Course Notes for Math 574: Adeles, Automorphic Forms, and Representations Spring 2002

Stephen D. Miller

            miller@math.rutgers.edu
     http://www.math.rutgers.edu/∼sdmiller

January 23, 2002

1 Introduction

1.1 Outline of Course

• p-adic numbers

• Adeles

• Tate’s thesis

• Automorphic forms on SL_{2}(ℝ)

• Automorphic representations of GL_{2}(ℝ)

• Adelic automorphic representations

• Study of congruence subgroups, Hecke operators, Atkin-Lehner theory • Jacquet-Langlands theory of automorphic L-functions, comparison with

Atkin-Lehner

1.2 Some Recommended Texts

(Listed in order of the chronology above)

• Koblitz [Kob]

• Robert [Rob]

• Ramakrishnan and Valenza [Ram-Val]

• Bump [Bump]

• Gelbart [Gelb]

• Iwaniec’s yellow/blue AMS book [Iwan2]

• Jacquet-Langlands (online) [Jac-Lan]

1.3 Goals and Emphasis of the Course

We want to become equally adept using the adelic and classical languages, and understand constructions and insights provided by each language in terms of the other; in short, to become bilingual. We assume some familiarity with the classical language, either from Iwaniec’s graduate courses or his texts [Iwan1] and [Iwan2].

1.4 A Note on these Notes

These notes often provide some unusual and unorthodox explanations of things which are typically explained in a different, and perhaps superior, way. This is being done for two reasons: to complement existing sources, and to develop a background with automorphic forms itself in mind. Furthermore, and more importantly, many technical details (such as convergence) will be omitted, as they are explained very well in other sources. Many arguments will be purely formal. This is being done to give a conceptual explanation of the techniques.

2 The p-adic Numbers ℚ_{p}

The p-adic numbers are themselves divergent series! They were first invented/discovered by Kurt Hensel about a century ago (see [Rosen] for a short biography). They are comparable to the decimal (or, more correctly base p) expansions of real numbers

1.351325 ⋯,

except that they are written

⋯ 5019325.135

from right to left instead of left to right. Truncating these infinite expansions gives rational numbers in either case, and both ℝ and ℚ_{p} are different completions of the rationals ℚ under different metrics. Roughly speaking, two numbers are “close” to each other in either case if a large initial stretch of digits agree. Addition and multiplication of real numbers is done from right to left (even though the expansions continue infinitely far to the right). Addition and multiplication in ℚ_{p} is done also from right to left; one has a well-defined starting point, but the process literally takes forever.

The p-adic integers ℤ_{p} are those p-adic numbers with nothing to the right of the decimal point. A more formal/algebraic definition is as the inverse limit

ℤ_{p} = lim_{←} ℤ/p^{n}ℤ = {∑_{n=0}^{∞} a_{n}p^{n}, 0 ≤ a_{n} < p} .

When one adds or multiplies p-adic integers, one must carry as with the usual decimal addition and mulitplication. The p-adic numbers are

ℚ_{p} = ∪_{k=0}^{∞}p^{−k}ℤ_{p} = { ∑_{n=−k}^{∞} a_{n}p^{n}, 0 ≤ a_{n} < p} .

Exercise 2.1. A. Show that it makes sense to reduce u ∈ ℚ_{p} (mod p^{k}) for any integer k.

B. Prove that if u_{1}, u_{2} ∈ ℚ_{p}, that

u_{1} = u_{2} ⇔ u_{1} ≡ u_{2} (mod p^{k}) for all k ∈ ℤ.

C. Check that the reduction is consistent with the addition and multiplication laws in ℤ_{p} and ℤ/p^{k}.

2.1 Division

Lemma 2.2. If x ≠ 0 is in ℚ_{p}, then it is invertible in ℚ_{p}.

Proof: By factoring out the leading term, we may assume

x = p^{k}u  (2.1)

where u = ∑ a_{n}p^{n} ∈ ℤ_{p} starts with 1 ≤ a_{0} ≤ p − 1. Since a_{0} is invertible in (ℤ/p)^{∗}, we may multiply u by an integer and assume a_{0} = 1. Writing u = 1 − py, y = − ∑_{n=0}^{∞} a_{n+1}p^{n} ∈ ℤ_{p}, we find that

u^{−1} = (1 − py)^{−1} = 1 + py + (py)^{2} + · · · .

This last expression is convergent because the powers of p increase, and so the n-th digit stabilizes after adding (py)^{n}. □

This proof introduces two important concepts. First, the integer k in (2.1) is called the order, ord(x). Secondly, the order gives a topology that makes our infinite sum converge quite easily. Under the p-adic absolute value |x|_{p} = p^{−ord(x)}, two numbers are indeed close when a large initial stretch of their digits agree. This metric has the ultrametric property

|x + y| ≤ max(|x|, |y|) ⟺ ord(x + y) ≥ min(ord(x), ord(y)).  (2.2)

Exercise 2.3. Prove (2.2), and that every convergent sequence is a also a cauchy sequence.

Exercise 2.4. Prove that any series ∑ a_{n} converges so long as a_{n} → 0.

Exercise 2.5. Show that ℤ_{p} and ℤ_{p}^{∗} are compact by using the convergent subsequence property.

This shows that any continuous image of ℤ_{p} is compact; in particular, continuous characters of ℤ_{p} and ℤ_{p}^{∗} take values in the unit circle {z ∈ ℂ | zz̄ = 1}.

In fact, the structure of ℚ_{p} can be described naturally as follows:

ℚ_{p} = {0} ∪ ℚ_{p}^{∗}

(by Lemma 2.2), and

ℚ_{p}^{∗} = ∐_{k∈ℤ}p^{k}ℤ_{p}^{∗},

where ℤ_{p}^{∗} are the units in ℤ_{p}; ℤ_{p}^{∗} = {|x| = 1}, a basic open set. All open sets are unions of the basic open neighborhoods that are translates ax + b of ℤ_{p} itself, so long as a ≠ 0. For example,

ℤ_{p}^{∗} = ∐_{j=1}^{p−1}(j + pℤ_{p}).

This gives a tree-like structure to ℚ_{p}.

2.2 Haar Measure and Integration

There is an additive measure dx, normalized to give ℤ_{p} measure 1, and a multiplicative measure d^{∗}x, which gives ℤ_{p}^{∗} measure 1. For example, I = pℤ_{p} has additive measure 1/p, since its p translates I + j, 1 ≤ j ≤ p form ℤ_{p}.

Lemma 2.6. The map x ↦ yx changes dx ↦ |y|dx, and does not change d^{∗}x.

Proof: This is clear if y is a unit (i.e. |y| = 1), and the example above can be iterated for the general case. □

In fact, the multiplicative measure is

d^{∗}x = \frac{dx}{|x|} \frac{p}{p − 1}.

Example 2.7. As an example of an integral, let us compute

∫_{ℤ_{p}} |x|^{s}dx = ∑_{k=0}^{∞} ∫_{p^{k}ℤ_{p}^{∗}} |x|^{s}dx

= \frac{p}{p − 1} ∑ p^{−ks} = \frac{p}{(p − 1)(1 − p^{−s})}.

This is a very typical example; it shows that p-adic integration is often just summation. An integrand like ours (which depends only on |x|) often turns out to be a geometric series.

2.3 Additive and Multiplicative Characters

There is a standard additive character on ℚ_{p} which is trivial on the integers ℤ_{p}:

e_{p}(x) = e^{−2πiq}, x ∈ q + ℤ_{p}

which is well defined because it only depends on the fractional part of the padic number x. As we will show momentarily, the general continuous additive character on ℚ_{p} is

ψ_{a}(x) = e_{p}(ax), a ∈ ℚ_{p}.

Its kernel is |a|ℤ_{p}, the conductor of ψ_{a}. We will also refer to |a| as the conductor here as well.

Theorem 2.8. If ψ is a continuous additive character of ℚ_{p}, then ψ(x) = e_{p}(ax) for some a ∈ ℚ_{p}.

We will use

Lemma 2.9. If a sequence of complex numbers x_{0}, x_{1}, … with x_{n+1} = x_{n}^{p} converges to 1, then x_{0} = e^{2πiα} for some α ∈ ℚ.

Proof of Lemma 2.9: Firstly, we must clearly have that |x_{n}| = 1, and that |α_{N}| < p^{−2} for N sufficiently large, where x_{n} = e^{2πiα_{n}}. We may also assume that |α_{n}| ≥ |α_{N}| for n ≥ N. But then |α_{N+1}| = |pα_{N}| > |α_{N}| unless α_{N} = 0. □

Proof of Theorem 2.8: We use that 1 is a topological generator of ℤ_{p}, i.e. the set ℕ = { 1, 1+1, 1+1+1, … } is dense in ℤ_{p}. Then ψ’s restriction to ℤ_{p} is determined purely by its values on ℤ. We have that ψ(n) = ψ(1)^{n}, and in particular

ψ(p^{k}) = ψ(1)^{p^{k}}.

Since ψ is continuous and |p^{k}| → 0 as k → ∞, ψ(1)^{p^{k}} → 1 as k → ∞, and Lemma 2.9 shows that ψ(1) = e^{2πij/p^{k}} for some k ≥ 0, (j, p) = 1. By using the character ψ (\frac{p^{k}}{j} x) instead, we may thus assume that ψ is trivial on ℤ_{p}. Now, ψ(1/p)^{p} = ψ(1) = 1, and so

ψ(1/p) = e^{2πia_{1}/p}.

Also,

ψ(1/p^{2})^{p^{2}} = 1,

and so

ψ(1/p^{2}) = e^{2πia_{2}/p^{2}},

with the compatibility condition that a_{2} ≡ a_{1} (mod p). Continuing we obtain a p-adic integer a such that

ψ(1/p^{k}) = e^{2πia/p^{k}}, k ≥ 0

and thus ψ(x) = e_{p}(−ax). □

We now turn to the multiplicative characters (Theorem 2.11), but first give an overview. Since ℚ_{p}^{∗} is isomorphic to a product of its “spine“ p^{ℤ} = {|x| | x ∈ ℚ_{p}^{∗}} with ℤ_{p}^{∗}, continuous characters of ℚ_{p}^{∗} will be of the form |x|^{s}χ(x), where χ(x) is a finite-order continuous character of ℤ_{p}^{∗}. By continuity, its kernel contains an open subgroup of the form 1 + p^{k}ℤ_{p}, which is called the conductor of χ for the smallest such value of k ≥ 0. Again, the conductor is often referred to just as p^{k}. χ is essentially a dirichlet character on ℤ/p^{k}. This is because

Exercise 2.10. Show that ℤ_{p}^{∗}/(1 + p^{k}ℤ_{p}) ≃ (ℤ/p^{k})^{∗}.

Theorem 2.11. All continuous multiplicative characters of ℚ_{p}^{∗} are of the form

|x|^{s}χ(x),  (2.3)

where χ is a finite order character of ℤ_{p}^{∗}.

Proof: Clearly as

ℚ_{p}^{∗} = {p^{k} | k ∈ ℤ} × ℤ_{p}^{∗},

such a decomposition as in (2.3) exists. We need only show that χ is indeed a finite-order character. To simplify, we argue here only for odd primes (the case p = 2 can be handled similarly). Recall that for all primes p > 2 there is an integer g which generates any (ℤ/p^{k})^{∗}. Such a g is a topological generator of ℤ_{p}^{∗}. Since

g^{(p−1)p^{k−1}} ≡ 1(mod p^{k}),

χ ((g^{p−1})^{p^{k}}) = χ(g^{p−1})^{p^{k}} → 1 as k → ∞. Thus Lemma 2.9 shows that χ(g) is the exponential of a rational number, and χ is of finite order. □

2.4 An Important Integral

Example 2.12. The “p-adic Gamma function”: If ψ is any additive character and χ any multiplicative character (both assumed to be continuous)

G(ψ, χ) := ∫_{ℚ_{p}^{∗}} ψ(x)χ(x)d^{∗}x.  (2.4)

This is in analogy to the classical situation of the field of real numbers, where Γ(s) = ∫_{0}^{∞} e^{−t}t^{s} \frac{dt}{t}. We can equivalently study

∫_{ℚ_{p}} ψ(x)χ(x)dx,  (2.5)

or by changing variables, the more-explicit

∫_{ℚ_{p}} e_{p}(x)χ(x)|x|^{s}dx,  (2.6)

where χ is now of finite order. Then (2.6) can be broken up as

∑_{k∈ℤ}p^{−ks} ∫_{|x|=p^{−k}^{x∈ℚp}} e_{p}(x)χ(x)dx,

and we shall now compute

∫_{|x|=p^{−k}^{x∈ℚp}} e_{p}(x)χ(x)dx  (2.7)

through some exercises:

Exercise 2.13. If ϕ is a homomorphism from a finite group G to ℂ, then

∑_{g∈G} ϕ(g) = \begin{cases}|G| & , if ϕ is trivial, \\ 0 & , otherwise. \end{cases}

Exercise 2.14. Assume that χ is trivial. Show that

∫_{|x|=p^{−k}^{x∈ℚp}} e_{p}(x)χ(x)dx = \begin{cases}\frac{p−1}{p}p^{−k} & , k ≥ 0, \\ −1 & , k = −1, \\ 0 & , otherwise. \end{cases}

Deduce that (2.6) equals

\frac{1 − p^{s}}{1 − p^{−s−1}} .

Hint: when k < 0 the integral is the sum ∑_{j∈(ℤ/p^{−k})^{∗}} e(−j/p^{−k}).

Now assume that χ is nontrivial of conductor p^{n}, n > 0. To simplify notation, let ν = −k so that ℓ = max(n, ν). Exercise 2.15. Show that if ν < n,

∫_{|x|=p^{ν}^{x∈ℚp}} e_{p}(x)dx = 0.

Hint: First express it in terms of the sum

∑_{j∈(ℤ/p^{n})^{∗}} e(−j/p^{ν})χ(j) = ∑_{ℓ∈ℤ/p^{n−ν}^{j∈(ℤ/p^{ν})^{∗}}} e (−\frac{j + p^{ν}ℓ}{p^{ν}}) χ(j + p^{ν}ℓ),

where we also use χ to refer to the associated dirichlet character. Then show that the inner sum

∑_{ℓ∈ℤ/p^{n−ν}} χ(j + p^{ν}ℓ) = χ(j) ∑_{ℓ∈ℤ/p^{n−ν}} χ(1 + p^{ν}ℓ) = 0

because {1 + p^{ν}ℓ} forms a subgroup of order p^{n−ν}, which is strictly smaller than the conductor p^{n}. Exercise 2.16. Show that if ν = n, then

∫_{|x|=p^{ν}^{x∈ℚp}} e_{p}(x)χ(x)dx = χ(−1)τ_{χ},

where the Gauss sum

τ_{χ} = ∑_{j∈(ℤ/p^{n})^{∗}} e(j/p^{n})χ(j).

Exercise 2.17. Use Exercise 2.13 to show that

∫_{|x|=p^{ν}^{x∈ℚp}} e_{p}(x)χ(x)dx = 0

also if ν > n.

We conclude:

Proposition 2.18. Let s ∈ ℂ and χ be a finite-order continuous character of ℚ_{p}^{∗}. Then the integral

∫_{ℚ_{p}} e_{p}(x)χ(x)|x|^{s}dx = \begin{cases}\frac{1−p^{s}}{1−p^{−s−1}} & , χ ≡ 1, \\ p^{ns}χ(−1)τ_{χ} & , otherwise. \end{cases}

3 Adeles

In this section we will introduce and motivate the adeles by their use in harmonic analysis.

3.1 The Adelization of the Unit Interval

To start let us recall our definition of the p-adic exponential function

e_{p}(x) = e^{−2πiq} = e(−q),

where q is a rational number such that x ∈ q + ℤ_{p}. We saw that e_{p}(x) is well-defined since the exponential function is trivial on integers. In fact, any periodic function can be extended to the p-adics in this fashion. By the same principle, we can also translate periodic functions by ℚ_{p} through the action

f(x) ↦ f(x − y), y ∈ ℚ_{p}.  (3.1)

This is again well-defined, because ℤ_{p}, the closure of ℤ in ℚ_{p}, acts trivially. Thus ℚ_{p} acts by shifting the argument of f by a rational number whose denominator is a power of p:

ℚ_{p}/ℤ_{p} ≃ \{\frac{a}{p^{k}} | (a, p) = 1, 0 ≤ a < p^{k}\}.

In fact, we may act by ℚ_{p_{1}} × ℚ_{p_{2}} or any finite product ∏_{p∈S} ℚ_{p}, or even

[∏_{p∈S} ℚ_{p}] × [∏_{p∉S} ℤ_{p}] , S finite.

To make this universal, we take the union of these sets over all finite sets of primes S. This is the finite adeles 𝔸_{f}, a restricted direct product of the ℚ_{p} with respect to the ℤ_{p}. The adeles

𝔸 = ℝ × 𝔸_{f} = {(a_{∞}; a_{2}, a_{3}, a_{5}, …) | almost all a_{p} ∈ ℤ_{p}}

are the restricted direct product of all ℚ_{p}, p ≤ ∞, using the convention that ℚ_{p} = ℝ. The “prime” p = ∞ is called the archimedean prime, because its valuation (the usual absolute value on ℝ) obeys Archimedes’ Axiom

for all |y| > |x| > 0, there exists n ∈ ℤ such that |nx| > |y|.  (3.2)

This fails for the “non-archimedean” valuations |·|_{p}, p < ∞, because |nx|_{p} ≤ |x|_{p} by the ultrametric property (2.2).

We can thus define an adelization any function f on ℝ/ℤ by

f_{𝔸}(a) = f_{𝔸}(α_{∞}; a_{f}) = f(a_{∞} − q)  (3.3)

where a = (a_{∞}; a_{f}) ∈ 𝔸 and

a_{p} ∈ q + ℤ_{p} for all p < ∞.  (3.4)

Thus 𝔸_{f} acts to shift periodic functions. One might think of ℝ as the melody, but 𝔸_{f} as its key.

Exercise 3.1. Prove that such a q as in (3.4) always exists.

Hint: Use the Chinese Remainder Theorem. Alternatively, first suppose that a_{f} ∈ ℤ_{p} except at one prime, and conclude the general case by summing the rational numbers obtained this way.

3.2 Diagonal Embedding of ℚ

On abstract grounds, each ℚ_{p} is a completion of ℚ, and hence ℚ embeds into each ℚ_{p}. To be more precise, let us recall from §2.1 that

Proposition 3.2. If m, n ∈ ℤ and p ∤ n ≠ 0, then \frac{m}{n} ∈ ℤ_{p}.

Proof: This follows from the proof of Lemma 2.2. □

It thus makes sense to diagonally embed ℚ ↪ 𝔸 because almost all components will be in ℤ_{p}. We will also refer to this image as ℚ.

Proposition 3.3. f_{𝔸}(a) as defined in (3.3) is invariant under the diagonallyembedded ℚ.

It is not surprising that 𝔸/ℚ is similar to ℝ/ℤ. After all, in this setting 𝔸_{f}, the non-archimedean component of 𝔸, serves to act on ℝ via the diagonal embedding of ℚ. Let

ℤ̂ = ∏_{p<∞} ℤ_{p} ≃ lim_{←N} ℤ/Nℤ.

We have the

Theorem 3.4. (Strong Approximation Theorem)

𝔸 = ℚ + ℝ + ℤ̂.

Corollary 3.5. A fundamental domain for 𝔸/ℚ is (ℝ/ℤ) + ℤ̂.

Since ℤ̂ acts trivially on periodic functions, these facts are readily visible from our viewpoint in the discussion above. Also, Corollary 3.5 shows us how to reverse the f ↦ f_{𝔸} correspondence: if a function on 𝔸/ℚ is trivial on the compact subgroup ℤ̂, it corresponds to a periodic function on ℝ/ℤ.

If we had replaced some of the ℤ_{p} in ℤ̂ = ∏_{p<∞} ℤ_{p} by their open, compact subgroups p^{k_{p}}ℤ_{p}, we would obtain NẐ and functions periodic on ℝ/Nℤ, where N = ∏ p^{k_{p}}. Indeed, the “missing” translations in ℤ̂ ∖ Nℤ̂ correspond to shifts by the integers 1, …, N − 1. Equivalently, one could view periodic functions on ℝ/Nℤ as vector-valued functions on ℝ/ℤ which transform according to a shift matrix. This viewpoint is very important when considering congruence subgroups of GL_{2}(ℤ) adelically.

4 Fourier Analysis on the Adeles

Suppose, completely formally, that f is a function on 𝔸 invariant under ℚ and ℤ̂, which is equivalently a function on ℝ/ℤ. By general principles we have a Fourier series

f(a) = ∑_{characters ψ of 𝔸 trivial on ℚ}∫_{𝔸/ℚ} f(a+x)ψ(x)^{−1}dx = ∑_{ψ} ψ(a) < f, ψ > .

(4.1)

It is worthwhile to understand how this corresponds to the usual Fourier series on ℝ/ℤ. First, we need to extend our classification of the characters of ℚ_{p} from §2.3 to 𝔸. One character is clearly given by

e_{𝔸}(x) = ∏_{p≤∞} e_{p}(x), e_{∞}(x) = e(x) = e^{2πix},

which is the adelization of e(x), the example we started with.

Proposition 4.1. The continuous additive characters of 𝔸 which are trivial on ℚ are precisely those given by e(qx) for some q ∈ ℚ.

Proof: Clearly these characters e(qx) are all trivial on ℚ. Conversely, any additive character can be decomposed as a product ψ = ∏_{p≤∞} ψ_{p} of continuous additive characters ψ_{p} of ℚ_{p}. When p = ∞, all characters are of the form e(c_{∞}x), and we have already seen in §2.3 that for p < ∞ all continuous additive characters are of the form e_{p}(c_{p}x). Let U = {|z − 1| < 1/10}, so that the only subgroup of ℂ^{∗} contained in U is the trivial subgroup {1}. Since ψ is continuous, the inverse image ψ^{−1}(U) is an open subset of 𝔸, and hence contains a basic open neighborhood of the form ℝ × Nℤ̂ for some integer N. Then the image ψ(Nℤ̂) is a subgroup contained in U, and hence trivial. We conclude that ψ_{p} = e(c_{p}x_{p}) is trivial on ℤ_{p} for almost all p, i.e. those not dividing N. Writing c_{p} = p^{k}u, where |u| = 1, we see that e(c_{p}u^{−1}) = e^{−2πip^{k}} = 1, and and thus these k ≥ 0, i.e. c_{p} ∈ ℤ_{p} for all p ∤ N. Thus, we conclude that globally ψ(x) = e_{𝔸}(cx), for c ∈ 𝔸.

Because e(c + q) = e(c · 1)e(q) = 1, we may use the strong approximation theorem to reduce to the case that c ∈ ℝ × ℤ̂. Thus e(c) = e^{2πic_{∞}} = 1, and c_{∞} ∈ ℤ. The condition e(cq) = 1 for any rational q, in particular q = p^{−k}, forces

e^{2πic_{∞}/p^{k}} = e^{2πic_{p}/p^{k}}.

Thus c_{p} ≡ c_{∞} (mod p^{k}) for all k, and c_{p} is the image of the integer c_{∞} for each p. We conclude that c is in the diagonal embedding of ℚ. □

With this description of the additive characters, we can understand the meaning of < f, ψ >, where ψ = e(qx).

< f, ψ >= ∫_{𝔸/ℚ} f(x)ψ(x)^{−1}dx

= ∫_{𝔸/ℚ} f(x)e(−qx)dx
= ∫_{ℝ/ℤ} ∫ℤ̂ f(x_{∞} + x_{f})e(−qx_{∞})e(−qx_{f})dx_{f}dx_{∞},

using the fundamental domain in Corollary 3.5. Since f is periodic on ℝ/ℤ, this is

= ∫_{ℝ/ℤ} f(x_{∞})e(−qx_{∞})dx_{∞} ∫ℤ̂ e(−qx_{f})dx_{f}.

Now, this inner integral factors as a product

∏_{p<∞} ∫_{ℤ_{p}} e_{p}(−q_{p}x_{p})dx_{p} = \begin{cases}1 & , q_{p} ∈ ℤ_{p} for all p \\ 0 & , otherwise \end{cases}

= \begin{cases}1 & , q ∈ ℤ, \\ 0 & , otherwise. \end{cases}

So < f, ψ > is the usual Fourier coefficient if q ∈ ℤ and is zero otherwise. Hence if the classical Fourier expansion of a periodic function f(x) = ∑_{n∈ℤ} a_{n}e(nx), then the adelic Fourier expansion of its adelization is

∑_{n∈ℤ}a_{n}e_{𝔸}(nx).

In other words, our original adelization (that of e^{2πix}) is in some sense the only example!

4.1 The Poisson Summation Formula

The classical Poisson Summation Formula

∑_{n∈ℤ}f(n) = ∑_{n∈ℤ} f̂(n)  (4.2)

can be proven by periodizing f(x) ↦ ∑_{n∈ℤ} f(x + n), and then expanding its Fourier series at x = 0. One needs, of course, to assume f decays rapidly enough so that the average converges, and that it is regular enough so that the right hand side of (4.2) does also. The space of Schwartz functions

S(ℝ) = {f ∈ C^{∞}(ℝ) | |x^{m}f^{(n)}(x)| → 0 as |x| → ∞ for all m, n ≥ 0}

is certainly sufficient for this purpose.

Exercise 4.2. Prove that the Fourier transform maps S(ℝ) to itself.

There is a p-adic analogue, the Schwartz-Bruhat space

S(ℚ_{p}) = {f ∈ C(ℚ_{p}) | f has compact support and is locally constant}.

Locally constant means that f(x) is constant on small open neighborhoods of x.

Proposition 4.3. If f ∈ S(ℚ_{p}), then f is a finite sum of the form

∑ c_{j} · χ_{a_{j}+b_{j}ℤ_{p}},

with c_{j} ∈ ℂ and a_{j} ∈ ℚ_{p}, b_{j} ∈ ℚ_{p}^{∗}.

Proof: For each a ∈ ℤ_{p} there is some N such that f(a) = f(x) whenever |x−a| < p^{−N}. The collection of such open sets covers the support of f, which is assumed compact. Thus there is a finite collection of a_{j} ∈ ℚ_{p} such that f is constant of the set a_{j} + p^{N+1}ℤ_{p}, and zero otherwise. We may assume that these sets are disjoint by the ultrametric property (2.2). □

One can define the global S(𝔸), the adelic Schwartz space, by first creating “pure tensors,” functions f = ∏_{p≤∞} f_{p} on 𝔸, with each f_{p} ∈ S(ℚ_{p}) and all but finitely many f_{p} = χ_{ℤ_{p}}, the characteristic function of ℤ_{p}. S(𝔸) is the span of these pure tensors.

Proposition 4.4. If f ∈ S(𝔸), then

∑_{q∈ℚ}f(q) = ∑_{q∈ℚ} f̂(q),  (4.3)

where

f̂(a) = ∫_{𝔸} f(x)e_{𝔸}(−ax)dx.  (4.4)

Remark 4.5. In (4.4) we could actually take any additive character ψ which is trivial on ℚ instead of the standard character e_{𝔸}(x).

Proof of Proposition 4.4: We follow the usual argument. Let

F (x) = ∑_{q∈ℚ}f(x + q) = ∑_{q∈ℚ}∫_{𝔸/ℚ} F (x)e(−qx)dx.

By “unfolding”

∫_{𝔸/ℚ} F (x)e(−qx)dx = ∫_{𝔸/ℚ} ∑_{r∈ℚ}f(x + r)e(−qx)dx

= ∑_{r∈ℚ}∫_{r+𝔸/ℚ} f(x)e(−qx)dx = ∫_{𝔸} f(x)e(−qx)dx = f̂(q).

(strictly speaking, we should replace be speaking of a translates of a fundamental domain rather than just r + 𝔸/ℚ). Now set x = 0 to obtain (4.3). □

4.2 Classical Interpretations

Let use see what the adelic Poisson Summation Formula (4.3) means classically in the crucial case when f is a pure tensor. In this situation, the Fourier transform factors as a product of local Fourier transforms

f̂(a) = ∏_{p≤∞} ∫_{ℚ_{p}} f_{p}(x)e_{p}(−a_{p}x_{p})dx_{p} = ∏_{p≤∞} f̂_{p}(a_{p}).

Exercise 4.6. Show that if f_{p} = χ_{ℤ_{p}}, then f̂_{p} = χ_{ℤ_{p}} also. Show that the adelic Fourier transform of a function f ∈ S(𝔸) remains in S(𝔸). (Hint: Use the fact that if f_{t}(x) = f(tx), then f̂_{t}(x) = \frac{1}{|t|} f̂(\frac{x}{t}).)

To simplify matters, let us first assume that all f_{p} = χ_{ℤ_{p}}, p < ∞. Then

f̂(a) = f̂_{∞}(a_{∞})χẐ(a_{f}) = \begin{cases}f̂_{∞}(a_{∞}) the usual F.T. & , a_{f} ∈ ℤ̂ \\ 0 & , otherwise. \end{cases}

Thus because q_{f} ∈ ℤ̂ ⇔ q ∈ ℤ, the adelic Poisson Summation Formula recovers the classical one.

Exercise 4.7. Show that if f_{p} = ∑_{j=1}^{p} a_{j} χ_{j+pℤ_{p}}, then

f̂_{p}(a) = \begin{cases}0 & , if a ∉ p^{−1}ℤ_{p}, \\ \frac{1}{p} ∑_{ℓ=1}^{p} â_{ℓ} & , if a ∈ \frac{ℓ}{p} + ℤ_{p} \end{cases}

where â_{ℓ} is the Discrete Fourier transform ∑_{j=1}^{p} a_{j}e(jℓ/p).

In the finite case, Poisson Summation reduces to the inversion formula a_{j} = \frac{1}{p} ∑_{ℓ=1}^{p} â_{ℓ}e(−jℓ/p).

4.3 The Riemann ζ-Function

First, we give a little background about distributions. They will not be important here per se, but will be later on in the course. All that is needed here is the Poisson Summation Formula itself.

In terms of distributions (=linear functions on spaces of functions), the classical Poisson Summation Formula shows that the delta function of the integers

δ_{ℤ}(x) = ∑_{n∈ℤ}δ_{n}(x)

is self-dual under the Fourier transform. Adelically, δ_{ℚ} = δ̂_{ℚ} also. There are many examples of self-dual functions, e.g. e^{−πx^{2}}, sech(πx), and Hermite polynomials. The functional equations of dirichlet L-functions can be derived from applying these distributional identities to multiplicative characters. We will develop the multiplicative theory of the adeles in the next section. For now we use the classical viewpoint to illustrate how the identity

δ_{ℤ} = δ̂_{ℤ}  (4.5)

implies the functional equation of the Riemann ζ-function. Indeed, it can be shown that the two are equivalent.

We will ingore the polar terms coming from δ_{0} in these sums; they can be dealt with by applying (4.5) to Schwartz functions e.g. e^{−πx^{2}} ala Riemann, or by careful distributional arguments. We apply (4.5) to f(x) = |x|^{−s}

< δ_{ℤ}, f >=< δ̂_{ℤ}, f >=< δ_{ℤ}, f̂ > by Parseval’s Theorem.  (4.6)

We compute that

f̂(ξ) = ∫_{ℝ} e(−ξx)|x|^{−s}dx = |ξ|^{s−1} ∫_{ℝ} e(x)|x|^{1−s}d^{∗}x.

This last integral is very important in distributional derivations of the functional equations of L-functions.

Exercise 4.8. 1. Show that for 0 < Re s < 1,

∫_{ℝ} e(x)|x|^{s}d^{∗}x = Γ_{ℂ}(s) cos (\frac{πs}{2}) ,

where

Γ_{ℂ}(s) = 2(2π)^{−s}Γ(s).

2. Use the duplication formula

Γ_{ℂ}(s) = Γ_{ℝ}(s)Γ_{ℝ}(s + 1),
Γ_{ℝ}(s) = π^{−s/2}Γ(s/2)

and the functional equation

Γ(s)Γ(1 − s) = π csc(πs)

to prove

∫_{ℝ} e(x)|x|^{s}d^{∗}x = Γ_{ℂ}(s) cos(πs/2) = \frac{Γ_{ℝ}(s)}{Γ_{ℝ}(1 − s)}.

Thus the Fourier transform of f(x) = |x|^{−s} is f̂(ξ) = \frac{Γ_{ℝ}(1−s)}{Γ_{ℝ}(s)}|ξ|^{1−s}, and Poisson Summation gives

∑ f(n) = ∑ f̂(n)

2ζ(s) = \frac{Γ_{ℝ}(1 − s)}{Γ_{ℝ}(s)}2ζ(1 − s),

the functional equation ξ(s) = Γ_{ℝ}(s)ζ(s) = ξ(1 − s).

5 Ideles

In this section we describe the multiplicative nature of the adeles and its invertible elements, the ideles

𝔸 = {a ∈ 𝔸 | a_{p} ∈ ℚ_{p}^{∗}, for all p ≤ ∞}.

Actually, the ideles predate the adeles; their name is derived from “ideals,” and the additive “adeles” are a backformation. We have already described the Haar measures for p < ∞; for p = ∞ it is the customary d^{∗}x_{∞} = \frac{dx}{|x|}.

Exercise 5.1. Prove that the ideles have measure zero under the additive Haar measure. (Hint: use the fact that ζ(1) = ∞.)

The topology on the ideles 𝔸^{∗} comes from the restricted direct product. There is an absolute value on the adeles, |x|_{𝔸} = ∏_{p≤∞} |x_{p}|_{p}. The change of coordinates x ↦ ax changes dx ↦ |a|dx. For example, multiplying by rationals does not change the additive Haar measure! There is a version of the Strong Approximation Theorem for the ideles, involving

ℤ̂^{∗} = ∏_{p<∞} ℤ_{p}^{∗} = lim_{←N}(ℤ/N)^{∗}.

Theorem 5.2. (Strong Approximation for Ideles)

ℚ^{∗} × ℤ̂ × ℝ^{∗} = 𝔸^{∗},

and a fundamental domain for

ℚ^{∗}\𝔸^{∗}

is given by (0, ∞) × ℤ̂^{∗}.

Proof: Given any idele a, all a_{p} ∈ ℤ_{p}^{∗} except for those p in a finite set S. We may factor

a = a_{∞} · [∏_{p∈S} a_{p}] · a_{r},

where a_{∞} ∈ ℝ, a_{r} = ∏_{p∉S} a_{p} ∈ ∏_{p∉S} ℤ_{p}^{∗}. For each p ∈ S, multiply a by the rational number |a_{p}|_{p}. Since p_{1} ∈ ℤ_{p_{2}}^{∗} for p_{1} ≠ p_{2}, this process will yield a rational multiple of a which lies in ℤ̂^{∗}. The description of the fundamental domain follows from the fact that ℚ^{∗} ∩ (ℝ^{∗}ℤ̂^{∗}) = {±1}. □

5.1 Character Theory

All continuous multiplicative characters of ℝ are of the form

|x|^{s}sgn(x)^{ϵ}, s ∈ ℂ, ϵ = 0 or 1.

Combining the multiplicative characters at the non-archimedean places (i.e. the ones from §2.3 related to dirichlet characters), we have a complete description of the multiplicative characters of 𝔸^{∗}. We need to know, in analogy with Proposition 4.1, the continuous characters of 𝔸^{∗} which are trivial on ℚ^{∗}. In view of the Strong Approximation Theorem, these characters are on (0, ∞) × ℤ̂^{∗}, and hence are products of |x|^{s} times continuous characters of finite order. As with the additive characters (Proposition 4.1), the continuity assumption ensures triviality on a subgroup of the form

[∏_{p∈S}(1 + p^{k_{p}}ℤ_{p})] ∏_{p∉S} ℤ_{p}^{∗}

for some finite set of primes S.

Exercise 5.3. Prove that

ℤ̂^{∗}/ (∏_{p∈S}(1 + p^{k_{p}}ℤ_{p})) ≃ (ℤ/N)^{∗}, N = ∏ p^{k_{p}}.

(Hint: see Exercise 2.10.)

In view of this exercise, finite order continuous characters are equivalent to characters of (ℤ/N)^{∗}, and hence to dirichlet characters. Let us now be more explicit in describing the correspondance between dirichlet characaters and finite order, continuous characters of ℚ^{∗}\𝔸^{∗}. Because the Chinese Remainder Theorem allows us to factor

(ℤ/N)^{∗} = ∏(ℤ/p^{k_{p}})^{∗},

any character χ of (ℤ/N)^{∗} factors into a product of characters of (ℤ/p^{k_{p}})^{∗}, so we will focus on the case that N = p^{k} is a prime power.

We will now describe the corresponding adelic character ω = ∏_{p≤∞} ω_{p}. First,

ω_{∞}(x) = \begin{cases}1 & , χ(−1) = 1, \\ sgn(x) & , χ(−1) = −1, \end{cases}

i.e. ω_{∞} is even or odd depending on whether not χ is.

For prime ℓ ≠ p, ω_{ℓ} will be trivial on ℤ_{ℓ}^{∗} and ω_{ℓ}(ℓ) = χ(ℓ).^{1} Thus ω_{ℓ}(ℓ^{k}u) = χ(ℓ)^{k} for all u ∈ ℤ_{ℓ}^{∗}, k ∈ ℤ. This is analogous to extending χ to the subset of the rationals

\{\frac{r}{s} ∈ ℚ | (p, rs) = 1\}

1We could also define ω_{ℓ} instead using the inverse; this is only a matter of convention.

by setting

χ(\frac{r}{s}) = χ(r)χ(s)^{−1}.

We now define ω_{p} in order to make ω globally invariant under ℚ^{∗}. In contrast to the other primes, we will have that ω_{p} is trivial on powers of p and on 1 + p^{k}ℤ_{p}. If u ∈ ℤ_{p}, we define

ω_{p}(u) = χ^{−1}(j),

where

j = u (mod p^{k}).

Exercise 5.4. Show that ω as defined here is indeed trivial on primes, and therefore invariant under multiplication by ℚ^{∗}.

Recall a dirichlet character χ_{1} (mod q_{1}) induces a character χ_{2} (mod q_{2}) if q_{1} | q_{2} and

χ_{2}(n) = \begin{cases}χ_{1}(n) & , (n, q_{2}) = 1 \\ 0 & , otherwise, \end{cases}

and is called imprimitive if it is not induced from any other character.

Exercise 5.5. Show that if χ_{1} and χ_{2} are dirichlet characters induced from the same primitive character, then they both correspond to the same adelic character ω, and moreover this is the only ambiguity.

This latter aspect (which also naturally removes the distinction of newforms in the setting of adelic modular forms) is very useful.

Exercise 5.6. Prove that if χ_{1} and χ_{2} are primitive dirichlet characters modulo q_{1} and q_{2}, respectively, and (q_{1}, q_{2}) = 1, then

τ_{χ_{1}χ_{2}} = χ_{1}(q_{2})χ_{2}(q_{1})τ_{χ_{1}}τ_{χ_{2}}.

6 Tate’s Thesis via Distributions

To obtain L-functions of primitive dirichlet characters χ, or equivalently finite order, continuous characters ω_{χ} of ℚ^{∗}\𝔸^{∗}, we integrate the identity

δ_{ℚ} = δ̂_{ℚ},

∫_{ℚ^{∗}\𝔸^{∗}} δ_{ℚ}(a)ω(a)|a|^{s−1}da = ∫_{ℚ^{∗}\𝔸^{∗}} δ̂_{ℚ}(a)ω(a)|a|^{s−1}da.  (6.1)

We are essentially taking spectral expansion of the self-dual δ_{ℚ} over ℚ^{∗}\𝔸^{∗}, and will find L-functions as coefficients. As usual, our argument is very formal; we will assume for simplicity that the conductor q of χ is prime, but this is easily removed using Exercise 5.6.

The left-hand side of (6.1) is just 1, since a fundamental domain for ℚ^{∗}\𝔸^{∗} is (0, ∞) × ℤ̂^{∗} (Theorem 5.2). Note that we need to use the additive Haar measure da instead of the seemingly more-natural multiplicative Haar measure d^{∗}a to use the delta functions here. The computation of the righthand side of (6.1) is more involved. Using the definitions of the Fourier transform and δ_{ℚ} = ∑_{q∈ℚ} δ_{q}, we find it is

∫_{ℚ^{∗}\𝔸^{∗}} ∑_{q∈ℚ}[∫_{𝔸} δ_{q}(x)e(−ax)] |a|^{s−1}ω(a)da = ∑_{q∈ℚ}∫_{ℚ^{∗}\𝔸^{∗}} e(−qa)|a|^{s−1}ω(a)da

= ∫_{ℚ^{∗}\𝔸^{∗}} |a|^{s−1}ω(a)da + ∫_{𝔸^{∗}} e(−a)|a|^{s−1}ω(a)da

We essentially ignored this first term before, when ω was trivial. However, using the fundamental domain in Theorem 5.2 for ℚ^{∗}\𝔸^{∗}, it is

∫_{0}^{∞} ∫ℤ̂^{∗} ω_{f}(x_{f})dx_{f}, ω = ω_{∞} · ω_{f}.

However, the inner integral is \frac{1}{N} ∑_{j=1}^{N} χ^{−1}(j) = 0 if χ is a nontrivial character of order N. So, (6.1) reads

1 = ∫_{𝔸^{∗}} e(−a)|a|^{s−1}ω(a)da = ∏_{p≤∞} ∫_{ℚ_{p}^{∗}} e_{p}(a)|a|_{p}^{s−1}ω(a_{p})da_{p}.

Exercise 6.1. Use Exercise 4.8 to show that

∫_{ℝ} e(x)|x|^{s}sgn(x)d^{∗}x = iΓ_{ℂ}(s + 1) sin(πs/2) = i\frac{Γ_{ℝ}(s + 1)}{Γ_{ℝ}(2 − s)}.

The nonarchimdean factors of the integral are the ones we considered in Proposition 2.18. Combining these together with the identities |τ_{χ}|^{2} = q,

τχ̄ = χ(−1)\overline{τ_{χ}} we get the correct functional equation of L(s, χ) = ∏_{p<∞} (1 − \frac{χ(p)}{p^{s}})^{−1} .

Theorem 6.2. Let Λ(s, χ) = Γ_{ℝ}(s + ϵ)L(s, χ), where χ(−1) = (−1)^{ϵ}. Then

Λ(s, χ) = \frac{τ_{χ}}{i^{ϵ}} q^{−s}Λ(1 − s, χ^{−1}).  (6.2)

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