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 ���������� �������� �������� �������� ���������� �������� 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