We can explain the $1/2$ observed in even rank

$d_X:=\Delta_{H^2}(X)\Delta_{\operatorname{Pic}}(X)$ modulo squares; $d_X\in\Z\setminus\{0\}$

Theorem (C-Elsenhans-Jahnel)

$$\begin{gathered}p\text{ good},\ p\nmid2d_X\Rightarrow\\\det(\operatorname{Frob}_p\mid T(1)\otimes\Bbb{Q}_\ell)=\left(\frac{d_X}{p}\right)=-1\Rightarrow\rho(X_p^{\mathrm{al}})\geq\rho(X^{\mathrm{al}})+2\end{gathered}$$

Corollary

If $\eta(X^{\mathrm{al}})=0$, then

$d_X$ nonsquare $\Rightarrow$ $L=\Bbb{Q}(\sqrt{d_X})$, $[L:\Bbb{Q}]=2$
$p$ good, inert in $L$ $\Rightarrow p\in\Pi_{\mathrm{jump}}(X)$, up to finitely many primes
$\displaystyle\liminf_{B\rightarrow\infty}\gamma(X,B)\geq1/2$
$E=\Bbb{Q}$ $\Rightarrow$ infinitely many integral rational curves on $X^{\mathrm{al}}$

Spoken: $d_X$ is a discriminant; it is defined on the next slide.

$\Delta_{\operatorname{Pic}}$ is the square class of the Picard representation determinant. Galois fixing every geometric class makes it one; an integral descent equality is unnecessary.

Inert primes in $L$ have density one half. The assumption $\eta=0$ makes the rank increase a jump. For $E=\Bbb{Q}$, Costa-Elsenhans-Jahnel supply infinitely many non-supersingular inert reductions, so the Li-Liedtke lifting theorem in "K3 surfaces" applies.

The rational-curves conclusion requires $E=\Bbb{Q}$. It is not certified for the numerical example. The negative sign alone proves nonsquareness.

The integer is the first factorization in C-Elsenhans-Jahnel, Example 2.37, attached there to Costa-Tschinkel, Example 3.3. When the Picard representation is trivial, $d_X=D_X$ modulo squares.

References: Costa-Tschinkel, "Variation of Neron-Severi ranks of reductions of K3 surfaces" (2014), arXiv:1405.2265, Example 3.3. The displayed factorization is Costa-Elsenhans-Jahnel, "On the distribution of the Picard ranks of the reductions of a K3 surface" (2020), arXiv:1610.07823v3, Example 2.37.

AUTHOR'S CALL: [s20-m03] Recommendation: Reveal the trivial Picard-representation premise with the second theorem.
Source: I15:L626-646.

NEEDS APPROVAL: [s21-m01] Recommendation: Keep the title "We can explain the 1/2".
Source: V:L672-702.

APPROVED (2026-09-14): [s21-m02] Invoke the general lifting theorem on slide 17 for rational curves, with no real or complex multiplication. The odd-rank corollary is not used here. The density bound remains unchanged. Costa-Elsenhans-Jahnel 2020, Corollary 2.16 and Theorem 3.1. 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"

NEEDS APPROVAL: [s21-m03] Use $L=\Bbb{Q}(\sqrt{d_X})$ and the inert-prime implication on the half-density slide (24); retain $\eta=0$ and the extra $E=\Bbb{Q}$ rational-curve hypothesis. The latter uses the general lifting theorem on slide 17, with Costa-Elsenhans-Jahnel, Theorem 3.1 and Lemma 3.3. It does not use the odd-rank corollary.

NEEDS APPROVAL: [s21-m04] Recommendation: Keep one factorization, visibly credited to Costa-Tschinkel; identify the 2014 paper and Example 3.3 in notes. E=Q for this example remains unverified.
GPT 6 astra: Remove the orphan integer from the candidate.
GPT 5.6 sol: Identify its source example before retaining the integer, or remove it.
Source: V:L697; CEJ 2020, Ex. 2.37.
SYNTHESIS: one identified example is selected from the three source rows in CEJ, Example 2.6.11 of arXiv:1610.07823 (published Example 2.37); V:L697-699. This does not certify E=Q for this surface; do not instantiate the rational-curve branch with an unverified endomorphism field.

NEEDS APPROVAL: [s21-m05] Review the existing half-density candidate with its updated lifting-theorem reference. Exact candidate text: "We can explain the $1/2$" | "Theorem (C-Elsenhans-Jahnel)" | "$$p\text{ good},\ p\nmid2d_X\Rightarrow\quad(\det(\operatorname{Frob}_p\mid T_\ell(1))=\left(\frac{d_X}{p}\right)=-1\Rightarrow\rho(X_p^{\mathrm{al}})\geq r+2)$$" | "Corollary" | "$d_X$ nonsquare $\Rightarrow$ $L=\Bbb{Q}(\sqrt{d_X})$, $[L:\Bbb{Q}]=2$" | "$p$ good, inert in $L$ $\Rightarrow p\in\Pi_{\mathrm{jump}}(X)$, up to finitely many primes" | "$\displaystyle\liminf_{B\rightarrow\infty}\gamma(X,B)\geq1/2$" | "$E=\Bbb{Q}$ $\Rightarrow$ infinitely many integral rational curves on $X^{\mathrm{al}}$" | "Example: Costa-Tschinkel" | "$$d_X=-1\cdot5\cdot151\cdot22490817357414371041\cdot387308497430\allowbreak 149337233666\allowbreak 358807996260\allowbreak 780875056740\allowbreak 850984213276\allowbreak 970343278935\allowbreak 342068889706\allowbreak 146733313789$$"