CLARKBARWICK

EULER’S GAMMAFUNCTION ANDTHE FIELDWITH ONEELEMENT

MIT–SPRING2017

Contents

Introduction 5

1 The analytic picture 7 1.1 The Mellin transform 7 1.2 Euler’s Gamma function 12 1.3 Ramanujan’s Master Theorem 17 1.4 Theta series 22 1.5 Pontryagin duality and Fourier analysis 31 1.6 Local zeta functions 43 1.7 Global L-functions 48

2 The field with one element 51 2.1 Borger’s picture 51 2.2 Bökstedt–Hesselholt–Madsen computations of THH 61 2.3 Regularised products and determinants 62

3 Connes–Consani on archimedean 퐿-factors 65 3.1 Hodge structures and 퐿-factors 65 3.2 Deligne cohomology and Beilinson’s theorem 67

Introduction

Euler’s Gamma function is the analytic continuation of the Mellin transform of the negative exponential function:

+∞ 푠 훤(푠) = ∫ 푡 exp(−푡) 푑 log 푡. The Gamma function has no 0 zeroes, and its only poles are This meromorphic function has profound arithmetic significance. For simple poles at nonpositive integers. instance, we have Riemann’s functional equation: if 휁 is the usual Riemann zeta function, we may set The Riemann zeta function is the analytic continuation of 푠 ∑ 푛−푠. 훯(푠) ≔ (2휋푠)−1/2훤 ( ) 휁(푠). 푛≥1 2 Then one has 훯(1 − 푠) = 훯(푠). So, in effect, the Gamma factor plays the role of Euler factor at the infinite place.

Our guiding question is imprecise, but tantalising. Why should 훤 carry this arithmetic significance? Why do factors that involve this ostensibly innocuous meromorphic function – whose definition lies wholly in the domain of classical analysis – serve as a means to complete the analytic functions of number theory? What about the 훤 function permits one to employ it to define the local factor of 휁 functions (and, more generally, 퐿 functions) at infinite places? The purpose of this course is to attempt to answer this question by connecting this humble function with a largely hypothetical object called the field with one element, F1. There is, of course, no field with only one element, but this object is nevertheless meant to serve as a kind of kōan whose contemplation leads to some appreciation of the analogous behaviour of function fields over finite fields and number fields. In effect, F1 is to be conceived of as a finite field, and number rings are to be conceived ofthe rings of functions on affine F1-varieties, and compactifying these varieties over F1 is to be interpreted as the addition of some infinite primes. Our aim is to offer some attempt at a description ofa precise form algebraic geometry over F1 – particularly as proposed in the distinct but related approaches of James 6

Borger and Alain Connes – that is sufficient to account for the 훤 factors corresponding to the infinite places. We will begin with a purely analytic discussion of the basic properties of the Gamma function and the Mellin transform; in particular, we will encounter a number of interesting functions, and we will prove Ramanujan’s Master Theorem. This much should be understandable to anyone with a basic un- derstanding of real and complex analysis. Our first effort to develop a conceptual explanation ofthe Gamma factors will then be Tate’s proof of the functional equations for Dedekind zeta functions and Hecke 퐿 functions. For this, some exposure to class field theory would be helpful. Deninger described Serre’s Gamma factors of motivic 퐿 functions in terms of regularized determinants and an arithmetic co- homology theory; we will explain his results, and we will enrich his description by passing to Connes and Consani’s description in terms of cyclic homology. This last clump of material is rather more involved, and we will be inexorably lead into deeper bits of algebra.

Please note that this text contains many incomplete or sketched proofs. This is deliberate: the reader is asked to supply these proofs as a means of coming to grips with the ideas herein. 1 The analytic picture

1.1 The Mellin transform

To define Euler’s 훤 function, we will employ an analytic tool – the multiplicative Fourier transform, or, as it is more traditionally called, the Mellin transform. In effect, the Mellin transform acts as the

Fourier transform on the multiplicative topological group R>0 relative to the multiplicative Haar measure 푑 log 푡.

1.1.1 Notation. We will use the Landau symbols. Suppose 푎 a point of some topological space, suppose 푔∶ 푉 ⧵ 푎 R>0 and 푓∶ 푉 ⧵ 푎 C functions defined in a punctured neighborhood 푉 ⧵ 푎. Then we will write Example. One has exp(푥) = 푓(푥) = 푂(푔(푥)) as 푥 푎 1 + 푥 + 푥2/2 + 푂(푥3) as 푥 0. if and only if for some neighborhood 푈 of 푎 and some 퐶 > 0, one has Example. A function 푓 on |푓(푥)| ≤ 퐶푔(푥) for every 푥 ∈ 푈. If the implicit constant 퐶 is liable to R>0 is bounded if and only if 푓(푥) = 푂(1) as 푥 + ∞. depend upon an auxiliary parameter 푡, then one writes 푓(푥) = 푂푡(푔(푥)) as 푥 푎. Example. If 푓 is analytic near the origin, one has 푓(푧) = One particular instance of this is the following. A sequence (휙푚)푚∈N 2 푁 푎0 + 푎1푧 + 푎2푧 + ⋯ + 푎푁푧 + of functions on 푉 ⧵ 푎 provide an asymptotic expansion 푁+1 푂푁(|푧| ) as 푧 0, where 푎0, 푎1,…, 푎푁 are the first 푁 푓(푥) ∼ ∑ 푎푚휙푚(푥) Taylor coefficients. Hence 푚∈N Taylor expansions are a kind of asymptotic expansion. of 푓 near 푎 if and only if, for any 푁 ∈ N, one has

푁−1 푓(푥) = ∑ 푎푚휙푚(푥) + 푂푁(휙푁(푥)) as 푥 푎. 푚=1 8 euler’s gamma function and the field with one element

Observe that no assumption about the convergence of the series ∑푚∈N 푎푚휙푚(푥) is made. We also write 푓(푥) = 표(푔(푥)) as 푥 푎 Example. One has log 푥 = 표(푥) as 푥 + ∞. if and only if 푓(푥)/푔(푥) 0 as 푥 푎. 2 We follow the analytic number theorists and write Example. One has 7푥 = 표(푥 ) as 푥 + ∞. 푓(푥) = 훺(푔(푥)) as 푥 푎 for the negation of the assertion 푓(푥) = 표(푔(푥)) as 푥 푎; that is, 푓(푥) = 훺(푔(푥)) as 푥 푎 if and only if

|푓(푥)| lim sup > 0. 푥 푎 푔(푥)

In more detail, we write Example. As a function on 푓(푥) = 훺+(푔(푥)) as 푥 푎 R, one has sin 푥 = 훺(1) as 푥 + ∞, and even sin 푥 = 훺 (1) as 푥 + ∞. if and only if ± 푓(푥) lim sup > 0, Example. As a function on R, 푥 푎 푔(푥) one has 1 + sin 푥 = 훺+(1) as 푥 + ∞, but 1 + sin 푥 ≠ we write 훺−(1) as 푥 + ∞. 푓(푥) = 훺−(푔(푥)) as 푥 푎 if and only if 푓(푥) lim sup < 0, 푥 푎 푔(푥) and we write

푓(푥) = 훺±(푔(푥)) as 푥 푎 if and only if both 푓(푥) = 훺+(푔(푥)) and 푓(푥) = 훺−(푔(푥)) as 푥 푎. Let us also employ the following notation for strips in the complex plane. When 훼, 훽 ∈ R ∪ {−∞, +∞}, we write

⟨훼, 훽⟩ ≔ {푠 ∈ C | ℜ(푠) ∈ [훼, 훽]} ; ⟩훼, 훽⟨ ≔ {푠 ∈ C | ℜ(푠) ∈ ]훼, 훽[} .

We may also have occasion to use the rather odd-looking notations ⟨훼, 훽⟨ and ⟩훼, 훽⟩ as well. the analytic picture 9

1.1.2 Definition. Suppose 푓∶ R>0 C a function that is absolutely integrable on [0, 푟] for any 푟 ∈ R>0, and assume that Note that |푀{푓}(푠)| is domi- nated by the sum −훼 −훽 푓(푡) = 푂(푡 ) as 푡 0, 푡 > 0 and 푓(푡) = 푂(푡 ) as 푡 + ∞. 1 +∞ ∫ |푓(푡)|푡ℜ(푠)−1 푑푡 + ∫ |푓(푡)|푡ℜ(푠)−1 푑푡, 0 1 Then the integral and this in turn is dominated by

1 +∞ 푀{푓}(푠) ≔ ∫ 푡푠푓(푡) 푑 log 푡 퐶 ∫ 푡ℜ(푠)−훼−1 푑푡 + 퐶 ∫ 푡ℜ(푠)−훽−1 푑푡. 0 1 R>0 The first summand converges converges on the strip ⟩훼, 훽⟨ and it defines a holomorphic function there. for ℜ(푠) > 훼, and the second We call the holomorphic function 푀{푓} the Mellin transform of 푓, and we converges for ℜ(푠) < 훽. call ⟩훼, 훽⟨ the strip of definition.

Hereisalist of examples of Mellin transforms. The verifications are left to the motivated reader.

1.1.3 Example. If 푎 > 0, then consider the ray [푎, +∞[ and its characteristic function 휒[푎,+∞[. Then the Mellin transform of 휒[푎,+∞[ is given by 푎푠 푀 {휒 }(푠) = − [푎,+∞[ 푠 with strip of definition 푠 ∈ ⟩−∞, −푎⟨.

1.1.4 Example. No polynomial admits a Mellin transform, because the integral never converges.

1.1.5 Example. The characteristic function of the open interval ]0, 1[ admits a Mellin transform; it is given by 1 푀 {휒 }(푠) = ]0,1[ 푠 with strip of definition 푠 ∈ ⟩0, +∞⟨.

1.1.6 Example. The Mellin transform of the function 푓(푥) = (1 + 푥)−1, is given by

푀{푓}(푠) = 휋 csc(휋푠) with strip of definition 푠 ∈ ⟩0, 1⟨.

1.1.7 Example. The Mellin transform of the function 푓(푥) = (1 − 푥)−1 is given by

푀{푓}(푠) = 휋 cot(휋푠) with strip of definition 푠 ∈ ⟩0, 1⟨. 10 euler’s gamma function and the field with one element

1.1.8 Example. The Mellin transform of the function 푓(푥) = tan−1(푥) is given by 휋 휋 푀{푓}(푠) = − 푠−1 sec ( 푠) 2 2 with strip of definition 푠 ∈ ⟩−1, 0⟨.

1.1.9 Example. The Mellin transform of the function 푓(푥) = cot−1(푥) is given by 휋 휋 푀{푓}(푠) = 푠−1 sec ( 푠) 2 2 with strip of definition 푠 ∈ ⟩0, 1⟨.

1.1.10 Example. The Mellin transform of the function

1 + 푥 푓(푥) = log | | , 1 − 푥 is given by 휋 푀{푓}(푠) = 휋푠−1 tan ( 푠) . 2 with strip of definition 푠 ∈ ⟩−1, 1⟨.

The Mellin transform enjoys some basic compatibilites with opera- tions on functions that we explore now. Of course it goes without saying that the Mellin transform is linear, but there are even better compatibili- ties worth studying.

1.1.11 Lemma. Suppose

푓∶ R>0 C a function with Mellin transform 푀{푓}. These identities hold in the appropriate strip of definition. (i) If 푎 > 0 and 푔(푥) = 푓(푎푥), then 푀{푔}(푠) = 푎−푠푀{푓}(푠). The proof of these claims – as well as the task of finding the (ii) If 푧 ∈ C and 푔(푥) = 푥푧푓(푥), then 푀{푔}(푠) = 푀{푓}(푧 + 푠). appropriate strip of definition – is left to the reader. (iii) If 푎 > 0 and 푔(푥) = 푓(푥푎), then 푀{푔}(푠) = 푎−1푀{푓}(푎−1푠).

(iv) If 푎 < 0 and 푔(푥) = 푓(푥푎), then 푀{푔}(푠) = −푎−1푀{푓}(푎−1푠).

(v) If 푔(푥) = 푓′(푥), then 푀{푔}(푠) = (1 − 푠)푀{푓}(푠 − 1).

푑 (vi) If 푔(푥) = log(푥)푓(푥), then 푀{푔} = 푑푠 푀{푓}(푠). the analytic picture 11

1.1.12. Consider the topological group isomorphism exp∶ R R>0; if Here of course R is considered 푓∶ R C, then it is easy to see that additively and R>0 is considered >0 multiplicatively. 푀{푓}(−2휋푖푦) = 퐹{푓 ∘ exp}(푦), where 퐹 is the Fourier transform.

1.1.13 Definition. If 휙 is a holomorphic function on a strip ⟩훼, 훽⟨, then for any 푥 ∈ R>0 and for any 푐 ∈ ]훼, 훽[, the formula This contour integral is indepen- dent of 푐 thanks to the Cauchy 1 푐+푖∞ integral formula. 푀−1{휙}(푥) ≔ ∫ 휙(푠)푥−푠 푑푠, 2휋푖 푐−푖∞ −1 defines a function 푀 {휙}∶ R>0 C, called the inverse Mellin trans- form of 휙.

1.1.14 Theorem (Mellin inversion). Suppose 푓∶ R>0 C a function with Mellin transform 푀{푓}. Then one has We omit this proof.

푓 = 푀−1{푀{푓}}.

A common situation is that the Mellin transform of a function admits an analytic continuation to a meromorphic function on the complex plane.

1.1.15 Notation. Let us write

−N ≔ {−1, −2, … }, and − N0 ≔ {0, −1, … }.

1.1.16 Proposition. Suppose 푓∶ R>0 C an infinitely differentiable function of rapid decay at infinity, and assume that we have an asymptotic By rapid decay, we mean that expansion sup |푥푚푓(푛)(푥)| < +∞ 푚 푥∈R>0 푓(푥) ∼ ∑ 푎푚푥 as 푥 0. 푚∈N0 for every 푚, 푛 ≥ 0. Then 푀{푓} admits a meromorphic continuation to the complex plane with simple poles at 푠 = −푛 ∈ −N0 of residue

Res−푛 푀{푓} = 푎푛.

Proof. One writes

1 +∞ 푀{푓}(푠) ≔ ∫ 푡푠푓(푡) 푑 log 푡 = ∫ 푡푠푓(푡) 푑 log 푡 + ∫ 푡푠푓(푡) 푑 log 푡, R>0 0 1 12 euler’s gamma function and the field with one element

and one sees readily that the second summand is entire. The first sum- mand can be rewritten, for any 푁 ∈ N0, as

1 1 푁−1 푁−1 푠 푠 푚 푎푚 ∫ 푡 푓(푡) 푑 log 푡 = ∫ 푡 (푓(푡) − ∑ 푎푚푡 ) 푑 log 푡 + ∑ . 0 0 푚=0 푚=0 푚 + 푠 Here the first summand converges on the half-plane ⟩−푁, +∞⟨. This integral is thus meromorphic on ⟩−푁, +∞⟨ with simple poles of residue

푎푚 at 푠 = −푚 ∈ {0, −1, … , −푁}. Since 푁 is arbitrary, we conclude.

The same argument provides us with the following more general statement:

1.1.17 Proposition. Suppose 푓∶ R>0 C an infinitely differentiable function of rapid decay at infinity, and assume also that there is a sequence

(훼푚)푚∈N of complex numbers such that

lim ℜ(훼푚) = +∞, 푚 +∞ and one has an asymptotic expansion

훼푚 푓(푥) ∼ ∑ 푎푚푥 as 푥 0. 푚∈N Then 푀{푓} admits a meromorphic continuation to the complex plane with simple poles at 푠 = −훼푛 (푛 ∈ N) of residue

Res 푀{푓} = 푎 . −훼푛 푛

1.2 Euler’s Gamma function

We are now prepared to introduce Euler’s function 훤.

1.2.1 Definition. If 푓(푥) = exp(−푥), then we define the Gamma function as the Mellin transform of 푓:

훤(푠) ≔ 푀{푓}(푠) = ∫ 푡푠 exp(−푡) 푑 log 푡 R>0 for 푠 ∈ ⟩0, +∞⟨.

1.2.2 Theorem. The function 훤 admits an analytic continuation to a meromorphic function on C with simple poles at 푠 = −푛 ∈ −N0 of residue

Figure 1.1: A graph of 훤(푠), for 푠 ∈ R. the analytic picture 13

(−1)푛 Res 훤 = . −푛 푛!

1.2.3 Example. One has

훤(1) = ∫ exp(−푡) 푑푡 = 1. R>0

1.2.4 Proposition (Functional Equation I). For any 푠 > 0, one has

훤(1 + 푠) = 푠훤(푠), and, consequently, 푛−1 훤(푛 + 푠) = 훤(푠) ∏(푘 + 푠) 푘=0 Proof. This is immediate from Lm. 1.1.11(ii).

1.2.5 Example. If 푛 ≥ 0 is an integer, one has Gauss preferred the normaliza- tion 훱(푠) = 훤(1 + 푠), so that 훤(1 + 푛) = 푛! 훱(푛) = 푛!.

1.2.6. One could instead have used Lm. 1.2.4 to analyticially continue 훤 from ⟩0, +∞⟨ strip-by-strip to C ⧵ −N0.

1.2.7 Definition. For 푢, 푣 ∈ ⟩0, +∞⟨, write

푣−1 푓푣(푥) = 휒]0,1[(푥)(1 − 푥) ; one defines the Beta function as

1 푢−1 푣−1 퐵(푢, 푣) ≔ 푀{푓푣}(푢) = ∫ 푥 (1 − 푥) 푑푥. 0

1.2.8 Proposition. For 푢, 푣 ∈ ⟩0, +∞⟨, one has

훤(푢)훤(푣) 퐵(푢, 푣) = . 훤(푢 + 푣)

Proof. One has

훤(푢)훤(푣) = ∬ exp(−푠 + 푡)푠푢−1푡푣−1 푑푢 푑푣, R>0×R>0 14 euler’s gamma function and the field with one element

and the change of variables 푢 = 푥푦 and 푣 = 푥(1 − 푦) can then be performed to obtain +∞ 1 훤(푢)훤(푣) = ∫ exp(−푥)푥푢+푣 푑 log 푥 ⋅ ∫ 푦푣−1(1 − 푦)푢−1 푑푦 0 0 = 훤(푢 + 푣)퐵(푢, 푣), as desired.

1.2.9 Proposition (Functional Equation II). If 푠 ∈ C ⧵ Z, then one has

훤(푠)훤(1 − 푠) = 퐵(푠, 1 − 푠) = 휋 csc(휋푠).

Proof. The first equality is the previous result. It is clear that 훤(1 + 푠)훤(−푠) = −훤(푠)훤(1 − 푠) and 휋 csc(휋(1 + 푠)) = −휋 csc(휋푠), so it suf- fices to verify the claim for 푠 ∈ ⟩0, 1⟩. In this strip, if 푓(푥) = (1 + 푥)−1, then we have seen that

푀{푓}(푠) = 휋 csc(휋푠), but on the other hand, 1 푀{푓}(푠) = ∫ 푥푠−1(1 − 푥)−푠 푑푥 = 퐵(푠, 1 − 푠), 0 as desired.

1.2.10 Example. We obtain computations directly from this: This offers an inefficient proof that ∫ exp(−푡2) 푑푡 = √휋. 1 R 훤 ( ) = √휋, 2 and, more generally, The value of the Gamma function at half-integers is 1 (2푚 − 1)!! 훤 (푚 + ) = √휋, of particular import, since it 2 2푚 appears in the formula for the volume of an 푛-ball, which we where the semifactorial is defined by the formula will discuss below. ⌈푘/2⌉−1 푘!! = ∏ (푘 − 2푗), 푗=0 so that

1.2.11 Definition. For any positive integer 푚 and any 푠 ∈ C ⧵ −N0, define 푚푠푚! 푚푠 훤 (푠) ≔ = . 푚 푠(1 + 푠) ⋯ (푚 + 푠) 푠 푠 푠 (1 + 1 ) ⋯ (1 + 푚 ) the analytic picture 15

1.2.12 Theorem (Euler Product Formula). For any 푠 ∈ C ⧵ −N0, we have

훤(푠) = lim 훤푚(푠). 푚 +∞

푚−1 1+푠 Proof. It is easy to see that 훤푚(1 + 푠) = 푠 ( 푚 ) 훤1+푚(푠); hence the limit on the right satisfies the functional equation for 훤. We are thus reduced to proving the claim for 푠 lying in the half plane ⟩0, +∞⟨. One has 푡 푚 exp(−푡) = lim (1 − ) , 푚 +∞ 푚 so the absolute convergence of both the integral and the limit permits us to write 푚 푡 푚 훤(푠) = lim ∫ (1 − ) 푡푠 푑 log 푡 푚 +∞ 0 푚 for 푠 ∈ ⟩0, +∞⟨. Integrating by parts, we obtain

푚 푚 푚 푡 푠 푚! 푠+푚−1 ∫ (1 − ) 푡 푑 log 푡 = 푚 ∫ 푡 푑푡 = 훤푚(푠), 0 푚 푠(푠 + 1) ⋯ (푠 + 푚 − 1)푚 0 as desired.

1.2.12.1 Corollary. In particular, for any 푠 ∈ C ⧵ −N0, we have

1 푠 1 (1 + 푛 ) 훤(푠) = ∏ 푠 . 푠 푛∈N 1 + 푛

1.2.13 Notation. For 푚 ∈ N0, we write ℎ푚 for the 푚-th harmonic number 푚 1 ℎ푚 = ∑ . 푘=1 푘 We let 훾 denote the Euler–Mascheroni constant, defined by

1 ℎ = 훾 + log 푁 + 푂 ( ) 푁 푁 for any positive integer 푁.

1.2.14 Theorem (Weierstraß Product Formula). For any element 푠 ∈

C ⧵ −N0, we have

푠 exp(−훾푠) exp ( 푛 ) 훤(푠) = ∏ 푠 . 푠 푛∈N 1 + 푛 16 euler’s gamma function and the field with one element

Proof. One has

푚 푚 1 1 푚푠 = exp(푠 log 푚) = exp (푠 log 푚 − 푠 ∑ ) ⋅ exp (푠 ∑ ), 푛=1 푛 푛=1 푛 whence 푚 푚 푠 1 1 exp ( 푛 ) 훤푚(푠) = exp (푠 log 푚 − 푠 ∑ ) ∏ 푠 . 푠 푛=1 푛 푛=1 1 + 푛 Letting 푚 + ∞, one obtains the Weierstraß Product Formula.

1.2.15 Proposition (Gauß Product Formula). One has

푚−1 푘 ∏ 훤 (푠 + ) = (2휋)(푚−1)/2푚1/2−푚푠훤(푚푠). 푘=0 푚

1.2.15.1 Corollary.

푚−1 푘 (2휋)(푚−1)/2 ∏ 훤 ( ) = . 푘=1 푚 √푚

The Psi function is the logarithmic derivative of the Gamma function.

1.2.16 Definition. One defines, for 푠 ∈ C ⧵ −N0 푑 훤′(푠) 훹(푠) ≔ log 훤(푠) = . 푑푠 훤(푠)

As a consequence of the Euler Product Formula, we deduce the following.

1.2.17 Proposition. One has, for any 푠 ∈ C ⧵ −N0, 푚 1 훹(푠) = lim (log 푚 − ∑ ) 푚 +∞ 푘=0 푘 + 푠

On the other hand, the Weierstraß Product Formula implies the following.

1.2.18 Proposition. One has, for any 푠 ∈ C ⧵ −N0, 푠 − 1 훹(푠) = −훾 + ∑ . 푛∈N 푛(푠 − 1 + 푛) the analytic picture 17

1.2.19 Example. It is obvious that 훹(1) = −훾. More generally,

훹(1 + 푚) = ℎ푚 − 훾.

1.2.20 Proposition (Functional Equation I). For any 푠 ∈ C ⧵ −N0, one has 1 훹(1 + 푠) = + 훹(푠). 푠

1.2.21 Proposition (Functional Equation II). For any 푠 ∈ C ⧵ Z, one has

훹(푠) − 훹(1 − 푠) = −휋 cot(휋푠).

Let 푣푛 be the volume of the unit 푛-ball. Then one has In the third equality, we’re simply using the fact that 푛 푛/2 2 +∞ 휋 = (∫ exp(−푥 ) 푑푥) ∫ 푓 푑휇 = ∫ 휇 {푥 ∈ 푋 | 푓(푥) > 푡} 푑푡 R 푋 0 2 2 for a positive function 푓 on a = ∫ exp (−푥1 − ⋯ − 푥푛 ) 푑푥1 ⋯ 푑푥푛 R푛 measure space (푋, 휇). 1 푛/2 = ∫ 푣푛 (− log 푡) 푑푡 0 +∞ 푛/2 = 푣푛 ∫ 푠 exp(−푠) 푑푠 0 푛 = 푣 훤 (1 + ). 푛 2

1.3 Ramanujan’s Master Theorem

1.3.1 Definition. Suppose 퐴, 푃, 훿 ∈ R are real numbers such that 퐴 < 휋 and 0 < 훿 ≤ 1. The Hardy class 퐻(퐴, 푃, 훿) consists of holomorphic functions 휙 on the half-plane ⟩−훿, +∞⟨ such that

휙(푠) = 푂(exp(−푃ℜ푠 + 퐴|ℑ푠|)).

1.3.2 Theorem (Ramanujan Master). Suppose 휙 ∈ 퐻(퐴, 푃, 훿). Then the power series 푓(푡) = ∑ (−푡)푚휙(푡) 푚∈N0 converges for 푡 ∈ ]0, exp(푃)[ and defines a real analytic functon 푓 there. The function 푓 extends to an analytic function on ]0, +∞[, and for any 푠 ∈ ⟩0, 훿⟨, we have 푀{푓}(푠) = 휋 csc(휋푠)휙(−푠). 18 euler’s gamma function and the field with one element

Proof. Write 훷(푡) ≔ ∑ (−푡)푚휙(푚); 푚∈N0 the growth conditions given ensure that 훷(푡) converges for 푡 ∈ ]0, exp(푃)[. The Cauchy Residue Theorem gives 1 푐+푖∞ 훷(푡) = ∫ 휋 csc(휋푠)휙(−푠)푡−푠 푑푠 2휋푖 푐−푖∞ for any 푐 ∈ ]0, 훿[. The integral on the right converges uniformly on compact subsets of ]0, +∞[, and now the claim now follows from the Mellin inversion formula.

Writing 휆(푠) = 휙(푠)훤(1 + 푠), we obtain the formula +∞ (−푡)푚 훤(푠)휆(−푠) = ∫ 푡푠 ∑ 휆(푚) 푑 log 푡. 0 푚! 푚∈N0 Much about this formula is interesting, but one way to think of it is to say that the Mellin transform of a power series interpolates the coefficients of that power series.

1.3.3 Example. Assume that 푓 admits an expansion of the form 휆(푚) 푓(푡) = ∑ (−푡)푚. 푚! 푚∈N0 Then one has 푀{푓}(푠) = 훤(푠)휆(−푠).

1.3.4 Example. Let’s write 1 퐵 푓(푡) ≔ = ∑ 푚 푡푚−1 exp(푡) − 1 푚! 푚∈N0 for 푡 > 0, where 퐵푚 is the 푚-th Bernoulli number, given by 푚 푘 1 푛 푘 푚 퐵푚 = ∑ ∑(−1) ( )푛 . 푘=0 1 + 푘 푛=0 푛 Clearly 푓 is of rapid decay at infinity, and so one knows that the Mellin transform 푀{푓} is meromorphic on C with simple poles at 푠 = 1 − 푚

(푚 ∈ N0) of residue 퐵 Res 푀{푓} = 푚 . 1−푚 푚! the analytic picture 19

Since exp(푡) > 1 if 푡 > 0, we can write 푓(푡) as a geometric series

푓(푡) = ∑ exp(−푚푡), 푚∈N so when we apply our rules for the Mellin transform, we obtain

푀{푓}(푠) = ∑ 훤(푠)푚−푠 = 훤(푠)휁(푠), 푚∈N where 휁(푠) = ∑ 푚−푠. 푚∈N Since we know that 훤 is nonvanishing and meromorphic on the plane with 푚 simple poles of residue (−1) /푚! at 푠 = −푚 ∈ −N0, it follows that 휁(푠) admits a meromorphic continuation to the plane with a unique simple pole of residue 1 at 푠 = 1. Moreover, the Ramanujan Master Theorem gives, for any −푚 ∈ −N0, 퐵 휁(−푚) = (−1)푚 푚+1 푚 + 1

1.3.5 Example. We can generalize the previous example in the following manner. The Hurwitz zeta function

휁(푠, 푞) = ∑ (푚 + 푞)−푠. 푚∈N Then if exp(−푞푡) 1 푓(푡) = − , 1 − exp(−푡) 푡 one has 푀{푓}(푠) = 훤(푠)휁(푠, 푞).

On the other hand, if 퐵푚(푞) denotes the 푚-th Bernoulli polynomial 푚 푚 푚−푘 퐵푚(푞) = ∑ ( )퐵푚푞 , 푘=0 푘 then near the origin, one has 푡 exp(푞푡) 퐵 (푞) = ∑ 푚 푡푚, exp(푡) − 1 푚! 푚∈N0 whence for −푚 ∈ −N, 퐵 (푞) 휁(1 − 푚, 푞) = − 푚 . 푚 20 euler’s gamma function and the field with one element

1.3.6 Definition. A Dirichlet series is a series

−푠 퐿(푠) = ∑ 푎푚푚 ; 푚∈N a generalized Dirichlet series is a series

−푠 퐿(푠) = ∑ 푎푚휆푚 푚∈N for some increasing sequence (휆푚) with 휆푚 + ∞ that grows at least as quickly as (푚푗) with 푗 > 0.

1.3.7 Theorem. Any generalized Dirichlet series 퐿(푠) = ∑푚∈N 푎푚 exp(−휆푚푠) admits an abscissa of convergence 휎푐 with the property that 퐿(푠) converges if 푠 ∈ ⟩휎푐, +∞⟨, and 퐿(푠) does not converge if 푠 ∈ ⟩−∞, 휎푐⟨. Further- more, if 푠0 ∈ ⟩휎푐, +∞⟨, then there is a neighbourhood of 푠0 on which 퐿(푠) converges uniformly.

1.3.8 Example. Suppose

−푠 퐿(푠) = ∑ 푎푚휆푚 푚∈N a generalized Dirichlet series. Now set, for 푡 > 0,

푓(푡) ≔ ∑ 푎푚 exp(−휆푚푡). 푚∈N Assume that 휙 has an asymptotic expansion

+∞ 푚 푓(푡) ∼ ∑ 푏푚푡 as 푡 0. 푛=−1 Then 푀{푓}(푠) = 훤(푠)퐿(푠), whence 퐿(푠) admits a meromorphic continuation to the complex plane with only one simple pole at 푠 = 1, and for any −푚 ∈ −N0,

푚 퐿(−푚) = (−1) 푚!푎푚.

1.3.9 Example. A character

휒∶ (Z/푘)× C×, the analytic picture 21

extends to Z/푘 by declaring 휒(푗) = 0 if 푗 is not a unit and then to Z via the obvious quotient map. The resulting map 휒∶ Z C is called a Dirichlet character. One defines the Dirichlet 퐿-series 퐿(푠, 휒) = ∑ 휒(푚)푚−푠, 푚∈N and one can follow the recipe above to write 푓(푡, 휒) = ∑ 휒(푚) exp(−푚푡). 푚∈N Then 푀{푓(•, 휒)}(푠) = 훤(푠)퐿(푠, 휒), whence 퐿(푠, 휒) admits a meromorphic continuation to the complex plane with only one simple pole at 푠 = 1, and for any −푚 ∈ −N,

퐵푚,휒 퐿(1 − 푚, 휒) = − , 푚 where 퐵푚,휒 is the generalized Bernoulli number, with

푡 exp(푚푡) 퐵푚,휒 ∑ 휒(푚) = ∑ 푡푚. exp(푚푡) − 1 푚! 푚∈N 푚∈N0

1.3.10 Example. Consider the Dirichlet series 푚 −푠 퐿(푠) = 퐿(푠, 휒4) = ∑ (−1) (2푚 − 1) . 푚∈N Then one has 1 1 퐸 푓(푡) = sech 푡 = ∑ 푚 푡푚, 2 2 푚! 푚∈N0 where 퐸푚 is the 푚-th Euler number. We thus obtain for any −푚 ∈ −N0, 퐸 퐿(−푚) = 푚 . 2

1.3.11 Theorem. Suppose 퐿(푠) = ∑푚∈N 푎푚 exp(−휆푚푠). Let

퐴(푡) = ∑ 푎푚. 푚≤푡

Then for any 푠 ∈ ⟩max{0, 휎푐}, +∞⟨, one has 퐿(푠) 푀{퐹}(푠) = , 푠 where 퐹(푡) = 퐴(1/푡). 22 euler’s gamma function and the field with one element

1.4 Theta series

1.4.1 Definition. Write

휃(푧) ≔ ∑ exp(푖휋푛2푧) = 1 + 2 ∑ exp(푖휋푛2푧). 푛∈Z 푛∈N This series converges in the upper half-plane H = {푧 ∈ C | ℑ(푧) > 0}, and it defines an analytic function there, called Jacobi’s 휃 function.

Let us contemplate a suitably normalized version of the Mellin trans- 1 form of the function 푓(푥) = 2 (휃(푖푥) − 1): set 푍(푠) ≔ 푀{푓}(푠/2).

2 One sees easily that the Mellin transform of 푔푛(푥) = exp(−휋푛 푥) can be computed as −푠 −2푠 푀{푔푛}(푠) = 휋 훤(푠)푛 , and absolute convergence permits us to obtain, for 푠 ∈ ⟩0, +∞⟨,

−푠 푀 { ∑ 푔푛}(푠) = 휋 훤(푠)휁(2푠). 푛∈N

But of course 푓 = ∑푛∈N 푔푛, and so we conclude

1.4.2 Proposition. One has

− 푠 푠 푍(푠) = 휋 2 훤 ( ) 휁(푠). 2

1.4.3 Theorem (Poisson Summation). For any (complex-valued) Schwartz function 휙 on R, one has

∑ 휙(푚) = ∑ 휙̂(푚). 푛∈Z 푛∈Z

1.4.3.1 Corollary (Functional equation). The Jacobi theta function satisfies1 1 Here, (푧/푖)1/2 = 1 푧 1/2 exp((1/2) log(푧/푖)), taking 휃 (− ) = ( ) 휃(푧). the principal branch of the log. 푧 푖 Proof. Consider the function 휙(푡) ≔ exp(−휋푡2), and observe that it is its own Fourier transform. Now for 푥 > 0, write

2 훾푥(푡) = 휙(√푥푡) = exp(−휋푥푡 ), the analytic picture 23

so that 훾푥(푛) = 푔푛(푥) in the notation above. Now

1 푦 1 푦 1 푦2 1 푦 훾̂(푦) = 휙̂( ) = 휙 ( ) = exp (−휋 ) = 훾 ( ). 푥 √푥 √푥 √푥 √푥 √푥 푥 √푥 푥 푥

Now we apply Poisson Summation to conclude.

1.4.3.2 Corollary (Functional equation). One has

푍(푠) = 푍(1 − 푠).

Proof. Using the functional equation for 휃, we obtain

1 1 1 푓(푥) = 푥−1/2푓 ( ) + 푥−1/2 − , 푥 2 2 and in the same manner as above, one may show that the Mellin trans- form of the function on the right is 푀{푓}(1/2 − 푠). Exercise. Check that the Mellin transform behaves as described. Let’s use the same strategy to give a functional equation for the 퐿- function of a nontrivial, primitive Dirichlet character 휒 of modulus 푘.

1.4.4 Definition. The exponent of 휒 is the quantity 휖 ∈ {0, 1} with the property that 휒(−1) = (−1)휖휒(1).

Then the corresponding theta series is When 푚 = 휖 = 0, we declare 푚휖 = 0. 푚2 휃(휒, 푧) = ∑ 휒(푚)푚휖 exp (푖휋 푧). 푚∈Z 푘

Let us run the same program as above. We study the Mellin transform of the function 1 푓(휒, 푥) = 휃(휒, 푖푥), 2 and with the same argument as above, we obtain the following.

1.4.5 Proposition. One has

푠 + 휖 푘 (푠+휖)/2 푠 + 휖 푀{푓(휒, •)} ( ) = 2 ( ) 퐿(휒, 푠)훤 ( ) 2 휋 2 24 euler’s gamma function and the field with one element

1.4.6 Definition. We define the completed 퐿-function of 휒:

푘 푠/2 푠 + 휖 훬(휒, 푠) ≔ ( ) 퐿(휒, 푠)훤 ( ) 휋 2

We want to extract a functional equation for 훬(휒, 푠) from a functional equation for the theta series. This latter functional equation involves an auxiliary term called the Gauß sum.

1.4.7 Definition. The Gauß sum for 휒 is given by

푘−1 푚 휏(휒) = ∑ 휒(푚) exp (푖2휋 ). 푚=0 푘

One has

푘−1 푘−1 |휏(휒)|2 = ∑ ∑ 휒(푛) exp(푖2휋푚푛/푘) exp(−푖2휋푚/푘) 푚=0 푛=0 푘−1 푘−1 = ∑ 휒(푛) ∑ exp(푖2휋푚(푛 − 1)/푘) 푛=0 푚=0 = 휒(1)푚 = 푚. with this in mind, here is our functional equation for the theta function.

1.4.8 Proposition (Functional equation). One has

1 휏(휒) 푧 휖+1/2 휃 (휒, − ) = ( ) 휃 (휒, 푧). 푧 푖휖√푚 푖

1.4.8.1 Corollary (Functional equation). One has

휏(휒) 훬(휒, 푠) = 훬(휒, 1 − 푠). 푖휖√푚

We have a higher-dimensional variant of all this as well.

1.4.9 Notation. For any finite Z/2-set 푆, and denote by 휎 the nontrivial involution. We will abuse notation and write 휎 also for complex conjuga- tion. We form the C-vector space

C푆 ≔ Map(푆, C) the analytic picture 25

with basis 푆, and we regard C푆 as an algebra under pointwise addition and multiplication. For 푧 ∈ C푆, we write

tr(푧) ≔ ∑ 푧(푠) and 푁(푧) ≔ ∏ 푧(푠). 푠∈푆 푠∈푆 There is an action of Z/2 given by conjugation: 푧 휎 ∘ 푧 ∘ 휎, and we may identify the R-algebra Map(푆, R) with the fixed point subalgebra for this action:

Map(푆, R) ≅ R푆 ≔ Map(푆, C)Z/2 ⊆ Map(푆, C).

This entitles us to speak of the upper half-space

1 H푆 ≔ {푧 ∈ C푆 | 푧 = 푧 ∘ 휎 and (푧 − 휎 ∘ 푧 ∘ 휎) > 0} . 2푖

1.4.10 Example. A standard example of a suitable Z/2-set 푆 is the set Hom(퐾, C) for a number field 퐾, with the obvious Galois action. Then one has isomorphisms

Hom(퐾,C) Hom(퐾,C) C ≅ 퐾 ⊗Q C and R ≅ 퐾 ⊗Q R.

1.4.11 Notation. There is also a hermitian form

(푧, 푤) ≔ tr(푧 ⋅ (휎 ∘ 푤)) = ∑ 푧(푠) ⋅ 휎(푤(푠)), 푠∈푆 which is invariant under the action of Z/2. This restricts to an inner product (•, •) on R푆. Let 휇 denote the Haar meeasure relative to that metric.

1.4.12 Definition. For every Z-structure 푊 ⊂ R푆, we define the theta series

휃푊(푧) = ∑ exp {푖휋(푤푧, 푤)} . 푤∈푊 This converges absolutely and uniformly on all compacta in H푆. 푆 More generally, suppose 푎, 푏 ∈ R , and suppose 푝∶ 푆 N0 such that 푝 ⋅ (푝 ∘ 휎) = 0. Set

푝 푝 휃푊(푎, 푏, 푧) = ∑ 푁 ((푎 + 푤) ) exp {푖휋 (((푎 + 푤)푧, 푎 + 푤) + 2(푏, 푤))} . 푤∈푊 26 euler’s gamma function and the field with one element

We now contemplate Schwartz functions 휙∶ R푆 C and their Fourier transforms

휙̂(푦) = ∫ 푓(푥) exp(−푖2휋(푥, 푦)) 푑휇, R푆 and we see much of the good behaviour we’ve become accustomed to.

1.4.13 Example. The function 휙(푥) = exp(−휋(푥, 푥)) is its own Fourier transform.

To prove a functional equation for 휃푊, we will need a Poisson Summa- tion formula.

1.4.14 Proposition (Poisson Summation). If 푊 ⊂ R푆 is a Z-structure, then let 푊∨ ≔ {푣 ∈ R푆 | ∀푤 ∈ 푊,(푤, 푣) ∈ Z}.

Then for any Schwartz function 휙∶ R푆 C, one has If 푊 is the Z-span of vectors 푤1,…, 푤푛, then covol(푊) ≔ ∑ 휙(푤) = covol(푊)−1 ∑ 휙̂(푣). | det(푤1,…, 푤푛)|. 푤∈푊 푣∈푊∨

1.4.14.1 Corollary (Functional equation). One has

1 √(푁(푧/푖)) 휃 (− ) = 휃 ∨ (푧). 푊 푧 covol(푊) 푊 More generally, one has

푝 1 tr(푝) −1 푝+1/2 푝 휃 (푎, 푏, − ) = (푖 exp(푖2휋(푎, 푏)) covol(푊)) 푁 ((푧/푖) ) 휃 ∨ (−푏, 푎, 푧). 푊 푧 푊

1.4.15 Notation. We define

(R푆) ≔ {푥 ∈ R푆 | 푥 = 푥 ∘ 휎 and 푥 > 0} . >0 One has a topological isomorphism

휙∶ (R푆) ∼ ∏ R >0 >0 푝∈푆/퐶2 defined by

푥 (∏ 푥(푠)) . 푠∈푝 푝∈푆/퐶2 the analytic picture 27

(We say that 푝 is real if it lies in the image of 푆퐶2 ; otherwise, we call it complex.) One now pulls back the Haar measure to obtain

푑 log 푥 ≔ 휙⋆ (∏ 푑 log 푥(푠)) . 푠∈푆

1.4.16 Definition. The Mellin transform corresponding to the 퐶2-set 푆 of a function 푓∶ (R푆) C푆 is the function >0

푠 푀푆{푓}(푠) ≔ ∫ 푁(푓(푥)푥 ) 푑 log 푥. 푆 푥∈(R )>0 In particular, if 푓(푥) = 푒푥푝(−푥), then the higher dimensional gamma function is given by 훤푆 ≔ 푀{푓}(푠).

1.4.17 Proposition. One obtains

훤푆(푠) = ∏ 훤푝(푠푝), 푝∈푆/퐶2 where {훤(푠푝) if 푝 is real; 훤푝(푠푝) ≔ { 21−tr(푠푝)훤(tr(푠 )) if 푝 is complex. { 푝

1.4.18 Definition. At last, we define the 퐿-function of the 퐶2-set 푆 by −푠/2 퐿푆(푠) ≔ 푁(휋 )훤푆(푠/2).

Using the proposition above, we find that

퐿푆(푠) = ∏ 퐿푝(푠푝), 푝∈푆/퐶2 where

−푠푝/2 {휋 훤(푠푝/2) if 푝 is real; 퐿푝(푠푝) = { 2(2휋)− tr(푠푝)/2훤(tr(푠 )/2) if 푝 is complex. { 푝 As a matter of notation, assume that 푛 ≔ #푆, and

푆 = 푟1(퐶2/퐶2) ∪ 푟2(퐶2/푒), 푆 so that 푛 = 푟1 + 2푟2. We may apply any function of C to a single complex number 푠 by writing 푓(푠) ≔ 푓(const푠). So, for instance,

(1−2푠)푟2 푟1 푟2 훤푆(푠) = 2 훤(푠) 훤(2푠) −푛푠/2 퐿푆(푠) = 휋 훤푆(푠/2) 28 euler’s gamma function and the field with one element

In particular, for the two 퐶2-orbits, 퐿 (푠) ≔ 퐿 (푠) = 휋−푠/2훤(푠/2) R 퐶2/퐶2 퐿 (푠) ≔ 퐿 (푠) = 2(2휋)−푠훤(푠), C 퐶2/푒 and one has 푟1 푟2 퐿푆(푠) = 퐿R(푠) 퐿C(푠) . Using what we have already seen about the gamma function, we obtain the following directly:

1.4.19 Theorem. One has

퐿R(1) = 1; −1 퐿C(1) = 휋 ; 푠 퐿 (2 + 푠) = 퐿 (푠); R 2휋 R 푠 퐿 (1 + 푠) = 퐿 (푠); C 2휋 C 휋 퐿 (1 − 푠)퐿 (1 + 푠) = sec ( 푠); R R 2 퐿C(1 − 푠)퐿C(푠) = 2 csc(휋푠); 푠 푟1+푟2 푠 푟2 퐿 (푠) = cos (휋 ) sin (휋 ) 퐿 (푠)푛퐿 (1 − 푠). 푆 2 2 C 푆

Now we apply this machinery to the 퐶2-set 푆 = Hom(퐾, C) for a number field 퐾 of degree 푛 over Q. Then R푆 is the Minkowski space, for 푆 which we have a selected isomorphism R ≅ 퐾 ⊗Q R. Note that if 푎 ◁ 푂퐾, then 푎 is a Z-structure on R푆 with covolume

covol(푎) = √푑푎, 2 where 푑푎 = 푁(푎) |푑퐾| is the absolute value of the discriminant. Observe that if 푥 ∈ 퐾×, then 푁((푥)) = |푁(푥)|.

1.4.20 Definition. The Dedekind zeta function of 퐾 is given by

−푠 휁퐾(푠) ≔ ∑ 푁(푎) , 0≠푎◁푂퐾 where 푁(푎) ≔ #(푂퐾/푎). This series converges on ⟩1, +∞⟨; moreover, one has −푠 −1 휁퐾(푠) = ∏ (1 − 푁(푝) ) . 0≠푝∈Spec 푂퐾 the analytic picture 29

For each ideal class 훷 ∈ Cl(퐾), one may define a partial zeta function

휁(훷, 푠) ≔ ∑ 푁(푎)−푠, 푎◁푂퐾, 푎∈훷 so that

휁퐾(푠) = ∑ 휁(훷, 푠). 훷∈Cl(퐾)

In effect, we are going to find functional equations for each partial zeta function, and then we are going to assemble these.

1.4.21 Notation. We write

푠/2 푍(훷, 푠) ≔ |푑퐾| 퐿푋(푠)휁(훷, 푠)

To express 푍(훷, 푠) as a suitable Mellin transform, we need to cut out the norm-one hypersurface

S푆 ≔ {푥 ∈ (R푆) | 푁(푥) = 1}, >0 so that (R푆) ≅ S푆 × R . Of course 푂× /휇 ⊂ S푆. We take 푑×푥 to be >0 >0 퐾 퐾 the unique Haar measure on S푆 such that

푑 log 푥 = 푑×푥 × 푑 log 푡.

By the Dirichlet unit theorem,

× 푆 log(푂퐾/휇퐾) ⊂ {푥 ∈ R | 푥 = 푥 ∘ 휎 and tr(푥) = 0} is a Z-structure. Let 퐹 be the inverse image of any fundamental domain of × 2 log(푂퐾/휇퐾).

1.4.22 Exercise. The domain 퐹 we constructed has volume

푟1+푟2−1 vol(퐹) = 2 푅퐾,. where × covol(log(푂퐾/휇퐾)) 푅퐾 ≔ √푟1 + 푟2 is the quantity known as Dirichlet regulator. 30 euler’s gamma function and the field with one element

1.4.23 Theorem. Write

1 1/푛 × vol(퐹) 푓퐹(푎, 푡) ≔ ∫ 휃푎(푖푥(푡/푑푎) ) 푑 푥 − . #휇퐾 퐹 #휇퐾 Then 푍(훷, 푠) = 푀{푓}(푠/2).

× Proof. Let 푅 denote the quotient of the action of 푂퐾 upon the ideal 푎, and form the sum over representatives

1/푛 푔(푥) = ∑ exp(−휋(푟푥/푑푎 , 푥)). 푟∈푅

Then one has the following Exercise!

푠 −푛푠 푠 |푑퐾| 휋 훤푋(푠)휁(훷, 2푠) = ∫ 푔(푥)푁(푥) 푑 log 푥. 푆 (R )>0 Consequently, we obtain

1/푛 × 푠 푍(훷, 2푠) = ∫ {∫ ∑ exp(−휋(푟푥(푡/푑푎) , 푥)) 푑 푥} 푡 푑 log 푡. 푆 R>0 S 푟∈푅 We’d like to connect this to the theta series, but this involves a sum over representatives of 푅 rather than a sum over 푎 itself. So:

1/푛 × 1/푛 × ∫ ∑ exp(−휋(푟푥(푡/푑푎) , 푥)) 푑 푥 = ∑ ∫ ∑ exp(−휋(푟푥(푡/푑푎) , 푥)) 푑 푥 S푆 × 휂2퐹 푟∈푅 휂∈|푂퐾| 푟∈푅 1 1/푛 × = ∑ ∫ ∑ exp(−휋(푟푥(푡/푑푎) , 푥)) 푑 푥 #휇퐾 × 휂2퐹 휂∈푂퐾 푟∈푅 1 1/푛 × = ∫ ∑ ∑ exp(−휋(푟푥(푡/푑푎) , 푥)) 푑 푥 #휇퐾 휂2퐹 × 휂∈푂퐾 푟∈푅 1 1/푛 × = ∫ {휃푎(푖푥(푡/푑푎) ) − 1} 푑 푥, #휇퐾 휂2퐹 as desired.

1.4.24 Theorem. The function 푍(훷, 푠) admits a meromorphic continuation to C with simple poles at 0 and 1 with

2푟1+푟2 2푟1+푟2 Res0 푍(훷, 푠) = − 푅퐾 and Res1 푍(훷, 푠) = 푅퐾. #휇퐾 #휇퐾 Furthermore, it satisfies the functional equation

푍(훷, 푠) = 푍(훷 ⊗ 휔, 1 − 푠), the analytic picture 31

where 휔 is the codifferent ideal

휔 = {푥 ∈ 퐾 | tr퐾|Q(푥푂퐾) ⊆ Z}.

Proof. In order to employ the functional equation for our 휃-function, the first task is to sort out what the dual lattice to 푎 is. Write 푏 = 푎−1 ⊗ 휔, so that 푏 = {푥 ∈ R푆 | 푥푎 ∈ 휔}. One notes that

Exercise. Show that 푑푏 = 1/푑푎. 푎∨ = {푥 ∈ R푆 | tr(푥푎) ⊆ Z} 푆 = {푥 ∈ R | ∀푟 ∈ 푎, tr퐾|Q(푥푟푂퐾) ⊆ Z} = {푥 ∈ R푆 | 푥푎 ∈ 휔} = 푏.

Note also that 휃푎∨ = 휃푎∨ , whence

1 1 −1/푛 푡푖푚푒푠 푓퐹(푎, ) = ∫ 휃푎(푖푥(푡푑푎) ) 푑 푥 푡 #휇퐾 퐹 1/2 1 (푡푑푎) 1/푛 × = ∫ 휃푏(푖푥(푡푑푎) ) 푑 푥 #휇퐾 covol(푎) 퐹−1 1/2 푡 1/푛 × = ∫ 휃푏(푖푥(푡/푑푏) ) 푑 푥 #휇퐾 퐹−1 1/2 = 푡 푓퐹−1 (푏, 푡).

Now the usual Mellin transform argument completes the proof.

1.4.25 Definition. The completed zeta function of 퐾 is given by

푠/2 푍퐾(푠) ≔ |푑퐾| 퐿푋(푠)휁퐾(푠)

1.4.26 Theorem. The function 푍퐾(푠) admits a meromorphic continuation to C with simple poles at 0 and 1 with

2푟1+푟2 # Cl(퐾) 2푟1+푟2 # Cl(퐾) Res0 푍(훷, 푠) = − 푅퐾 and Res1 푍(훷, 푠) = 푅퐾, #휇퐾 #휇퐾 Furthermore, it satisfies the functional equation

푍퐾(푠) = 푍퐾(1 − 푠).

1.5 Pontryagin duality and Fourier analysis 32 euler’s gamma function and the field with one element

Before we go into details about Tate’s thesis, it’s good to have a crash course on locally compact abelian (LCA) groups and Pontryagin duality.

1.5.1 Example. Here are some LCA groups.

• finite abelian groups – i.e., finite direct sums of cyclic groups Z/푝푣 – with the discrete topology

• Z with the discrete topology

• R with the usual topology

• any finite dimensional R-vector space – these are called vector groups

• Q with the discrete topology (but not the subspace topology!)

• A closed subgroup of an LCA group is LCA.

• A quotient of an LCA group is LCA.

• The circle T = R/Z is LCA.

• The adic circle T = Q/Z is LCA.

• Finite products and coproducts of LCA groups exist and coincide; we write ⊕ for the common operation. In particular, LCA is an additive category.

• If {퐴훼}훼∈훬 is any collection of LCA groups, then the product

∏ 퐴훼 훼∈훬 is LCA.

• The limit of any diagram of continuous homomorphisms of LCA groups is LCA.

• If 훷 is the poset of natural numbers ordered by divisibility, then we obtain Ẑ ≔ lim Z/푚; 푚∈훷op we can look 푝-typically as well:

Z ≔ lim Z/푝푛. 푝 op 푛∈N0

These are rings. Exercise. Characterise all compact hausdorff rings. the analytic picture 33

• One has a topological isomorphism

̂ Z ≅ ∏ Z푝. 푝∈훱

• We can also form Ẑ× ≔ lim (Z/푚)×; 푚∈훷op we can look 푝-typically as well:

Z× ≔ lim (Z/푝푛)×. 푝 op 푛∈N0

• We have the solenoid S ≔ lim R/푚Z 푚∈훷op and the 푝-solenoid S ≔ lim R/푝푛Z. 푝 op 푛∈N0 • One has a topological isomorphism

S푝 ≅ (R × Z푝)/Z.

• The filtered colimit of any diagram of open continuous homomorphisms of LCA groups is LCA.

• We have the finite adèles

−1̂ Afin ≔ colim푛∈훷 푛 Z.

and the 푝-adics Q ≔ colim 푝−푛Z . 푝 푛∈N0 푝

• The adèles are given by

A ≔ R ⊕ Afin.

• One has topological isomorphisms

Q/Z ≅ colim푚∈훷 Z/푚

and Q /Z ≅ colim Z/푝푛, 푝 푝 푛∈N0 both of which are discrete. 34 euler’s gamma function and the field with one element

• One has topological isomorphisms

S ≅ A/Q.

S푝 ≅ (R ⊕ Q푝)/Z[1/푝].

• If {퐴훼}훼∈훬 is any collection of LCA groups, then the coproduct

∐ 퐴훼 = colim푆∈ℱ(훬) ⨁ 퐴훼, 훼∈훬 훼∈푆

where ℱ(훬) is the filtered poset of finite subsets of 훬, is LCA. Warning: the natural map ∐훼∈훬 퐴훼 ∏훼∈훬 퐴훼 is continuous, and it is a set-theoretic inclusion, but it is not the inclusion of a subspace!

• Assume {퐴훼}훼∈훬 is any collection of LCA groups, 퐽∞ ⊂ 훬 some finite subset, and 퐾훽 ⊆ 퐴훽 a compact subgroup for each 훽 ∈ 훬 − 퐽∞. Then the restricted product

{퐾훽}∏∐ 퐴 ≔ colim (∏ 퐴 × ∏ 퐾 ), 훼 푆∈ℱ(훬,퐽∞) 훼 훽 훼∈훬 훼∈푆 훽∈훬−푆

where ℱ(훬, 퐽∞) is the filtered poset of finite subsets of 훬 that contain 퐽∞, is LCA. This is also the sum

{퐾훽}∏∐ 퐴 ≅ ⨁ 퐴 ⊕ (( ∏ 퐴 ) × ( ∐ (퐴 /퐾 ))) . 훼 훼 훽 (∏ (퐴 /퐾 )) 훽 훽 훽∈훬−퐽∞ 훽 훽 훼∈훬 훼∈퐽∞ 훽∈훬−퐽∞ 훽∈훬−퐽∞

{퐾훽} Second warning: the natural map ∏∐훼∈훬 퐴훼 ∏훼∈훬 퐴훼 is con- tinuous, and it is a set-theoretic inclusion, but it is not the inclusion of a subspace!

• One has topological isomorphisms

{Z푝} Afin ≅ ∏∐ Q푝 푝∈훱

and {OQ }푣 finite A ≅ 푣 ∏∐ Q푣, 푣

where the restricted product is over the set of places, 퐽∞ is the set of infinite places, and O ⊂ Q is the ring of integers. Q푣 푣 the analytic picture 35

• More generally, for any number field 퐾, one defines

{O퐾 }푣 finite A퐾 ≅ 푣 ∏∐ 퐾푣, 푣

where 퐽∞ is the set of infinite places.

• The multiplicative from of this are the idèles

× {O퐾 }푣 finite × I퐾 ≅ 푣 ∏∐ 퐾푣 . 푣

Warning: the natural map I퐾 A퐾 is continuous, and it is a set- theoretic inclusion, but it is not the inclusion of a subspace! This is because multiplicative inverse on the suitable subspace of the adèles may not be continuous; we can repair this, however, and we can see that the

inclusion I퐾 A퐾 × A퐾 given by

푥 (푥, 푥−1)

is the inclusion of a subspace.

• One has a topological isomorphism

× ̂× IQ ≅ Q ⊕ R>0 ⊕ Z .

• If 퐾 is a number field, then the 퐾-solenoid is

S퐾 ≔ A퐾/퐾.

1.5.2 Definition. The Pontryagin dual of an LCA group 퐴 is the LCA group 퐴̂ ≔ Hom(퐴, T). The assignment 퐴 퐴̂ is a functor LCAop LCA.

1.5.3 Theorem (Pontryagin). The assignment 퐴 퐴̂ is an equivalence LCAop ∼ LCA, and it is its own inverse.

퐴 퐴̂ Z/푝 Z/푝 finite finite ZT 36 euler’s gamma function and the field with one element

discrete compact RR vector group 푉 푉∨ ̂ product ∏훼∈훬 퐴훼 coproduct ∐훼∈훬 퐴훼 ̂ limit lim 퐴훼 colimit colim 퐴훼 closed subgroup 퐵 ≤ 퐴 quotient 퐴̂/퐵⟂ Q/Z Ẑ

Q푝/Z푝 Z푝 torsion profinite ⟂ {퐾훽} {퐾훽 } ̂ ∏∐훼∈훬 퐴훼 ∏∐훼∈훬 퐴훼 Afin Afin Q푝 Q푝 number field 퐾 A퐾/퐾

1.5.4 Example.

We now give an abstract proof of Pontryagin duality in a manner that does not use the classification of LCA groups.

1.5.5 Definition. Suppose 퐴 a topological abelian group. We will say that 퐴 is admissible if it can be exhibited as a topological subgroup of a product ∏훼∈훬 퐵훼 of LCA groups. Suppose 퐴 a topological abelian group. Then we say that a topology 휏 on |퐴| is 퐴-characteristic if (|퐴|, 휏) is an admissible topological group, and

Hom(퐴, T) = Hom((|퐴|, 휏), T).

1.5.6 Proposition. On any admissible topological abelian group 퐴, there is both a coarsest and finest 퐴-characteristic topology.

Proof. The existence of the coarsest 퐴-characteristic topology is an exercise; let 퐶 be the corresponding admissible topological abelian group. ′ Now let {퐴훼}훼∈훬 be the collection of all admissible topological abelian ′ ′ groups such that |퐴훼| = |퐴| and the topology on 퐴훼 is 퐴-characteristic. Form the pullback ′ 퐹 ∏훼∈훬 퐴훼

퐶 퐶훬. 훥 the analytic picture 37

Of course |퐹| = |퐴|, and it is now a simple matter to see that the topology on 퐹 is 퐴-characteristic.

1.5.7 Definition. A topological abelian group is said to be T-cogenerated if the topology on 퐴 is the finest 퐴-characteristic topology. We write TAb for the category of T-cogenerated topological abelian groups and continuous homomorphisms.

1.5.8 Exercise. Every LCA group is T-cogenerated.

Here’s an auxiliary category.

1.5.9 Definition. Denote by D the following category. The objects are pairs (퐴, 퐵, 휂) consisting of (discrete) abelian groups 퐴 and 퐵 and a pairing 휂∶ 퐴 ⊗ 퐵 T that is nondegenerate. A morphism Nondegeneracy means that for (휙, 휓)∶ (퐴, 퐵, 휂) (퐴′, 퐵′, 휂′) is a homomorphism 휙∶ 퐴 퐴′ and a every 푎 ∈ 퐴, there is 푏 ∈ 퐵 such that 휂(푎, 푏) ≠ 1, and for every ′ homomorphism 휓∶ 퐵 퐵 such that 푏 ∈ 퐵, there is 푎 ∈ 퐴 such that 휂(푎, 푏) ≠ 1. 휂(휙(푎), 푏′) = 휂(푎, 휓(푏′)).

It is not difficult to see that this is an additive category. Furthermore, itis symmetric monoidal relative to the tensor product

′ ′ ′ ′ ′ ′ (퐴, 퐵, 휂) ⊗ (퐴 , 퐵 , 휂 ) = (퐴 ⊗ 퐴 , Hom(퐴, 퐵 ) ×Hom(퐴⊗퐴′,T) Hom(퐴 , 퐵), 휉).

Now the object 퐷 ≔ (T훿, Z, id) is of particular import.

The following is now completely formal.

1.5.10 Proposition. The category D is self-dual. More precisely, it is symmetric monoidal with respect to the tensor product above, it has an internal Hom Hom, and moreover the natural morphism

퐴 Hom(Hom(퐴, 퐷), 퐷) is an isomorphism.

1.5.11 Theorem. The category TAb is equivalent to D above, and more- over the functor Hom(−, T) on TAb coincides with the functor Hom(−, 퐷) on D. 38 euler’s gamma function and the field with one element

Proof. The functor TAb D is given by the assignment

퐴 (|퐴|, Hom(퐴, T), ev).

Its inverse is given as follows: for any (퐴, 퐴′, 휂), give 퐴 the subspace topology of 퐴̂′. Now define D TAb as the assignment

(퐴, 퐴′, 휂) (|퐴|, 휏), where, 휏 is the finest 퐴-characteristic topology on |퐴|.

1.5.12 Notation. For any LCA group 퐴, denote by ℳ(퐴) the set of all regular, countably additive, complex Borel measures on 퐴. Denote by ℋ(퐴) the set of all Haar measures – i.e., nontrivial, biinvariant, regular, countably additive, Borel measures – on 퐴. Recall that ℋ(퐴) is an R>0- torsor.

1.5.13 Definition. Suppose 퐴 an LCA group and 휇 ∈ ℳ(퐴). Then denote by 휇̂∶ 퐴̂ C the function

휇̂(휒) ≔ ∫ 휒(푎) 푑휇(푎). 퐴 This is a uniformly continuous, bounded function, the Fourier–Stieltjes trasform of 휇. If 휆 ∈ ℋ(퐴), and if 휇 is absolutely continuous with respect to 휆, then by Radon–Nikodym, 푑휇 = 푓 푑휆 for some 푓 ∈ 퐿1(퐴, 휆). In this case, we write 푓̂for 휇̂. This is the Fourier transform of 푓. In the other direction, if 휇 ∈ ℳ(퐴̂). Then denote by 휇̌∶ 퐴 C the function 휇̌(푎) ≔ ∫ 휒(푎) 푑휇(휒). 퐴 This is a bounded function, the inverse Fourier–Stieltjes trasform of 휇. If 휆 ∈ ℋ(퐴̂), and if 휇 is absolutely continuous with respect to 휆, then by Radon–Nikodym, 푑휇 = 푓 푑휆 for some 푓 ∈ 퐿1(퐴̂, 휆). In this case, we write 푓̌ for 휇̌. This is the inverse Fourier transform of 푓.

1.5.14 Theorem. The assignment 휇 휇̂ is an isomorphism

̂ ℳ(퐴) ≅ 풞푢(퐴), ̂ where 풞푢(퐴) denotes the space of uniformly continuous, bounded functions on 퐴̂; in fact, it is an algebra isomorphism for the convolutions. the analytic picture 39

The assignment 푓 푓̂is an isomorphism

1 ̂ 퐿 (퐴, 휆) ≅ 풲0(퐴), ̂ ̂ where 풲0(퐴) ⊂ 풞0(퐴) is a dense subalgebra called the Wiener algebra.

1.5.15 Lemma. If 휆 ∈ ℋ(퐴), then there is a 휇 ∈ ℋ(퐴̂) such that for any 푓 ∈ 퐿1(퐴, 휆) ∩ 퐿2(퐴, 휆), one has

̂ ̂ 2 ̂ ̂ 푓 ∈ 풞0(퐴) ∩ 퐿 (퐴, 휇), and ‖푓‖2 ≤ ‖푓‖2.

Dually, for any 휙 ∈ 퐿1(퐴̂, 휇) ∩ 퐿2(퐴̂, 휇), one has

̌ 2 ̌ 휙 ∈ 풞0(퐴) ∩ 퐿 (퐴, 휆), and ‖휙‖2 ≤ ‖휙‖2.

1.5.16 Definition. If 휆 and 휇 are as in the previous Lemma, then since 퐿1(퐴, 휆) ∩ 퐿2(퐴, 휆) ⊂ 퐿2(퐴, 휆) is dense (and similarly for 퐿2(퐴, 휆)), the Fourier transform 푓 푓̂ and the inverse Fourier transform 휙 휙̌ extend to continuoun linear maps

퐿2(퐴, 휆) 퐿2(퐴̂, 휇) and 퐿2(퐴̂, 휇) 퐿2(퐴, 휆), respectively.

1.5.17 Theorem (Fourier Inversion/Plancherel). For any 휆 ∈ ℋ(퐴), then there is a unique 휇 ∈ ℋ(퐴̂) such that: (1) for any 푓 ∈ 퐿1(퐴, 휆) ∩ 퐿2(퐴, 휆), one has

̂ ̂ 2 ̂ ̂ 푓 ∈ 풞0(퐴) ∩ 퐿 (퐴, 휇), and ‖푓‖2 ≤ ‖푓‖2;

(2) dually, for any 휙 ∈ 퐿1(퐴̂, 휇) ∩ 퐿2(퐴̂, 휇), one has

̌ 2 ̌ 휙 ∈ 풞0(퐴) ∩ 퐿 (퐴, 휆), and ‖휙‖2 ≤ ‖휙‖2;

(3) for any 푓 ∈ 퐿2(퐴, 휆), 푓̌̂ = 푓;

(4) for any 휙 ∈ 퐿2(퐴̂, 휇), 휙̌̂ = 휙;

(5) (Plancherel) The assignments 푓 푓̂and 휙 휙̌ are isometries, so that 퐿2(퐴, 휆) ≅ 퐿2(퐴̂, 휇). 40 euler’s gamma function and the field with one element

The pair (휆, 휇) of the theorem are called dual Haar measures. We’ll get in the habit of writing 휆 for a Haar measure on 퐴 and 휆̂ for its dual Haar measure on 퐴̂. Here’s a standard identity

1.5.18 Theorem (Parseval). For any 푓, 푔 ∈ 퐿2(퐴, 휆), one has

∫ 푓푔 푑휆 = ∫ 푓푔̂ ̂ 푑휆̂. 퐴 퐴̂

Now in order to get off the ground with a general Poisson summation formula, we must isolate a class of functions on any LCA group that are of rapid decay. These are called the Schwartz–Bruhat functions. Again, normally one proceeds by means of a classification result. We do things differently here, following Scott Osborne’s paper.

1.5.19 Definition. A function 푓 ∈ 퐿∞(퐴, 휆) is of brisk decay if there exists a compactum 퐾 ⊂ 퐴 such that for every integer 푛 ≥ 1 there is a constant 퐶푛 > 0 such that for any integer 푚 ≥ 1, one has

퐶푛 ‖푓| 푚 ‖ < . (퐴⧵퐾 ) ∞ 푚푛

1.5.20. Easy observations include:

(1) Brisk decay is independent of the choice of 휆.

(2) If 퐾 is the compactum for a function 푓 of brisk decay, then 푓 van- ishes ae away from the subgroup generated by 퐾.

(3) Functions of brisk decay are translation-invariant.

(4) Functions of brisk decay are closed under convolution.

1.5.21 Exercise. Prove that every function of brisk decay lies in 퐿푝(퐴, 휆) for every 푝 ≥ 1. (Hint: bound the integral of |푓| over 퐾푚 ⧵ 퐾푚−1.)

1.5.22 Definition. We say that a function 푓 ∈ 퐿∞(퐴, 휆) is of rapid decay – or is a Schwartz–Bruhat function – if and only if 푓 is of brisk decay on 퐴 and 푓̂ is of brisk decay on 퐴̂. We write 풮(퐴) for the collection of Schwartz–Bruhat functions. the analytic picture 41

1.5.23 Lemma. The vector space 풮(퐴) is a Fréchet space that is dense in 2 퐿 (퐴, 휆). I believe 풮(퐴) is also nuclear, but I didn’t put any thought into this. 1.5.24 Lemma. The Fourier transform gives a topological isomorphism 풮(퐴) ≅ 풮(퐴̂).

1.5.25 Lemma. Every Schwartz–Bruhat function lies in 풞0(퐴).

1.5.26 Example. On a finite group, all functions are Schwartz-Bruhat functions.

1.5.27 Example. On Z푚, the Schwartz-Bruhat functions are precisely the 푚 functions 푓 such that for any 푘 ∈ Z>0, the quantity

sup |푛푘푓(푛)| < +∞. 푛∈Z푚

1.5.28 Example. On T푚, the Schwartz-Bruhat functions are precisely the smooth functions.

1.5.29 Example. On R푚, the Schwartz-Bruhat functions are precisely the usual Schwartz functions.

1.5.30 Exercise. Describe the Schwartz–Bruhat functions on any group of the form 퐹 × Z푚 × T푛 × R푝, where 퐹 is finite abelian.

1.5.31 Example. One has

풮(퐴) = colim(푈,퐾) 풮(푈/퐾), where the colimit is over subgroups 퐾 < 푈 < 퐴 with 푈 open and compactly generated, 퐾 compact, and 푈/퐾 a Lie group (and in particular of the form 퐹 × Z푚 × T푛 × R푝).

1.5.32 Example. If 퐴 is totally disconnected, then 풮(퐴) is precisely the collection of locally constant functions with compact support.

1.5.33 Theorem (Poisson summation). Suppose

0 퐴′ 퐴 퐴″ 0 42 euler’s gamma function and the field with one element

a short exact sequence of LCA groups.

(1) For any 휆 ∈ ℋ(퐴) and 휆′ ∈ ℋ(퐴′), there exists a unique 휆″ ∈ ℋ(퐴″) such that for any 푓 ∈ 풮(퐴),

∫ 푓(푥) 푑휆(푥) = ∫ ∫ 푓(푦푧) 푑휆′(푦) 푑휆″(푧). 푥∈퐴 푧∈퐴″ 푦∈퐴′

″ (2) For any 푓 ∈ 풮(퐴), define a function 휋∗(푓) on 퐴 by integration along the fibers ″ 휋∗(푓)(푧) ≔ ∫ 푓(푥푦) 푑휆 (푦), 푦∈퐴′ where 푥 is any lift of 푧 to 퐴. Then on 퐴̂″, which is canonically identi- fied with (퐴′)⟂, we have

̂ ̂ 휋∗(푓) = 푓|(퐴′)⟂ .

(3) For any 푥 ∈ 퐴,

∫ 푓(푥푦) 푑휆′(푦) = ∫ 푓̂(휒)휒(푥) 푑휆̂″(휒) 푦∈퐴′ 휒∈퐴̂″

Proof. The proof of point (1) is straightforward.2 2 And consequently it is an For 휒 ∈ (퐴′)⟂ and for any 푥 ∈ 퐴 and 푦 ∈ 퐴′, one has 휒(푥푦) = 휒(푥). Exercise. Hence

̂ ″ 휋∗(푓)(휒) = ∫ 휋∗(푓)(푧)휒(푥) 푑휆 (푧) 퐴″ = ∫ ∫ 푓(푦푧)휒(푦푧) 푑휆′(푦) 푑휆″(푥퐴′) 퐴″ 퐴′ = ∫ 푓(푥)휒(푥) 푑휆(푥), 퐴 where the last identification follow from point (1). This proves point (2).

Finally, Fourier Inversion gives, for any 푥 ∈ 퐴 with image 푧 = 휋(푥), ̂

′ ̂

∫ 푓(푥푦) 푑휆 (푦) = 휋∗(푓)(푧) = 휋∗(푓)(푧) 푦∈퐴′ ̂ ̂ = 푓|(퐴′)⟂ (푧) = ∫ 푓̂(휒)휒(푥) 푑휆̂(휒), 휒∈(퐴′)⟂ whence we obtain (3). the analytic picture 43

1.5.33.1 Corollary. If 훬 ⊂ 퐴 is a discrete cocompact subgroup, then for any 푓 ∈ 풮(퐴), one has 1 ∑ 푓(푥) = ∑ 푓̂(휒). 푥∈훬 covol(퐴/훬) 휒∈훬⟂

1.6 Local zeta functions

We now assume that 푘 is a local field of characteristic 0.

1.6.1. We define an absolute value | ⋅ | on 푘 by choosing a Haar measure 휆, and a measurable subset3 푈 of finite measure, and declaring 3 A compact neighborhood of 0 will do. 휇(푥푈) |푥| ≔ . 휇(푈) The resulting absolute value does not depend on 휆 or 푈. We have three options: • If 푘 ≅ R, then | ⋯ | is the ordinary abolute value.

• If 푘 ≅ C, then | ⋯ | is the square of the ordinary absolute value.

• If 푘 is nonarchimedean, let 표 ⊂ 푘 be the ring of integers, 푝 ◁ 표 the maximal ideal, 휋 a uniformizing parameter, and 퐹 the finite residue field of order 푞. Then |휋| = 1/푞.

1.6.2. We have seen that 푘+ is Pontryagin self-dual; the isomorphism 푘+ ≅ 푘̂+ is not unique, but for any character 휒 ∈ 푘̂+, we may obtain such an isomorphism by 푥 휒(푥). For each such 휒, there exists a unique Haar measure 휇+ on 푘+ such that 휇̂+ = 휇+ under this identification, and moreover, 휇+ does not depend on 휒. We therefore dub this the canonical additive Haar measure.

4 ̂+ 4 1.6.3. It is convenient to fix a character 푋푘 ∈ 푘 – and hence an though perhaps unnecessary? isomorphism 푘+ ≅ 푘̂+ – once and for all. We have cases I’m still not clear on how much really depends on these choices, (1) If 푘 ≅ R, then select which do seem in some sense uniform. 1 푋R ∶ R R R/Z ≅ 푆 . (−1)

(2) If 푘 ≅ Q푝, then select 푋 ∶ Q Q /Z ⊂ Q/Z ⊂ 푆1. Q푝 푝 푝 푝 44 euler’s gamma function and the field with one element

−1 (3) If 푘 ≅ F푝((푡)), then take the coefficient of 푡 as a map F푝((푡)) F푝 and select 푋 ∶ F ((푡)) F ⊂ 푆1. F푝((푡)) 푝 푝

(4) If 푘 ⊂ 푘0 is an extension of one of the above “prime” cases, then select 푋 ≔ 푋 ∘ tr . 푘 푘0 푘/푘0

1.6.4. Now we look to 푘×. We have a exact sequence

× 1 푈푘 푘 푉푘 1, where × 푉푘 ≔ {푣 ∈ R>0 | ∃푥 ∈ 푘 푣 = |푥|} × is the valuation group of 푘 , which is R>0 itself if 푘 is archimedean and 푞Z otherwise, and × 푈푘 ≔ {푥 ∈ 푘 | |푥| = 1} is compact. This sequence is (noncanonically) split by a homomorphism 푥 푥,̃ which in the nonarchimedean case is determined by a uniformis- ing parameter. Now we wish to specify a canonical multiplicative Haar measure. To × this end, if 푔 ∈ 풞00(푘 ), then the function 푔(푥)/|푥| is a function in + × 풞00(푘 − {0}). So we define a functional 훷 on 풞00(푘 ) by 푔(푥) 훷(푔) ≔ ∫ 푑휇+. 푘+−{0} |푥|

1.6.5 Exercise. Show that the functional 훷 is invariant under translation.

Consequently, 훷 corresponds to a Haar measure which we might as well call log |휇+|. Now we normalise log |휇+|: • If 푘 is archimedean, then 휇× = log |휇+|.

• If 푘 is nonarchimedean, then 푞 휇× = log |휇+|. 푞 − 1

1.6.6 Exercise. If 푘 is the completion of a number field 퐾 at a finite place × 푝, compute the volume of 푈푘 under 휇 . Answer:

× 1 휇 (푈푘) = , √푑푘 the analytic picture 45

where 푑푘 is the discriminant of 푘, which is 1 for all 푝 except those that divide the discriminant of 퐾.

1.6.7 Definition. A quasicharacter on 푘× is a homomorphism 푘× C×.

1.6.8. Any quasicharacter 휓 on 푘× factors as

휓(푥) = 휒(푥)| ̃ 푥|푠, where 휒 is the character on 푈푘 obtained as the restriction of 휓, and 푠 is determined by 휓 completely if 푘 is archimedean, and modulo 2휋푖/ log 푞 if 푘 is nonarchimedean.

The assignment 휓 (휒, 푠) identifies the set 푄푘 of quasicharacters on × 푘 with 푈̂푘 × 푆푘, where

{C if 푘 is archimedean; 푆푘 = { C/(2휋푖/ log 푞)Z if 푘 is nonarchimedean. {

5 5 We endow 푄푘 with the structure of a complex manifold by declaring with infinitely many compo- 6 ̂ 푠 nents that for any character 휒 ∈ 푈푘, the map 푠 휒(푥)| ̃ 푥| should be a local 6 Note that 푈̂푘 is discrete. isomorphism. Also note that the quantity ℜ(휓) ≔ ℜ(푠) is uniquely determined by 휓.7 We call ℜ(휓) the exponent of 푠. 7 Why?

In effect, 푄푘 is going to be our analogue of the complex plane, and we’re again going to work with “strips”

⟩푎, 푏⟨ ≔ {휓 ∈ 푄푘 | ℜ(휓) ∈ ]푎, 푏[}.

1.6.9 Definition. Suppose now 푓 ∈ 풮(푘+). For any quasicharacter 휓 with ℜ(휓) > 0, set

푧(푓, 휓) ≔ ∫ 푓(푥)휓(푥) 푑휇×(푥). 푘×

1.6.10 Lemma. The function 푧(푓, 휓) is well defined and holomorphic on the region ⟩0, +∞⟨.

Now to get our local functional equation, we need the analogue of

푠 1 − 푠 on 푄푘. This is the following. 46 euler’s gamma function and the field with one element

1.6.11 Notation. Define an involution 휓 휓̂ on 푄푘, where |푥| 휓̂(푥) ≔ . 휓(푥)

Note that ℜ(휓̂) = 1 − ℜ(휓).

1.6.12 Exercise. Use Fubini to prove that for 휓 ∈ ⟩0, 1⟨ and 푓, 푔 ∈ 풮(푘+), we have 푧(푓, 휓)푧(푔 ̂, 휓̂) = 푧(푔, 휓)푧(푓̂, 휓̂).

̂ + For each character 휒 ∈ 푈푘, if we can find one function 푓휒 ∈ 풮(푘 ) such that:

̂ (1) 휓 푧(푓휒, 휓̂) is not identically 0 on ({휒} × 푆푘) ∩ ⟩0, 1⟨, and

(2) the function 푧(푓, 휓) 휌(휓) ≔ 푧(푓̂, 휓̂)

admits a meromorphic continuation to all of {휒} × 푆푘, then for any 푔 ∈ 풮(푘+), the function 푧(푔, 휓) admits a meromorphic continuation to all of 푄, and moreover

푧(푔, 휓) = 휌(휓)푧(푔 ̂, 휓̂).

We’ll construct our functions 푓휒 case by case for the “prime fields.” The appropriate mods involving the trace are left as an Exercise.

(1) Suppose 푘 = R. There are two options for 휒:

2 푠 (1.a) For 휒 = 1, take 푓1(푥) = exp(−휋푥 ), so that when 휓(푥) = |푥| , we get −푠/2 푧(푓1, 휓) = 휋 훤(푠/2), and ̂ (푠−1)/2 푧(푓1, 휓̂) = 휋 훤(1 − 푠/2). Hence 휋 휌(푠) = 21−푠휋−푠 cos( 푠)훤(푠), 2 which we’ve seen is meromorphic on C. the analytic picture 47

2 (1.b) For 휒 = −1, take 푓−1(푥) = 푥 exp(−휋푥 ), so that when 휓(푥) = sgn(푥)|푥|푠, we end up with 휋 휌(푠) = 푖21−푠휋−푠 sin( 푠)훤(푠), 2 which again is meromorphic on C.

(2) Suppose 푘 = Q푝. Then we have two options for 휒:

(2.a) 휒 = 1 (unramified case). Choose 푓1 to be the indicator function of Z푝. Then

̂ 푓1(푥) = ∫ exp(−2휋푖{푥푦}) 푑휇(푦) = 푓1. Z푝

Now let’s perform our Mellin integrals for 휓 = | ⋅ |푠 (for 푠 ∈ ⟩0, 1⟨:

푠 × 푧(푓1, 휓) = ∫ 푓1(푥)|푥| 푑휇 (푥) × Q푝 푝 = ∫ |푥|푠−1푑휇(푥) 푝 − 1 Z푝−{0} +∞ −푟푠 1 = ∑ 푝 = −푠 , 푟=0 1 − 푝 and, on the other side,

̂ ̂ 1−푠 × 푧(푓1, 휓̂) = ∫ 푓1(푥)|푥| 푑휇 (푥) × Q푝 1 = , 1 − 푝푠−1

which is not identically zero on ({1} × 푆푘) ∩ ⟩0, 1⟨. So

1 − 푝(푠 − 1) 휌(푠) = , 1 − 푝푠

which meromorphically continues to the component {1} × 푆푘.

(3) 휒 ≠ 1 (ramified case). Now 휒 factors through some (Z/푝푛)× 푆1, and assume that 푛 is minimal with this property. Set

{0 if |푥| > 푝푛; 푓휒(푥) ≔ { exp(2휋푖{푥}) if |푥| ≤ 푝푛. { 48 euler’s gamma function and the field with one element

Basic properties of Fourier transforms give

{ 푛 ̂ 0 if |1 − 푥| > 푝 ; 푓휒(푥) = { 푝푛 if |1 − 푥| ≤ 푝푛. {

Now we have 휓(푥) = 휒(푥)| ̃ 푥|푠, and we compute8 8 Here’s our old friend the Gaußsum! 푛 푛푠+1−푛 푝 −1 푝 푛 푧(푓휒, 휓) = ∑ 휒(푟) exp(2휋푖푟/푝 ), 푝 − 1 푟=1 and 푝 푧(푓 , 휓) = , 휒 푝 − 1 and now 푝푛−1 휌(휓) = 푝푛(푠−1) ∑ 휒(푟) exp(2휋푖푟/푝푛). 푟=1

No problem continuing this to {1} × 푆푘.

(4) When 퐾 = F푝((푡)), we’ll run essentially the same program, with the indicator function in the unramified case and, in the ramified case, a rescaled version thereof multiplied by the character.

1.7 Global L-functions

+ ̂+ Now we let 퐾 be a global field. Our identifications 퐾푣 ≅ 퐾푣 give us a selected identification ̂ A퐾 ≅ A퐾. There’s a unique norm on this ring induced by the finite ones. Similarly, + the Tamagawa measure on A퐾 is the measure 휇 given by forming the + × product of the 휇푣 ’s, and the Tamagawa measure on I퐾 is the measure 휇 × given by forming the product of the 휇푣 ’s. On 퐾 ⊂ A퐾, we install the counting measure. We have the following consequence of Poisson summation:

1.7.1 Proposition. 휇(A퐾/퐾) = 1.

For the idèles, we have a canonical norm coming from all the local norms, and to embed 푘× nicely, we first need to pass to the idèles of norm 1: 1 1 I퐾 I퐾 푉퐾 1. the analytic picture 49

Now if 퐾 is a number field, 푉퐾 = R>0, and if 퐾 is a function field, then Z 푉퐾 = 푞 . In the first case, choose the measure on 푉퐾 to be log 휇, and in the second, choose the measure to be (log 푞) ⋅ #. Now select the measure 1 × 휇 on I퐾 to be compatible with the other two in the usual way.

× 1 1.7.2 Proposition. 퐾 ⊂ I퐾 is discrete and cocompact.

̂× × 1.7.3 Definition. A homomorphism 휒∶ I퐾/퐾 C is called a qua- sicharacter. We’ll call the space of all these H.

H decomposes (topologically) as the product (퐾×)⟂ × C; any 휓 ∈ H can be written uniquely as 푥 휒(푥)|푥|푠, where 휒 ∈ (퐾×)⟂ and 푠 ∈ C is an action of C that gives H a complex manifold structure. |푥| Again any 휙 gives rise to 휓̂, given by 푥 휓(푥) .

1.7.4 Definition. Suppose 푓 ∈ 풮(A퐾), and suppose 휒 ∈ H. Then we define 푍(푓, 휒) ≔ ∫ 푓(푥)휒(푥) 푑휇×(푥). I퐾

1.7.5 Theorem. The integral 푍(푓, 휓) converges for ℜ(휓) > 1. It extends to a meromorphic function on H, and the only poles are at (1, 0) with residue − covol(퐾×)푓(0) and at (1, 1) with residue covol(퐾×)푓̂(0). Finally, we have the simple functional equation

푍(푓, 휓) = 푍(푓̂, 휓̂).

High points of proof. We can decompose that integral

+∞ × 푍(푓, 휒) ≔ ∫ 푓(푥)휒(푥) 푑휇 (푥) = ∫ 푍푡(푓, 휒) 푑 log 푡, I퐾 0 where 1 푍푡(푓, 휒) = ∫ 푓(푡푦)휓(푡푦) 푑휇 (푦). 1 I퐾

Now let’s analyse 푍푡 not by decomposing it into local pieces, but instead selecting a fundamental domain 퐸 for 푘× and writing

1 푍푡(푓, 휓) = ∫ ( ∑ 푓(훼푡푦)) 휓(푡푦) 푑휇 (푦). 퐸 훼∈퐾× 50 euler’s gamma function and the field with one element

On the other hand, Poisson summation yields

∫ ( ∑ 푓(훼푡푦)) 휓(푡푦) 푑휇1(푦) = ∫ ( ∑ 푓̂(훼푦/푡)) 휓̂(푦/푡) 푑휇1(푦), 퐸 훼∈퐾 퐸 훼∈퐾 whence one obtains

1 ̂ ̂ 1 푍푡(푓, 휓) + 푓(0) ∫ 휓(푡푦) 푑휇 (푦) = 푍1/푡(푓, 휓̂) + 푓(0) ∫ 휓̂(푦/푡) 푑휇 (푦) 퐸 퐸 It turns out that there are two options for those integrals: either the quasicharacter is unramified or it isn’t. If it is, then we get

∫ 휓(푡푦) 푑휇1(푦) = 푡푠휇1(퐸); 퐸 if not, then we get ∫ 휓(푡푦) 푑휇1(푦) = 1. 퐸 So the ramified case is going to be easier: you take 푍(푓, 휓) and split it into two pieces, using what you’ve learned.

+∞ +∞ ̂ 푍(푓, 휒) = ∫ 푍푡(푓, 휒) 푑 log 푡 + ∫ 푍푢(푓,휒 ̂) 푑 log 푢. 1 1 That’s literally it in that case. The unramified case is only mildly more annoying. 2 The field with one element

2.1 Borger’s picture

Jim Borger has proposed a fascinating picture of F1, which offers an amazing insight into this object. To describe it, it’s convenient to start with an understanding of plethystic algebra.

2.1.1. Our rings and algebras will all be commutative with 1.

2.1.2 Definition. Let 푘 denote a ring. An affine 푘-algebra scheme is an endofunctor 푋∶ Alg푘 Alg푘 such that the composite

Alg푘 Alg푘 Set is corepresentable. If 푋 is in addition a comonad, then we say that 푋 is a 푘-plethory.

2.1.3. An affine 푘-algebra scheme is 푋 = Spec 푅 for a 푘-algebra 푅 that has been equipped with the following structure:

• a cozero 휖+ ∶ 푅 푘;

+ • a coaddition 훥 ∶ 푅 푅 ⊗푘 푅; • an antipode 휎∶ 푅 푅;

• a coone 휖× ∶ 푅 푘;

× • a comultiplication 훥 ∶ 푅 푅 ⊗푘 푅;

• a ring map 푘 Hom푘(푅, 푘); 52 euler’s gamma function and the field with one element

subject to a whole pile of axioms. The structure of a plethory on 푋 then suppiles one more piece of data on 푅, a noncommutative operation

∘∶ 푅 ×푘 푅 푅 that corepresents the comonad structure. This is called the composition product or plethysm.

2.1.4 Example. The constant endofunctor at the zero ring is an affine 푘-algebra scheme; it is corepresented by Spec 푘.

2.1.5 Example. The identity endofunctor 퐼∶ Alg푘 Alg푘 is an affine 푘-algebra scheme; it is corepresented by Spec 푘[푡]. Since it is obviously a comonad, this is also a plethory.

2.1.6 Example. The assignment 푆 W(푆) of the ring of “big” Witt vectors is a Z-plethory. It is corepresented by the algebra 훬 of symmetric Aut N functions over Z. This is defined as the subring of Z⟦푥1, 푥2, …⟧ consisting of those power series in which the degree of the monomials is bounded. The theorem of elementary symmetric functions shows that

훬 = Z[휆1, 휆2, … ], where 휆 = ∑ 푥 ⋯ 푥 푛 푖1 푖푛 푖1<⋯<푖푛 are the usual elementary symmetric functions. (We usually let 휆0 = 1.) We could also freely generate 훬 under the complete symmetric functions

휎 = ∑ 푥 ⋯ 푥 푛 푖1 푖푛 푖1≤⋯≤푖푛

Of special relevance are the Adams symmetric functions

푛 푛 휓푛 ≔ 푥1 + 푥2 + ⋯ ; we may freely generate 훬 by elements 푤1, 푤2, ⋯ ∈ 훬 determined by the relations 푛/푑 휓푛 = ∑ 푑푤푑 , 푑|푛 the field with one element 53

since 1 ∑ (−1)푚휆 푇푚 = exp (− ∑ 휓 푡푚) = ∏ (1 − 푤 푡푚). 푚 푚 푚 푚 푚∈N0 푚∈N 푚∈N This is how one obtains the usual description of elements of W(푆) in terms of Witt components (푤1, 푤2,…). The plethysm is composition of symmetric functions; this corresponds to the Artin–Hasse map W(푆) W(W(푆)). We will describe all these structures very precisely with 퐾-theory soon.

2.1.7 Example. If 퐺 is a group, then the endofunctor 퐹퐺 ∶ 푆 푆퐺 has a comonad structure, and it is corepresented by 푘[퐺], which is the free polynomial algebra on the underlying set of 퐺. This is not the group-algebra, but it is the symmetric algebra generated by the group algebra 푘퐺.

2.1.8 Definition. If 푃 is a 푘-plethory, then a 푃-algebra is a coalgebra for the comonad 푃.

2.1.9 Example. Every 푘-algebra is an 퐼-algebra for the identity plethory 퐼 represented by 푘[푥] in a unique way.

2.1.10 Example. If 퐺 is a group, then an 퐹퐺-algebra is a 푘-algebra with an action of 퐺.

2.1.11 Definition. Suppose 퐾 an idempotent complete symmetric monoidal category with finite colimits (such that the symmetric monoidal structure preserves finite colimits separately in each variable). We write Alg(퐾) for the category of idenmpotent (commutative) 퐾-algebras. A 퐾-biring is a comonad 푋∶ Alg(퐾) Alg(퐾) such that the composite

Alg(퐾) Alg(퐾) Cat is corepresentable. If 푋 is a comonad, then we say that 푋 is a 퐾-plethory.

[...]

2.1.12 Definition. A quotient of the plethory W corepresented by 훬 is the subplethory W훹 corepresented by the subring 훹 ⊂ 훬 generated by the elements 휓푛 for 푛 ∈ N. 54 euler’s gamma function and the field with one element

2.1.13. The plethory W훹 is freely generated by the elements 휓푛 for 푛 ∈ N; that is, for any ring 푅, one has

N W훹(푅) = 푅 as a ring. For any 푛 ∈ N, define

N N 휓푛 ∶ 푅 푅 by

휓푛(a) = 휓푛(푎1, 푎2, 푎3, … ) = (푎푛, 푎2푛, 푎3푛, … ).

The comonad structure W훹 W훹 ∘ W훹 is then given by

(a) (휓1(a), 휓2(a), 휓3(a) … ).

A 훹-algebra is thus a ring 푅 with an action of N×. Of course and 훬-algebra gives rise to a 훹-algebra structure.

To some degree, the structure of a 훬-algebra can be recovered from the strecture of a 훹-algebra.

2.1.14 Lemma (Newton formula). For any 푘 ≥ 1, one has

푘 푘−푚 푘+1 ∑ (−1) 휆푘−푚휓푘 = (−1) 푘휆푘. 푚=1

2.1.14.1 Corollary. If 푅 is flat over Z, then for any 훹-algebra structure on 푅, there is at most one 훬-algebra structure that lifts it.

Proof. The lemma above permits one to write each 휆푘 as a homogeneous polynomial of degree 푘 with Q coefficients in 휓1,…, 휓푘, where |휓푗| = 푗.

2.1.15 Theorem (Wilkerson). If 푅 is flat over Z, then any 훹-algebra structure on 푅 in which for every prime 푝,

푝 휓푝(푥) ≅ 푥 mod 푝푅 admits a unique 훬-algebra structure lifting it. the field with one element 55

Proof. We’re just after existence. Uniqueness we already have. On 푅 ⊗ Q, define 휆푘 for 푘 ≥ 1 by means of generating functions:

푑 −푡 log ( ∑ 휆 (푥)푡푚) = ∑ (−1)푛휓 (푥)푡푛. 푑푡 푚 푛 푚∈N0 푛∈N

This is1 a 훬-structure on 푅 ⊗ Q. 1 Exercise.

We aim to show that for any prime 푝 and any 푥 ∈ 푅 ⊗ Z(푝), the map 휆푘 ∶ 푅 ⊗ Q 푅 ⊗ Q carries 푥 to 푅 ⊗ Z(푝). This is obvious for 푘 = 1, since 휆1(푥) = 푥. Now assume the claim for all 휆푚 for 푚 < 푘. Suppose 푝 ∤ 푘. Then we can divide by 푘, so the Newton formula plus the induction hypothesis does the job. If 푘 = 푝, we observe that

푝 푝+1 휓푝(푥) − 푥 = 푝 ((−1) 휆푝(푥) + 푃(휆1(푥), … , 휆푝−1(푥))) for some polynomial 푃 with Z coefficients. By assumption, the left hand side lies in 푝(푃 ⊗ Z(푝)), so

푝+1 (−1) 휆푝(푥) + 푃(휆1(푥), … , 휆푝−1(푥)) ∈ 푅 ⊗ Z(푝).

Now the induction hypothesis does the job. If 푘 = 푚푝 for some 푚, then we observe that

(푝+1)(푚+1) 휆푘(푥) = (−1) 휆푚(휆푝(푥)) + 푄(휆1(푥), … , 휆푘−1) for some polynomial 푄 with Z coefficients. Again with the induction hypothesis.

2.1.16. Wilkerson’s theorem says that if 푅 is flat over Z, then a 훬-algebra structure on 푅 is exactly the same as a compatible family of lifts of frobenius for each prime 푝. Incidentally, one can show that if 푅 is a reduced 훬-algebra, then it is automatically flat over Z.

2.1.17 Definition (Borger). An F1-algebra is a 훬-algebra. We write CAlg for the category of such. F1

2.1.18. We have an adjunction

− ⊗ Z∶ CAlg(F ) CAlg(Z)∶ 푈, F1 1 56 euler’s gamma function and the field with one element

where − ⊗ Z is the functor that forgets the 훬-algebra structure, and 푈 is F1 W. The functor − ⊗ Z preserves both limits and colimits, since it admits F1 a right adjoint (W) and a left adjoint.

We think of the 훬-algebra structure as descent data to F1. When a ring 푅 admits a 훬 structure, we say that 푅 is defined over F1 or descends to F1.

2.1.19 Example. The ring Z is defined over F1 in a unique fashion, in which each 휓푝 = id.

2.1.20 Example. For any commutative monoid 푋, the monoid algebra Z푋 푝 has is defined over F1: the descent data are given by 휓푝(푥) = 푥 for any 푥 ∈ 푋. In this way, one has

푛 Z[푡]/(푡 − 1) ≅ F 푛 ⊗ Z. 1 F1

푛 The F1-algebra F1푛 is the field with 1 elements, which we regard as a field extension of F1. Pushing this farther, we can write

Z(Q/푍푍) ≅ F ⊗ Z. 1 F1

2.1.21 Example. There is actually another F1-structure on Z[푥] (different from Z[N0]). This is the one that has the potential to be more interesting from our point of view.

Consider the category Rep(SL2(C)) of finite dimensional representations of Rep(SL2(C)); denote by 푉 the standard 2-dimensional representation. We have the representation ring 퐾0(Rep(SL2(C))), which is a 훬-algebra in the canonical manner. We may compute it by identifying a representation 푎 0 with its character on the torus elements ( ), so that [푉] = 푎 + 푎−1. 0 푎−1 −1 We find that 퐾0(Rep(SL2(C))) is then the fixed points of the ring Z[푎, 푎 ] −1 −1 under the 퐶2 action 푎 푎 . Of course, if 푥 = [푉] = 푎 + 푎 , then we obtain an isomorphism

퐾0(Rep(SL2(C))) ≅ Z[푥], but the 훬-algebra structure turns out to be different. Indeed, to see this, consider the 2푛-dimensional tensor power 푉⊗푛, which corresponds to [푉⊗푛] = 푥푛 = (푎 + 푎−1)푛 ∈ Z[푎, 푎−1], and the the field with one element 57

(푛 + 1)-dimensional representation Sym푛(푉), which corresponds to

[Sym푛(푉)] = 푎푛 + 푎푛−2 + ⋯ + 푎2−푛 + 푎푛 ∈ Z[푎 + 푎−1].

If we substitute 푎 = exp(푖휃), then we get 푥 = 2 cos 휃, and we find that

푛 [Sym (푉)] = 푈푛(푥/2), where 푈푛 are the Chebyshev polynomials of the second kind. Equivalently, the Adams operations are given by

휓푝(푥) = 2푇푝(푥/2), where 푇푝 are the Chebyshev polynomials of the first kind.

2.1.22. One might like to declare that the “correct” polynomial ring in one variable over F1 is the free F1-algebra generated by one element. The problem is that it’s not so clear what “free” means, and to this end the problem is that it’s not so clear what the “underlying set” of an F1-algebra ought to be. To illustrate, consider a finite Galois extension 퐹 ⊂ 퐸 with group 퐺. Then an 퐹-algebra is the “same” as an 퐸-algebra 푅 equipped with a semilinear action of 퐺. The underlying set of the 퐹-algebra is, of course, not the underlying set of 푅, but rather the underlying set of 푅퐺. Under the analogy between 훬 and 퐸휎[퐺], it’s not a priori obvious what should play the role of 푅퐺. In the particular case in which 푅 is flat, a reasonable answer might be fixed points for the action of N×.

To describe the theory of more general objects over F1, we can extend the comonad W as follows.

2.1.23 Definition. Consider Shv(Z), the category of étale sheaves (of sets) on the category Aff/Z of finitely generated (over Z) affine schemes. Since W is a monad on Aff/Z, the left Kan extension of W is again a monad on Shv(Z), and we write Shv(F1) for the resulting category of algebras, and we call the objects thereof F1-sheaves. Observe that Shv(F1) can equally well be considered the category of coalgebras over the comonad given by precomposition with W. Fur- thermore, it is not at all difficult to check that precomposition with W preserves filtered colimits, whence Shv(F1) is a topos, which we call the 58 euler’s gamma function and the field with one element

large étale topos of F1. Furthermore, the adjunction

∗ 푣 ∶ Shv(F1) Shv(Z)∶ 푣∗ is a geometric morphism of topoi. In fact, it is an essential geometric ∗ morphism – the left adjoint 푣 admits a further left adjoint 푣!. However, this geometric morphism is not étale, since the left adjoint 푣! is not conservative; indeed, this would follow if and only if the assignment 푅 W(푅) is conservative.

We have to contend with the theory of modules over F1-algebras. Luckily, we can be surprisingly concrete.

2.1.24 Definition (Beck). If 퐶 is a category that admits all finite limits, then for any object 퐴 ∈ 퐶, one may define an 퐴-module to be an abelian group object in the category 퐶/퐴, so that

Mod(퐴) ≔ Ab(퐶/퐴).

If 푀 is an 퐴-module in this sense, then a derivation from 퐴 to 푀 is a morphism 퐴 푀 in 퐶/퐴; the set

Der(퐴, 푀) ≔ Mor (퐴, 푀) 퐶/퐴 admits an abelian group structure since 푀 does.

2.1.25 Exercise. This recovers the usual notion of a module when 퐶 is the category of rings. Indeed, for any object 퐵 ∈ Ab(퐶/퐴), the kernel ker[퐵 퐴] inherits an 퐴-module structure in the usual sense. On the other hand, if 푀 is an 퐴-module in the usual sense, endow 퐴 ⊕ 푀 with the ring structure of square-zero extension of 퐴. Show that these assignments define inverse equivalences of categories.

2.1.26 Proposition. If 퐶 is presentable, then the forgetful functor

푈∶ Mod(퐴) 퐶/퐴 admits a left adjoint 훺. Consequently, one has a universal derivation

푑∶ 퐴 훺퐴 that induces an isomorphism

MorMod(퐴)(훺퐴, 푀) ≅ Der(퐴, 푀). the field with one element 59

2.1.27 Definition. It is a simple matter to see that the comonad W extends to a comonad on the category CAlgnu(Z) of non-unital rings. In particular, any abelian group can fed into W as a ring with zero multiplication. If 퐴 is a ring and 푀 is an 퐴-module, then we write W(푀) for the W(퐴)-module given by the set 푀N with componentwise addition and scalar multiplication given by

(푎푚)푘 ≔ 휓푘(푎)푚푘. If 퐴 is an 훬-ring, then a 훬-module over 퐴 is an 퐴-module 푀 and an 퐴-linear map 휆∶ 푀 W(푀) such that 휖 ∘ 휆 = id, and the square

푀 W(푀)

W(휆) W(푀) WW(푀) 훥

2.1.28 Proposition (Hesselholt). Suppose 퐴 an F1-algebra. Then the category Mod(퐴) is equivalent to the category of 훬-modules on 퐴.

Proof. Formal. The assignment 퐵 ker[퐵 퐴] lifts to an equivalence from Mod(퐴) to 훬-modules on 퐴.

2.1.28.1 Corollary. Under this equivalence, a derivation from an F1- algebra 퐴 to an 퐴-module 푀 is a map 푑∶ 퐴 푀 such that

1. 푑(푎 + 푏) = 푑(푎) + 푑(푏);

2. 푑(푎푏) = 푎푑(푏) + 푏푑(푎);

3. 푛/푘−1 휆푛(푑(푎)) = ∑ 휆푘(푎) 푑(휆푘(푎)). 푘|푛

For any F1-algebra 퐴 and any 퐴-module 푀, a derivation 퐴 푀 is in particular a derivation 퐴 ⊗ Z 푀 ⊗ Z; hence we obtain a F1 F1 comparison homomorphism

훺퐴⊗ Z 훺퐴 ⊗F Z. F1 1

2.1.28.2 Corollary. The comparison homomorphism above is an isomor- phism. 60 euler’s gamma function and the field with one element

2.1.29. Perhaps even more down to earth is the following description of

퐴-modules over an F1-algebra 퐴: consider the twisted monoid algebra 퐴휓N×; then Mod(퐴) is equivalent to the category of left 퐴휓N×-modules.

2.1.30 Example. Let’s consider F1 itself. Recall that in F1, the Adams × operations 휓푝 act trivially on Z. So an F1-module is a ZN -module, i.e., an abelian group 푀 along with commuting operations 휆푝 ∶ 푀 푀 for every 푝 ∈ 훱.

2.1.31 Example. Suppose 푋 a commutative monoid. Recall that N× acts 푝 × × × on 푋 via 휓푝(푥) = 푥 ; we thus form the semidirect product 푋 ⋊휓 N . It is As a set, 푋 ⋊휓 N is 푋 × N , easy to see that an F 푋-module is nothing more than an abelian group with but the monoid action is twisted: 1 (푥, 푚)(푦, 푛) = (푥휓 (푦), 푚푛). × 푚 an action of 푋 ⋊휓 N . In particular, if 푋 = 퐶푛, then we find than an F1-module consists of an abelian group 푀, an automorphism 휎 of 푀 of order dividing 푛, and 푝 commuting operations 휆푝 ∶ 푀 푀 such that 휆푝 ∘ 휎 = 휎 ∘ 휆푝. Similarly, if 푋 = Z, then we find that a quasicoherent sheaf on G is 푚,F1 an abelian group 푀 with an automorphism 휎 and commuting operations 푝 휆푝 ∶ 푀 푀 such that 휆푝 ∘ 휎 = 휎 ∘ 휆푝.

2.1.32 Example. A module over the Chebyshev line, F1[푥] is a mod- ule over Z[푥]휓N×; i.e., it’s an abelian group 푀 with an endomorphism

푥∶ 푀 푀 and commuting operators 휆푝 ∶ 푀 푀 such that

휆푝 ∘ 푥 = 2푇푝(푥/2) ∘ 휆푝.

We must compute the F1푛 -points of various sheaves over F1.

2.1.33 Example. If 푋 is a monoid and 푆푋 ≔ Spec(F1푋) then of course

푆 (F 푛 ) ≅ Hom (F 푋, F 퐶 ) ≅ Mor (푋, 퐶 ). 푋 1 F1 1 1 푛 Mon 푛,+ In particular, we find that 1 A (F 푛 ) = 퐶 F1,toric 1 푛,+ and G (F 푛 ) = 퐶 . 푚,F1 1 푛

2.1.34 Example. If A1 denotes Spec of the Chebyshev line, then F1 1 A (F 푛 ) = Hom (Z[푥], Z[퐶 ]), F1 1 훹 푛 the field with one element 61

an element of which picks out an element 푓 ∈ Z[푡]/(푡푛 − 1) such that for any 푝 ∈ 훱, 푝 푛 푓(푡 ) = 2푇푝(푓(푡)/2) mod (푡 − 1).

2.1.35 Definition. Consider the poset 훷 of natural numbers ordered by divisibility. A truncation set is a sieve 푆 ⊆ 훷 (equivalently, a subfunctor of the terminal presheaf 훷 Set. If 푛 ∈ N, then let 푆/푛 denote the pullback of 푆 under the multiplication by 푛 map 푛∶ 훷 훷; that is,

푆/푛 ≔ {푑 ∈ 훷 | 푛푑 ∈ 푆}.

2.2 Bökstedt–Hesselholt–Madsen computations of THH

Fix a smooth scheme 푋 over a perfect field 푘. Recall that

TR푛(푋, 푝) ≔ THH(푋)⟨푝푛−1⟩.

• The cyclotomic structure on THH endows TR∗(푋, 푝) with the structure of a 푝-typical Witt complex.

2.2.1 Theorem (Hesselholt–Madsen). There is a canonical ring isomor- phism ∼ • 푊•(푋) TR0(푋, 푝) of cyclotomic Green functors.

Consequently, we obtain a unique morphism of 푝-typical Witt com- plexes ∗ • 휂∶ 푊•훺푋 TR∗(푋, 푝).

2.2.2 Theorem (Hesselholt). The map 휂 induces an isomorphism

푘 ∼ • 푊•훺푋 TR푘(푋, 푝) for 푘 ≤ 1.

2.2.3 Theorem (Hesselholt–Madsen). For each 푛 ≥ 1, the 푊푛(푘)-module 푛 TR2(푘, 푝) is free of rank 1, and the canonical graded 푊푛(푘)-algebra homo- morphism 푛 푛 Sym(TR2(푘, 푝)) TR∗(푘, 푝) is an isomorphism. 62 euler’s gamma function and the field with one element

2.2.4 Theorem (Hesselholt). For each 푛 ≥ 1, the canonical graded 푊푛(푘)- algebra homomorphism

∗ ∗ 푛 푛 푊 훺 ⊗ ∗ 푓 TR (푘, 푝) TR (푋, 푝) 푛 푋 푓 푊푛(푘) ∗ ∗ is an isomorphism.

2.3 Regularised products and determinants

2.3.1 Definition. Suppose, for any 푛 ∈ N, that 휆푛 is a complex number with chosen argument 훼푛. Assume that 푁 ∈ N such that 휆푛 ≠ 0 for any 푛 ≥ 푁. Consider the Dirichlet series

−푠 ∑ |휆푛| exp(−푖푠훼푛), 푛≥푁 which converges on some half plane ⟩푀, +∞⟨. Assume also that this sum admits an analytic continuation to a holomorphic function 휁푁(푠) on a half-plane ⟩−휀, +∞⟨. The the regularised product

푁−1 ′ ∏ (휆푛, 훼푛) ≔ ( ∏ ) exp(−휁푁(0)). 푛∈N 푛=1

We say the regularised product converges when the assumptions above hold.

2.3.2. We shall always take the branch of the argument lying in (−휋, 휋]. The existence and value of the regularised product is independent ofthe enumeration, so we are free to write ∏푛∈푆 휆푛 for a countable set 푆.

2.3.3 Definition. Suppose 푉 a C-vector space of countable dimension, and suppose that 훩 is an endomorphism of 푉. Assume the following:

푁 • 푉 is the direct sum ⨁휆∈C 푉휆, where 푉휆 = ker(훩 − 휆) for 푁 ≫ 0 is finite-dimensional.

• Let (휆푛) be the sequence of eigenvalues of 훩, with multiplicity. The regularised product

∏ 휆푛 푛 converges. the field with one element 63

Then we define the regularised determinant as

det(훩) ≔ ∏ 휆푛. ∞ 푛

2.3.4 Exercise. If 0 푉′ 푉 푉″ 0 is an exact sequence of vector spaces and endomorphisms, then

det(훩) = det(훩′) det(훩″), ∞ ∞ ∞ in the sense that if the right side converges, then so does the left, and then the values are equal.

2.3.5 Example. If 훾 ∈ C× and 푧 ∈ C, then let us compute

∏ 훾(푧 + 푛). 푛∈Z The Hurwitz zeta function is defined on ⟩1, +∞⟨ by 1 휁(푠, 푧) ≔ ∑ , (푧 + 푛)푠 푛∈N0 and it admits an analytic continuation to C with a pole at 1. We can introduce 1 휁 (푠, 푧) ≔ ∑ , 훾 (훾(푧 + 푛))푠 푛∈N0 and we can compute that

−1 푑 푑 ∏ 훾(푧 + 푛) = (푧훾) exp(− 휁훾(0, 푧)) exp(− 휁−훾(0, −푧)). 푛∈Z 푑푠 푑푠

2.3.6 Proposition. Let 푉 be an anticommutative graded C-algebra such that 푉푗 ⊂ 푉 is finite dimensional for all 푗. Suppose 훩 a graded C-linear derivation, and suppose that there exists a unit 훽 ∈ 푉−2 such that 2휋푖 훩(푣) = 훽. log 푞 Then −푠훩 det(푠 − 훩|푉2∗+푗) = det(1 − 푞 |푇푗) ∞

Now we define the graded derivation 훩 on TP∗(푋) for 푋 smooth over F푞 as follows. For the periodicity class 훽 ∈ TP−2(푋), we set 2휋푖 훩(푣) = 훽. log 푞 64 euler’s gamma function and the field with one element

훩 ∗ and we define 훩 on TP0 and TP1 so that 푞 = Fr푞 . (This can actually be done in more than one manner, but this point isn’t important.) We thus find that

−푠 ∗ det(푠 − 훩|TP2∗+푗(푋) ⊗푊 C) = det(1 − 푞 Fr푞 |TP푗 ⊗푊 C) ∞ 3 Connes–Consani on archimedean 퐿-factors

3.1 Hodge structures and 퐿-factors

3.1.1 Definition. Denote by S the real algebraic group C×; that is, S is the Weil restriction of the complex algebraic group G푚 to R ⊂ C. Denote by 푤∶ G푚 S the canonical morphism that on real points is the inclusion R× C×.

Now a Hodge structure is a finite rank abelian group HZ along with an action 휎 of the real algebraic group S on HR ≔ HZ ⊗ R. We will say that 푘 (HZ, 휎) is pure of weight 푘 if the action of 휎푤(푡) on HR is action by 푡 for any 푡 ∈ R×.

× × 3.1.2. An action of S(C) ≅ C × C on HC ≔ HZ ⊗ C is specified by the decomposition

푝,푞 푝,푞 −푝 −푞 HC = ⨁ H ,H = {푥 ∈ HC | ∀(푢, 푣) ∈ S(C), (푢, 푣)푥 = 푢 푣 푥}. 푝,푞∈Z

This representation is real just in case H푞,푝 = H푝,푞. Hence a Hodge structure can be defined as such a decomposition. This Hodge structure is of weight 푘 if and only if H푝,푞 = 0 unless 푝 + 푞 = 푘.

This decomposition also specifies a filtration of HC, called the Hodge filtration: 푝+1 푝 ⋯ 퐹 H ⊂ 퐹 H ⊂ ⋯ ⊂ HC, given by 퐹 푝H ≔ ⨁ H푟,푠. 푟≥푝,푠∈Z 66 euler’s gamma function and the field with one element

The Hodge structure H is pure of weight 푘 just in case 퐹 푞H ∩ 퐹 푝H = 0 whenever 푝 + 푞 = 푘 + 1.

3.1.3 Example. Define a Hodge structure

Z(1) ≔ (2휋√−1)Z = ker [exp∶ C C×] with Z(1)−1,−1 = Z(1); this is called the Tate Hodge structure, pure of weight −2. Its tensor powers are Hodge structures

Z(푛) ≔ (2휋√−1)푛Z ⊂ C with Z(푛)−푛,−푛 = Z(푛); these are pure of weight −2푛.

3.1.4 Example. If 푋 is a compact Kähler manifold, then the holomorphic • Poincaré lemma guarantees a quasi-isomorphism 훺푋 ≔ 훺푋 ≃ C푋. Now the “foolish” filtration

≥푛 ≥푛−1 ⋯ 훺푋 훺푋 ⋯ 훺푋 gives rise to a spectral sequence

푝,푞 푞 푝 푝+푞 퐸1 = H (푋, 훺푋) ⟹ H (푋, C) whose abutment is the Hodge filtration on H∗(푋, C). From Hodge theory, we know that this spectral sequence degenerates, whence we obtain a decomposition H푘(푋, C) = ⨁ H푞(푋, 훺푝). 푝+푞=푘 Moreover, one has H푝(푋, 훺푞) = H푞(푋, 훺푝). Thus the singular cohomology H푘(푋, Z) is a Hodge structure pure of weight 푘. This is all neatly summarized in the statement that the singular cohomol- ogy of compact Kähler manifolds (and thus of smooth projective varieties over C) is “really” valued in the category of Hodge structures.

3.1.5 Notation. Recall

−푠/2 훤R(푠) ≔ 휋 훤(푠/2) −푠 훤C(푠) ≔ 2(2휋) 훤(푠), so that

훤C(푠) = 훤R(푠)훤R(푠 + 1). connes–consani on archimedean 푙-factors 67

× × If HC is a complex representation of C × C (i.e., a C-Hodge struc- 푝,푞 ture), then we write ℎ(푝, 푞) = dimC H , and 훤 ≔ ∏ 훤 (푠 − min{푝, 푞})ℎ(푝,푞). HC C 푝,푞

If HR is a Hodge structure, then one defines H푛,+ ≔ {푥 ∈ H푛,푛 | 푥 = (−1)푛푥}; H푛,− ≔ {푥 ∈ H푛,푛 | 푥 = −(−1)푛푥} and

푛,+ ℎ(푛, +) ≔ dimC H ; 푛,− ℎ(푛, −) ≔ dimC H . Now

훤 ≔ ∏ 훤 (푠 − 푛)ℎ(푛,+)훤 (푠 − 푛 + 1)ℎ(푛,−) ∏ 훤 (푠 − 푝)ℎ(푝,푞). HR R R C 푛 푝<푞 In particular, suppose 푋 a smooth projective variety over a global field 퐾, and let 푣 be an archimedean place of 퐾. If 퐾푣 ≅ C, then 푤 an H (푋(퐾푣) , C) admits a C-Hodge structure, and we write 푤 퐿 (ℎ (푋), 푠) ≔ 훤 푤 an (푠). 푣 H (푋(퐾푣) ,C) 푤 an If 퐾푣 ≅ R, then H (푋(퐾푣(푖)) , C) admits a Hodge structure, and we write 푤 퐿 (ℎ (푋), 푠) ≔ 훤 푤 an (푠). 푣 H (푋(퐾푣(푖)) ,C)

3.2 Deligne cohomology and Beilinson’s theorem

Deligne cohomology can be thought of as a systematic way of packaging both ordinary cohomology and the intermediate Jacobians.

3.2.1 Definition. For any integer 푝, the Deligne complex Z(푝)D is a sheaf of complexes (or, better, simplicial abelian groups, or, still better, spectra) on the site of complex analytic manifolds defined as the homotopy fiber product: ℎ ≥푝 Z(푝)D ≔ Z(푝) ×C 훺 . Its cohomology groups on a compact complex analytic manifold 푋 are the Deligne cohomology groups:

푞 푞 HD(푋, Z(푝)) ≔ H (푋, Z(푝)D). 68 euler’s gamma function and the field with one element

푝 3.2.2. Equivalently, Z(푝)D is the fiber of the natural map Z(푝) C/퐹 훺. One can form an explicit complex of sheaves of abelian groups on a complex analytic manifold that represents Z(푝)D: 1 푝−1 0 Z푋(푝) 풪푋 훺푋 ⋯ 훺푋 0.

3.2.3 Example. Of course Z(0)D = Z, and so 푞 푞 HD(푋, Z(0)) ≅ H (푋, Z).

3.2.4 Example. The map

0 Z(1) 풪푋 0 exp exp × 1 1 풪푋 1

∼ × is an equivalence exp∶ Z푋(1)D 풪푋[−1], and so 푞 푞−1 × HD(푋, Z(1)) ≅ H (푋, 풪푋). 2 In particular, note that HD(푋, Z(1)) ≅ Pic(푋), and we have an exact sequence 1 2 0 퐽 (푋) HD(푋, Z(1)) NS(푋) 0.

3.2.5 Example. There is an equivalence

× 1 Z푋(2)D ≃ [푑 log∶ 풪푋 훺푋] [−1]. 2 One can show that HD(푋, Z(2)) is the group of line bundles with holomor- phic connection.

3.2.6. The long exact sequence of the fiber gives H푞−1(푋, C) H푞(푋, C) H푞−1(푋, Z(푝)) H푞 (푋, Z(푝)) H푞(푋, Z(푝)) . 퐹 푝H푞−1(푋, C) D 퐹 푝H푞(푋, C) When 푞 < 2푝, the map on the right is an injection, so we have a short exact sequence

푝 푞−1 푞 푞 0 퐽 H (푋, Z(푝)) HD(푋, Z(푝)) H (푋, Z(푝)) 0. When 푞 = 2푝, we get a short exact sequence

푝 2푝 푝 0 퐽 (푋) HD (푋, Z(푝)) Hdg (푋) 0. connes–consani on archimedean 푙-factors 69

If 푋 is a smooth projective variety over a global field 퐾, then for any archimedean place 푣 of 퐾, we write

푞 푞 {HD(푋푣, R(푝)) if 푣 is complex; HD(푋푣, R(푝)) ≔ { 푞 H (푋 ⊗ 퐾 (푖), R(푝))퐶2 if 푣 is real, { D 푣 퐾푣 푣 where 퐶2 acts via de Rham conjugation.

3.2.7 Theorem (Beilinson).

푤 −1 푤+1 ord푠=푚퐿푣(H (푋), 푠) = dimR HD (푋푣, R(푤 + 1 − 푚)).

3.2.8 Definition. Suppose 푋C a smooth complex projective variety. ∞ an ∞ an Consider the Fréchet algebras 퐶 (푋C , R) and 퐶 (푋C , C); using a topologised version of the Hochschild complex, THcts ⊗ C, we obtain an identification

cts ∞ an TP(푋C) ⊗ C ≃ TP (퐶 (푋C , C)) ⊗ C, and we define

cts ∞ an TPR(푋C) ⊗ C ≔ TP (퐶 (푋C , R)) ⊗ C.

Now we have a natural 휆-algebra structure on all these invariants, and we define 훩0 as the generator of the 휆 operations, so that

훩0 푘 = 휆푘.

Now we select the map

훩0 (2휋푖) ∶ TPR(푋C) ⊗ C TP(푋C) ⊗ C.

Recall that we have the map TC(푋C) TP(푋C). We form the homotopy pullback

an ℎ P (푋C) ≔ ((TC(푋C) ⊗ C) × 훩 TPR(푋C) ⊗ C) [1]. TP(푋C)⊗C, (2휋푖) 0

For a smooth real projective variety 푋R, define

an an 퐶2 P (푋R) ≔ P (푋C) , where 퐶2 acts on the coefficients. 70 euler’s gamma function and the field with one element

There are two domains of interest:

퐸푑 = {(푛, 푗) | 푛 ≥ 0, 0 ≤ 2푗 − 푛 ≤ 2푑} and

퐴푑 = {(푞, 푚) | 0 ≤ 푞 ≤ 2푑, 푚 ≤ 푞/2}

3.2.9 Proposition. There are isomorphisms

an 훩0=푗 2푗+1−푛 휋푛(P (푋C)) ≅ H퐷 (푋C, R(푗 + 1)) and an 훩0=푗 2푗+1−푛 휋푛(P (푋R)) ≅ H퐷 (푋C, R(푗 + 1)) for (푛, 푗) ∈ 퐸푑 (else the left side vanishes).