Torsion-free cokernel

$X/\Bbb{Q}$ K3; $p>2$ a prime of good reduction.

Theorem (Elsenhans—Jahnel)

The specialization map

$$\operatorname{Pic}(X^{\mathrm{al}}) \hookrightarrow \operatorname{Pic}(X_p^{\mathrm{al}})$$

has torsion-free cokernel for $p \neq 2$.

Thus, if $\rho(X_p^{\mathrm{al}}) = \rho(X^{\mathrm{al}})$ every invertible sheaf lifts.

For example, if $\rho(X_p^{\mathrm{al}}) = 2$, Elsenhans—Jahnel approach is

compute $\operatorname{Pic}(X_p^{\mathrm{al}})$
estimate the degree of a hypothetical effective divisor of the lift
use Gröbner bases to verify that such a divisor does or does not exist

This approach is only practical if one can compute $\operatorname{Pic}(X_p^{\mathrm{al}})$ and if the obtained estimates are low.

Elsenhans-Jahnel: this becomes the proof of a generic example using a single prime.

The cokernel is torsion-free, so equality of ranks forces the specialization map to be an isomorphism of lattices.

Integral information descends. An effective divisor of the predicted degree can be tested with Groebner bases.

Here $X$ is a K3 surface over $\Bbb{Q}$ and $p>2$ is a prime of good reduction. The published theorem allows ramification degree $e<p-1$; over $\Bbb{Q}$, $e=1$.

For reduction rank two, the Elsenhans-Jahnel approach computes the lattice, bounds a possible effective lift, and checks it with Groebner bases.

Elsenhans and Jahnel had already given the first explicit degree-two examples over Q with geometric Picard rank one in 2008. Their torsion-free specialization theorem was published in 2011. Here "generic" means geometric Picard rank one.

Imagine: only in the late 2000s did we have explicit generic examples over Q even in degree two.

First degree-two examples: Elsenhans—Jahnel, "K3 surfaces of Picard rank one and degree two", ANTS VIII, LNCS 5011 (2008), 212-225; author's text, introduction and Corollary 30. The one-prime method here is their later "The Picard group of a K3 surface and its reduction modulo p", arXiv:1006.1972, posted 10 June 2010; Algebra & Number Theory 5 (2011), 1027-1040.

APPROVED (2026-09-14): [s13-m01] Keep slide 13 in section 1.2. Contact-sheet verdict: APPROVED; no note supplied.

APPROVED (2026-09-14): [s13-m02] Keep "Theorem (Elsenhans-Jahnel)"; open with "The specialization map". Contact-sheet verdict: APPROVED; no note supplied.

NEEDS APPROVAL: [s13-m03] Suggestion: Introduction: "For rho(X_p^{al}) = 2, the Elsenhans-Jahnel approach:" followed by the existing three steps. Source: V:L479-498.
Restored the V:L489 source introduction as the visible baseline. The shorter recommendation above remains open; both jurors are preserved in PLAN-remaining.md D09.

APPROVED (2026-09-14): [s13-m04] Keep the refinement folded into section 1.2; keep slide 13. Contact-sheet verdict: APPROVED; no note supplied.

SUPERSEDED (2026-09-14): [s13-m05] s12-c01 requires the authors-only label "Theorem (Elsenhans-Jahnel)". Full locator: "The Picard group of a K3 surface and its reduction modulo p", Algebra & Number Theory 5 (2011), 1027-1040, Theorem 1.4 and Remarks 1.5(a), p. 1028. The old dated/numbered label is withdrawn.

APPLIED (2026-09-14): [s13-m06] s13-c01 authorizes the checked correction: add "$X/\Bbb{Q}$ K3; $p>2$ a prime of good reduction" before the theorem. Published Theorem 1.4 and Remarks 1.5(a), p. 1028, allow e<p-1 and specialize to e=1 over Q. The previous arXiv-based provenance was too narrow; the slide does not claim the result at arbitrary ramified places.

APPLIED (2026-09-15): [s13-m07] s13-c01: retain the first explicit degree-two K3 example over Q with geometric Picard rank one. The visible credit names only Elsenhans-Jahnel, per the standing citation rule. The ANTS VIII paper (2008), introduction and Corollary 30, constructs the examples after recalling the earlier degree-four examples. The separate torsion-free specialization theorem is Theorem 1.4 and Remarks 1.5(a) in the 2011 paper. Here generic means geometric Picard rank one.

AUTHOR'S CALL: [s13-m90] Keep the odd-prime restriction in the opening scope, in the theorem conclusion, or both? Both formulations are preserved pending your choice.