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 conjugate 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:
(r +r2 −1)
ζK 1
(0) = −
hK R K
.
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(K̄/K). We define the Cartier dual of T by T ∗ = HomZp (T, µp∞ ) with
the adjoint Galois action. Let F be a Selmer structure on T . Recall that F consists
of choices of local Selmer structures
HF1 (Kv , T ) ⊆ H 1 (Kv , T )
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 F ∗ on
T ∗ , and the Selmer group S(K, T ∗ ) is defined to be the subgroup of H 1 (K, T ∗ ) of
cocycles whose local restrictions lie in HF1 ∗ (Kv , T ∗ ) for every place v.
Kolyvagin has shown that in some situations 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 F on T . If a is a fractional ideal of K, we define modified
Selmer structures on T and T ∗ by
(
HF1 (Kv , T ) v - a;
Fa =
H 1 (Kv , T ) v | a;
and
(
HF1 ∗ (Kv , T ∗ ) v - a;
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 SF (K, T ) (resp. SF ∗ (K, T ∗ ), resp. SFa (K, T ), resp. SFa∗ (K, T ∗ ))
as S(K, T ) (resp. S(K, T ∗ ), resp. Sa (K, T ), resp. S a (K, T ∗ )). For any place v we
will write Hs1 (Kv , T ) for the singular quotient H 1 (Kv , T )/HF1 (Kv , T ).
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 F
and a as above there is an exact sequence
(1) 0 → S(K, T ) → Sa (K, T ) → ⊕ Hs1 (Kv , T ) → S(K, T ∗ )∨ → S a (K, T ∗ )∨ → 0.
v|a
This exact sequence suggests the following approaching to bounding the order of
the Selmer group S(K, T ∗ ): first, one bounds the order of S a (K, T ∗ ) for “sufficiently
divisible” ideals a. In practice it is often not difficult to show that one can choose
an ideal a so that S a (K, T ∗ ) vanishes. For such an a, (1) yields an isomorphism
S(K, T ∗ )∨ ∼
= ⊕ Hs1 (Kv , T ) / img Sa (K, T ).
v|a
In particular, if one can exhibit elements of Sa (K, T ) which fill up ⊕v|a Hs1 (Kv , T )
with finite index η, then one concludes that #S(K, T ∗ )|η. (More generally, if
S a (K, T ∗ ) is non-zero but computable, one has #S(K, T ∗ )|η · #S a (K, T ∗ ).) An
explicitly given submodule of Sa (K, T ) which has finite index in ⊕v|a Hs1 (Kv , T ) is
called a Kolyvagin system.
In practice, one can bound S a (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
F on T momentarily.
To exhibit a Kolyvagin system, we need some method of constructing elements
of H 1 (K, T ) with controlled ramification. We will use the Kummer map
κ : K × → H 1 (K, µn )
σ(α1/n )
.
α 7→ σ 7→
α1/n
L-FUNCTIONS AND CYCLOTOMIC UNITS
3
Here α1/n is a fixed choice of nth root of α in K̄; changing the choice of α1/n changes
κ by a coboundary. The Kummer map is compatible with changing n; in particular,
if we take the inverse limit over powers of p we obtain a map κ : K × → H 1 (K, T ).
The same construction yields local Kummer maps
κv : Kv× → H 1 (Kv , T )
for each v. In fact, κv induces an isomorphism
d
×
1
κv : K
v → H (Kv , T )
d
×
×
where K
v is the p-adic completion of Kv .
Consider the exact sequence
ord
×
0 → OK
→ Kv× −→v Z → 0
v
with p-adic completion
d
d
×
× ordv
0→O
Kv → Kv −→ Zp → 0.
(2)
We define the local Selmer structure HF1 (Kv , T ) by requiring that the exact sequence
0 → HF1 (Kv , T ) → H 1 (Kv , T ) → Hs1 (Kv , T ) → 0
corresponds to (2) under the isomorphism κv of the middle terms. (It is straightforward 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
∼
=
Hs1 (Kv , T ) −→ Zp
fitting into a commutative diagram
(3)
K×
κ
/ H 1 (K, T )
/ H 1 (Kv , T )
s
∼
=
ordv
/ Zp
Z
In particular, it follows from (3) that for any α ∈ K × , κ(α) ∈ H 1 (K, T ) satisfies
the local condition given by F at every place which is relatively prime to α. In the
notation introduced earlier, this means that
κ(α) ∈ S(α) (K, T ).
We now use κ to produce a Kolyvagin system for T . To motivate our construction, 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 ).
(To see this, note that H 1 (K, T ∗ ) = Hom(GK , Qp /Zp ). The local conditions F ∗
correspond to allowing only those homomorphisms which factor through Gal(L/K)
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 isomorphism (4) follows.) Furthermore, for any ideal a, the restricted Selmer group
S a (K, T ∗ ) corresponds to the dual of the quotient of A by the subgroup generated
by the prime factors of a. In particular, S a (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 Tchebatorev 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
observed above, we have S a (K, T ∗ ) = 0. In particular, by (1) we have
∼ ⊕ H 1 (Kv , T )/ img Sa (K, T ).
(5)
S(K, T ∗ )∨ =
v|a
s
We will now compute this directly, without any reference to (4).
ni
By definition, lpi is principal; choose a generator αi and set ci = κ(αi ) ∈
Sli (K, T ). Since ordli (αi ) = pni , by (3) we have
(6)
H 1 (Kl , T )/(Zp · img ci ) ∼
= Z/pni 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
⊕ H 1 (Kv , T )/ img C ∼
= Z/pn1 Z ⊕ · · · ⊕ Z/pnr Z ∼
= A.
v|a
s
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
(r +r −1)
the p-part of the normalized L-value ζK 1 2 (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 Euler system: Cyclotomic units
We continue to let T = Zp (1), and we now restrict our attention to the field
K = Q(ζp )+ , the maximal real subfield of the pth 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 unit cm = ζm − 1, regarded as
an element of Q(ζm )× . (In fact, cm is only actually a unit in the ring of integers of
Q(ζm ) when m is composite, but this is irrelevant.) For any prime l we have the
fundamental distribution relation
(
cm
l | m;
(8)
NQ(ζml )/Q(ζm ) cml =
−1
(1 − Frobl )cm l - m;
here NQ(ζml )/Q(ζm ) is the norm and Frobl ∈ Gal(Q(ζm )/Q) is the Frobenius at l.
Note that Gal(Q(ζm )/Q) acts multiplicatively, so that
cm
∈ Q(ζm )× .
(1 − Frob−1
l )cm =
Frob−1
l 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
q
qqq
q
q
qqq
Q(ζmp∞ )
Q(ζmlpn )
q
s
q
s
q
s
qq
ss
q
s
q
s
qqq
sss
Q(ζmlp2 )
Q(ζmpn )
q
s
q
s
q
s
qq
ss
q
s
q
s
qqq
sss
Q(ζmp2 )
Q(ζpn )
Q(ζmlp )
q
s
q
s
q
s
q
s
qqq
sss
sss
qqq
Q(ζp2 )
Q(ζmp )
s
s
s
s
sss
sss
Q(ζp )
Q(ζp∞ )
Here m is an integer not divisible by p and l 6= p is a prime not dividing m. For
fixed m, the distribution relation (8) shows that the cyclotomic units
(9)
{cmpn }n≥1
form a norm compatible system in the middle cyclotomic tower above. Let Um
denote the inverse limit of the Q(ζmpn )× with respect to the norm:
Um = lim Q(ζmpn )× .
←−
n≥1
Let cm∞ ∈ Um be the element (9).
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 : Uml → Um of inverse systems.
The distribution relation (8) in this context yields the relation
(10)
Nml/m cml∞ = (1 − Frob−1
l )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
of dm in H 1 (K, T /pk T ) under the Kummer map lies in S(m) (K, T /pk T ).
Fixing an integer k, one can choose primes l1 , . . . , lr so that the images of the
Gal(Q(ζp+ )/Q)-conjugates of
dl1 , dl1 l2 , dl1 l2 l3 , . . . , dl1 ···lr
under the Kummer map form a Kolyvagin system in S(l1 ···lr ) (K, T /pk T ). The Euler
system formalism allows one to compute the index of the module generated by this
Kolyvagin system in
⊕ Hs1 (Kv , T /pk T ).
v|l1 ···lr
Taking k sufficiently large and combining all of this, one obtains the following result.
×
Theorem. #S(K, T ∗ ) | [OK
: CK ] where CK is the group of cyclotomic units of
K.
See [4, Chapter 8] for the precise definition of CK and see [4, Chapter 15, Section
3] for the details of the argument (although Washington 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) → Z×
p.
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(Q̄/Q) and G = Gal(Q(ζp∞ )/Q). There is a canonical isomorphism
G∼
= Z×
p such that the composition
ε : GQ G ∼
= Z×
p
is the cyclotomic character. Set ∆ = Gal(Q(ζp )/Q); there is a canonical isomorphism ∆ ∼
= (Z/pZ)× , and the composition
ω : GQ ∆ ∼
= (Z/pZ)×
is the Teichmüller character.
There is a canonical injection (Z/pZ)× ,→ Z×
p , which yields a canonical injection
∆ ,→ G. We let Q∞ be the fixed field of Q(ζp∞ ) by ∆:
Q(ζp∞ )
ww
w
ww Γ
ww
ww
Q(ζp )
Q∞
v
∆ vvv
Γ
v
vv
vv
Q
∆
L-FUNCTIONS AND CYCLOTOMIC UNITS
7
Define a character hεi by
hεi = ω −1 ε : GQ Γ ∼
= 1 + pZp ⊂ Z×
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 Q̄p . There is a p-adic meromorphic function
ζ p (s) on the disk
(11)
1
{s ∈ Cp : |s| < p1− p−1 },
analytic except for a pole at s = 1, such that for any integer k ≤ 0,
ζ p (k) = Lp (k, ω −k+1 ).
(Here for any integer m and Dirichlet character χ, we write Lm (s, χ) for the Dirichlet 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 ζ p (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 ζ p (s) to define a p-adic meromorphic function
p
ζm
(s) on the disk (11) with
(12)
p
ζm
(k) = Lpm (k, ω −k+1 )
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
(13)
p
p
ζml
(k) = (1 − ω k−1 (l)l−k )ζm
(k).
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
fm (T ) =
1
· gm (T )
T −p
satisfies
p
ζm
(s) = fm (1 + p)s − 1
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 Λ̃ to be the localization of Λ at γ − p − 1. We can now
define an element Zm ∈ Λ̃ by
Zm = fm (γ − 1) =
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 hεi : Γ → Zp by linearity to a map Λ → Zp which we still write as hεi.
More generally, for any integer k we can extend the character hεik : Γ → Z×
p to a
8
TOM WESTON, UNIVERSITY OF MICHIGAN
map hεik : Λ → Zp . If k ≤ 0, we use the fact that hεi(γ) = 1 + p to compute
hεik (Zm ) = hεik fm (γ − 1)
= fm hεik (γ) − 1
= fm (1 + p)k − 1
p
= ζm
(k).
We can use the arithmetic of Λ̃ to rewrite the relation (13) in a simpler form.
This relation says that
(14)
hεik Zml = (1 − ω k−1 (l)l−k )hεik Zm .
Consider Frobl ∈ Γ = Gal(Q∞ /Q); by definition, this element satisfies ω(Frobl ) =
−1
ω(l) and ε(Frobl ) = l. Thus hεi(Frob−1
and therefore
l ) = ω(l)l
.
(15)
1 − ω k−1 (l)l−k = hεik 1 − ω −1 (l) Frob−1
l
It follows from (14) and (15) that in the Iwasawa algebra Λ̃ we have the relation
Zml = 1 − ω −1 (l) Frob−1
Zm
l
where we regard Frob−1
as an element of Λ̃ in the obvious way.
l
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 −1 ; fix a generator ξ. For
each m define zm = Zml ⊗ ξ ∈ Λ̃ ⊗ Zp (ω −1 ). One easily computes that we now have
(16)
zml = 1 − Frob−1
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 padic 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
(r +r −1)
S(K, T ∗ ) and the (appropriately normalized) special value ζK 1 2 (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 étale cohomology
of X. (For example, one could take X to be an elliptic curve 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 Euler product involving characteristic polynomials 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 conjectured 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 LT ∗ . This p-adic L-function should live
in the tensor product of a certain extended Iwasawa algebra Λ and a Dieudonné
module D (depending on T ). Perrin-Riou has defined Λ and D, together with a
map
ψ : lim H 1 K(ζpn ), T ) → Λ ⊗ D.
←
−
n
Perrin-Riou conjectures that there is an element z1 of this inverse limit such that
ψ(z1 ) = LT ∗ .
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 ) = LT ∗ ,m
where LT ∗ ,m is the p-adic L-function with Euler factors at primes dividing m removed. (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 LT ∗ , and thus to special values of L(T ∗ , s). See
[3, Chapter 8] for more details.
Although this may seem like an extraordinarily ambitious program, Kato has
succeeded in proving a great deal of it in the case of the Tate module of an elliptic
curve.
References
[1] Bernadette Perrin-Riou, La fonction L p-adique de Kubota-Leopoldt, pp. 65–93 in: Arithmetic
geometry, Contemporary Mathematics 174, American Mathematical Society, 1994.
[2] Ken Ribet, A modular construction of unramified p-extensions of Q(µp ), Inventiones Mathematicae 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.