Euler's Gamma Function and the Field with One Element

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 cohomology 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 dominated 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⟨.

Load more