A SIMPLE PROOF OF NON-VANISHING MODULO p OF DIRICHLET L–FUNCTIONS

HARUZO HIDA

Studying the p–primary part p^{e_{n}} of the class number h_{n} of the cyclotomic field ℚ(μ_{ℓ^{n}}) is a classical problem for primes p, ℓ. When p = ℓ, there is a formula by Iwasawa:

e_{n} = λn + μp^{n} + ν

for three constants λ, μ and ν. In 1979, Ferrero and Washington proved that μ = 0.

When p ≠ ℓ, in 1978, Washington proved that e_{n} is bounded independently of n. The two cases: p = ℓ (μ = 0) and p ≠ ℓ (boundedness) are philosophically close, although technically there is a difference. Washington achieved this by showing that Dirichlet L–values L(0, χ) does not vanish modulo p for almost all odd Dirichlet characters χ of ℓ– power conductor. A new proof of this fact later given by Sinnott (in the mid 1980’s) found to be more amenable to generalization. We describe briefly a “modular” version of the proof by Sinnott in the case of p ≠ ℓ, and then we will discuss in the colloquium talk how to generalize it to more general Hecke L–values of CM fields. We assume all primes to be odd.

1. Statement of the theorem

We consider the group scheme 𝔾_{m} = Spec(ℤ[t, t^{−1}]). We fix a Dirichlet character λ of (ℤ/Nℤ)^{×} with λ(−1) = −1 and two embeddings

ℂ ↩^{i_{∞}} \overline{ℚ} ↪^{i_{p}} \overline{ℚ}_{p}. We regard λ to have values in any one of the three fields. Consider a rational function:

Φ(t) = Φ_{λ}(t) = ∑_{n=1}^{∞} λ(n)t^{n} = \frac{∑_{a=1}^{N} λ(a)t^{a}}{1 − t^{N}} ∈ ℤ[t, t^{−1}]_{(p,t−1)} = O_{𝔾_{m},1},

where O_{𝔾_{m},1} is the stalk at the closed point (p, t−1). As Euler observed in 1735,

Φ(1) = L(0, λ).

The Number Theory Seminar talk at UC Irvine, May 8th 2003.

Thus L(0, λ) is a p–adic integer in the ring ℤ_{p}[λ] generated by the values of λ. Writing 𝔓 for the maximal ideal of the p–adic valuation ring of \overline{ℚ}_{p}, the theorem of Washington can be stated as follows:

Theorem 1.1. For almost all character χ : ℤ_{ℓ}^{×} → μ_{ℓ^{∞}}, L(0, λχ) ≢ 0 mod 𝔓.

By Kummer’s class number formula, writing h_{n}^{+} for the class number of the maximal real field of ℚ(μ_{ℓ^{n}}) and h_{n}^{−}h_{n}^{+} for the class number of ℚ(μ_{ℓ^{n}}), we have

h_{n}^{−} = 2ℓ^{n} ∏_{χ mod ℓ^{n}:χ(−1)=−1} (−\frac{1}{2}L(0, χ)) .

Then the boundedness of the p–primary part of h_{n}^{−} is clear, and that of h_{n} = h_{n}^{+}h_{n}^{−} actually follows from this (by an argument of Washington).

The modular proof of this theorem has several steps. (Step1) Hecke operators: Introduction of Hecke operators U(ℓ) acting

on rational functions on 𝔾_{m/\overline{𝔽}_{p}} and functions on μ_{ℓ^{∞}}. (Step2) Measure associated to Hecke eigenforms: Fix an isomorphism

ℤ_{ℓ} ≅ ℤ_{ℓ}(1) = \underleftarrow{lim}_{n}μ_{ℓ^{n}} given by a ↦ ζ^{a}. For Hecke eigenform

f|U(ℓ) = af with unit eigenvalue a ∈ \overline{𝔽}_{p}^{×}, construction of a measure df on ℤ_{ℓ}^{×} with ∫_{a+ℓ^{n}ℤ_{ℓ}} df ≑ f(ζ_{n}^{a}) for the image ζ_{n}^{a} of ζ^{a}

in μ_{ℓ^{n}}.

(Step3) Evaluation formula: For a character χ : Γ = ℤ_{ℓ}^{×}/μ_{ℓ−1} → \overline{𝔽}_{p}^{×}:

L(0, λχ^{−1}) ≑ ∫_{ℤ_{ℓ}^{×}} χdΦ = ∫_{Γ} χdΨ

for Ψ(ζ) = ∑_{ε∈μ_{ℓ−1}} Φ(ζ^{ε}).

(Step4) Zariski density: Regard Ψ as induced from the rational function Ψ̃(t_{ε}) = ∑_{ε∈μ_{ℓ−1}/{±1}} (Φ(t_{ε}) + Φ(t_{ε}^{−1})) on G = 𝔾_{m}^{μ_{ℓ−1}/{±1}} by pull-back under the embedding i : μ_{ℓ^{∞}} ↪ G given by ζ ↦ (ζ^{ε}). The Zariski density of i(μ_{ℓ^{∞}}) in a big subvariety in G defined by the equations of ε ∈ μ_{ℓ−1} (for example, if ℓ = 5, −ε = ε^{2} = −1 for ε = \sqrt{−1} gives t_{ε^{2}}t_{1} = 1) implies the vanishing of Ψ̃ and hence Φ (a contradiction) if ∫_{ℤ_{ℓ}^{×}} χdΦ = 0 for infinitely many characters. The Zariski density is borrowed from Sinnott.

We are going to describe each steps.

NON-VANISHING MODULO p OF DIRICHLET L–FUNCTIONS 3

2. Hecke operators

A little more generally, we start with 𝔾_{m} over a general ring A. Define a Hecke operator U(ℓ) acting on functions on 𝔾_{m} by

ϕ|U(ℓ)(t) = \frac{1}{ℓ} ∑_{ζ∈μ_{ℓ}} ϕ(ζt^{1/ℓ}).

Expanding ϕ as ∑_{n≫−∞} a(n, ϕ)t^{n}, we find a(n, ϕ|U(ℓ)) = a(nℓ, ϕ) and Φ|U(ℓ) = λ(ℓ)Φ. Hence Φ is a Hecke eigen function in O_{𝔾_{m},1}.

3. Measure associated to an Hecke eigenform

Pick an eigen function ϕ ∈ O_{𝔾_{m},1/A} of U(ℓ) for a ℤ_{p}–algebra A. Write ϕ|U(ℓ) = aϕ, and assume that a ∈ A^{×}. We define an ℓ–adic measure dϕ on ℤ_{ℓ} in the following way: for a function f : ℤ/ℓ^{n}ℤ → A,

∫ fdϕ = (ℓa)^{−n} ∑_{x∈ℤ/ℓ^{n}ℤ}f(x)ϕ(ζ_{n}^{x}),

where ζ_{n} = (ζ mod ℓ^{n}ℤ_{ℓ}(1)) which is a primitive ℓ^{n}–th root of unity. This is well defined independently of the choice of ℤ/ℓ^{n}ℤ through which f factors. Formally, we could have defined

∫ fdϕ(t) = (ℓa)^{−n} ∑_{x∈ℤ/ℓ^{n}ℤ}f(x)ϕ(ζ_{n}^{x}t) ∈ O_{𝔾_{m},1/A},

and then, ∫ fdϕ is its evaluation at t = 1: ∫ fdϕ(1). We then have for a primitive Dirichlet character χ : (ℤ/ℓ^{n}ℤ)^{×} → A^{×}

∫ χdϕ(t) = (ℓa)^{−n} ∑_{x} ∑_{m} χ(x)a(m, ϕ)(ζ_{n}^{x}t)^{m}

= (ℓa)^{−n} ∑_{m} ( ∑_{x∈ℤ/ℓ^{n}ℤ}χ(x)ζ_{n}^{mx}) t^{m} = G(χ)(ℓa)^{−n} ∑_{m} χ^{−1}(m)a(m, ϕ)t^{m},

where G(χ) is the Gauss sum: G(χ) = ∑_{x∈ℤ/ℓ^{n}ℤ} χ(x)ζ_{n}^{x} ≠ 0.

4. Evaluation formula

Applying the computation in the previous section to ϕ = Φ = ∑_{m=0}^{∞} λ(m)t^{m} and evaluating the result at t = 1, we find

∫ χdΦ = (ℓλ(ℓ))^{−n}G(χ)Φ_{χ^{−1}λ}(1) = (ℓλ(ℓ))^{−n}G(χ)L(0, χ^{−1}λ).

If a(m, ϕ) = 0 for m divisible by ℓ, the measure dϕ is supported on ℤ_{ℓ}^{×}. Thus taking N so that ℓ|N, we may assume that dΦ is supported on ℤ_{ℓ}^{×}.

Since any character χ : ℤ_{ℓ}^{×} → μ_{ℓ^{∞}} factors through Γ = ℤ_{ℓ}^{×}/μ_{ℓ−1}, we want to have a measure φ supported on Γ = ℤ_{ℓ}^{×}/μ_{ℓ−1} so that we have

∫_{Γ} χdφ = ∫_{ℤ_{ℓ}_{×}} χdΦ for all character χ of Γ.

The measure φ is not associated to a rational function like Φ, but if we allow functions on μ_{ℓ^{∞}}, φ is associated to a function Ψ close to Φ. Noting μ_{ℓ−1} ⊂ ℤ_{ℓ} acts such functions by ϕ(ζ) ↦ ϕ(ζ^{s}) (s ∈ ℤ_{ℓ}), we find that φ = dΨ for Ψ given by

Ψ(ζ) = ∑_{ε∈μ_{ℓ−1}} Φ(ζ^{ε}) = ∑_{ε∈μ_{ℓ−1}/{±1}}(Φ(ζ^{ε}) + Φ(ζ^{−ε})) if ℓ ∤ N.

5. Application of Zariski density

We now assume that A = \overline{𝔽}_{p}. We first assume that ∫_{Γ} χdφ = 0 (that is, L(0, χ^{−1}λ) ≡ 0 mod 𝔓) for all χ : Γ → μ_{ℓ^{∞}}. Then by orthogonality relation of characters, we find Ψ(ζ) = 0 for all ζ ∈ μ_{ℓ^{∞}}. Thus if the image of μ_{ℓ^{∞}} in 𝔾_{m}^{μ_{ℓ−1}/\{±1\}} by ζ ↦ (ζ^{ε})_{ε} is Zariski dense (over \overline{𝔽}_{p}), we find that Φ(t^{ε}) + Φ(t^{−ε}) = 0, which is impossible by a simple form of Φ. The Zariski density is proven by Sinnott using Artin’s theorem of algebraic independence of characters {ζ ↦ ζ^{a_{j}}}_{j=1,2,...,m} for a_{j} ∈ ℤ_{ℓ} linearly independent over End(𝔾_{m}) = ℤ.

Now we assume a weaker condition that we have an infinite sequence of characters {χ_{j}}_{j} of order ℓ^{n_{j}} with ∫_{Γ} χ_{j}dφ = 0, which implies by variable change: ∫_{Γ} χ_{j}(x)dφ(ax) = 0. For any field k of characteristic different from ℓ, we have the following formula for a primitive ℓ^{n}-th root of unity ζ_{n} with n ≥ 2 not in k:

Tr_{k[ζ_{n}]/k}(ζ_{n}^{x}) = 0 ⇔ ζ_{n}^{x} ∈ k.

This follows from the minimal polynomial of ζ_{n}: X^{ℓ^{n−1}(ℓ−1)}+X^{ℓ^{n−1}(ℓ−2)}+ ⋯ + 1 = 0 over ℤ. For simplicity, we assume that λ has values in 𝔽_{p}^{×}. Then applying the Frobenius automorphism F (x) = x^{p}, we find for χ = χ_{j} and n = n_{j}

0 = (∫_{Γ} χ(x)dφ(ax))^{p^{n}} = (∑_{u} χ(u)Ψ(ζ_{n}^{au}))^{p^{n}}

= ∑_{u} χ^{p^{n}}(u)Ψ(ζ_{n}^{aup^{n}}) = χ^{p^{n}}(p^{n})^{−1} ∫_{Γ} χ^{p^{n}}dφ(ax)

for all n. Thus taking the trace from the field 𝔽_{p}[χ_{j}] generated by the values of χ_{j} to 𝔽_{p}[μ_{ℓ}], we find that

∑_{u∈χ_{j}^{−1}(𝔽_{p}[μ_{ℓ}])}χ_{j}(u)Ψ(ζ_{n_{j}}^{au}) = 0

for all a ∈ Γ. The above sum only involves u ∈ Γ^{ℓ^{n_{j} −m}}/Γ^{ℓ^{n_{j}}}. Writing the order of the ℓ–primary part of (𝔽_{p}[μ_{ℓ}])^{×} as ℓ^{m} and taking n_{j} so that n_{j} ≥ 2m, we can identify the multiplicative group Γ^{ℓ^{n_{j} −m}}/Γ^{ℓ^{n_{j}}} with the additive one ℤ/ℓ^{m}ℤ by ℤ/ℓ^{m}ℤ ∋ v ↦ 1 + ℓ^{n_{j}−m}v = u. We can then write χ_{j}(u) = ζ_{m}^{b_{j}v} for b_{j} ∈ (ℤ/ℓ^{m}ℤ)^{×} and ζ_{n}^{au} = ζ_{n}^{a}ζ_{m}^{v}. Since {χ_{j}} is infinite, we may assume that b_{j} is a constant b. Then we have for any a ∈ ℤ_{ℓ}^{×}

∑_{v mod ℓ^{m}} ζ_{m}^{bv}Ψ(ζ_{n_{j}}^{a}ζ_{m}^{v}) = ∑_{v mod ℓ^{m}} ζ_{m}^{bv}Ψ|ζ_{m}^{v}(ζ_{n_{j}}^{a}) = 0,

where for f ∈ \overline{𝔽}_{p}(𝔾_{m}), f|x ∈ \overline{𝔽}_{p}(𝔾_{m}) is defined by f|x(t) = f(tx) for x ∈ 𝔾_{m}(\overline{𝔽}_{p}). Since i(⋃_{j} μ_{ℓ^{n_{j}}}^{×}) is still dense in the closure V of i(μ_{ℓ^{∞}}) in G, we conclude ∑_{v mod ℓ^{n_{0}}} ζ_{n_{0}}^{av}Φ|ζ_{n_{0}} = 0 on 𝔾_{m}, whose Taylor expansion at t = 1 does not vanish as easily computed.