − (l)l− = ε 1 ω− (l) Frob− . − h i − l  It follows from (14) and (15) that in the Iwasawa algebra Λ˜ we have the relation 1 1 Zml = 1 ω− (l) Frob− Zm − l  1 ˜ where we regard Frobl− as an element of Λ in the obvious way. 1 We can improve the situation slightly by twisting these elements. Let Zp(ω− ) 1 be a free Zp-module of rank 1 with GQ-action given by ω− ; fix a generator ξ. For ˜ 1 each m define zm = Zml ξ Λ Zp(ω− ). One easily computes that we now have ⊗ ∈ ⊗ 1 (16) zml = 1 Frob− zm. − l  This relation precisely matches the relation (10) of the Euler system cm . (The norm does not appear here as it is implicit in certain identifications of∞ Iwasawa algebras.) Even though the constructions of the cm (via cyclotomic units) and the zm (via p- adic zeta functions) were completely different, one can not help but believe that the similarity in (10) and (16) is not merely a coincidence. In fact, it is possible to give a construction of p-adic zeta functions (and thus the zm) in terms of cyclotomic units. This is accomplished via a method of Coleman for turning norm compatible families of units (such as the cm) into power series; with a few appropriate modifications, these yield precisely the power series gm(T ) used to define Zm above. See [1] for details. We have now seen that the Euler system of cyclotomic units is directly connected with p-adic zeta functions (and therefore special values of L-functions) via Coleman power series. When all of this is sorted out, the Kolyvagin system coming from the cyclotomic units now gives a direct relation between the order of the Selmer group (r1+r2 1) S(K,T ∗) and the (appropriately normalized) special value ζK − (0); in the end, one recovers a (slightly weak form of the) Dirichlet class number formula. The advantage of this approach is that there is some hope that it may generalize to other L-functions.

6. Generalizations We conclude by briefly indicating a very speculative conjectural framework for proving results about Selmer groups and special values of L-functions. Let X be a smooth projective variety over Q and let T be a piece of the p-adic ´etalecohomology of X. (For example, one could take X to be an and T to be its p-adic L-FUNCTIONS AND CYCLOTOMIC UNITS 9

Tate module.) We make the assumption that the minus eigenspace for the action of complex conjugation on T has rank 1. Let T ∗ be the Cartier dual of T . It is a standard conjecture that an involving characteristic polyno- mials of Frobenius elements acting on T ∗ defines an L-function L(T ∗, s) admitting a meromorphic continuation to the entire complex plane. Perrin-Riou has conjec- tured that one can p-adically interpolate the special values of the L-functions of T ∗ and its twists to form a p-adic L-function T ∗ . This p-adic L-function should live in the tensor product of a certain extendedL Iwasawa algebra Λ and a Dieudonn´e module (depending on T ). Perrin-Riou has defined Λ and , together with a map D D 1 ψ : lim H K(ζpn ),T ) Λ . → ⊗ D ←−n Perrin-Riou conjectures that there is an element z1 of this inverse limit such that

ψ(z1) = T . L ∗ z1 alone is not enough to yield an Euler system. However, Rubin has extended all of this to the case of L-functions with Euler factors removed: he conjectures that there are elements zm such that

ψ(zm) = T ,m L ∗ where T ,m is the p-adic L-function with Euler factors at primes dividing m re- L ∗ moved. (In the cyclotomic case, these elements are given by the zm constructed in the previous section.) As in our computation in the cyclotomic case, for a prime l not dividing mp, zm and zml will then be related by an Euler factor in such a way that they form an Euler system. This Euler system would give rise to a Kolyvagin system in the usual way, and this would finally yields a bound on the Selmer group

S(Q,T ∗) related to z1, and thus to T ∗ , and thus to special values of L(T ∗, s). See [3, Chapter 8] for more details. L Although this may seem like an extraordinarily ambitious program, Kato has succeeded in proving a great deal of it in the case of the of an elliptic curve. References

[1] Bernadette Perrin-Riou, La fonction L p-adique de Kubota-Leopoldt, pp. 65–93 in: , Contemporary Mathematics 174, American Mathematical Society, 1994. [2] , A modular construction of unramified p-extensions of Q(µp), Inventiones Mathe- maticae 34 (1976), pp. 151–162. [3] Karl Rubin, Euler systems, Princeton University Press, 2000. [4] Lawrence Washington, Introduction to cyclotomic fields, Second Edition, Springer-Verlag, 1997.