So far we have been trying to improve the inequality $\rho(X^{\mathrm{al}})\leq\rho(X_p^{\mathrm{al}})$.
Can we use the inequality to our advantage?
If there are infinitely many $p$ primes such that
$$\rho(X^{\mathrm{al}})<\rho(X_p^{\mathrm{al}})\text{ and }\rho(X_p^{\mathrm{al}})\neq22,$$
then $X^{\mathrm{al}}$ contains infinitely many rational curves.
The set $\{p:\rho(X_p^{\mathrm{al}})\neq22\}$ has positive density (density 1 after finite extension).
$\rho(X^{\mathrm{al}})$ odd $\Rightarrow$ infinitely many integral rational curves on $X^{\mathrm{al}}$.
APPROVED (2026-09-14): [s15-m01] Historical approval, superseded by s17-c02: generalize the odd-rank statement to the two verified cases; label "Theorem (Li-Liedtke; C-Elsenhans-Jahnel)". Li-Liedtke supplies odd rank; Costa-Elsenhans-Jahnel supplies even rank, no real or complex multiplication, and a nontrivial jump character. Author, verbatim: "We should write the "Corollary (Li-Liedtke)" more generically, so we can use it immediately when we show the density is at least 1/2. We can add our names to it also. In particular, this should help with the delivery in slide "We can explain the 1/2", and now the cororllary is obvious"
REFUSED (2026-09-14), CHECK FAILED: [s15-m02] the proposed universal Kummer-to-SO assertion is false on the transcendental representation. Costa-Elsenhans-Jahnel 2020, Example 2.36(b): rank-18 Kummer surfaces from quadratic-conjugate elliptic factors have a nontrivial jump character. No universal assertion added. Author, verbatim: "I am also unsure what is the purpose of Slide 17, in particular given Slide 16, some of teh questions are already answered in the previous slide. On slide 15, not sure we should write "The jump criterion on the slides that immediately follow is a consequence of that one fact.". Intead, we should point, there is an easy way to explain some jumps, O vs SO . And maybe there one should point out that for kummer varieties we always land in SO (check this for me please)"
APPLIED (2026-09-15): [s15-m03] s17-c02: the final box is "Corollary (Li-Liedtke)". $\rho(X^{\mathrm{al}})$ odd $\Rightarrow$ infinitely many integral rational curves on $X^{\mathrm{al}}$. Jun Li and Christian Liedtke, "Rational curves on K3 surfaces", Inventiones Mathematicae 188 (2012), 713-727; arXiv:1012.3777, introduction and Theorem 3.3. The introduction says integral; the proof produces integral rational curves of arbitrarily large degree.
NEEDS APPROVAL: [s15-m05] Recommendation: Keep the source Bogomolov-Zarhin box; speak the credit "Positive density: Joshi-Rajan; density one after finite extension: Bogomolov-Zarhin."
Source: Bogomolov-Zarhin 2009, Thm. 0.1 and following note.
NEEDS APPROVAL: [s15-m07] Review the current candidate below; s17-c02 settles the odd-rank corollary and its required cross-references. The title choice remains open under s15-m08. Exact candidate text: "K3 surfaces" | "So far we have been trying to improve the inequality $\rho(X^{\mathrm{al}})\leq\rho(X_p^{\mathrm{al}})$." | "Can we use the inequality to our advantage?" | "Theorem (Li-Liedtke)" | "If there are infinitely many $p$ primes such that" | "$$\rho(X^{\mathrm{al}})<\rho(X_p^{\mathrm{al}})\text{ and }\rho(X_p^{\mathrm{al}})\neq22,$$" | "then $X^{\mathrm{al}}$ contains infinitely many rational curves." | "Theorem (Bogomolov-Zarhin)" | "The set $\{p:\rho(X_p^{\mathrm{al}})\neq22\}$ has positive density (density 1 after finite extension)." | "Corollary (Li-Liedtke)" | "$\rho(X^{\mathrm{al}})$ odd $\Rightarrow$ infinitely many integral rational curves on $X^{\mathrm{al}}$."
NEEDS APPROVAL: [s15-m08] s17-c03. Title candidates: "Rational curves"; "Infinitely many rational curves"; "What do rank jumps give us?". Recommendation: "Rational curves". The current title remains until the author chooses.