alternative lecture notes - Rational points and algebraic cycles

gives rise to an exact sequence
1 → π et
1 (X, ¯x 0 ) → π et
1 (X, ¯x 0 ) → Gal(k/k) → 1.
A k-point ι of X would give rise to a homomorphism Gal(k/k) → π1 et (X, ῑ), and there is an
isomorphism π1 et (X, ῑ) ≃ π1 et (X, ¯x 0 ) lying over Gal(k/k) that is well-defined up to conjugation
by an element of π1 et (X, ¯x 0 ).
Conjecture 1.5 (Grothendieck section conjecture). If X is a smooth projective curve over
a number field of genus at least 2, then the section obstruction is complete.
If true, the map from k-points to equivalence classes of sections is a bijection.
Also, if true, the question of deciding whether X has a k-point is decidable (see proof of
Ambrus Pal).
Let X be a geometrically connected variety over a number field k. If X(k v ) = ∅, then
X(k) = ∅. One can compute a finite set S of places such that X(k v ) is nonempty for all
v /∈ S.
Lind: The variety 2y 2 = x 4 − 17 has Q p -points for all p, but no Q-point.
Let A k = ∏ ′
v k v := {(x v ) ∈ ∏ v k v : x v ∈ O v almost always}. Then k ↩→ A k , so if
X(A k ) = ∅, then X(k) = ∅.
Brauer–Manin obstruction:
X(k) ⊆ X Br (A k ) ⊆ X(A k ).
Given u ∈ H 2 (X, G m ) and (x v ) ∈ X(A k ), we obtain x ∗ vu ∈ H 2 (K v , G m ) inv
↩→ Q/Z, and
((x v ), u) := ∑ inv(x ∗ v, u) ∈ Q/Z; the Hasse–Brauer–Noether theorem implies that if (x v )
comes from a k-point, then ((x v ), u) = 0. Define the Brauer set
This contains X(k).
X(A k ) Br := {(x v ) : ((x v ), u) = 0 for all u ∈ H 2 (X, G m )}.
Descent obstruction: Let G be an affine algebraic group over k. Let X be a base variety
over k. A G-torsor over X is a surjective morphism f : Y → X with a G-action over X
(i.e., G × Y → Y respects the projection to X) such that the map G × Y → Y × X Y is an
isomorphism. Isomorphism types are classified by Het(X, 1 G).
Given the class u ∈ Het(X, 1 G) of some torsor f : Y → X and given (x v ) ∈ X(A k ),
we obtain x ∗ vu ∈ H 1 (k v , G). We then ask: does the collection of elements (x ∗ vu) come
from an element of H 1 (k, G)? Let X(A k ) f be the set of (x v ) ∈ X(A k ) satisfying this
condition. Let X(A k ) desc be the intersection of X(A k ) f over all torsors under all affine
algebraic groups. Sometimes one restricts the set of groups considered, to define other sets:
X(A k ) conn , X(A k ) fin . Harari: X(A k ) conn = X(A k ) Br under certain reasonable conditions.
2. July 4 (Schlank)
Given a smooth geometrically connected variety X, a choice of base point gives
1 → π et
1 (X) → π et
1 (X) → Γ k → 1
where Γ k := Gal(k/k). Rational points give sections Γ k → π1 et (X) up to conjugation by
elements of π1 et (X).
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