The Phenomenon of Trivial Zeroes

Lectures by R. Greenberg Notes by E. Ghate

March 8, 2001

1 The Kubota-Leopoldt p-adic L-function

Let p denote an odd prime.

Let ψ be an even Dirichlet character of conductor f_{ψ}. Assume that ψ ≠ ψ_{0} and that p^{2} | f̸_{ψ}.

Let ω denote the Teichmuller character mod p. Thus if a ∈ and (a, p) = 1, then ω(a) is the unique (p − 1)^{th} root of unity such that ω(a) ≡ a mod p. Note that ω is an odd character.

Let χ = ωψ^{−1}. So χ is odd.

The Kubota-Leopoldt p-adic L-function attached to ψ is a p-adically continuous function which is defined on the unit disk |s|_{p} ≤ 1 in _{p}, takes values in _{p}, and is uniquely determined by the interpolation property

L_{p}(1 − n, ψ) = (1 − ψ_{n}(p)p^{n−1})L_{∞}(1 − n, ψ_{n})

for all n ≥ 1. Here ψ_{n} is the primitive Dirichlet character associated to ψω^{−n} and L_{∞}(s, ψ_{n}) is the usual complex Dirichlet L-function attached to ψ_{n}. We remark that L_{∞}(1−n, ψ_{n}) is always non-zero since 1−n and ψ_{n} have opposite parity. Moreover, it is expressible in a well known way in terms of generalized Bernoulli numbers. In fact the existence of L_{p}(s, ψ) follows from the Kummer-Carlitz congruences for these Bernoulli numbers.

When n = 1 we have

L_{p}(0, ψ) = (1 − ψ_{1}(p))L_{∞}(0, ψ_{1}) = (1 − χ^{−1}(p))L_{∞}(0, χ^{−1}).

Since χ is odd, L_{∞}(0, χ^{−1}) ≠ 0. Thus we see that the p-adic L-function L_{p}(s, ψ) vanishes at s = 0 if and only if χ^{−1}(p) = 1. Since the vanishing occurs because of an Euler factor, we refer to the zero of L_{p}(s, ψ) at s = 0, when it arises, as a trivial zero.

Let O_{ψ} = _{p}[ψ]. Iwasawa showed that, under the hypothesis that p^{2} | f̸_{ψ}, there is a power series

g_{ψ}(T) ∈ O_{ψ}[[T]]

such that

L_{p}(s, ψ) = g_{ψ}((1 + p)^{s} − 1)).

Note that if s ∈ _{p}, then (1 + p)^{s} ∈ 1 + p _{p}, so that (1 + p)^{s} − 1 ∈ p _{p} and the power series converges. In particular we may consider L_{p}(s, ψ) as a p-adically analytic, and not just continuous, function defined on the unit disk in _{p}. In particular it makes sense to consider its derivative at a point in _{p}.

Theorem 1.1 (Ferrero-Greenberg). Say that L_{p}(s, ψ) has a trivial zero at s = 0. Then it is a simple zero. In other words:

L′_{p}(0, ψ) ≠ 0.

Sketch of Proof of Theorem 1.1. For simplicity we shall assume that χ is a quadratic character. Since χ is odd, χ cuts out an imaginary quadratic field which we call K. The condition that χ^{−1}(p) = 1 just says that p = 𝔭𝔭̄ is split in K. Let h denote the class number of K. Then 𝔭^{h} = (π) for some π ∈ O_{K}. Let

q := \frac{π}{π̄} ∈ K.

Note that since π is only determined up to a unit of K, q is determined up to a unit of K.

Define the p-adic logarithm map log_{p} : _{p}^{×} → _{p} as follows. If x ∈ p _{p} then set

log_{p}(1 + x) = ∑_{j=1}^{∞}(−1)^{j−1} \frac{x^{j}}{j} .

This induces a bijection between 1 + p _{p} and p _{p}. Now extend log_{p} to all of _{p}^{×} by setting log_{p}(ζ) = 0 if ζ ∈ μ_{p−1} and log_{p}(p) = 0.

Write (with hindsight)

L′_{p}(0, ψ) = −L_{χ} · L_{∞}(0, χ^{−1})  (1.2)

for some number L_{χ}, which we call the L-invariant of χ. We claim that

L_{χ} = \frac{log_{p}(q)}{ord_{p}(q)}.  (1.3)

Note that the right hand side of (1.3) is well defined even though q is determined up to a unit of K. The theorem follows from (1.3). Indeed

ord_{p}(q) = ord_{p}(π) − ord_{p}(π̄) = h − 0 = h,

and

log_{p}(q) = log_{p}(π̄^{−2}),

since ππ̄ = p^{h} up to a unit in O_{K} implies that log_{p}(π) = − log_{p}(π̄). Thus

L_{χ} = \frac{log_{p}(π̄^{−2})}{h}  (1.4)

has non-zero numerator, so that L_{χ} ≠ 0, proving the theorem.

Let us now sketch a proof of (1.3). There is a formula for the derivative of L_{p}(s, ψ) at s = 0 that is analogous to the formula of Lerch for the derivative of L_{∞}(s, ψ) at s = 0. Let Γ_{p} denote the p-adic Γ function of Morita. Then we have

L′_{p}(0, ψ) = ∑_{a=1}^{f} ψ_{1}(a) log_{p} Γ_{p} (\frac{a}{f_{ψ_{1}}} ) − (1 − ψ_{1}(p))L(0, ψ_{1}) log_{p}(f_{ψ_{1}}). (1.5)

Since ψ_{1}(p) = 1 by assumption, the last term on the right hand side of (1.5) disappears. We collect the remaining terms according to the sign of ψ_{1}(a) and write

L′_{p}(0, ψ) = log_{p} (\frac{∏_{ψ_{1}(a)=1} Γ_{p} (\frac{a}{f_{ψ_{1}}} )}{∏_{ψ_{1}(a)=−1} Γ_{p} (\frac{a}{f_{ψ_{1}}} )}) .

Now a formula of Gross and Koblitz implies that the numerator and denominator in the expression above are essentially products of a Gauss sum and a (p − 1)^{th} root of −p. This gives (1.3) (FILL IN DETAILS!!)

2 Iwasawa modules

We turn now from the ‘analytic side’ to the ‘algebraic side’.

Recall that χ is an odd (not necessarily quadratic) Dirichlet character. Assume for simplicity that χ takes values in ( /p )^{×} so that χ cuts out a cyclic extension K of degree dividing p − 1. Thus we may consider χ as a character

χ : Gal(K/ ) → μ_{p−1}.

Let _{∞} be the cyclotomic _{p}-extension of . Thus _{∞} = ∪_{n=1}^{∞} _{n} where n is the unique cyclic of order p^{n} extension sitting inside (μ_{p^{n+1}}).

Write Δ = Gal(K/ ) and Γ = Gal( _{∞}/ ) →^{∼} _{p}. Consider the element σ ∈ Gal( (μ_{p^{∞}})/ ) defined by

σ(ζ) = ζ^{1+p}

for ζ ∈ μ_{p^{∞}}. Let γ ∈ Γ denote the restriction of σ to _{∞}. Then Γ is topologically generated by γ.

Now let K_{n} = K _{n} for each n ≥ 1. Let K_{∞} = ∪_{n=1}^{∞}K_{n} = K _{∞}. K_{∞} is called the cyclotomic _{p}-extension of K. Since K is a prime to p extension, it is linearly disjoint with every _{n} over . Thus we have

[figure: K_{∞}; Γ; Δ; K; Δ; _{_{∞}}; Γ; ]

and Gal(K_{∞}/ ) = Γ × Δ.

Now let L_{n} be the p-Hilbert class field of K_{n}. L_{n} is the maximal unramified abelian p-extension of K_{n}. Furthermore

Gal(L_{n}/K_{n}) = Cl(K_{n})_{p}

is the p-primary part of the class group of K_{n}. Let L_{∞} = ∪_{n=1}^{∞}L_{n}. Then L_{∞} is the maximal abelian unramified pro-p extension of K_{∞}. Set

X := Gal(L_{∞}/K_{∞}) = lim_{←} Gal(L_{n}K_{∞}/K_{∞}) = lim_{←} Gal(L_{n}/K_{n}),

where the last equality holds since for n sufficiently large L_{n} is linearly disjoint from K_{∞} over K_{n}, the former extension of K_{n} being unramified, and the latter being totally ramified, for n sufficiently large, at some prime of K_{n} above p.

We now have the following tower of number fields:

[figure: L_{∞}; X; K_{∞}; Γ×Δ; ]

Since X is a pro-p group it is a _{p}-module. Moreover Δ×Γ acts on X. Write

X = ⨁_{ϕ∈Δ̂} X^{ϕ}

where

X^{ϕ} = {x ∈ X | δ(x) = ϕ(δ) · x for all δ ∈ Δ}

is the eigenspace of Δ corresponding to ϕ. Since the action of Δ and Γ commute, each X^{ϕ} is a Γ module.

Theorem 2.1 (Iwasawa, Ferrero-Washington). The following are true:

1. X is a finitely generated _{p}-module.

2. If ϕ is odd, then X^{ϕ} is a free _{p}-module, say

X^{ϕ} = _{p}^{λ_{ϕ}}.

3. If ϕ = 1 is the trivial character, then X^{ϕ} = 0.

The above theorem gives some information about X^{ϕ} if ϕ is odd, and in the case when ϕ is the (even) trivial character. Not much is known about the other even eigenspaces of Δ. However we make the following conjecture.

Conjecture 2.2. If ϕ ≠ 1 and ϕ is even then X^{ϕ} is a finite module.

Remark 1. We remark that if F = (\sqrt{d}) is a real quadratic field (BUT THERE IS NO F IN THE NOTATION ABOVE) then the conjecture says that the p-parts of the class numbers of the fields in the tower defining the cyclotomic _{p} extension of F are bounded independently of n. We also remark that in the case when K = (μ_{p}) is the p^{th} cyclotomic field, then the conjecture follows from Vandiver’s Conjecture which in fact says that X^{ϕ} = 0 for all even ϕ.

By continuity X^{ϕ} is in fact a Λ-module where Λ = _{p}[[Γ]] denotes the completed group algebra of Γ over _{p}. In fact if we identify Λ with _{p}[[T]] then T acts as γ − 1 (and so topologically nilpotently).

3 Main Conjecture

We now return to the situation where χ is an odd Dirichlet character cutting out a cyclic extension K of of degree dividing p − 1.

Let f_{χ}(T) denote the characteristic polynomial of γ − 1 acting on X^{χ}. Say

f_{χ}(T) = a_{0} + a_{1}T + ⋯ + a_{λ_{χ}−1}T^{λ_{χ}−1} + T^{λ_{χ}}.

Then (WHY??) p|a_{i} for 0 ≤ i ≤ λ_{χ} − 1. Thus f_{χ}(T) is a distinguished polynomial.

Theorem 3.1 (Iwasawa Main Conjecture, Mazur-Wiles’ Theorem). Let p be an odd prime. Assume that ψ is an even Dirichlet character such that χ = ωψ^{−1} cuts out a cyclic extension K of of degree dividing p − 1. Then

g_{ψ}(T) = f_{χ}(T) · u_{χ}(T)

for some unit power series u_{χ}(T) ∈ Λ = _{p}[[T]].

Corollary 3.2. Let χ be a quadratic Dirichlet character so that K is an imaginary quadratic field. If p splits in K then

1. T | f_{χ}(T).

2. λ_{χ} = 1 ⟺ π̄ ∉ ( _{p}^{×})^{p}. Here p = 𝔭𝔭̄ and 𝔭^{h} = (π) where h is the class number of K and π ∈ K.

Proof. Let us write g_{ψ}(T) = ∑_{j=0}^{∞} b_{j}T^{j} with b_{j} ∈ O_{ψ}. Since χ(p) = 1 we have L_{p}(0, ψ) = g_{ψ}(0) = b_{0} = 0. So T |g_{ψ}(T) and the first statement follows immediately from the main conjecture.

As for the second statement recall that

f_{χ}(T) = a_{0} + a_{1}T + ⋯ + a_{λ_{χ}−1}T^{λ_{χ}−1} + T^{λ_{χ}}

with a_{0} = 0. Thus λ_{χ} = 1 ⟺ a_{1} = 1 ⟺ b_{1} ∈ _{p}^{×} where the last implication follows from the he main conjecture and the fact that p divides all the a_{i}.

Now

L′_{p}(0, ψ) = \frac{d}{ds}g_{ψ}((1 + p)^{s} − 1)|_{s=0} = log_{p}(1 + p) · b_{1}.  (3.3)

By (3.3) and (1.2) we have

b_{1} = \frac{L′_{p}(0, ψ)}{log_{p}(1 + p)} = \frac{−L_{χ} · L_{∞}(0, χ)}{log_{p}(1 + p)}.

Since L_{∞}(0, χ) = \frac{2h}{w} where w is the number of roots of unity in K and L_{χ} is given by (1.4) we get

b_{1} = \frac{log_{p}(π̄^{−2})}{h} \frac{−2h}{w · log_{p}(1 + p)} = \frac{4 log_{p}(π̄)}{w log_{p}(1 + p)}.

Now say that a ∼ b for two elements a and b in \overline{}_{p} if ord_{p}(a) = ord_{p}(b). Then since ord_{p}(4) = ord_{p}(w) = ord_{p}(p − 1) = 0 (note that even though w = 6 when K = (\sqrt{−3}), p ≠ 3 since p splits in K) we get

b_{1} ∼ \frac{log_{p}(π̄)}{log_{p}(1 + p)} ∼ \frac{log_{p}(π̄^{p−1})}{log_{p}(1 + p)}.

Since π̄ ∈ _{p}^{×}, we have π̄^{p−1} ∈ 1 + p _{p}. Write π̄ = 1 + pc with c ∈ _{p}. Then since log_{p}(1 + x) ∼ x for x ∈ p _{p} we have

b_{1} ∼ \frac{log_{p}(1 + cp)}{log_{p}(1 + p)} ∼ \frac{cp}{p} = c.

Thus λ_{χ} = 1 ⟺ c ∈ _{p}^{×}.

Now

( _{p}^{×})^{p} = μ_{p−1} × (1 + p^{2} _{p})

since μ_{p−1}^{p} = μ_{p−1} and log_{p}(1+p _{p})^{p} = p log(1+p _{p}) = p^{2} _{p} = log_{p}(1+p^{2} _{p}). Therefore c ∈ _{p}^{×} ⟺ π̄^{p−1} = 1 + pc ∉ ( _{p}^{×})^{p} ⟺ π̄ ∉ ( _{p}^{×})^{p}.

Example. Suppose that K = (\sqrt{−3}). Then h = 1. Note χ(p) = 1 ⟺ p ≡ 1 mod 3. In such case we may write p = ππ̄ = a^{2} + 3b^{2} with π = a + b\sqrt{−3}. Thus the corollary above says that λ_{χ} = 1 ⟺ π̄ ∉ ( _{p})^{×} or equivalently if (a − b\sqrt{−3})^{p−1} ≢ 1 mod a + b\sqrt{−3}. The last statement is verifiable by computation. For instance, p = 13 splits in K and in this case λ_{χ} > 1 since π̄^{p−1} ≡ 1 mod π^{2}. One may also check (HOW?) that p ≡ 2 mod 3 ⟺ λ_{χ} = 0.

4 A direct proof of Corollary 3.2

We now give a proof of Corollary 3.2 which does not use properties of the Kubota-Leopoldt p-adic L-function via the main conjecture. We will assume however, for simplicity, that the imaginary quadratic field K has class number h = 1.

Recall that p = 𝔭𝔭̄ splits in K. Let K_{∞} denote the cyclotomic _{p}- extension of K. The primes 𝔭 and 𝔭̄ are totally ramified in K_{∞}/K since p is totally ramified in the cyclotomic _{p}-extension _{∞}/ . Let us write 𝔭_{∞} and 𝔭̄_{∞} for the primes of K_{∞} lying over 𝔭 and 𝔭̄ respectively. Write D(𝔭_{∞}), respectively D(𝔭̄_{∞}), for the corresponding decomposition subgroups of Gal(L_{∞}/K_{∞}).

Since K is imaginary quadratic we may write X = X^{1} ⊕ X^{χ} where we have written 1 for the trivial character. But X^{1} = 0 (WHY??). So X = X^{χ}. Thus Δ =< δ > acts on X by χ. In particular δ acts on X by χ(δ) = −1. On the other hand since δ permutes 𝔭 and 𝔭̄, and therefore 𝔭_{∞} and 𝔭̄_{∞}, we have δ(D(𝔭_{∞})) = D(𝔭̄_{∞}). This shows that the two decomposition subgroups are equal. We shall simply denote them by D.

Proposition 4.1. We have D =^{∼} _{p}.

Proof. Let K_{∞}^{−} = ∪_{n=1}^{∞}K_{n}^{−} denote the anti-cyclotomic _{p}-extension of K. It is characterized amongst the (uncountably many) _{p}-extensions of K by the fact that K_{n}/ is Galois and the non-trivial element δ ∈ Δ = Gal(K/ ) acts by −1. By contrast we shall write K_{∞}^{+} for the cyclotomic _{p}-extension K_{∞} since Δ acts trivially on Gal(K_{n}^{+}/ ). To distinguish the Galois groups of these two _{p}-extension we write Γ^{+} for Gal(K_{∞}^{+}/K) and Γ^{−} for Gal(K_{∞}^{−}/K). Since K_{∞}^{−} and K_{∞}^{+} are disjoint over K we have Gal(K_{∞}^{−}K_{∞}^{+}/K_{∞}^{+}) =^{∼} Γ^{−}.

Let L′_{∞} denote the fixed field of D and let X′ = X/D. We thus have the following picture.

[figure: L_{∞}; D; K_{∞}^{−}K_{∞}^{+}; Γ^{−}; L′_{∞}; X′; K_{∞}^{+}; Γ^{+}; K]

We now claim that

1. K_{∞}^{−}K_{∞}^{+} ⊂ L_{∞}, and,

2. K_{∞}^{−}K_{∞}^{+} ⊄ L′_{∞}.

The proposition follows from these two statements. The first implies that Γ^{−} =^{∼} _{p} is a subgroup of the finitely generated _{p}-module X and the latter implies it is in fact a subgroup of D. Since D is a decomposition group (with trivial inertia subgroup) it is pro-cyclic. Hence D =^{∼} _{p}.

So it is enough to prove the two statements above.

For the first it suffices to show that pro-p abelian extension K_{∞}^{−}K_{∞}^{+}/K_{∞}^{+} is unramified since L_{∞} is the maximal pro-p unramified abelian extension of K_{∞}^{+}. In a _{p}-extension only the primes lying over p can be ramified. So it suffices to show that 𝔭_{∞} and 𝔭̄_{∞} are unramified in K_{∞}^{−}K_{∞}^{+}. We show that 𝔭_{∞} is unramified since the proof for 𝔭̄_{∞} is exactly the same. The idea is to show this locally by completing everything at (suitable choices of primes lying above) 𝔭_{∞}. To ease notation we simply write 𝔭 for a prime that lies above 𝔭 in an extension of K.

Let _{p}^{cycl} denote the cyclotomic _{p}-extension of _{p} and let _{p}^{unr} denote the unramified _{p}-extension of _{p}. We have

(K_{∞}^{−}K_{∞}^{+})_{𝔭} = (K_{∞}^{−})_{𝔭}(K_{∞}^{+})_{𝔭} = (K_{∞}^{−})_{𝔭} _{p}^{cycl},

since clearly (K_{∞}^{+})_{𝔭} = _{p}^{cycl}.

Let UN (M/N) denote the universal norm group for an extension of fields M/N.

Lemma 4.2. We have

UN ((K_{∞}^{−})_{𝔭}/ _{p}) = ⟨\frac{π}{π̄} ⟩ × μ_{p−1} if h = 1,
UN ( _{p}^{cycl}/ _{p}) = < p > ×μ_{p−1}.

By Lemma 4.2 and local class field theory we see that (K_{∞}^{−})_{𝔭} ≠ _{p}^{cycl}.

Lemma 4.3. If F_{∞} is a _{p}-extension of _{p} distinct from _{p}^{cycl} then

F_{∞} _{p}^{cycl} = _{p}^{unr} _{p}^{cycl}.

By Lemma 4.3 we have (K_{∞}^{−})_{𝔭} _{p}^{cycl} = _{p}^{unr} _{p}^{cycl}, and so it is an unramified extension of _{p}^{cycl} = (K_{∞}^{+})_{𝔭} as desired.

For the second statement note that L′_{∞} is the maximal abelian unramified pro-p extension of K_{∞}^{+} in which the primes lying above the primes of K_{∞}^{+} lying above p are completely split. Thus (L′_{∞})_{𝔭} = (K_{∞}^{+})_{𝔭} = _{p}^{cycl} proving that K_{∞}^{−}K_{∞}^{+} ⊄ L′_{∞} since the containment fails even locally.

We now show that Proposition 4.1 implies the first statement in Corollary 3.2. Let D(𝔭) be the decomposition group at 𝔭 in the extension L_{∞}/K. There is an exact sequence

0 → D = _{p} → D(𝔭) → D(𝔭_{∞}/𝔭) = _{p} → 0

where D(𝔭_{∞}/𝔭) = Γ^{+} is the decomposition group in the extension K_{∞}^{+}/K. On the other hand D(𝔭) maps surjectively to the decomposition group at 𝔭, namely Gal((K_{∞}^{−}K_{∞}^{+})_{𝔭}/K_{𝔭}), in the extension K_{∞}^{−}K_{∞}^{+}/K. By Lemma 4.3 this group is Gal( _{p}^{unr} _{p}^{cycl}/ _{p}) =^{∼} _{p} × _{p}. Thus D(𝔭) is an extension of _{p} by _{p} which has _{p}^{2} as a quotient and so D(𝔭) =^{∼} _{p} × _{p}. We conclude that Γ^{+} acts trivially on D =^{∼} _{p}, so that T has 0 as an eigenvalue. In particular the characteristic polynomial of T factors as

f_{χ}(T) = T · f_{X′}(T)

where X′ = X/D. This proves the first statement of Corollary 3.2.

We now turn to the proof of the second statement of the same corollary. For this we will need a few more propositions.

Proposition 4.4. We have X′ =^{∼} _{p}^{λ_{χ}−1}.

Proof. This follows from Theorem 2.1 and Proposition 4.1.

Let T X′ denote the image of X′ under the operator T. Denote by L′_{0} the fixed field of T X′. So we have the following picture.

[figure: L′_{∞}; T X′; L′_{0}; X′/T X′; K_{∞}^{+}; Γ^{+}; K]

Proposition 4.5. The abelian group X′/T X′ has finite cardinality (a power of p) and

|\frac{X′}{T X′} | ∼ a_{1}.

Recall that a ∼ b for two elements a and b in \overline{}_{p} if ord_{p}(a) = ord_{p}(b).

Proof. If G is an arbitrary group and N a normal subgroup such that N ⊂ Z(G) and G/N is (pro)-cyclic, then it is a simple group theoretic fact that G is abelian. Now X′/T X′ is the maximal quotient of X′ on which T acts by 0, that is, on which γ acts by the identity. Applying the group theoretic fact with N = X/T X′ and G/N = Γ^{+} we see that L′_{0} is an abelian extension of K.

The maximal pro-p abelian extension which is unramified outside 𝔭 and 𝔭̄ is K_{∞}^{−}K_{∞}^{+}. We do not prove this here, but its proof uses our assumption that the class number h of K is one. So we must have K_{∞}^{+} ⊂ L′_{0} ⊂ K_{∞}^{−}K_{∞}^{+}. But L′_{0} ≠ K_{∞}^{−}K_{∞}^{+}, since otherwise K_{∞}^{−}K_{∞}^{+} ⊂ L′_{∞} contradicting the second statement in the proof of Proposition 4.1. Since Gal(K_{∞}^{−}K_{∞}^{+}/K_{∞}^{+}) = Γ′ and the only non-zero closed subgroups of _{p} are of finite index, we see that L′_{0}/K_{∞}^{+} is a finite extension. Thus X′/T X′ has finite cardinality.

The formula for the valuation of the cardinality now follows easily. One has

|\frac{X′}{T X′} | = det(T : X′ → X′) = ±f_{X′}(0) = ±a_{1}

as desired

The following proposition gives another formula for the valuation of a_{1}.

Proposition 4.6. We have

|\frac{X′}{T X′} | ∼ \frac{log_{p}(π/π̄)}{log_{p}(1 + p)}.

The following lemma will be used in the proof of Proposition 4.6.

Lemma 4.7. Say that F_{∞} = ∪_{n=1}^{∞}F_{n} is a _{p}-extension of _{p}. If F_{∞}∩ _{p}^{cyl} = F_{n} for some n then

p^{n} ∼ \frac{log_{p}(q)}{log_{p}(1 + p)}

where

UN (F_{∞}/ _{p}) = < q > × μ_{p−1}.

Proof of Lemma 4.7. Note UN ( _{p}^{cylc}/ _{p}) =< p > ×μ_{p−1}. By local class field theory theory the norm subgroup of the intersection F_{n} of F_{∞} and _{p}^{cyl} corresponds to the group generated by the individual universal norm subgroups, namely

< p, q > ×μ_{p−1}.

Thus

\frac{_{p}^{×}}{< p, q > ×μ_{p−1}} →^{∼} Gal(F_{n}/ _{p}).

Since

log_{p} (\frac{_{p}^{×}}{< p > ×μ_{p−1}} ) →^{∼} p _{p},

we have

[F_{n} : _{p}] = p^{n} = |\frac{p _{p}}{log_{p}(q)}| ∼ \frac{log_{p}(q)}{p} = \frac{log_{p}(q)}{log_{p}(1 + p)},

as desired.

Proof of Proposition 4.6. We have already shown that L′_{0}/K_{∞}^{+} is a finite subextension of K_{∞}^{−}K_{∞}^{+}. Thus we may write L′_{0} = K_{n}^{−}K_{∞}^{+} for a unique n. Since K_{∞}^{−} ∩ K_{∞}^{+} = K we have [L′_{0} : K_{∞}^{+}] = [K_{n}^{−} : K] = p^{n}. Now

(L′_{0})_{𝔭} = (K_{n}^{−})_{𝔭}(K_{∞}^{+})_{𝔭} = (K_{n}^{−})_{𝔭} _{p}^{cycl}.

On the other hand since the primes lying above 𝔭 are totally split in the extension L′_{0}/K_{∞}^{+} we have

(L′_{0})_{𝔭} = (K_{∞}^{+})_{𝔭} = _{p}^{cycl}.

Comparing the above two equations shows that (K_{n}^{−})_{𝔭} _{p}^{cycl} = _{p}^{cycl} so that

(K_{∞}^{−})_{𝔭} ∩ _{p}^{cycl} = (K_{n}^{−})_{𝔭}.

The proposition now follows from Lemma 4.7, noting that the generator q in the universal norm subgroup for the localization of the anti-cyclotomic p-extension in _{p} is given by q = \frac{π}{π̄} (see Lemma 4.2).

The second statement in Corollary 3.2 follows from Propositions 4.5 and 4.6. Indeed we have already remarked that λ_{χ} = 1 ⟺ a_{1} = 1, which by the two propositions just mentioned is equivalent to log_{p}(q)/ log_{p}(1+p) ∈ _{p}^{×}. As is evident from the algebraic manipulations carried out towards the end of the original proof of Corollary 3.2, this is equivalent to π̄ ∉ ( _{p})^{×}.

This completes the direct proof of Corollary 3.2 (under the assumption that the class number of K is one).

5 p-adic L-functions of elliptic curves

Let E be an elliptic curve defined over with good reduction at p. Assume that the reduced elliptic curve Ẽ is ordinary, which can be defined by either one of the following two equivalent conditions:

1. p | a̸_{p} where a_{p} = 1 + p − Ẽ( _{p}); this means that if α and β are the two roots of the polynomial X^{2} − a_{p}x + p then

ord_{p}(α) = 0 and ord_{p}(β) = 1.

2. Ẽ[p^{∞}] = _{p}/ _{p} (as compared with E[p^{∞}] = ( _{p}/ _{p})^{2}); thus the Tate module V_{p} has a filtration of G_{p}-equivariant subspaces

0 ⊂ F^{1}V_{p} ⊂ V_{p}  (5.1)

where F^{1}V_{p} = ker(V_{p}(E) → V_{p}(Ẽ)) is the kernel of the reduction map on the Tate modules, such that I_{p} acts by the cyclotomic character χ_{p} on F^{1}V_{p} and acts trivially on V_{p}/F^{1}V_{p}.

Then Mazur and Swinnerton-Dyer have constructed a p-adic L-function L_{p}(s, E) attached to E which is characterized by the following three properties:

• L_{p}(s, E) = g_{E}((1 + p)^{s−1} − 1) where g_{E}(T) ∈ \frac{1}{p^{a}} _{p}[[T]] (actually it is conjectured that g_{E} never has denominators, that is a = 0, and this is known if E[p] is irreducible).

• If L(s, E) is the usual Hasse-Weil L-function attached to E and Ω_{E} the Neron period of E then one has the interpolation property

L_{p}(1, E) = (1 − \frac{β}{p} )^{2} \frac{L(1, E)}{Ω_{E}}.

• If Γ =< γ > is the Galois group of the cyclotomic _{p} extension of then for each finite order Dirichlet character ϕ ∈ Γ̂ one has

g_{E}(ζ − 1) = α^{−m}G(ϕ^{−1})\frac{L(1, E, ϕ)}{Ω_{E}}

where ζ = ϕ(γ), the conductor of ϕ is p^{m}, G(ϕ^{−1}) denotes the Gauss sum of ϕ^{−1}, and L(s, E, ϕ) is the L-function of E twisted by ϕ.

We see that if E has good ordinary reduction at p then L_{p}(s, E) does not have a trivial zero at s = 1 since the Hasse-Weil bounds imply that β ≠ p.

On the other hand let us assume that E has multiplicative reduction at p (that is the conductor of E is divisible by p but not by p^{2}). Then |a_{p}| = 1. In this case Mazur, Tate and Teitelbaum have constructed a p-adic L-function L_{p}(s, E), the interpolation property at s = 1 now being

L_{p}(1, E) = (1 − \frac{a_{p}}{p} ) \frac{L(1, E)}{Ω_{E}}.

Thus L_{p}(s, E) has a trivial zero at s = 1 exactly when a_{p} = 1, that is when E has split multiplicative reduction at p. Over _{p}, E is a Tate curve and

E( _{p}) =^{∼} \overline{}_{p}^{×}/q_{E}  (5.2)

where the Tate period q_{E} = p^{a}u ∈ _{p}^{×} with a ≥ 1 and u ∈ _{p}^{×}. Define

L_{E} = \frac{log_{p}(q_{E})}{ord_{p}(q_{E})}

which we call the L-invariant of E. We have the following theorem.

Theorem 5.3 (Greenberg-Stevens). Assume that p ≥ 5 and E has split multiplicative reduction at p. Then

L′_{p}(1, E) = L_{E} · \frac{L(1, E)}{Ω_{E}}.

We point out the obvious similarity of this formula with (1.2) and (1.3).

6 p-adic L-functions attached to motives

Let V = {V_{ℓ}} denote a compatible system of ℓ-adic representations arising from a motive over . The V_{ℓ} are _{ℓ} vector spaces with an action of the full Galois group G_{}of and have dimension d independent of ℓ.

To V one may attach a complex analytic L-function

L(s, V) = ∏_{p} L^{(p)}(s, V)

where

L^{(p)}(s, V) = det (1 − Frob_{p}|_{V_{ℓ}^{Ip}})^{−1}

is the local factor at p (and is conjecturally independent of ℓ).

Let V^{∗} = V_{ℓ}^{∗} denote the dual of V, where

V_{ℓ}^{∗} = Hom(V_{ℓ}, _{ℓ}(1))

and G_{}acts on _{ℓ}(1) by the cyclotomic character. One may attach Γ-factors Γ_{V} (s) and Γ_{V^{∗}}(s) so that the completed L-functions satisfy a functional equation relating L(s, V) and L(2 − s, V^{∗}). We assume that L(1, V) is a critical values in the sense of Deligne, that is neither Γ_{V} (s) and Γ_{V^{∗}}(s) has a pole at s = 1. Then Deligne constructs a period Ω_{V} such that

\frac{L(1, V)}{Ω_{V}} ∈ \overline{}.

We now assume that V has ‘good reduction at p’, that is the inertia subgroup I_{p} acts trivially on V_{ℓ} when ℓ ≠ p. Then we may write

L^{(p)}(V, s) = ∏_{j=1}^{d}(1 − β_{j}p^{−s})^{−1}.

for the eigenvalues β_{j} of Frob_{p}. In such case V^{∗} also has good reduction at p - denote the eigenvalues of Frob_{p} on V_{ℓ}^{∗} by β_{j}^{∗} for j = 1, …, d. Note that

β_{j}^{∗} = pβ_{j}^{−1}  (6.1)

for each j = 1, …, d.

Let us assume that V_{p} is Hodge-Tate. We will not define what this means. What is relevant here is that one can associate to V_{p} certain Hodge numbers h_{n} with n ∈ such that ∑ h_{n} = d.

We also assume that V is ordinary at p. That is we assume that ‘the Newton polygon is equal to the Hodge polygon’, or, more concretely,

|{β_{j} | ord_{p}(β_{j}) = n}| = h_{n},

for each n ∈ . (WHY ARE THE EIGENVALUES OF INTEGRAL VAL-UATION?)

There is a second way to define what it means for V to be ordinary at p. In this definition one requires V_{p} to have a decreasing filtration F^{n}V_{p} consisting of G_{p}-stable spaces such that

I_{p} acts on \frac{F^{n}V_{p}}{F^{n+1}V_{p}} by χ_{p}^{n}.  (6.2)

Is is known that if V is ordinary at p in the latter sense then V_{p} is Hodge-Tate with

h_{n} = dim_{_{p}} (\frac{F^{n}V_{p}}{F^{n+1}V_{p}} )

for all n and V_{p} is ordinary at p in the former sense. Also, if V arises from the étale cohomology groups of a smooth projective variety over then it is known that the two definitions of ordinarity at p are equivalent.

One might ask whether, under the hypothesis of good reduction at p and ordinarity at p (in the second sense defined above) it is possible to recover the eigenvalues of Frob_{p} purely from V_{p} (one knows of course how to recover these eigenvalues from V_{ℓ} for ℓ ≠ p where I_{p} acts trivially). This is indeed possible. By (6.2) I_{p}. acts trivially on

\frac{F^{n}V_{p}}{F^{n+1}V_{p}} ⊗ χ_{P}^{−n}

Let α_{i}(n) for i = 1, … r_{n} denote the eigenvalues of Frob_{p} on the twisted space above. Note that since G_{p} is compact and these eigenvalues must be bounded, and in fact must therefore be units. Then we have

{β_{j} | ord_{p}(β_{j}) = n} = {p^{n}α_{i}(n) | i = 1, …, r_{n}},

for each n.

Under the assumptions of good reduction at p and ordinarity at p we conjecture the existence of a p-adic L-function L_{p}(s, V) where

L_{p}(s, V) = g_{V} ((1 + p)^{s−1} − 1)

for g_{V} (T) ∈ O[[T]] for some p-adic ring O/ _{p} (in general g may be a quotient of power series but we ignore this point here) such that L_{p}(s, V) satisfies the interpolation property

L_{p}(1, V) = ∏_{j} (1 − \frac{β_{j}}{p} ) ∏_{k} (1 − \frac{β_{k}^{∗}}{p} ) \frac{L(1, V)}{Ω_{V}}.  (6.3)

Here the product on β_{j} is over all j such that ord_{p}(β_{j}) ≥ 1 and similarly the product on β_{k}^{∗} runs over all k such that ord_{p}(β_{k}^{∗}) ≥ 1. In particular (6.1) implies that there are exactly d Euler factors in the interpolation property above.

7 Trivial Zeroes

Let V be a motive over and let p be a prime of good ordinary reduction. The interpolation property (6.3) implies that the p-adic L-function L_{p}(s, V) attached to V may have a ‘trivial-zero’ at s = 1 arising from the vanishing of one of the Euler factors in the interpolation property. We have:

Conjecture 7.1. Assume that V has good ordinary reduction at p. L_{p}(s, V) has a trivial zero at s = 1 if and only if either

\frac{F^{0}V_{p}}{F^{1}V_{p}} or \frac{F^{0}V_{p}^{∗}}{F^{1}V_{p}^{∗}}

contains the trivial representation.

The above conjecture is true in many interesting examples where the padic L-function has been already constructed. We describe three examples now.

Example 1) This is the Kubota-Leopoldt case which we have already discussed in some detail. Recall that χ was an odd quadratic character and p was an odd prime such that χ(p) = 1. The fact that p was split implied that the the Kubota-Leopoldt p-adic L-function L_{p}(s, ωχ) had a simple zero at s = 0.

Consider the motive V defined by V_{ℓ} = _{ℓ}(χ) by which we mean the one dimensional vector space over _{ℓ} on which Gal(\overline{}/ ) acts by χ. V_{p} has the following filtration of G_{p} = Gal(\overline{}_{p}/ _{p}) stable spaces:

F^{0}V_{p} = V_{p}
F^{1}V_{p} = 0

and F^{0}V_{p}/F^{1}V_{p} = _{p} is the trivial G_{p}-representation since Frob_{p} acts by χ(p) = 1.

Example 2) We consider the case of an elliptic E curve over with split multiplicative reduction at p. Recall that in this case the p-adic L-function of Mazur, Tate and Teitelbaum L_{p}(s, E) has a simple zero as s = 1.

Let q_{E} be the associated Tate period. By (5.2) we see that the p^{∞}-torsion over \overline{}_{p} is given by

E[p^{∞}] = < μ_{p^{∞}}, ^{p^{∞}} \sqrt{q_{E}} > .

Thus, there is an exact sequence:

1 → μ_{p^{∞}} → E[p^{∞}] → _{p}/ _{p} → 1.  (7.2)

Let V_{ℓ} = V_{ℓ}(E) be the usual Tate module of E. By (7.2) we have the following sequence of Gal(\overline{}/ )-modules:

1 → _{p}(1) → V_{p}(E) → _{p} → 1.

Thus V_{p} = V_{p}(E) has the following filtration of G_{p}-stable subspaces:

F^{0}V_{p} = V_{p}
F^{1}V_{p} = _{p}(1)
F^{2}V_{p} = 0

and again F^{0}V_{p}/F^{1}V_{p} = _{p} is the trivial G_{p}-representation.

Example 3) We consider the case of the adjoint motive attached to an elliptic curve over with good ordinary reduction at p. In this case a p-adic L-function has been constructed by Coates and Schmidt and it is known to have a trivial zero at s = 1.

We have (see (5.1)) that V_{p}(E) restricted to G_{p} can be represented matricially by

(\begin{matrix}χ_{p}ϕ^{−1} & ∗ \\ 0 & ϕ \end{matrix})

Thus the action of G_{p} on Sym^{2}(V_{p}(E)) is given by

(\begin{matrix}χ_{p}^{2}ϕ^{−2} & ∗ & ∗ \\ 0 & χ_{p} & ∗ \\ 0 & 0 & ϕ^{2} \end{matrix})

Let V be the adjoint motive attached to E. Thus V_{p} = Sym^{2}(V_{p}(E))(−1) and the action of G_{p} is given by

(\begin{matrix}χ_{p}ϕ^{−2} & ∗ & ∗ \\ 0 & 1 & ∗ \\ 0 & 0 & χ_{p}^{−1}ϕ^{2} \end{matrix})

Thus V_{p} has a natural filtration of G_{p}-stable subspaces:

F^{−1}V_{p} = V_{p} ⊃ F^{0}V_{p} ⊃ F^{1}V_{p} ⊃ F^{2}V_{p} = 0

with F^{0}V_{p}/F^{1}V_{p} = _{p} the trivial G_{p}-representation as predicted by Conjecture 7.1.

8 Selmer groups and the Main conjecture

Let V be a motive over with ordinary reduction at p. (EXPLAIN WHY WE DROP THE GOOD REDUCTION AT p HYPOTHESIS) One may (conjecturally) attach a p-adic L-function L_{p}(s, V) to V where

L_{p}(s, V) = g_{V} ((1 + p)^{s−1} − 1)

for some power series g_{V} (T).

One may also attache a Selmer group (over _{∞}) to V. This is a cohomologically defined _{p}-module which we shall denote by Sel_{V} ( _{∞}). The main conjecture predicts that

(g_{V} (T)) = characteristic ideal of Sel_{V} ( _{∞})

where Sel_{V} ( _{∞}) is the Pontryagin dual of Sel_{V} ( _{∞}).

Thus a trivial zero for the p-adic L-function should have algebraic consequences reflected in the structure of the Selmer group. This has already been discussed in some detail in Sections 1 to 4 in the Kubota-Leopoldt situation where the (dual) of the Selmer group turns out to the Iwasawa module X.

In this section we define the Selmer group in the Kubota-Leopoldt case as well as in the other examples involving elliptic curves considered in the previous section. We then describe properties of these Selmer groups that hold independently of, but are predicted by, the main conjecture.

Example 1) Recall the set up in the Kubota-Leopoldt case: χ is an imaginary quadratic character and p is an odd prime such that χ(p) = 1. Also Γ = Gal( _{∞}/ ) and Δ = Gal(K/ ) where K is the imaginary quadratic field corresponding to χ. We set as usual K_{∞} = K _{∞}.

Set A = _{p}(χ)/ _{p}(χ). Then the inflation-restriction sequence yields an isomorphism:

H^{1}(Gal(\overline{}/ _{∞}), A) →^{∼} H^{1}(Gal(\overline{}/K_{∞}), A)^{Δ}

since Δ has order 2 and p is an odd prime. But Gal(\overline{}/K_{∞}) acts trivially on A so

H^{1}(Gal(\overline{}/K_{∞}), A)^{Δ} = Hom_{Δ}(Gal(\overline{}/K_{∞})^{ab}, A).

Recall that L_{∞} is the maximal unramified abelian pro-p extension of K_{∞}. Let I_{v} and D_{v} denote the inertia, respectively decomposition, subgroups of Gal(\overline{}/ _{∞}) at a prime v of _{∞}. Define the Selmer group attached to A by

Sel_{A}( _{∞}) := ker (H^{1} (Gal(\overline{}_{∞}/ _{∞}), A) → ∏_{v} H^{1} (I_{v}, A))
= Hom_{Δ}(X, A)
= Hom(X^{χ}, A)
= Hom(X, A)

since X^{χ} = X. Taking Pontryagin duals we get

Sel_{A}( _{∞}) = X.

Note that Gal( _{∞}/ ) acts on Sel_{A}( _{∞}) making it, and therefore X as well, into a Γ-module.

Recall that L′_{∞} was the maximal unramified abelian pro-p extension of K_{∞} in which every prime of K_{∞} splits completely and X′ = Gal(L′_{∞}/K_{∞}). So if we define the strict Selmer group attached to A by using the decomposition subgroups D_{v} instead of the inertia subgroups I_{v} then one sees:

Sel_{A}^{str}( _{∞}) := ker (H^{1} (Gal(\overline{}_{∞}/ _{∞}), A) → ∏_{v} H^{1} (D_{v}, A))
= Hom_{Δ}(X′, A)
= Hom(X′, A).

Recall that D = Gal(L_{∞}/L′_{∞}) so that X′ = X/D. We conclude that

\frac{Sel_{A}( _{∞})}{Sel_{A}^{Str}( _{∞})} = Hom(D, A) = \begin{matrix} & p \\ & p \end{matrix}

with trivial Γ action (since Γ acts trivially on D, as we showed earlier, and Γ acts trivially on A).

Taking Pontryagin duals we see that X = Sel_{A}( _{∞}) has _{p} as a submodule on which Γ acts trivially. Thus we recover what we had proved earlier about X (see Corollary 3.2) rephrased in the language of Selmer groups.

Example 2) Let E be an elliptic curve with good ordinary reduction or split multiplicative reduction at p. Set

E[p^{∞}] := \frac{V_{p}(E)}{T_{p}(E)}

and let F^{+}E[p^{∞}] =

\begin{cases}ker(E[p^{∞}] → Ẽ[p^{∞}]) & if E has good odinary reduction at p, \\ μ_{p^{∞}} & if E has split multiplicative reduction at p. \end{cases}

Let π denote the unique prime of _{∞} lying above p. Define the Selmer group attached to E[p^{∞}] by

Sel_{E[p^{∞}]}( _{∞}) = ker (H^{1} (Gal(\overline{}_{∞}/ _{∞}), E[p^{∞}])

⟶ H^{1} (I_{π}, \frac{E[p^{∞}]}{F^{+}E[p^{∞}]}) × ∏_{v |̸p} H^{1} (I_{v}, E[p^{∞}]) ).

As usual Gal( _{∞}/ ) acts on Sel_{E[p^{∞}]}( _{∞}) and its Pontryagin dual.

Define the strict Selmer group Sel_{E[p^{∞}]}^{str}( _{∞}) attached to E[p^{∞}] as in Example 1 replacing the inertia groups throughout by decomposition groups. (It turns out that the only thing that matters is the replacement of I_{π} by D_{π}; replacing the other inertia groups by the corresponding decomposition groups has no effect).

Now suppose that we are in the good ordinary reduction case. Then it is known that the strict Selmer group coincides with Selmer group and that both these groups are isomorphic to the p-primary part of the classical Selmer group Sel_{E}( _{∞}) attached to E.

On the other hand if we are in the split multiplicative reduction case then the strict Selmer group coincides with the classical Selmer group, but Sel_{E[p^{∞}]}^{str}( _{∞}) ⊂ Sel_{E[p^{∞}]}( _{∞}) and

\frac{Sel_{E[p^{∞}]}( _{∞})}{Sel_{E[p^{∞}]}^{str}( _{∞})} =^{∼} \begin{matrix} & p \\  & p \end{matrix}

with trivial Γ-action. Taking Pontryagin duals we see that Sel_{E[p^{∞}]}( _{∞}) has a copy of _{p} as a submodule with trivial Γ-action. This is exactly what is predicted by the trivial zero of the p-adic L-function via the main conjecture.

Example 3) The adjoint case is similar to other two cases.

9 L-invariants

We start with some general remarks regarding how one should think of the L-invariant of an arbitrary _{p}-extension of _{p}.

Let _{p}^{unr} denote the unramified _{p}-extension of _{p} and let _{p}^{cycl} denote the cyclotomic _{p}-extension of _{p}. These are linearly disjoint and their composition contains all _{p}-extensions of _{p}. We have natural projections induced by restriction:

Gal( _{p}^{unr} _{p}^{cycl}/ _{p}) → Gal( _{p}^{unr}/ _{p}) →^{∼} _{p},
Gal( _{p}^{unr} _{p}^{cycl}/ _{p}) → Gal( _{p}^{cycl}/ _{p}) →^{∼} p _{p},

where the former identification with _{p} is via the p-adic ordinal ord_{p} and the latter identification is via the p-adic logarithm log_{p}. Thus the function ord_{p} and log_{p} give coordinates on _{p}^{2}, which may be identified with the Galois group of the composition of all _{p}-extensions of _{p}.

Now let K_{∞} be an arbitrary _{p} extension of _{p}. Say that the universal norm subgroup

UN (K_{∞}/ _{p}) = < q > × μ_{p−1}

with q ∈ _{p}^{×}. We have the reciprocity map

Rec : _{p}^{×} → Gal( _{p}^{unr} _{p}^{cycl}/ _{p})

and local class field theory says that Rec(q) generates Gal( _{p}^{unr} _{p}^{cycl}/K_{∞}). We define the L invariant attached to the _{p}-extension K_{∞} of _{p} by

L = \frac{log_{p}(q)}{ord_{p}(q)}.

Thus the L-invariant should be thought of as the ‘slope of the line corresponding to K_{∞}’.

Now let V be a motive over with ordinary reduction at p and let L_{p}(s, V) be the associated p-adic L-function. We have seen in various examples that whenever L_{p}(s, V) has a simple zero at s = 1 then there is a formula of the form

L′_{p}(1, V) = L_{V} · \frac{L(1, V)}{Ω_{V}}

for an explicit number L_{V}, the so called L-invariant of V. We now show that, at least in our three running examples, one may define L_{V} cohomologically.

Example 1) Recall that A = _{p}(χ)/ _{p}(χ) where χ is an odd character and p is odd prime with χ(p) = 1. Let

D := H^{1}(Gal(\overline{}/ ), A)_{div}.

Then D →^{∼} H^{1}(Gal(\overline{}/K), A)_{div}^{Δ} = Hom_{Δ}(Gal(\overline{}/K)^{ab}, A)_{div} where the isomorphism is induced as usual from the the inflation-restriction sequence and the equality follows from the fact that Gal(\overline{}/K) acts trivially on A. Let K_{∞}^{−} denote the anti-cyclotomic _{p}-extension of K. Since the non-trivial element of Δ acts by −1 on Gal(K_{∞}^{−}/K) and by −1 on A we see that

D =^{∼} Hom(Gal(K_{∞}^{−}/K), A) = _{p}/ _{p}.

On the other hand the action of Gal(\overline{}_{p}/ _{p}) on A is trivial. So

H^{1}(Gal(\overline{}_{p}/ _{p}), A) = Hom(Gal( _{p}^{unr} _{p}^{cycl}/ _{p}), _{p}/ _{p}) = ( _{p}/ _{p})^{2}

since every homomorphism of Gal(\overline{}_{p}/ _{p}) to _{p}/ _{p} must factor through the Galois group of the maximal abelian pro-p extension of _{p}.

Now under the restriction map D maps to the above cohomology group and the image of D may be identified with

Hom(Gal(K_{∞}/ _{p}), _{p}/ _{p})

for a _{p}-extension K_{∞} of _{p}. This _{p}-extension must be the completion of K_{∞}^{−} at 𝔭. If the class number of K is one then 𝔭 = (π). By Lemma 4.2 we see that q_{χ} = π/π̄ generates the torsion-free part of the universal norm subgroup. The L-invariant of K_{∞} is just

L = \frac{log_{p}(q_{χ})}{ord_{p}(q_{χ})}

which coincides with, and gives a cohomological interpretation of, the Linvariant L_{χ} attached to χ defined earlier.

Example 2) Let E be an elliptic curve over with split multiplicative reduction at p. Let us assume that L(1, E) ≠ 0. Then it is known that the classical Selmer group of E over is finite (since both E( ) and the Tate-Shafarevich group are finite).

Let H_{∗}^{1}(Gal(\overline{}/ ), E[p^{∞}]) be the set of all cocycles that are locally trivial at all primes except possibly at p. (IS THIS CORRECT?) Set

D := H_{∗}^{1}(Gal(\overline{}/ ), E[p^{∞}])_{div}.

It is a fact that D =^{∼} _{p}/ _{p}. On the other hand one has maps

D → H^{1}(Gal(\overline{}_{p}/ _{p}), E[p^{∞}]) → H^{1}(Gal(\overline{}_{p}/ _{p}), _{p}/Z_{p}) = ( _{p}/ _{p})^{2}

where the second map is induced by the exact sequence (7.2) and the equality has already been encountered in Example 1 above.

The image of D = _{p}/ _{p} is non-zero and as in Example 1 coincides with Hom((Gal(K_{∞}/ _{p}), _{p}/ _{p}) for some _{p}-extension K_{∞} of _{p}. It is a fact that the Tate period q_{E} satisfies

q_{E} ∈ UN (K_{∞}/ _{p}) =< q > ×μ_{p−1}, say.

Thus the L-invariant of K_{∞} is just

L = \frac{log_{p}(q)}{ord_{p}(q)} = \frac{log_{p}(q_{E})}{ord_{p}(q_{E})} = L_{E}.

This gives the desired cohomological interpretation of L_{E}.

Example 3) Let E be an elliptic curve with good ordinary reduction or multiplicative reduction at p. Let A = Ad(V_{p}(E))/Ad(T_{p}(E)) =^{∼} ( _{p}/ _{p})^{3}. Let H_{∗}^{1}(Gal(\overline{}/ ), A) be all those cocycles which are locally trivial except possibly at p. A result of Flach says that Sel_{A}( ) is finite. This implies that

H_{∗}^{1}(Gal(\overline{}/ ), A)_{div} = ( _{p}/ _{p})^{2}.

We have seen that there is a filtration

A = F^{−1}A ⊃ F^{0}A ⊃ F^{1}A ⊃ F^{2}A = 0

with F^{0}A/F^{1}A = _{p}/ _{p} with trivial G_{p}-action. Define

F^{0}H_{∗}^{1}(Gal(\overline{}/ ), A)

to be the submodule of H_{∗}^{1}(Gal(\overline{}/ ), A) consisting of all cocycles which take vales in F^{0}A. Then

D := F^{0}H_{∗}^{1}(Gal(\overline{}/ ), A)_{div} = _{p}/ _{p}.

On the other hand there are natural maps

D → H^{1}(Gal(\overline{}_{p}/ _{p}), F^{0}A) → H^{1}(Gal(\overline{}_{p}/ _{p}), F^{0}A/F^{1}A) = ( _{p}/ _{p})^{2}.

Again the image of D coincides with Hom((Gal(K_{∞}/ _{p}), _{p}/ _{p}) for some p-extension K_{∞} of _{p}. Let q generate the torsion free part of the universal norm subgroup of K_{∞}. Set

L_{Ad(E)} = \frac{log_{p}(q)}{ord_{p}(q)}.

It is a fact that if E has multiplicative reduction at p then

L_{Ad(E)} = L_{E}.

Moreover Greenberg-Tilouine can prove the derivative formula (THOUGH IN THESE NOTES WE HAVE NOT DEFINED THE ADJOINT MOTIVE IN THE MULTIPLICATIVE REDUCTION CASE, WE HAVE NOT DE-FINED THE CORRESPONDING p-ADIC L-FUNCTION, AND WE HAVE NOT SHOWN THAT IT HAS A SIMPLE ZERO).

IF E has complex multiplication by K and good ordinary reduction at p then

L_{Ad(E)} = L_{χ}

where χ is the quadratic character corresponding to K.

Thus we see that L_{Ad(E)} collapses to L_{χ} or L_{E} in the above cases. When E has good ordinary reduction at p but does not have CM then it is not known that the cohomologically defined number L_{Ad(E)} is the one that appears in the derivative formula. So far one has not been able to check this numerically either.