Étale Cohomology
Galois action on algebraic varieties over finite fields — Weil conjectures via ℓ-adic cohomology
Variety over 𝔽_p
Elliptic curve
ℙ¹
𝔸¹
Hyperelliptic
p =
5
Extension degree n (𝔽_pⁿ):
1
Point Counts |X(𝔽_pⁿ)|
Weil conjectures (Deligne 1974):
Z(X/𝔽_p,t) = exp(Σ|X(𝔽_pⁿ)|tⁿ/n)
= P₁(t)P₃(t)… / P₀(t)P₂(t)…
Rationality: Z ∈ ℚ(t) ✓ (Dwork)
Functional equation: symmetry ✓
Riemann hypothesis: |αᵢ|=p^{i/2} ✓
Frobenius φ: x↦xᵖ generates Gal(𝔽̄_p/𝔽_p)