Absolute zeta functions and the absolute automorphic forms

Transcription

1 Absolute zeta functions and the absolute automorphic forms Nobushige Kurokawa April, 2017, (Shanghai) 1 / 44

2 1 Introduction In this report we study the absolute zeta function ζ f (s) associated to a certain absolute automorphic form f(x) on the group Γ = R >0 = {x R x > 0}. We require f(x) to satisfy the following automorphy : f( 1 x ) = Cx D f(x), where D Z with C = ±1. To explain our problem we first recall the history of absolute zeta functions briefly. 2 / 44

3 Soulé (2004; C. Soulé, Les variétés sur le corps à un élément. Mosc. Math. J, 4 (2004), ) introduced the absolute zeta function (the zeta functions over F 1 ) of a suitable scheme X as the limit of the congruence zeta function where ζ X/Fp (s) = exp ζ X/F1 = lim p 1 ζ X/Fp (s), ( m=1 ) X(F p m) m p ms ; see also Kurokawa (2005; N. Kurokawa, Zeta functions over F 1. Proc. Japan Acad. Ser. A., 81 (2005), ) and Deitmar (2006; A. Deitmar, Remarks on zeta functions and K-theory over F 1. Proc. Japan Acad. Ser. A., 82 (2006), ). 3 / 44

4 Connes-Consani (2010 and 2011; A. Connes and C. Consani, Schemes over F 1 and zeta functions. Compositio Mathematica, 146 (2010), ; A. Connes and C. Consani, Characteristic 1, entropy and the absolute point. Noncommutative geometry, arithmetic, and related topics, 2011, ) interpreted it as ( ) f(x)x s 1 ζ X/F1 (s) = exp dx log x 1 when f(x) = X(F x ) Z[x]. 4 / 44

5 In this report we follow Kurokawa-Ochiai (2013; N. Kurokawa and H. Ochiai, Dualities for absolute zeta functions and multiple gamma functions. Proc. Japan Acad. Ser. A., 89 (2013), 75 79) and Deitmar-Koyama-Kurokawa (2015; A. Deitmar, S. Koyama and N. Kurokawa, Counting and zeta functions over F 1. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 85 (2015), 59 71) using the zeta-regularization process: ( ) ζ f (s) = exp w Z X/F 1 (w, s) w=0 with Z f (w, s) = 1 Γ(w) 1 f(x)x s 1 (log x) w 1 dx. 5 / 44

6 We notice that Z f (w, s) = 1 Γ(w) 0 f(e t )e st t w 1 dt is the Mellin transform frequently used in the theory of zeta functions, where f(e t ) is usually an automorphic form. We call ζ f (s) and Z f (w, s) as the absolute Hasse zeta function and the absolute Hurwitz zeta function respectively. 6 / 44

7 for f (x) = f( 1 x ). 7 / 44 2 Polynomial counting functions Let f(x) be a suitable function satisfying f( 1 x ) = Cx D f(x) (C = ±1, D Z). We construct the absolute Hurwitz zeta function Z f (w, s) and the absolute Hasse zeta function ζ f (s) as and Z f (w, s) = 1 Γ(w) 1 ( ζ f (s) = exp w Z f (w, s) respectively. Moreover we define f(x)x s 1 (log x) w 1 dx ε f (s) = ζ f ( s) ζ f (s) w=0 )

8 Theorem 1. Let f(x) Z[x] be a monic polynomial satisfying f( 1 x ) = Cx D f(x) (C = ±1, D Z). Then: (1) ζ f (s) is a rational function in s. (2) ζ f (s) has the functional equation ζ f (D s) C = ( 1) χ(f) ζ f (s), where χ(f) = f(1) is the Euler-Poincaré characteristic. Moreover ε f (s) = ( 1) χ(f). (3) ζ f (s) is holomorphic in Re(s) > deg(f) 1 except for the simple pole at s = deg(f). 8 / 44

9 (4) The regularized Euler constant ( ζ γ f (s) (f) = lim s deg(f) is expressed by the formula γ (f) = C 1 0 ζ f (s) + 1 s deg(f) ( ) dx f 0 (x) C x, where f 0 (x) = f(x)x r with r = ord x=0 f(x). ) 9 / 44

10 Proof of Theorem 1 (1) Let f(x) = k a(k)x k. Then Z f (w, s) = k a(k)(s k) w and ζ f (s) = k (s k) a(k). 10 / 44

11 (2) The functional equation f( 1 x ) = Cx D f(x) is equivalent to Hence, ζ f (D s) C = k = k a(d k) = Ca(k). ( (D s) k ) Ca(k) = (k s) a(k) = ( 1) f(1) k k ( (D k) s (s k) a(k) ) a(d k) = ( 1) χ(f) ζ f (s). This shows ε f (s) = ( 1) χ(f). 11 / 44

12 (3) From we see that f(x) = x n + k<n a(k)x k ζ f (s) = 1 s n (s k) a(k) k<n has a simple pole at s = n with residue R(f) = k<n(n n k) a(k) = k a(n k). k=1 12 / 44

13 (4) From the expression for ζ f (s) in (3) we have ζ f (s) ζ f (s) + 1 s n = a(k) s k. k<n Hence γ (f) = k<n = k<n a(k) n k Ca(D k). n k We know that C = f 0 (0) = a(r), D = n + r, n f 0 (x) = a(k)x k r. k=r 13 / 44

14 Thus n 1 γ (f) = = C = C k=r Ca(n + r k) n k n k=r+1 1 ( 0 a(k) k r f 0 (x) C ) dx x. QED 14 / 44

15 We remark that the so called limit formula for ζ f (s) is ( lim ζ f (s) R(f) ) = γ(f), s deg(f) s deg(f) where R(f) is the residue and γ(f) is the analogue of the Euler constant. The regularized Euler constant γ (f) = is nothing but γ (f) = γ(f) R(f). ( ζ ) f (s) lim s deg(f) ζ f (s) + 1 s deg(f) 15 / 44

16 3 Examples In this section we illustrate by examples that the absolute zeta functions ζ X/F1 (s) of some typical Z-schemes over F 1 are identified with the zeta functions ζ f (s) of certain cyclotomic absolute automorphic forms f such that f(x) = X(F x ) for primes powers x. We list the f(x), ζ X/F1 (s) = ζ f (s), and the functional equation for ζ X/F1 (s). 16 / 44

17 Affine space A n f(x) = x n, f( 1 x ) = x 2n f(x), χ(f) = 1, ζ A n /F 1 (s) = ζ f (s) = 1 s n, ζ A n /F 1 (2n s) = ζ A n /F 1 (s). 17 / 44

18 Projective space P n f(x) = x n + x n = xn+1 1 x 1, f( 1 x ) = x n f(x), χ(f) = n + 1, ζ P n /F 1 (s) = 1 (s n) s, ζ P n /F 1 (n s) = ( 1) n+1 ζ P n /F 1 (s). 18 / 44

19 Grassmannian space Gr(n, m) f(x) = (xn 1) (x n m+1 1) (x m, 1) (x 1) f( 1 x ) = x m(n m) f(x), ( ) n χ(f) =, m ζ Gr(n,m)/F1 (s) is a rational function satisfying ζ Gr(n,m)/F1 (m(n m) s) = ( 1)( n m ) ζ Gr(n,m)/F1 (s). 19 / 44

20 Multiple gamma functions of oder r f(x) = (x m(1) 1) (x m(r) 1), f( 1 x ) = ( 1)r x m f(x), m = m(1) + + m(r), { 1 r = 0, χ(f) = 0 r 1, ζ f (s) = (s m(i)) ( 1)r I +1 I {1,...,r} = Γ r (s m, m), m(i) = m(i), i I ζ f ( m s) ( 1)r = ζ f (s). 20 / 44

21 General linear group GL(n) f(x) = x n(n 1) 2 (x 1)(x 2 1) (x n 1), f( 1 x ) = ( 1)n x n(3n 1) 2 f(x), χ(f) = 0 (for n 1), ζ GL(n)/F1 (s) is the rational function Γ n (s n 2, (1, 2,..., n)), n(3n 1) ζ GL(n)/F1 ( s) ( 1)n = ζ 2 GL(n)/F1 (s). 21 / 44

22 Special linear group SL(n) f(x) = x n(n 1) 2 (x 2 1) (x n 1), f( 1 x ) = ( 1)n 1 x n(3n 1) 2 +1 f(x), χ(f) = 0 (for n 2), ζ SL(n)/F1 (s) is the rational function Γ (n 1) (s (n 2 1), (2, 3,..., n)), n(3n 1) ζ SL(n)/F1 ( 1 s) ( 1)n 1 = ζ 2 SL(n)/F1 (s). 22 / 44

23 Symplectic group Sp(n) (size 2n) f(x) = x n2 (x 2 1)(x 4 1) (x 2n 1), f( 1 x ) = ( 1)n x n(3n+1) f(x), χ(f) = 0 (for n 1), ζ Sp(n)/F1 (s) is the rational function Γ n (s (2n 2 + n), (2, 4,..., 2n)), ζ Sp(n)/F1 (n(3n + 1) s) ( 1)n = ζ Sp(n)/F1 (s). Remarks. (1) ζ Sp(n)/F1 (s) = ζ GL(n)/F1 ( s n 2 ), (2) ζ Sp(n)/Fq (s) = ζ ( s n GL(n)/Fq 2 2 ) for a prime power q. 23 / 44

24 4 Easy examples of Euler constants Theorem 2. (1) γ (P n /F 1 ) = H n, where H n = n is the harmonic number. Especially γ (P n /F 1 ) = γ log n + O( 1 n ) as n, where γ is the original Euler constant. (2) γ (G n m/f 1 ) = H n. In particular as n. γ (G n m/f 1 ) = γ + log n + O( 1 n ) 24 / 44

25 Proof of Theorem 2 (1) Since ( ) f P n(x) = x n + x n = xn+1 1, x 1 γ (P n /F 1 ) = = ( ) dx f P n(x) 1 0 x n 1 k 1 k=1 = H n. 25 / 44

26 (2) Since f G n m (x) = (x 1) n n ( ) = ( 1) n k n x k, k k=0 1 ( ) dx γ (G n m/f 1 ) = ( 1) n f G n m (x) ( 1) n 0 x ( ) n n ( 1) k+1 k = k k=1 = H n. 26 / 44

27 The last equality is shown as follows: 1 ( ) dx γ (G n m/f 1 ) = ( 1) n f G n m (1 x) ( 1) n 1 x = 1 0 = H n. 0 x n 1 x 1 dx QED 27 / 44

28 5 Cyclotomic absolute automorphic forms Theorem 3. Let f(x) = x l (xm(1) 1) (x m(a) 1) (x n(1) 1) (x n(b) 1) for an integer l 0, and positive integers m(i), n(j). Put where deg(f) = l + m n, deg(f) = deg(f) + l m = n = = 2l + m n, a m(i), i=1 b n(j). j=1 28 / 44

29 Define ( ζ f (s) = exp w Z f (w, s) w=0 ) with Z f (w, s) = 1 Γ(w) Then the following properties hold. (1) 1 f(x)x s 1 (log x) w 1 dx. f( 1 x ) = Cx D f(x) with C = ( 1) a b and D = deg(f). (2) ζ f (s) is a meromorphic function on C written explicitly by multiple gamma functions of order b. Moreover, zeros and poles of ζ f (s) belong to Z. (3) ζ f (s) is non-zero holomorphic in Re(s) > deg(f) 1 except for the simple pole at s = deg(f). 29 / 44

30 (4) ζ f (s) has the functional equation ζ f (D s) C = ε f (s)ζ f (s), where ε f (s) is written explicitly by multiple sine functions of order b. (5) ζ f (s) is a rational function if and only if f(x) Z[x]. (6) When ζ f (s) is a rational function (i.e., f(x) Z[x] by (5)) ζ f (D s) C = ( 1) χ(f) ζ f (s). 30 / 44

31 We refer to f(x) and ζ f (s) in Theorem 3 as cyclotomic absolute automorphic forms and cyclotomic absolute zeta functions respectively. In fact, { a f(x) = x l i=1 (xm(i) 1) b j=1 (xn(j) 1) = { f(x) = x l M m=1 l Z 0, m(i), n(j) Z >0, a, b Z 0 Φ m (x) c(m) l Z 0, c(m) Z, M Z 0 where Φ m (x) is the m-th cyclotomic polynomial. }, } 31 / 44

32 To prove Theorem 3 we use the multiple gamma function Γ r (s, n) which is essentially the absolute zeta function associated to the cyclotomic absolute automorphic form In this case f(x) = 1 (x n(1) 1) (x n(r) 1). ζ f (s) = Γ r (s + n, n), ζ f ( n s) ( 1)r = ζ f (s)ε f (s), ε f (s) = S r (s + n, n), where S r (s, n) = Γ r (s, n) 1 Γ r ( n s, n) ( 1)r is the multiple sine function of Kurokawa-Koyama(2003; N. Kurokawa and S. Koyama, Multiple sine functions. Forum mathematicum, 15 (2003), ). 32 / 44

33 6 Remarks on Euler constants We notice some examples of regularized Euler constants for cyclotomic absolute automorphic forms f γ (f) = ( ζ ) f (s) lim s deg(f) ζ f (s) + 1. s deg(f) 33 / 44

34 Theorem 4. Let f 1 (x) = f 2 (x) = f 3 (x) = f 4 (x) = f 5 (x) = 1 x 1 = 1 Φ 1 (x), 1 x + 1 = 1 Φ 2 (x) = x 1 x 2 1, 1 x 2 + x + 1 = 1 Φ 3 (x) = x 1 x 3 1, 1 x 2 x + 1 = 1 Φ 6 (x) = (x2 1)(x 3 1) (x 1)(x 6 1), x x 4 + x = Φ 4 (x) Φ 3 (x)φ 6 (x) = x4 1 x / 44

35 Then: (1) γ (f 1 ) = γ, where γ is the original Euler constant. (2) γ (f 2 ) = 1 0 (f 2(x) 1) dx x = log 2 / Q. (3) γ (f 3 ) = 1 0 (f 3(x) 1) dx x = 1 2 log 3 + π 6 / Q. 3 (4) γ (f 4 ) = 1 0 (f 4(x) 1) dx x = π 3 3 / Q (5) γ (f 5 ) = 1 0 (f 5(x) 1) dx x = 1 4 log 3 π 12 3 / Q Remark f 5 = 1 2 (f 3 + f 4 ), γ (f 5 ) = 1 2 (γ (f 3 ) + γ (f 4 )). 35 / 44

36 We remark that ζ f1 (s) = Γ 1 (s + 1) = Γ(s + 1) 2π and ( ζ γ f1 (s) (f 1 ) = lim s 1 ζ f1 (s) + 1 ) s + 1 ( Γ = lim s 0 Γ (s) + 1 ) s = Γ Γ (1) = γ. Manin (1995; Manin, Yu. I.: Lectures on zeta functions and motives (according to Deninger and Kurokawa). Asterisque 228 (1995), ) indicated that ζ f1 (s) would be essentially the absolute zeta function ζˇp /F 1 (s) for dual infinite dimensional projective space over F 1 (p.134). 36 / 44

37 In this sence γ = lim n γ (ˇP n /F 1 ) = lim n γ (P n /F 1 ) = lim n ( H n ). More generally: Theorem 5. Let g r (x) = 1 (x 1) r γ (g r ) = γ for a positive integer r. Then r ζ k (1), k=2 where ζ k (s) = n=1 Especially, γ (g r ) γ + Q. ( n + k 2 k 1 ) n s. 37 / 44

38 Examples γ (g 1 ) = γ γ (g 2 ) = γ γ (g 3 ) = γ γ (g 4 ) = γ + 1 γ (g 5 ) = γ / 44

39 We remark that Euler (1776; L. Euler, Evolutio formulae integralis 1 dx( 1 x + 1 ) a termino x = 0 usque ad x = 1 extensae, lx Nova Acta Acad. Scient. Imp. Petrop. 4 (1789), 3 16 (Presented Feb. 29, 1776). (Opera Omnia, Series I, Vol. 18, pp )[E629]) obtained the following formula: γ = = n=2 1 n=2 ( ( n 1 n log n 1 k ( 1)k k 1 k=1 n log ζ G n 1 m /F 1 (n). ) ) 39 / 44

40 Euler s calculation (1) 1 0 ( 1 1 x + 1 ) dx = log x = = = = log x + 1 x (1 x) log x dx log x + 1 (log x) n n=0 n! (1 x) log x n=2 n=2 1 n! 1 0 (log x) n 1 1 x dx 1 n! ( 1)n 1 (n 1)!ζ(n) ( 1) n n ζ(n). n=2 dx 40 / 44

41 Euler s calculation (2) n=2 ( 1) n n ζ(n) = = ( 1) n n m n ( 1 m log(1 + 1 ) m ) n=2 m=1 m=1 = lim M m=1 = lim M = γ. M ( 1 m log(1 + 1 ) m ) ( M ) log(m + 1) 41 / 44

42 Euler s calculation (3) γ = = = = = Connes Consani = ( 1 1 x + 1 log x 0 1 n=2 log x + 1 x (1 x) log x dx 0 n=2 ) dx log(1 (1 x)) + 1 x dx (1 x) log x 1 (1 x)n 1 dx n log x 1 n 1 (x 1) n 1 x n 1 dx log x 1 n log ζ Gm n 1 /F 1 (n). n=2 42 / 44

43 We can generalize this formula in many ways using absolute automorphic forms and absolute zeta functions. For example: Theorem 6. γ = n=2 H n n + 1 log ζ Gm n 1 /F 1 (n), where H n is the harmonic number. In the proof we use the limit formula for the absolute Hurwitz zeta function Z f (w, s) around w = 2 for the absolute automorphic form f(x) = 1 (x 1) / 44

44 Celebrating Alain Connes 70 = ζ SL(2) n /F 1 (3(n + 24)) 1. n=0 44 / 44