跳到主要内容

Chapter 30: φ(30) = [18,2,1] — Motives, Cohomology, and Arithmetic Collapse

30.1 The Abundant Trinity​

With φ(30) = [18,2,1], we see eighteen (abundant threshold) with duality and unity. The number 30 = 2×3×5 is the primorial of the first three primes, representing maximal early smoothness. This perfectly captures how motivic cohomology unifies eighteen cohomology theories through dual realizations into one universal framework.

Definition 30.1 (Primorial Structure):

30=2⋅3⋅5=p1⋅p2⋅p3=3#30 = 2 \cdot 3 \cdot 5 = p_1 \cdot p_2 \cdot p_3 = 3\#

The product of first three primes, maximally smooth at this scale.

30.2 The Dream of Motives​

Grothendieck's Vision: A universal cohomology theory from which all others derive.

Definition 30.2 (Motive): For a smooth projective variety X, its motive h(X) should satisfy:

Htheoryi(X)=Hom(h(X),Ltheoryi)H^i_{\text{theory}}(X) = \text{Hom}(h(X), \mathbb{L}^i_{\text{theory}})

where L\mathbb{L} is the Lefschetz motive.

30.3 The Eighteen Cohomology Theories​

From [18], eighteen cohomology theories unified by motives:

  1. Betti cohomology: HB∗(X;Z)H^*_B(X; \mathbb{Z})
  2. de Rham cohomology: HdR∗(X)H^*_{dR}(X)
  3. Dolbeault cohomology: Hp,q(X)H^{p,q}(X)
  4. Étale cohomology: Heˊt∗(X;Zℓ)H^*_{ét}(X; \mathbb{Z}_\ell)
  5. ℓ-adic cohomology: Heˊt∗(X;Qℓ)H^*_{ét}(X; \mathbb{Q}_\ell)
  6. p-adic cohomology: Crystalline
  7. Crystalline cohomology: Hcrys∗(X/W)H^*_{crys}(X/W)
  8. Rigid cohomology: Overconvergent
  9. Deligne cohomology: HD∗(X;Z(n))H^*_{\mathcal{D}}(X; \mathbb{Z}(n))
  10. Motivic cohomology: HM∗(X;Z(n))H^*_{\mathcal{M}}(X; \mathbb{Z}(n))
  11. Syntomic cohomology: p-adic periods
  12. Absolute Hodge: HAH∗(X)H^*_{AH}(X)
  13. K-theory: K∗(X)K_*(X)
  14. Chow groups: CH∗(X)CH^*(X)
  15. Galois cohomology: H∗(GK,M)H^*(G_K, M)
  16. Group cohomology: For arithmetic groups
  17. Intersection cohomology: IH∗(X)IH^*(X)
  18. Perverse cohomology: Via derived categories

30.4 The Dual Nature [2]​

The [2] represents fundamental duality:

Homology/Cohomology:

Hi(X)⊗Hi(X)→ZH_i(X) \otimes H^i(X) \to \mathbb{Z}

Poincaré Duality: For compact oriented manifold:

Hi(X)≅Hn−i(X)H^i(X) \cong H_{n-i}(X)

30.5 The Unity [1]: Universal Cohomology​

The [1] represents how all cohomologies derive from one source:

Principle 30.1 (Motivic Unity):

Hany∗(X)=HM∗(X)⊗MRtheoryH^*_{\text{any}}(X) = H^*_{\mathcal{M}}(X) \otimes_{\mathcal{M}} R_{\text{theory}}

All cohomologies are realizations of motivic cohomology.

30.6 L-Functions and Motives​

Definition 30.3 (Motivic L-Function): For motive M:

L(M,s)=∏pdet⁡(1−Frobp⋅p−s∣MℓIp)−1L(M, s) = \prod_p \det(1 - \text{Frob}_p \cdot p^{-s} | M_\ell^{I_p})^{-1}

Special Case:

ζ(s)=L(h(Spec Z),s)\zeta(s) = L(h(\text{Spec } \mathbb{Z}), s)

The Riemann zeta is the L-function of the trivial motive!

30.7 The Standard Conjectures​

Grothendieck's Conjectures:

  1. Hodge Standard: Lefschetz operator properties
  2. Lefschetz Standard: Hard Lefschetz theorem
  3. Künneth Standard: Künneth projectors algebraic
  4. Numerical Standard: Numerical ≡ homological equivalence

These would construct the category of motives.

30.8 Period Matrix and RH​

Definition 30.4 (Period Matrix): For a motive M:

Ω(M)=(∫γ1ω1⋯∫γ1ωn⋮⋱⋮∫γnω1⋯∫γnωn)\Omega(M) = \begin{pmatrix} \int_{\gamma_1} \omega_1 & \cdots & \int_{\gamma_1} \omega_n \\ \vdots & \ddots & \vdots \\ \int_{\gamma_n} \omega_1 & \cdots & \int_{\gamma_n} \omega_n \end{pmatrix}

Deligne's Theorem: RH for varieties over finite fields relates to Hodge structure of periods.

30.9 Mixed Motives​

Definition 30.5 (Mixed Motive): Has weight filtration:

0=W−1M⊆W0M⊆W1M⊆⋯⊆M0 = W_{-1}M \subseteq W_0M \subseteq W_1M \subseteq \cdots \subseteq M

with pure graded pieces GriWM\text{Gr}_i^W M.

For Zeta: Lives in mixed motivic world via:

ζ(s)↔ExtMM1(Q(0),Q(n))\zeta(s) \leftrightarrow \text{Ext}^1_{MM}(\mathbb{Q}(0), \mathbb{Q}(n))

30.10 Beilinson's Conjectures​

Conjecture 30.1 (Beilinson): For smooth projective X/ℚ:

ords=mL(hi(X),s)=dim⁡QK2m−i−1(X)Q(m)\text{ord}_{s=m} L(h^i(X), s) = \dim_\mathbb{Q} K_{2m-i-1}(X)^{(m)}_\mathbb{Q}

Special values of L-functions = ranks of K-groups!

30.11 The Motivic Galois Group​

Definition 30.6: The Tannakian Galois group:

Gmot=Aut⊗(ω:Mot→VecQ)G_{\text{mot}} = \text{Aut}^{\otimes}(\omega : \text{Mot} \to \text{Vec}_\mathbb{Q})

Acts on all motivic cohomology.

Dream: RH equivalent to property of GmotG_{\text{mot}}.

30.12 Periods and Transcendence​

Kontsevich-Zagier: Periods are integrals:

P=∫σωP = \int_{\sigma} \omega

where σ is semi-algebraic set, ω algebraic form.

Examples: π, ζ(3), log(2), multiple zeta values

Connection: Zeros of ζ related to algebraic independence of periods.

30.13 The Hodge Conjecture​

Conjecture 30.2 (Hodge): On smooth projective X:

H2p(X;Q)∩Hp,p(X)=span of algebraic cyclesH^{2p}(X; \mathbb{Q}) \cap H^{p,p}(X) = \text{span of algebraic cycles}

Relation to RH: Both about identifying special subspaces in cohomology.

30.14 Arithmetic Cohomology​

Definition 30.7 (Arithmetic Cohomology): For scheme X/ℤ:

Hari(X,Z(n))=ExtDMMi(h(X),Z(n))H^i_{\text{ar}}(X, \mathbb{Z}(n)) = \text{Ext}^i_{DMM}(h(X), \mathbb{Z}(n))

Captures both geometric and arithmetic information.

For Spec ℤ: Related to values ζ(n).

30.15 Synthesis: Motivic Unity​

The partition [18,2,1] reveals motivic structure:

  1. [18] - Complete Theories: Eighteen cohomologies
  2. [2] - Duality: Homology/cohomology, Poincaré
  3. [1] - Unity: Universal motivic theory

Key insights:

  • 30 = primorial: First three primes product
  • Universal cohomology: Motives unify all
  • L-functions: Attached to motives
  • Standard conjectures: Would complete theory
  • Period matrices: Connect to RH
  • Mixed motives: Zeta lives here
  • Beilinson: Special values = K-theory
  • Galois group: Universal symmetry
  • Periods: Transcendental invariants
  • Hodge conjecture: Millennium problem
  • Arithmetic cohomology: Over Spec ℤ
  • Realizations: Each cohomology a view
  • Tannakian formalism: Category theory
  • Deep unity: Geometry, arithmetic, analysis
  • Ultimate goal: Motivic proof of RH

The theory of motives represents Grothendieck's vision of a universal mathematical framework where all cohomological phenomena are unified. The Riemann Hypothesis, in this view, becomes a statement about the motivic L-function of the simplest arithmetic object - Spec ℤ itself.

Chapter 30 Summary:

  • Motives provide universal cohomology theory
  • Eighteen cohomology theories unified by motivic framework
  • Partition [18,2,1] shows abundant-dual-unity structure
  • L-functions attach to motives, including ζ(s)
  • Periods and transcendence connect to zeros
  • RH becomes property of motivic Galois group
  • Ultimate unification of arithmetic and geometry

"In the garden of motives, eighteen flowers bloom - each a different cohomology theory, each revealing its own aspect of the hidden unity, all sustained by the same universal root that nourishes the zeros of zeta in the soil of absolute arithmetic."