Berthelot-Ogus-Raynaud

Theorem (Berthelot-Ogus-Raynaud)

$\operatorname{Pic}(X_p^{\mathrm{al}})_{\mathbf{Q}}\subset H_{K'}$
$\operatorname{sp}\bigl(\operatorname{Pic}(X^{\mathrm{al}})_{\mathbf{Q}}\bigr)=\operatorname{Pic}(X_p^{\mathrm{al}})_{\mathbf{Q}}\cap F^1_{K'}$

SOURCE: O:L1240-1242

What it does. The room should leave knowing this statement and not its proof. It is the only thing black-boxed in the lecture.

PANEL FLAG: what the "Raynaud 1979" half of this attribution supplies could not be established from the source or the surrounding files. The lifting criterion itself is Berthelot-Ogus 1983, section 3 (the 1978 reference is their book, not the criterion), and that has been corrected here. The Raynaud citation is carried over from the source deck unchanged, pending the author saying what it is for.

EDITORIAL: the "Geometric version" paragraph is not on the frame (nyc-jnts.tex:1240-1242 is the theorem block only); it states the form of the criterion the K3 examples use.

Spoken: For classes defined over Q_p and over F_p.

Spoken: Quoted, not unpacked: a class in the special fibre comes from characteristic zero exactly when it lies in the filtration step.

Spoken: The K3 examples use eigenvalues -p and zeta_8 p, so they need the geometric version; and local liftability alone is not descent, so the hypothesis that the classes are defined over the base fields is part of the statement.

Spoken: Everything that follows is the work of making both sides of that intersection computable.

TRANSCRIBED FROM: frobenious-dist/nyc-jnts.tex:1240-1242