Improving upper bounds: two specializations

$$\operatorname{Pic}(X^{\mathrm{al}}) \hookrightarrow \operatorname{Pic}(X_p^{\mathrm{al}}) \quad \text{and} \quad \rho(X^{\mathrm{al}}) \leq \rho(X_p^{\mathrm{al}})$$

Kloosterman—van Luijk

If $p$ and $q$ are two primes of good reduction, and

$$\begin{gathered} \rho(X_p^{\mathrm{al}}) = \rho(X_q^{\mathrm{al}}) = 2r, \\ \operatorname{disc} \operatorname{Pic}(X_p^{\mathrm{al}}) \neq \operatorname{disc} \operatorname{Pic}(X_q^{\mathrm{al}}) \quad \text{in } \Bbb{Q}^{\times}/(\Bbb{Q}^{\times})^2. \end{gathered}$$

then

$$\rho(X^{\mathrm{al}}) < 2r.$$

van Luijk (2005): first explicit K3 surfaces $X/\Bbb{Q}$ with $\rho(X^{\mathrm{al}})=1$.

Does this always work?

Imagine taking until 2005 to write down a generic K3 surface over Q and prove it has geometric Picard rank one.

The same comparison as for elliptic curves, now using discriminants instead of endomorphism algebras.

Compare square classes: the ratio of the discriminants must be a nonsquare.

There are algorithms in principle: Charles; Poonen-Testa-van Luijk; Hassett-Kresch-Tschinkel; Shioda; Lairez-Sertoz.

$\#\operatorname{Br}$ is a square; the sign and $q$-power are known factors. An equal-rank specialization identifies $L=\operatorname{Pic}(X^{\mathrm{al}})$ with a finite-index sublattice of $M=\operatorname{Pic}(X_p^{\mathrm{al}})$. If $[M:L]=n$, then $\operatorname{disc}L=n^2\operatorname{disc}M$.

These were the first explicit examples over Q with geometric rank one; earlier existence arguments were ineffective. The paper was posted in 2005 and published in 2007.

Reference: van Luijk, "K3 surfaces with Picard number one and infinitely many rational points", arXiv:math/0506416, posted 21 June 2005; Algebra & Number Theory 1 (2007), 1-15.

Van Luijk supplies the two-prime discriminant comparison; Kloosterman computes the discriminant square classes using Artin-Tate.

APPROVED (2026-09-14): [s11-m01] Keep the primes p and q. Author, verbatim: "p and q is the natural story"

APPROVED (2026-09-14): [s11-m02] Keep the historical sentence; add the spoken remark below. Author, verbatim: "that is true. he wrote a paper about it! We should put on the speaker notes, imagine, taking this long to try to prove that there are generic K3 surfaces over Q"

REFUSED (2026-09-14): [s11-m03] Replace the closing failure warning with "Does this always work?" Slide 27 answers it; no added conclusion or reveal after the hypotheses. Author, verbatim: "We know that, but we should not foreshadow it right! We should isntead aks does this always work"

APPLIED (2026-09-14): [s11-m04] s11-c02/c03 replace the historical sentence with "van Luijk (in 2005) proved rho(X^{al}) = 1 for explicit K3 surfaces X/Q." The posting year fits the requested historical parenthesis; the journal year stays in the notes. This supersedes the earlier two-date citation proposal. Source: arXiv:math/0506416 submission history, 21 June 2005; published introduction and Theorem 3.1, Algebra & Number Theory 1 (2007), 1-15. Earlier existence results were ineffective.

NEEDS APPROVAL: [s11-m05] Recommendation: Keep the finite-index explanation spoken: for $L\subset M$ of index $n$, $\operatorname{disc}L=n^2\operatorname{disc}M$. Both review jurors preferred a visible lemma; this recommendation retains the source frame structure and asks the author about placement. Checked Kloosterman, arXiv:math/0502439, Proposition 4.2, p. 6, and the basis-change determinant identity.

APPLIED, NEEDS APPROVAL (2026-09-14): [s11-m06] Suggestion: Use the corrected explanation: "#Br is a square; the sign and q-power are known factors." Applied to plan rationale and notes; finite-index placement remains s11-m05. Source: Costa-Tschinkel, Conj. 2.1; redteam-astra.md, F06.

APPLIED, NEEDS APPROVAL (2026-09-14): [s11-m07] Suggestion: Retain the existing corrections "in Q^times/(Q^times)^2" and "rho(X^{al}) < 2r". Source frames V:L466-477 and O:L1167-1182 omitted the square class and wrote Pic < 2r; van Luijk 2007, Remark 3.2, pp. 8-9, supports the correction. No new body edit.
Corrected transcription: these already-present deviations are retained and remain marked for the author, rather than silently represented as the literal V/O frame.

NEEDS APPROVAL: [s11-m08] Years exception: use the posting year 2005; switch to the publication year 2007 if preferred. Exact candidate: "van Luijk (2005): first explicit K3 surfaces $X/\Bbb{Q}$ with $\rho(X^{\mathrm{al}})=1$." The added "first" makes the historical point visible; the surprise stays in the existing speaker note. Checked arXiv:math/0506416, submitted 21 June 2005, and Algebra & Number Theory 1 (2007), 1-15, introduction. This is a paper date; Elsenhans and Jahnel's later introduction dates the construction to 2004.