NONVANISHING MODULO p OF VALUES OF L-FUNCTIONS

HARUZO HIDA

One of the general principles of the nature of values of arithmetically defined L-function is that the value gives

the size of a finite (or infinite) group related to the L-function.

Here are examples:

1. This is found by Dirichlet. Take the quadratic residue symbol χ(x) = (\frac{−D}{x}) for a positive square free integer D, the value L(0, χ) is equal to the class number of the field F = ℚ[\sqrt{−D}], where the class group is defined by the group of fractional ideals of F modulo principal ideals, and its order is the class number.

2. The second example is due to Kummer. For a prime power ℓ^{n}, consider all characters χ : (ℤ/ℓ^{n}ℤ)^{×} → ℂ^{×} with χ(−1) = −1 (odd characters), and the cyclotomic field F_{n} = ℚ(μ_{ℓ^{n}}) generated by the group of ℓ^{n}-th roots of unity. The class number h_{n}^{+} of the maximal real subfield F_{n}^{+} = F_{n} ∩ ℝ is a factor of the class number h_{n} of F_{n}. Then we have

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

3. The third example is a conjecture of Birch and Swinnerton-Dyer. Pick an elliptic curve E defined by an equation y^{2} = 4x^{3}−g_{2}x−g_{3} with g_{2}, g_{3} ∈ ℤ. For a ring R, define

E(R) = {(x, y) ∈ R^{2}|y^{2} = 4x^{3} − g_{2}x − g_{3}} ⊔ {∞}.

Then E(R) has a natural structure of an abelian group. Define L(s, E) = ∏_{p}(1 − a_{p}p^{−s} + p^{1−2s})^{−1}, where 1 − a_{p} + p = |E(𝔽_{p})|. Then writing the period of E as Ω = ∫_{E(ℝ)} \frac{dx}{y}, if L(s, E) has zero of order r, then rank E(ℚ) = r and the leading coefficient of the Taylor expansion of L(s, E)/Ω at s = 1 gives essentially the size of the Tate-Shafarevich group �(E) of E (made out of Galois cohomology groups associated to E).

The Perspective Seminar talk at UCLA, February 14th 2005.

Using L-values as a tool, one can get information on the group (for example, the class group). 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 proven by p-adic analysis of the L-values:

e_{n} = λn + μp^{n} + ν

for three constants λ, μ and ν. In 1979, Ferrero and Washington proved that μ = 0. Thus λ > 0 ⇔ p is irregular in Kummer’s sense.

When p ≠ ℓ, in 1978, Washington proved that e_{n} is bounded independently of n. 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 describe how to count the number of χ with L(0, χ) ≡ 0 mod p. We assume all primes to be odd.

1. Statement of the theorem

Consider 𝔾_{m} = Spec(ℤ[t, t^{−1}]). For any function λ : ℤ/Nℤ → ℂ with ∑_{j=0}^{N−1} λ(j) = 0, we consider the Dirichlet L-function:

L(s, λ) = ∑_{n=1}^{∞} λ(n)t^{n}n^{−s}|_{t=1}.

If we formally make s = 0, we get

L(0, λ) = ∑_{a=1}^{N} λ(a)t^{a} ∑_{j=0}^{∞} t^{Nj}|_{t=1} = \frac{∑_{a=1}^{N} λ(a)t^{a}}{1 − t^{N}} |_{t=1} =: Φ(t)|_{t=1}.

Thus Φ is a rational function of t (a fraction N(t)/D(t) for polynomials N(t) and D(t)). Since 1 − t divides only once 1 − t^{N} and ∑_{a=1}^{N} λ(a)t^{a}|_{t=1} = 0, the value at right-hand side makes sense, and Euler’s result in 1735 combined with the analytic continutation, the formula is valid. We suppose that λ is a character of (ℤ/Nℤ)^{×} with λ(−1) = −1. Take the integer ring \overline{ℤ} of the algebraic closure \overline{ℚ} of ℚ and fix a prime 𝔓 over p of \overline{ℤ}. Thus 𝔽 = \overline{ℤ}/𝔓 is an algebraic closure of 𝔽_{p}. We move around characters χ of (ℤ/ℓ^{n}ℤ)^{×} for n = 1, 2, …, ∞; so, we think of all continuous characters of ℤ_{ℓ}^{×}.

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

Here is an art of counting such χ.

NON-VANISHING MODULO p OF L-VALUES 3

2. Hecke operators

For any rational function ϕ of t, 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 an eigen function (a Hecke eigen function).

3. Measure associated to an Hecke eigenform

Pick an eigen function ϕ ∈ 𝔽(t) of U(ℓ). Write ϕ|U(ℓ) = aϕ, and assume that a ∈ 𝔽^{×}. We fix a primitive root ζ_{n} ∈ μ_{ℓ^{n}} so that ζ_{n+1}^{ℓ} = ζ_{n} for all n. We define an ℓ–adic measure dϕ on ℤ_{ℓ} in the following way: for a function f : ℤ/ℓ^{n}ℤ → 𝔽,

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

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^{1/ℓ}),

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

∫ 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).

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

To deal with the above sum which is no longer a rational function, we need a non-elementary input of algebraic geometry. To avoid this and to give some flavour of the counting technique, we pretend that Ψ is a rational function. Define ϕ|[ψ] = ∑_{ζ∈μ_{ℓ^{n}}} ψ(ζ)ϕ(tζ) for a function ψ : μ_{ℓ^{n}}(𝔽) → 𝔽, and prove

Theorem 4.1. Let ϕ ∈ 𝔽_{P} (t) be an eigenfunction with ϕ|U(ℓ) = aϕ (a ≠ 0) rational over a finite field 𝔽_{P} ⊂ 𝔽. Write P = 1 + uℓ^{m} for u prime to ℓ and suppose m > 0. Suppose that ϕ|[ψ] ≠ 0 for all characters ψ of μ_{ℓ^{∞}}(𝔽) ∩ 𝔽_{P}^{×} and put N_{1} = max_{ψ}(deg(div_{0}(ϕ|[ψ]))). Then the zero set Z = {χ ∈ Hom(Γ, μ_{ℓ^{∞}}(𝔽))| ∫_{Γ} χdϕ = 0} is finite. Moreover, |Z| is bounded by max(ℓ^{m}N_{1}, ℓ^{2m+1}).

Proof. Write 𝔽_{P} [χ] the subfield of 𝔽 generated by the values of a character χ ∈ Γ̂_{L}. We put 𝔊_{χ} = Gal(𝔽_{P} [χ]/𝔽_{P}). We have

Tr_{𝔽_{P} [χ]/𝔽_{P}}(χ(γ)) = \begin{cases}0 & if χ(γ) ∉ 𝔽_{P}, \\ |𝔊_{χ}| & if χ(γ) ∈ 𝔽_{P}. \end{cases} (4.1)

This follows from the fact that the ℓ^{n}-power roots of unity ζ has minimal equation over ℚ(μ_{ℓ}) given by X^{ℓ^{n−1}(ℓ−1)} + X^{ℓ^{n−1}(ℓ−2)} + ⋯ + 1 (so Tr_{𝔽_{P} [χ]/𝔽_{P}}(ζ) = 0). Since ϕ(t^{P}) = ϕ(t)^{P} for the Frobenius generator t ↦ t^{P} of 𝔊_{χ}, by computation, we get ∫_{Γ} χdϕ = 0 ⇔ ∫_{Γ} χ^{σ}dϕ = 0 for all σ ∈ 𝔊_{χ}.

Suppose ∫_{Γ} χdϕ = 0 and ord(χ) = ℓ^{n−1} with n ≥ 2m + 1; thus χ = Γ_{n−1} = Γ/Γ^{ℓ^{n−1}} ≅ μ_{ℓ^{n−1}}(𝔽). Then we have

0 = ∑_{σ∈𝔊_{χ}} χ^{σ}(z)^{−1} ∫_{Γ} χ^{σ}dϕ = ∑_{σ∈𝔊_{χ}} ∫_{Γ} χ(z^{−1}γ)^{σ}dϕ = ∑_{γ∈Γ_{n−1}} χ^{σ}(z^{−1}γ)ϕ(ζ_{n}^{γ})

= ∑_{γ∈Γ_{n−1}} Tr_{𝔽_{P} [χ]/𝔽_{P}}(χ(z^{−1}γ))ϕ(ζ_{n}^{γ}) =^{(4.1)} |𝔊_{χ}| ∑_{γ∈zχ^{−1}(𝔽_{P})}χ(z^{−1}γ)ϕ(ζ_{n}^{γ}),

where ζ_{n} is a generator of μ_{ℓ^{n}}(𝔽). Since γ ∈ χ^{−1}(𝔽_{P}) ≅ ℤ/ℓ^{m}ℤ, we have γ = 1 + ℓ^{n−m}u ∈ Γ_{n−1} for u ∈ ℤ_{ℓ}. Thus ψ_{χ}(ζ_{m}^{u}) = χ(1 + ℓ^{n−m}u) is a character, because (1 + ℓ^{n−m}u)(1 + ℓ^{n−m}v) = 1 + ℓ^{n−m}(u + v) in Γ_{n−1} (2n − 2m > n ⇔ n > 2m). Then if z ≡ 1 mod ℓ^{m},

0 = ∑_{γ∈χ^{−1}(𝔽_{P})}χ(γ)ϕ_{n}(ζ_{n}^{zγ}) ∑_{u∈ℤ/ℓ^{m}ℤ}ψ_{χ}(ζ_{m}^{u})ϕ_{n}(ζ_{n}^{z}ζ_{m}^{u}) = ϕ|[ψ_{χ}](ζ_{n}^{z}).

Thus ϕ|[ψ_{χ}] has ℓ^{n−m} zeros: {ζ_{n}^{z}|z ∈ Γ^{ℓ^{m−1}}}; so,

ℓ^{n−m} ≤ deg(div_{0}(ϕ|[ψ_{χ}])) ≤ N_{1}.

From this the bound follows.