Alain Connes

56 Pages·2017·1.69 MB·English

The Riemann-Roch
strategy
A. Connes
(Collaboration with C. Consani)
1
RH equivalent
RH problem is equivalent to an inequality for real valued
functions f on R∗ of the form
+
(cid:90) (cid:90)
∗
RH ⇐⇒ s(f, f) ≤ 0 , ∀f | f(u)d u = f(u)du = 0.
−1 −1
s(f, g) := N(f (cid:63) ˜g), ˜g(u) := u g(u )
(cid:63) = convolution product on R∗
+
∞ 2
(cid:90) ∞ u h(u) − h(1)
(cid:88) ∗
N(h) := Λ(n)h(n) + d u + c h(1)
1 u2 − 1
n=1
1
c = (log π + γ).
2
2
Explicit Formula
F : [1, ∞) → R continuous and continuously differen-
tiable except for finitely many points at which both
(cid:48)
F(u) and F (u) have at most a discontinuity of the first
−1/2−(cid:15)
kind, and s.t. for some (cid:15) > 0 : F(u) = O(u )
(cid:90) ∞
s−1
Φ(s) = F(u) u du
1
∞
1 1 1
(cid:88) (cid:88) (cid:88) −m/2 m
Φ( )+Φ(− )− Φ(ρ− ) = log p p F(p )+
2 2 2
ρ∈Zeros p m=1
3/2
γ log π (cid:90) ∞ t F(t) − F(1)
+( + )F(1) + dt
2 2 1 t(t2 − 1)
3
Weil’s formulation
h ∈ S(C ) a Schwartz function with compact support :
K
−1
(cid:90) (cid:48) h(u )
ˆ ˆ (cid:88) (cid:88)ˆ (cid:88) ∗
h(0) + h(1) − h(χ˜, ρ) = d u
K∗ |1 − u|
Z v v
χ∈C(cid:100)K,1 χ˜
(cid:82) (cid:48)
where the principal value is normalized by the addi-
K∗
v
tive character α and for any character ω of C
v K
(cid:90)
ˆ z ∗ ˆ ˆ
h(ω, z) := h(u) ω(u) |u| d u, h(t) := h(1, t)
4
The adele class space
and the explicit formulas
Let K be a global field, the adele class space of K is the
quotient X = A /K× of the adeles of K by the action
K K
of K× by multiplication.
(cid:90)
Tξ(x) := ξ(ux) = k(x, y)ξ(y)dy
k(x, y) = δ(ux − y),
(cid:90) (cid:90)
Tr (T) := k(x, x)dx = δ(ux − x)dx
distr
1 (cid:90) 1
= δ(z)dz =
|u − 1| |u − 1|
5
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6
Critical zeros as absorption spectrum
The spectral side now involves all non-trivial zeros and
the geometric side is given by :
(cid:18)(cid:90) (cid:19) (cid:90) h(w−1)
∗ (cid:88) ∗
Tr h(w)ϑ(w)d w = d w
distr
K× |1 − w|
v v
(A. Connes, Selecta 1998, R. Meyer, Duke, 2005)
7
The limit q → 1 and
the Hasse-Weil formula
C. Soul´e : ζ (s) := lim Z(X, q−s)(q − 1)N(1), s ∈ R
X
q→1
−s −s
where Z(X, q ) denotes the evaluation at T = q of
the Hasse-Weil exponential series
 
r
T
(cid:88) r
Z(X, T) := exp N(q )
 
r
r≥1
For the projective space Pn : N(q) = 1 + q + . . . + qn
1
n+1
ζ (s) = lim (q − 1) ζ (s) =
Pn(F1) q→1 Pn(Fq) (cid:81)n(s − k)
0
8
The limit q → 1
The Riemann sums of an integral appear from the right
hand side :
∂ ζ (s) (cid:90) ∞
s N −s ∗
= − N(u) u d u
ζ (s) 1
N
Thus the integral equation produces a precise equation
for the counting function N (q) = N(q) associated to
C
the hypothetical curve C :
∂ ζ (s) (cid:90) ∞
s Q
−s ∗
= − N(u) u d u
ζQ(s) 1
9
The distribution N (u)
This equation admits a solution which is a distribution
(cid:80)
and is given with ϕ(u) := n Λ(n), by the equality
n<u
d
N(u) = ϕ(u) + κ(u)
du
where κ(u) is the distribution which appears in the ex-
plicit formula
2
(cid:90) ∞ (cid:90) ∞ u f(u) − f(1) 1
∗ ∗
κ(u)f(u)d u = d u+cf(1) , c = (log π+γ)
1 1 u2 − 1 2
The conclusion is that the distribution N(u) is positive
on (1, ∞) and is given by
 
ρ+1
d u
(cid:88)
N(u) = u − order(ρ) + 1
 
du ρ + 1
ρ∈Z
10