Math 726: L-functions and modular forms Fall 2011

Lecture 36 : The p-adic L-function attached to Dirichlet characters.

Instructor: Henri Darmon Notes written by: Francesc Castella

A natural extension of this course would go on discussing the construction of p-adic avatars of each of the types of L-functions that have been studied so far. But given the limited time, we will have to content ourselves just with the following treatment of the p-adic analogue of the Dirichlet L-functions.

Let D > 1 be an integer prime to p, and let

χ : (ℤ/Dℤ)^{×} → ℂ^{×}

be a (non-trivial) primitive Dirichlet character of conductor D. Fix once an for all algebraic closures \overline{ℚ} and \overline{ℚ}_{p} of ℚ and ℚ_{p}, respectively, and a choice of embeddings ι_{∞} : \overline{ℚ} ↪ ℂ and ι_{p} : \overline{ℚ} ↪ \overline{ℚ}_{p}. Since the values of χ are all algebraic (in fact, roots of unity), we can thus alternately view χ as taking values in \overline{ℚ}_{p}^{×} ⊂ ℂ_{p}^{×}, as convenient.

Let ζ_{D} ∈ ℂ denote the primitive D-th root of unity ζ_{D} := exp(2πi/D). With a slight abuse of notation, we also let ζ_{D} denote the primitive D-th root of unity in ℂ_{p}^{×} given by ι_{p}ι_{∞}^{−1}(ζ_{D}). Recall that the Gauss sum attached to the pair χ and ζ_{D} is given by

τ(χ) = ∑_{a=0}^{D−1} χ(a)ζ_{D}^{−a},

and that the Dirichlet L-function of χ is defined for s ∈ ℂ with ℜ(s) > 1 by the infinite series

L(s, χ) = ∑_{n=1}^{∞} χ(n)n^{−s}.

The algebraicity of the values of L(s, χ) at the negative integers will follow from the following integral representation.

Lemma 1.

L(s, χ) = \frac{1}{Γ(s)} ∫_{0}^{∞} f_{χ}(t)t^{s} \frac{dt}{t},

where

f_{χ}(t) = \frac{1}{τ(χ^{−1})} ∑_{a=0}^{D−1} \frac{χ^{−1}(a)}{ζ_{D}^{−a}e^{t} − 1}.  (1)

Proof. It follows from a direct calculation, using the expression χ(a) = τ(χ) ∑_{b=0}^{D−1} ζ_{D}^{ba}χ̄(b). Since the ideas needed are quite analogous to those used in the proof of the integral representation for the Riemann zeta function, the details are left as an exercise to imitate correctly the proof of Proposition 1 from Lecture 31.

Corollary 1. For all n ∈ ℤ_{≥0}, the value

L(−n, χ) = (−1)^{n} (\frac{d}{dt})^{n} f_{χ}(t)|_{t=0} belongs to ℚ(ζ_{D}, χ).  (2)

Proof. See the proof of Corollary 1 from Lecture 31. The details are left as an exercise.

It will be convenient to rewrite (2) as

L(−n, χ) = (\frac{d}{dt})^{n} g_{χ}(t)|_{t=0}  (3)

where

g_{χ}(t) = f_{χ}(−t) = \frac{−1}{τ(χ^{−1})} ∑_{a=0}^{D−1} \frac{χ^{−1}(a)}{ζ_{D}^{a}e^{t} − 1}.

(The proof of this equality is not as trivial as it may seem.)

Remark 1. Since we assume χ ≠ 1, the function g_{χ}(t) in (1), which plays the role analogous to that of \frac{1}{e^{t}−1} for the Riemann zeta function (or the trivial character), is continuous at t = 0. (Recall or note that the same is not true for \frac{1}{e^{t}−1}, and we had to consider \frac{1}{e^{t}−1} − \frac{a}{e^{at}−1} for some a ∈ ℤ_{p}^{×} in order to remove the pole at the origin.)

Continuing with the parallel with the Riemann zeta function, set 1 + T = e^{t} and define

F_{χ}(T) := \frac{−1}{τ(χ^{−1})} ∑_{a=0}^{D−1} \frac{χ^{−1}(a)}{ζ_{D}^{a}(1 + T) − 1},

which will thus play the role here of \frac{1}{T} − \frac{a}{(1+T)^{a}−1}.

Lemma 2. The power series F_{χ}(T) belongs to O_{ℂ_{p}}[[T]].

Proof. First note that τ(χ^{−1}) is a unit in O_{ℂ_{p}}, since we are assuming p ∤ D. The claim thus follows from the expansion

\frac{1}{ζ_{D}^{a}(1 + T) − 1} = \frac{1}{ζ_{D}^{a} − 1 + ζ_{D}^{a}T} = \frac{1}{(ζ_{D}^{a} − 1)(1 + \frac{ζ_{D}^{a}T}{ζ_{D}^{a}−1})}

= \frac{1}{ζ_{D}^{a} − 1} (1 − \frac{ζ_{D}^{a}T}{ζ_{D}^{a} − 1} + (\frac{ζ_{D}^{a}T}{ζ_{D}^{a} − 1})^{2} − ⋯) .

Corollary 2. There exists a measure μ_{χ} ∈ D(ℤ_{p}, ℂ_{p}) satisfying

A_{μ_{χ}}(T) = F_{χ}(T).

It is characterised by the property that

∫_{ℤ_{p}} x^{n}dμ_{χ}(x) = L(−n, χ), for all n ∈ ℤ_{≥0}.

Proof. Repeating the same steps as in the proof of Theorem 3 in Lecture 34, we obtain

∫_{ℤ_{p}} x^{n}dμ_{χ}(x) = (\frac{d}{dt})^{n} (∫_{ℤ_{p}} e^{tx}dμ_{χ}(x))|_{t=0} = g_{χ}^{(n)}(t)|_{t=0}.

The result thus follows from (3).

As it was the case in the construction of Kubota-Leopoldt p-adic L-function, in order to obtain from μ_{χ} a well-behaved function of the p-adic variable s, we must consider its restriction to ℤ_{p}^{×}. Set

μ_{χ}^{[p]} := res_{ℤ_{p}^{×}}(μ_{χ}).

Proposition 1. The Amice transform of μ_{χ}^{[p]} is given by

A_{μ_{χ}^{[p]}}(T) = \frac{−1}{τ(χ^{−1})} ∑_{a=0}^{D−1} (\frac{χ^{−1}(a)}{ζ_{D}^{a}(1 + T) − 1} − \frac{χ^{−1}(a)}{ζ_{D}^{ap}(1 + T)^{p} − 1}) .

Proof. After Lemma 3 from Lecture 35, and using once more the fact that ℤ_{p}^{×} = ℤ_{p} ∖ pℤ_{p}, we can write

A_{μ_{χ}^{[p]}}(T) = A_{μ_{χ}}(T) − A_{1_{pℤp}·μ_{χ}}(T) = A_{μ_{χ}}(T) − A_{φψμ_{χ}}(T).

(Or if one likes it, A_{μ_{χ}^{[p]}}(T) = A_{(ψφ−φψ)μ_{χ}}(T).) Since by Lemma 4 in Lecture 35, the action of φ is described by T ↦ (1 + T)^{p} − 1, it remains to study the action of ψ on A_{μ_{χ}}(T). We are going to apply Lemma 5 from Lecture 35, and for that purpose we write

A_{μ_{χ}}(T) = \frac{−1}{τ(χ^{−1})} ∑_{a=0}^{D−1} \frac{χ^{−1}(a)}{ζ_{D}^{a}(1 + T) − 1}

= \frac{1}{τ(χ^{−1})} ∑_{a=0}^{D−1} χ^{−1}(a) ∑_{n=0}^{∞} ζ_{D}^{an}(1 + T)^{n}

= \frac{1}{τ(χ^{−1})} ∑_{n=0}^{∞}(∑_{a=0}^{D−1} χ^{−1}(a)ζ_{D}^{an})(1 + T)^{n}.

Then by the afore mentioned Lemma, we have

A_{ψμ_{χ}}(T) = \frac{1}{τ(χ^{−1})} ∑_{n=0}^{∞}(∑_{a=0}^{D−1} χ^{−1}(a)ζ_{D}^{apn})(1 + T)^{n}

= \frac{χ(p)}{τ(χ^{−1})} ∑_{n=0}^{∞}(∑_{a=0}^{D−1} χ^{−1}(a)ζ_{D}^{an})(1 + T)^{n}  (changing ap by a new a)
= χ(p)A_{μ_{χ}}(T).  (4)

In particular, it follows that

ψμ_{χ} = χ(p)μ_{χ}.  (5)

Finally, putting all this together we obtain

A_{μ_{χ}^{[p]}}(T) = A_{μ_{χ}}(T) − A_{φψμ_{χ}}(T)  (6)
= A_{μ_{χ}}(T) − χ(p)A_{φμ_{χ}}(T)  (by (4))
= A_{μ_{χ}}(T) − χ(p)A_{μ_{χ}}((1 + T)^{p} − 1)  (by the action of φ)

= \frac{−1}{τ(χ^{−1})} ∑_{a=0}^{D−1} \frac{χ^{−1}(a)}{ζ_{D}^{a}(1 + T) − 1} − χ(p)\frac{−1}{τ(χ^{−1})} ∑_{a=0}^{D−1} \frac{χ^{−1}(a)}{ζ_{D}^{a}(1 + T)^{p} − 1}

= \frac{−1}{τ(χ^{−1})} ∑_{a=0}^{D−1} (\frac{χ^{−1}(a)}{ζ_{D}^{a}(1 + T) − 1} − \frac{χ^{−1}(a)}{ζ_{D}^{ap}(1 + T)^{p} − 1}) ,

where the last equality follows from the change of variable ap goes to a new a. The result follows.

Define

L_{p}(s, χ) := ∫_{ℤ_{p}^{×}} x^{−s}dμ_{χ}(x).

Theorem 1. For every n ∈ ℤ_{≥0},

L_{p}(−n, χ) = (1 − χ(p)p^{n})L(−n, χ).

Proof. This follows from a computation in complete analogy with the proof of Theorem 1 of Lecture 35:

L_{p}(−n, χ) = ∫_{ℤ_{p}^{×}} x^{n}dμ_{χ}(x) = ∫_{ℤ_{p}} x^{n}dμ_{χ}(x) − ∫_{pℤ_{p}} x^{n}dμ_{χ}(x)
= ∫_{ℤ_{p}} x^{n}dμ_{χ}(x) − ∫_{ℤ_{p}} x^{n}d(φψμ_{χ})(x)
= ∫_{ℤ_{p}} x^{n}dμ_{χ}(x) − ∫_{ℤ_{p}}(px)^{n}d(ψμ_{χ})(x)
= ∫_{ℤ_{p}} x^{n}dμ_{χ}(x) − χ(p)p^{n} ∫_{ℤ_{p}} x^{n}dμ_{χ}(x)
= (1 − χ(p)p^{n})L(−n, χ),

where the penultimate equality follows from (5).

We end our discussion today by recalling the following classical formula of Dirichlet, whose p-adic analogue will be proven in the remaining last lecture.

Theorem 2 (Dirichlet). If χ is a primitive Dirichlet character of conductor D > 1, then

L(1, χ) = \frac{−1}{τ(χ)} ∑_{a=0}^{D−1} χ^{−1}(a) · log(1 − ζ_{D}^{a}),  (7)

where log denotes the principal branch of the logarithm:

log(z) = log |z| + iθ, if z = |z|e^{iθ} with −π ≤ θ < π.

Proof. Indeed, writting

L(s, χ) = ∑_{n=1}^{∞} \frac{χ(n)}{n^{s}} = \frac{1}{τ(χ̄)} ∑_{a=0}^{D−1} χ^{−1}(a) ∑_{n=1}^{∞} \frac{(ζ_{D}^{a})^{n}}{n^{s}}

and setting s = 1, we obtain

L(1, χ) = \frac{1}{τ(χ̄)} ∑_{a=0}^{D−1} χ^{−1}(a) ∑_{n=1}^{∞} \frac{(ζ_{D}^{a})^{n}}{n} = \frac{1}{τ(χ̄)} ∑_{a=0}^{D−1} χ^{−1}(a) · log(1 − ζ_{D}^{a}).

Remark 2. Notice how the formula (7) shows that the nature of the special value L(1, χ) depends critically on the parity of χ.

• If χ is odd (i.e. χ(−1) = −1), then for every pair a, a′ ∈ {0, 1, …, D−1} with a+a′ = 0 (mod D), the log | |-part in χ^{−1}(a) log(1−ζ_{D}^{a}) and χ^{−1}(a′) log(1−ζ_{D}^{a′}) cancel each other, and as a result we see from (7) that

the value L(1, χ) lies in \frac{πi}{τ(χ)}ℤ[χ];

that is, the special value L(χ, 1) is, up to a period (πi in essence), algebraic.

• If χ is even (i.e. χ(−1) = 1), then for every pair a, a′ ∈ {0, 1, …, D−1} with a+a′ = 0 (mod D), the arguments of χ^{−1}(a) log(1−ζ_{D}^{a}) and χ^{−1}(a′) log(1−ζ_{D}^{a′}) cancel each other, and (7) reduces to

L(1, χ) = \frac{−1}{τ(χ^{−1})} ∑_{a=0}^{⌈(D−1)/2⌉} χ^{−1}(a)\{log(1 − ζ_{D}^{a}) + log(1 − ζ_{D}^{−a})\}

= \frac{−1}{τ(χ^{−1})} ∑_{a=0}^{⌈(D−1)/2⌉} χ^{−1}(a) log((1 − ζ_{D}^{a})(1 − ζ_{D}^{−a}));

that is, (if D has at least two distinct prime factors) the special value L(1, χ) is a linear combination of logarithms of cyclotomic units in the real subfield ℚ(ζ_{D}+ζ_{D}^{−1}) ⊂ ℚ(ζ_{D}). (If D is prime, we get the logarithms of D-units in this field.)