Math 726: L-functions and modular forms Fall 2011

Lecture 37 : The proof of a p-adic class number formula.

Instructor: Henri Darmon Notes written by: Francesc Castella

We will end this course with the proof of a p-adic analogue of the class number formula of Dirichlet’s that was proven in the last Lecture. Before we can state it, we need to introduce a p-adic analogue of the classical logarithm.

Lemma 1. There exists a unique function

log_{p} : 𝔾_{m}(O_{ℂ_{p}}) ⟶ ℂ_{p}

satisfying the following two properties:

1. log_{p}(1 + t) = ∑_{n=1}^{∞}(−1)^{n+1} \frac{t^{n}}{n}, for all t ∈ O_{ℂ_{p}}^{×} with |t| < 1;

2. log_{p}(ab) = log_{p}(a) + log_{p}(b), for all a, b ∈ O_{ℂ_{p}}^{×}.

Proof. Consider the reduction map

𝔾_{m}(O_{ℂ_{p}}) ⟶ 𝔾_{m}(\overline{𝔽}_{p}).

Given x ∈ \overline{𝔽}_{p}, it lies in a finite extension of 𝔽_{p}, and so there exists a q = p^{m} such that x^{q} = x. The Teichmüller lift of x is then defined by

ξ_{x} := lim_{j→∞} x̃^{q^{j}} ∈ O_{ℂ_{p}}

where x̃ denotes an arbitrary lift of x in O_{ℂ_{p}} under the reduction map. Note that

• the expression ξ_{x} defines an element in O_{ℂ_{p}}, since {x̃^{q^{j}}}_{j} is a Cauchy sequence and O_{ℂ_{p}} is complete;

• the value ξ_{x} ∈ O_{ℂ_{p}} is independent of the chosen lift x̃; and

• it satisfies ξ_{x}^{q} = ξ_{x}, as is readily seen from its definition.

Now any x ∈ O_{ℂ_{p}}^{×} can be written as

x = ξ_{x} · (1 + t), for some t ∈ O_{ℂ_{p}} with |t| < 1,

(indeed, x and ξ_{x} have the same reduction) and we are led to set

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

This gives a function with the desired properties, and uniqueness follows immediately.

In order to evaluate the expression

L_{p}(1, χ) = ∫_{ℤ_{p}^{×}} x^{−1}dμ_{χ}(x)

we would need to consider the “measure” x^{−1}μ_{χ} on ℤ_{p}. This has a singularity at x = 0, leading to difficulties when we try to integrate the constant function 1 = \binom{x}{0} against it. But for n ≥ 1, the integration of \binom{x}{n} against x^{−1}μ_{χ} on ℤ_{p} causes no problem, and in fact

∫_{ℤ_{p}} \binom{x}{n}x^{−1}dμ_{χ}(x) = \frac{1}{n} ∫_{ℤ_{p}} \binom{x − 1}{n − 1}dμ_{χ}(x).  (1)

These considerations lead us to introduce the regularised Amice transform

Ã_{x^{−1}μ_{χ}}(T) = ∑_{n=1}^{∞} (∫_{ℤ_{p}} \binom{x}{n}x^{−1}dμ_{χ}(x)) T^{n} = ∫_{ℤ_{p}} [(1 + T)^{x} − 1] x^{−1}dμ_{χ}(x).

(Notice that the “regularisation” consists in removing the constant term in what would correspond to the usual Amice transform.) We observe the following:

• As we see from (1), writting Ã_{x^{−1}μ_{χ}}(T) = ∑_{n=1}^{∞} λ_{n}T^{n}, the coefficients λ_{n} lie in \frac{1}{n}O_{ℂ_{p}}, and hence Ã_{x^{−1}μ_{χ}}(T) ∉ ℂ_{p} ⊗ O_{ℂ_{p}}[[T]] since the λ_{n} have unbounded denominators. Nevertheless, the power series Ã_{x^{−1}μ_{χ}}(T) still converges for |T | < 1.

• Ã_{x^{−1}μ_{χ}}(T) is a “primitive” of A_{μ_{χ}}(T), in the sense that

(1 + T)\frac{d}{dT} Ã_{x^{−1}μ_{χ}}(T) = A_{μ_{χ}}(T).  (2)

Indeed:

(1 + T)\frac{d}{dT} Ã_{x^{−1}μ_{χ}}(T) = (1 + T)\frac{d}{dT} ∑_{n=1}^{∞} (\frac{1}{n} ∫_{ℤ_{p}} \binom{x − 1}{n − 1}dμ_{χ}(x)) T^{n}

= (1 + T) ∑_{n=1}^{∞} ∫_{ℤ_{p}} \binom{x − 1}{n − 1}dμ_{χ}(x)T^{n−1}

= ∑_{n=0}^{∞} ∫_{ℤ_{p}} [\binom{x − 1}{n} − \binom{x − 1}{n − 1}] dμ_{χ}(x)T^{n}

= ∑_{n=0}^{∞} ∫_{ℤ_{p}} \binom{x}{n}dμ_{χ}(x)T^{n}

= A_{μ_{χ}}(T).

Theorem 1. There exists a constant C ∈ ℂ_{p} such that

Ã_{x^{−1}μ_{χ}}(T) = \frac{−1}{τ(χ^{−1})} ∑_{a=0}^{D−1} χ^{−1}(a) · log_{p}(ζ_{D}^{a}(1 + T) − 1) + C.  (3)

Proof. Letting H(T) denote the RHS of (3) (the constant is irrelevant here), we see that

• H(T) defines a convergent power series for |T | < 1, since

log_{p}(ζ_{D}^{a}(1 + T) − 1) = log_{p}(ζ_{D}^{a} − 1 + ζ_{D}^{a}T)

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

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

• H(T) satisfies

(1 + T)\frac{d}{dT} H(T) = A_{μ_{χ}}(T).

This follows from a straighforward computation that is left as an exercise. These two facts together with (2), imply that Ã_{x^{−1}μ_{χ}}(T) and H(T) differ by a constant.

It remains to compute Ã_{res_{ℤ^{×}_{p}}(x^{−1}μ_{χ})}(T). In fact, since res_{ℤ_{p}^{×}}(x^{−1}μ_{χ}) is a honest measure, the regularisation process is no longer neeeded, and so reflected in the following. Proposition 1.

A_{res_{ℤ^{×}_{p}}(x^{−1}μ_{χ})}(T) = \frac{−1}{τ(χ^{−1})} (1 − \frac{χ(p)}{p}) ∑_{a=0}^{D−1} χ^{−1}(a) · log_{p}(ζ_{D}^{a}(1 + T) − 1).

Proof. First, from Lemma 2 in Lecture 35, giving the formula for the Amice transform of a measure restricted to pℤ_{p} in terms of that of the original measure, we have

Ã_{res_{pℤp}(x^{−1}μ_{χ})}(T) = \frac{1}{p} ∑_{ξ∈μ_{p}} Ã_{x^{−1}dμ_{χ}}(ξ(1 + T) − 1).

Using this, we compute

A_{res_{ℤ^{×}_{p}}(x^{−1}μ_{χ})}(T) = Ã_{x^{−1}μ_{χ}}(T) − Ã_{res_{pℤp}(x^{−1}μ_{χ})}(T)

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

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

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

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

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

where the penultimate equality follows after replacing ap by a in the right terms in the preceding equality, so that χ(a) gets replaced by χ(a)χ^{−1}(p). The result follows.

As an immediate consequence, the proof of the p-adic analogue of Dirichlet’s class number formula follows. (Cf. Theorem 2 from Lecture 36.)

Theorem 2. Let χ be a primitive Dirichlet character of conductor D > 1 prime to p. Then

L_{p}(1, χ) = \frac{−1}{τ(χ^{−1})} (1 − \frac{χ(p)}{p}) ∑_{a=0}^{D−1} χ^{−1}(a) · log_{p}(1 − ζ_{D}^{a}).

Proof. Indeed, Proposition 1 gives the last of the following equalities

L_{p}(1, χ) = ∫_{ℤ_{p}^{×}} x^{−1}dμ_{χ}(x) = A_{res_{ℤ^{×}_{p}}(x^{−1}μ_{χ})}(T)|_{T =0}

= \frac{−1}{τ(χ^{−1})} (1 − \frac{χ(p)}{p}) ∑_{a=0}^{D−1} χ^{−1}(a) · log_{p}(1 − ζ_{D}^{a}).

Some concluding remarks are in order. The above class number formula can be seen as a manifestation of a rather elusive connection between two types of objects of a completely different nature, namely special values of L-functions –the values of the Dirichlet L-function L(s, χ), as interpolated by L_{p}(s, χ)–, and special elements of arithmetic content –the cyclotomic units 1 − ζ_{D}^{a}, in the form of their p-adic logarithms.

In the literature one can find p-adic analogues of essentially each of the types of Lfunctions that have been treated in this course: constructions due to Mazur and Swinnerton-Dyer et.al., corresponding to the Hecke L-functions L(f, s); to Hida et.al., corresponding to Rankin-Selberg L-functions like L(f ⊗ g, s); etc.. And for each of these other types of padic L-functions, the connections that arise with arithmetic in the form of generalised class number formulae is currently a broad and deep area of great mathematical interest, with many of its gems still awaiting further exploration.