Computing Picard Lattices of K3 Surfaces

Lecture 1: Reduction methods

Edgar Costa (MIT)

ICERM: Arithmetic, Geometry and Computations on K3 surfaces
September 14, 2026

Supported by the Simons Foundation

Slides available at edgarcosta.org

The geometric Picard group

X/k a K3 surface; k⊂ℂ a number field.
k^al an algebraic closure of k, X^al:=X×_k k^al.

Pic (X^al) ≃ℤ^ρ, ρ:=ρ(X^al)

Pic (X^al) ≃ℤ⟨algebraic curves in X^al ⟩/ ⟨linear equivalences ⟩⊂H₂(X_ℂ, ℤ)

Curves on X^al modulo linear equivalence.
For K3 surfaces ⇒ linear/algebraic/numerical equivalence agree.

ρ, and more precisely the Picard lattice, is a coarse invariant.

The intersection pairing

Pic (X^al) with the intersection pairing (D,D') ↦D ·D'.

Even symmetric bilinear form: D ·D = 2g(D) − 2 for every curve D.

disc Pic (X^al):=det (D_i·D_j) in a ℤ-basis.

Signature (1,ρ(X^al)−1): Hodge index theorem.

Gal (k^al/k) permutes curves, respects linear equivalence and preserves the pairing.

Pic inside H²

Over ℚ^al, viewing X also as a complex manifold,

Pic (X^al)≃H^1,1(X_ℂ) ∩H²(X_ℂ, ℤ)⊂H²(X_ℂ, ℤ) ≃(−E₈)² ⊕U³ ≃ℤ²²

dim H^1,1(X_ℂ)=20; 1≤ρ(X^al)≤20.
The degree of "difficulty" is negatively correlated with ρ(X^al); ρ(X^al)=1 is generic.

T(X) := Pic (X^al)^⊥ ⊂H²(X_ℂ,ℤ): the transcendental lattice.
T(X)_ℚ: minimal rational sub-Hodge structure containing H^2,0(X_ℂ) after complexification.
H²(X_ℂ,ℚ) ≃Pic (X^al)_ℚ ⊕T(X)_ℚ
The "new and interesting" Galois representations arise from T(X).

In characteristic p the bound 20 fails: ρ(X_p^al) can be as large as 22.

Computing Pic as a Galois module

Goal

From the equations of X, compute Pic (X^al) ⊂H₂(X_ℂ,ℤ) as a Gal (k^al/k)-module.

"The evaluation of ρ for a given surface presents in general grave difficulties." (Zariski)

Corollary

The Picard Galois module gives the algebraic Brauer group for studying rational points.

H¹(Gal (k^al/k), Pic X^al)≃Br ₁(X)/Br ₀(X)X(k)⊂X(𝔸_k)^Br ⊂X(𝔸_k)

Picard lattice, over finite fields

P₂(t) = det (t − Frob _q ∣H²_et(X_p^al,ℚ_ℓ)); q=pⁿ, ℓ≠p.
q⁻²²P₂(qt) monic; roots ζ_i, |ζ_i| = 1.

q⁻²²P₂(qt) = h(t)∏_iΦ_{k_i}(t)^γ_iΦ_k the k-th cyclotomic polynomial; h has no cyclotomic factor

Example: X := Z(y⁴ − x³z + yz³ + zw³ + w⁴) ⊂ℙ³, p = 89.

p⁻²²P₂(pt) = (t−1)¹⁺⁴(t+1)(t⁴+1)h(t), deg h = 12

H²:=H²_et(X₈₉^al,ℚ_ℓ(1))=P_Φ₁⊕P_Φ₂⊕P_Φ₈⊕P_h

dim (H²)^{Frob ₈₉⁸=1}=(1+4)+1+4=10

Away from small p, naive point counting is impractical, more on Kedlaya's talk

Algebraic classes are Frobenius invariant, so the example bounds the invariant classes above by ten; Tate then supplies the matching lower bound.

The twist (1) divides Frobenius eigenvalues by 89; these are the normalized roots introduced above.

P_{Φ_k}=ker Φ_k(Frob ₈₉) and P_h=ker h(Frob ₈₉) on this H². Their dimensions are 5,1,4,12.

Since 1,2,8 divide 8, the first three summands give 10 classes fixed by Frob ₈₉⁸. The remaining 12 contribute none.

"What the characteristic polynomial gives you" explains how these invariant classes determine Picard ranks.

Elsenhans and Jahnel adapt Harvey's p-adic method to double covers, including degree-two K3 surfaces: "Point counting on K3 surfaces and an application concerning real and complex multiplication", LMS Journal of Computation and Mathematics 19 (Special issue A) (2016), 12-28, Section 1.2 and Algorithm 4.6.

In practice we compute a p-adic approximation to the Frobenius matrix. Kedlaya's lecture will tell us more.

We return to this quartic in Lecture 2, where the obstruction cuts the upper bound from 10 to 4.

Historical verdicts below retain the numbering before s05-c01. Current slide 5 is the example; current slide 6 is the theorem and even-rank statement. The new reversal mark records the current order; existing IDs and recorded verdicts are preserved.

APPROVED (2026-09-14): [s06-m01] slides 5 and 6 are swapped: slide 5 is the theorem and slide 6 is the finite-field example. They stay separate; the merge is refused. Author: "We might need to swap 5 and 6, first the theorical result, then the example. but we want separate slides."

APPROVED (2026-09-14): [s06-m02] Choose the polynomial convention for all three lectures and the Artin-Tate recall. Options: P_2 characteristic with roots q and q*zeta; P_2 reciprocal with roots 1/q and zeta/q; or separate names. Recommendation: use P_2 for det(t-Frob) and chi(t)=t^22 P_2(1/t) for the K3 reciprocal polynomial. Applied: P_2(t)=det(t-Frob); chi(t)=t^22 P_2(1/t) for the K3 reciprocal polynomial.<br />

APPROVED (2026-09-14): [s06-m03] Keep Pic(X_89^{al})_Q. Contact-sheet verdict: APPROVED; no note supplied.

APPROVED (2026-09-14): [s06-m04] Keep the even-rank fact on slide 5. Contact-sheet verdict: APPROVED; no note supplied.

APPROVED (2026-09-14): [s06-m05] Keep the example and its required definitions. Contact-sheet verdict: APPROVED; no note supplied.

APPROVED (2026-09-14): [s06-m06] Choose the title now that the example and theorem stay separate. Options: keep "Picard lattice, over finite fields"; or use "Example: Picard ranks at p=89". Recommendation: use "Example: Picard ranks at p=89". Author: "We should have a note that Kedlaya's Lecture will tell us more" Applied: the existing title is preserved; the Kedlaya pointer is added below the machinery citation.<br />

APPROVED (2026-09-14): [s06-m07] Keep slide 6 as the worked example. Contact-sheet verdict: APPROVED; no note supplied.

APPROVED (2026-09-14): [s06-m08] the proposed merge of slides 5 and 6 is refused; they stay two separate slides. Author: "We might need to swap 5 and 6, first the theorical result, then the example. but we want separate slides."

NEEDS APPROVAL (2026-09-14): [s06-m09] Suggestion: Keep the spoken pointer "Kedlaya's lecture will tell us more." Exact title and date remain unverified (reference-years.md, entry 20).

PROPOSED (2026-09-14): [s06-m10] Suggestion: Use "[Abbott-Kedlaya-Roe 2010; C 2015; C-Harvey-Kedlaya 2019; Pancratz-Tuitman 2015]". Author order settled by arXiv:1307.1250, title page; reference-years.md, entries 16-19. Approved slide body awaits this citation edit.
Inherited source error: V:L437 prints Tuitman--Pancratz. The title page of arXiv:1307.1250 lists Sebastian Pancratz before Jan Tuitman. The visible source frame is retained pending approval of this single citation candidate.

NEEDS APPROVAL (2026-09-14): [s06-m11] Recommendation: Keep the spoken definition $P_{\zeta_k}=\ker\Phi_k(F)$, with $F$ the Tate-twisted action on rational Picard classes (cohomological Frobenius divided by 89). Dimensions 5,1,4; checked O:L1384-1396 and the displayed factors. Definition applied in notes; its placement remains open.

APPLIED, NEEDS APPROVAL (2026-09-14): [s06-m12] characteristic polynomial variable lowercased to t per the author, "let's use lower case t or x for our characteristic polynomials. in particular, avoiding u in slide 14".

APPLIED, CONFIRM REVERSAL (2026-09-14): [s06-m13] Applied s05-c01: slide 5 is now the p = 89 example; slide 6 is the theorem. This reverses the earlier order, not the decision to keep two slides. Earlier instruction (2026-09-14): "We might need to swap 5 and 6, first the theorical result, then the example." Full earlier instruction: "We might need to swap 5 and 6, first the theorical result, then the example. but we want separate slides." Latest instruction (2026-09-14): "This slide should be swapped with the next one, as we need to introduce P_2(t)". Recommendation: confirm the example-first order. Existing mark IDs follow their original subjects and are not renumbered.

APPLIED, SOURCE-SETTLED (2026-09-14): [s06-m14] Applied s06-c01 and s06-c02 on current slide 5: remove the Tate-class identification and Picard-rank count before the theorem; replace the columns by the four cohomology summands. Here $H^2=H^2_{\mathrm{et}}(X_{89}^{\mathrm{al}},\Bbb{Q}_\ell(1))$, and $P_{\Phi_k}=\ker\Phi_k(\operatorname{Frob}_{89})$, $P_h=\ker h(\operatorname{Frob}_{89})$. The Tate twist divides the untwisted eigenvalues by 89. O:L1384-1394 gives multiplicities 1,1,4,4 and the complete cyclotomic dimension 10; V:L404 gives total degree 22. Combining the two Phi_1 factors gives dimensions 5,1,4,12. Since 1,2,8 divide 8 and h has no cyclotomic factor, exactly 10 classes are fixed by Frobenius^8 on this twisted H^2. O:L622-624 and I15:L214-219 justify that these multiplicities give invariant dimensions. Ito-Ito-Koshikawa, arXiv:1809.09604v2, Section 10.3 (pp. 59-60), identifies the twisted cohomology decomposition and excludes roots of unity on the complement. This supersedes the Picard-only notation in s06-m03 and s06-m11.

AUTHOR'S CALL: [s06-m90] Keep the displayed invariant-space dimension, the following count of invariant classes, or both? Both formulations are preserved pending your choice.

AUTHOR'S CALL: [s06-m91] Slide order: theorem or example first? Decision s05-m01 records your instruction "first the theorical result, then the example. but we want separate slides", which puts the theorem first. The deck currently shows the example "Picard lattice, over finite fields" first and the theorem "What the characteristic polynomial gives you" second, with P_2 defined on the example and only recalled on the theorem. A later reversal is recorded in artifacts/plan/lecture1.md and in the tracker, but your confirmation of it was never established. Which do you want: theorem first, or example first? Nothing is waiting on this.

AUTHOR'S CALL: [s06-m92] Keep "Picard lattice, over finite fields", or choose "Frobenius at p=89" or "Cyclotomic factors and invariant classes"?

What the characteristic polynomial gives you

Tate conjecture (proved)

X_p/𝔽_q an abelian surface or a K3 surface; q=pⁿ, ℓ≠p.
For abelian surfaces: ρ:=rk (Pic /Pic ⁰).

ρ(X_p) = ord _{t = q} P₂(t)
ρ(X_p^al) = ∑_ζ ord _{t = qζ} P₂(t), where ζ runs over all roots of unity.
For K3 surfaces ⇒ ρ(X_p^al) is even.

Artin-Tate for K3 surfaces over 𝔽_q

lim _t→q(P₂(t))/((t−q)^ρ) =(−1)^ρ−1q^21−ρ#Br (X_p) disc (Pic (X_p))

#Br (X_p)∈ℚ^×2 and P₂(t) give disc (Pic (X_p)) mod ℚ^×2

APPROVED (2026-09-14): [s05-m01] slides 5 and 6 are swapped: slide 5 is the theorem and slide 6 is the finite-field example. They stay separate; the merge is refused. Author: "We might need to swap 5 and 6, first the theorical result, then the example. but we want separate slides." Author: "we should present the full formula of Artin-Tate and then the conclusion"

APPROVED (2026-09-14): [s05-m02] Choose the polynomial convention for all three lectures and the Artin-Tate recall. Options: P_2 characteristic with roots q and q*zeta; P_2 reciprocal with roots 1/q and zeta/q; or separate names. Recommendation: use P_2 for det(t-Frob) and chi(t)=t^22 P_2(1/t) for the K3 reciprocal polynomial. Applied: P_2(t)=det(t-Frob); chi(t)=t^22 P_2(1/t) for the K3 reciprocal polynomial.<br />

APPROVED (2026-09-14): [s05-m03] Choose which definitions accompany the example and which content slide 5 may refer forward to. Options: restore zeta, counting-range and Tate-kernel displays; keep the example alone; or add only its required definitions. Recommendation: keep the example and required definitions, then rewrite the reference in slide 5 to match. Author: "Here is also where, we should note that the rank tehre must be even." Applied: the example keeps its required definitions; slide 5 states the even-rank fact.<br />

APPROVED (2026-09-14): [s05-m04] Remove the visible Costa-Tschinkel label; keep the formula unchanged and its provenance in the speaker notes. Author, verbatim: "we should remove [Costa-Tschinkel, Conj. 2.1], that is Artin--Tate formula for K3 surfaces"

APPROVED (2026-09-14): [s05-m05] Theorem (many people). Credit history and Kuga-Satake in the speaker notes; supersedes the visible credit correction in e384285. Author, verbatim: "We should not specify whod id what on the slide, the point is that it was many people, and I should say in the speaker notes that Kuga--Satake plays a crucial role"

REFUSED (2026-09-14): [s05-m06] Refuse the citation strip on behalf of the author: s05-m04 and s05-m05 remove slide 5 attributions. Tate 1966, Milne 1975 and Liu-Lorenzini-Raynaud 2005, corr. 2018 stay in the speaker notes. The perfect-square Brauer order stays visible. Author on s05-m04, verbatim: "we should remove [Costa-Tschinkel, Conj. 2.1], that is Artin--Tate formula for K3 surfaces" Author on s05-m05, verbatim: "We should not specify whod id what on the slide, the point is that it was many people, and I should say in the speaker notes that Kuga--Satake plays a crucial role"

APPLIED, SOURCE-SETTLED (2026-09-14): [s05-m07] Applied the missing X_p, P_2 and abelian Picard-rank definitions to the approved new theorem expansion. Checked I15:L203-224 and Milne, Abelian Varieties, Section 17. The full Artin-Tate formula, credits policy and arithmetic exponent are unchanged.

APPLIED, NEEDS APPROVAL (2026-09-14): [s05-m08] characteristic polynomial variable lowercased to t per the author, "let's use lower case t or x for our characteristic polynomials. in particular, avoiding u in slide 14".

APPLIED, CONFIRM REVERSAL (2026-09-14): [s05-m09] Applied s05-c01: slide 5 is now the p = 89 example; slide 6 is the theorem. This reverses the earlier order, not the decision to keep two slides. Earlier instruction (2026-09-14): "We might need to swap 5 and 6, first the theorical result, then the example." Full earlier instruction: "We might need to swap 5 and 6, first the theorical result, then the example. but we want separate slides." Latest instruction (2026-09-14): "This slide should be swapped with the next one, as we need to introduce P_2(t)". Recommendation: confirm the example-first order. Existing mark IDs follow their original subjects and are not renumbered.

PROPOSED LABEL (2026-09-14): [s05-m10] Applied s05-c05 on current slide 6. Author: "Theorem (many people) -> Tate Conjecture (now proved in long series of papers), can we perhaps do better here?" Recommendation: use "Tate conjecture (proved)". The draft displays this candidate for confirmation. It names the result and its status without repeating the proof history. Source: I15:L214-219; Ito-Ito-Koshikawa, arXiv:1809.09604v2, Section 1.2 and Section 6.4 (Remark 6.9), for all characteristics. This supersedes the label in s05-m05; its spoken Kuga-Satake reminder remains.

APPLIED, SOURCE-SETTLED (2026-09-14): [s05-m11] Applied s05-c02, s05-c03 and s05-c04 on current slide 6. Artin-Tate returns the arithmetic discriminant $\operatorname{disc}(\operatorname{Pic}(X_p))$, with $X_p/\Bbb{F}_q$, not the geometric discriminant. Costa-Tschinkel, arXiv:1405.2265v1, Section 2, Conjecture 2.1 and equation (8), explicitly puts $X_{\mathbb{F}_q}$ on the left of (8); the setup says "Let X be a smooth projective surface over" $\mathbb{F}_q$. The source's lattice notation is rendered as Pic for this K3 lecture. The consequence includes the square Brauer order; the redundant legend is removed. The full formula keeps $q^{21-\rho}$ after converting the reciprocal polynomial. Milne's 1975a article page, "The condition p != 2", removes the characteristic restriction and explains the square-order input. Slide 12 must first pass to an extension for the geometric discriminant. Bibliographic locators belong in this review metadata and the plan, not the spoken notes.

Why the reduction rank is even

X_p/𝔽_q K3; P₂(t)=det (t−Frob ∣H²_et(X_p^al,ℚ_ℓ))

q⁻²²P₂(qt)=h(t)∏_iΦ_{k_i}(t)^γ_i

h∈ℚ[t]: no cyclotomic factor
|z|=1 ⇒ conj (z)=z⁻¹
Real roots: +1,−1, already in Φ₁,Φ₂
Roots of h: nonreal pairs; deg h even

Weil + Tate

ρ(X_p^al)=∑_iγ_ideg Φ_{k_i}=22−deg h∈2ℤ

Reduction to finite characteristic

Take f ∈ℤ[x,y,z,w] and X := Z(f) ⊂ℙ³_ℚ.

We may consider the surface X_p := Z(f mod p) ⊂ℙ³_{𝔽_p}.

Theorem

If X and X_p are smooth then the specialization map is injective

Pic (X^al) ↪Pic (X_p^al) and ρ(X^al) ≤ρ(X_p^al).

Goal

For a given f and p, improve the inequality ρ(X^al) ≤ρ(X_p^al).

Parity reasons might already force the inequality to not be sharp.

Endomorphisms of the transcendental lattice can complicate things even further.

Pic plays the role of End (A)

Pic for a K3 surface plays a similar role as End (A) for an abelian variety A.

Pic (A)/Pic ⁰(A) = NS (A)

ρ(A):=rk (Pic (A)/Pic ⁰(A)); also after base change.

(Pic (A)/Pic ⁰(A))_ℚ ≃{φ∈End (A)_ℚ : φ^† = φ}, † the Rosati involution

"Compute the Picard lattice of a K3 surface" ↭ "compute End (A)".
We also stratify moduli of abelian varieties via End (A).

A an abelian surface, char k≠2 ⇒ ρ(Kum (A)^al) = ρ(A^al)+16.
Kum (A) is the K3 surface obtained by resolving A/{±1}.

Proving that an elliptic curve does not have CM

End _ℚ E^al = ℚ or ℚ(√(−d)) (CM)

End _ℚ E^al ↪End _ℚ E_p^al ↩ℚ(Frob _p).
a_p := Tr (Frob )
p∤a_p ⟺End _ℚ E_p^al is a quadratic field

If E has CM by ℚ(√(−d)), then
a_p ≡0 mod p⟺p inert or ramified in ℚ(√(−d))⟺End _ℚ E^al ≄ End _ℚ E_p^al

If E is non-CM, then End _ℚ E_p^al ∩End _ℚ E_q^al ≃ℚ with prob. 1;
and we expect Prob (a_p ≡0 mod p) ∼1/√(p)

Two universal examples: 11.a2 and 27.a3

E: y² + y = x³ − x² − 10x − 20 (LMFDB label: 11.a2)

End _ℚ E₃^al ≃ℚ(√(−11))
End _ℚ E₁₃^al ≃ℚ(√(−1))
⇒End _ℚ E^al = ℚ

E: y² + y = x³ − 7 (LMFDB label: 27.a3)

p = 2 mod 3 ⇒a_p = 0 ⇒End _ℚ E_p^al is a quaternion algebra
p = 1 mod 3 ⇒End _ℚ E_p^al ≃ℚ(√(−3))
⇒End _ℚ E^al = ℚ(√(−3))

Improving upper bounds: two specializations

Pic (X^al) ↪Pic (X_p^al) and ρ(X^al) ≤ρ(X_p^al)

Kloosterman—van Luijk

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

ρ(X_p^al) = ρ(X_q^al) = 2r,disc Pic (X_p^al) ≠disc Pic (X_q^al) in ℚ^×/(ℚ^×)².

then

ρ(X^al) < 2r.

van Luijk (2005): first explicit K3 surfaces X/ℚ with ρ(X^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.

#Br is a square; the sign and q-power are known factors. An equal-rank specialization identifies L=Pic (X^al) with a finite-index sublattice of M=Pic (X_p^al). If [M:L]=n, then disc L=n²disc 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.

Let's apply it to a K3 surface with a ℤ/5 automorphism

X : x³ z + 3x² y² + 5xw³ + y³ w + 3yz³ − 5z² w² = 0 ⊂ ℙ³

p	ρ(X_p^al)	disc Pic (X_p^al) mod ℚ^×2
11	18	−55
13	18	−85

Theorem (Artin-Tate)

P₂(t) ⇝disc Pic (X_p) mod (ℚ^×)².

55/85 = 11/17 ∉(ℚ^×)², hence ρ(X^al) ≤17.
Symplectic order-5 action ⇒ ρ(X^al) ≥17 [Garbagnati-Sarti].

The prime-field ranks are 1 at 11 and 5 at 13. Extend to 𝔽_11³⁰ and 𝔽_13⁴, respectively ⇒ ρ(X_{𝔽_p^r})=ρ(X_p^al)=18 for (p,r)=(11,30),(13,4). Apply Artin-Tate to Frobenius to the powers 30 and 4, when all divisor classes are defined, to obtain the geometric discriminant square classes.

The ratio is 11/17, a nonsquare. The upper bound 17 and the symplectic lower bound 17 give ρ(X^al)=17.

Garbagnati-Sarti, "Symplectic automorphisms of prime order on K3 surfaces", Journal of Algebra 318 (2007), 323-350, arXiv:math/0603742, Proposition 1.1.

Take σ(x:y:z:w)=(x:ζy:ζ²z:ζ⁴w), with ζ of order 5. Every monomial has weight 2 mod 5, and det (σ)=ζ². The residue 2-form is fixed because det (σ)/ζ²=1.

There are 16 coinvariant classes and an invariant polarization, giving ρ≥17.

This is the Artin-Tate formula from Tate 1966 and Milne 1975. Apply it after extending the field so every divisor class is defined.

CROSS-LECTURE: [s12-m01] Suggestion: Use P_2(t)=det(t-Frob); chi(t)=t^22 P_2(1/t) for K3 surfaces in all three lectures. Settled by s05-m02 and s06-m02; retain this CROSS-LECTURE tag as the follow-up. No Lecture 2 or 3 edits here.

APPROVED (2026-09-14): [s12-m02] Verified: the order-5 action is symplectic; rho(X^{al}) >= 17 [Garbagnati-Sarti 2007, Prop. 1.1]. Author, verbatim: "is it symplectic? if so we should reference the paper"

NEEDS APPROVAL: [s12-m03] Suggestion: Heading: "Theorem (Artin-Tate)". The formula was already established on slide 6; no repeated "a theorem here". Source: I15:L203-224.
Restored the source attribution as the visible review baseline; heading approval remains open.

APPROVED (2026-09-14): [s12-m04] Reveal theorem, extension and conclusion at 0/1/2, in the present order. Contact-sheet verdict: APPROVED; no note supplied.

APPROVED (2026-09-14): [s12-m05] Let's apply it to a K3 surface with a Z/5 automorphism. Author, verbatim: "The title should be, let's apply it to a K3 surface with a Z/5 automorphism"

APPROVED (2026-09-14): [s12-m06] Keep disc Pic in the Artin-Tate recall. Author, verbatim: "Let's use Pic, not NS"

APPLIED (2026-09-14): [s12-m07] s12-c01 supersedes the dated visible citation proposal. Keep "Theorem (Artin-Tate)" and [Garbagnati-Sarti] visible. Full references remain in the notes and provenance. Checked I15:L203-224 and Garbagnati-Sarti, arXiv:math/0603742, Proposition 1.1, p. 3.

APPLIED, SOURCE-SETTLED (2026-09-14): [s12-m08] Applied square-class heading and legend; retained -55 and -85. Checked NSranks/nsranks k3.ipynb:496-498 and independently powered the stored Frobenius polynomials in order5_3.data. These values are not asserted to be Gram determinants.

APPLIED, NEEDS APPROVAL (2026-09-14): [s12-m09] characteristic polynomial variable lowercased to t per the author, "let's use lower case t or x for our characteristic polynomials. in particular, avoiding u in slide 14".

APPLIED, SOURCE-SETTLED (2026-09-14): [s12-m10] Applied the extension-field subscript in the new Artin-Tate explanation. Prime-field ranks remain 1 and 5; geometric ranks are 18 after degrees 30 and 4. Checked and independently factored NSranks/data/17/order5_3.data, rows 11 and 13.

Torsion-free cokernel

X/ℚ K3; p>2 a prime of good reduction.

Theorem (Elsenhans—Jahnel)

The specialization map

Pic (X^al) ↪Pic (X_p^al)

has torsion-free cokernel for p ≠2.

Thus, if ρ(X_p^al) = ρ(X^al) every invertible sheaf lifts.

For example, if ρ(X_p^al) = 2, Elsenhans—Jahnel approach is

compute Pic (X_p^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 Pic (X_p^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 ℚ and p>2 is a prime of good reduction. The published theorem allows ramification degree e<p−1; over 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.

Endomorphisms of the transcendental Hodge structure

E:=End _Hdg(T)={a∈End _ℚ(T):a_ℂ(T^i,j)⊂T^i,j}

T minimal rational sub-Hodge structure of H² with H^2,0⊂T_ℂ

0≠α∈E⇒α(H^2,0)=H^2,0⇒im α=T⇒α⁻¹∈E

Theorem (Zarhin)

E: a totally real field or a totally imaginary quadratic extension of one, i.e., a CM field

d:=[E:ℚ], m:=dim _E T; dm=22−ρ(X^al)

E totally real; V:=T(1)⊗ℚ_ℓ; g=Frob _p^a in connected monodromy

V⊗ℚ_ℓ^al=⨁_{σ:E↪ℚ_ℓ^al}V_σ, dim V_σ=m, g|V_σ∈SO(V_σ)

m odd⇒dim ker (g−1)≥d⇒ρ(X_p^al)≥ρ(X^al)+d

We are still working with a projective K3 surface X/k, with k⊂ℂ a number field. For reduction, p is a finite place of good reduction and ℓ differs from its residue characteristic.

T is the smallest rational sub-Hodge structure of H²(X_ℂ,ℚ) whose complexification contains H^2,0(X_ℂ). That line has complex dimension one. Endomorphisms preserve the Hodge decomposition; their kernels and images are rational sub-Hodge structures.

If α kills H^2,0, minimality gives ker α=T, hence α=0. Otherwise its image contains that line, so minimality gives im α=T. Finite dimension gives ker α=0; the inverse also preserves the Hodge decomposition.

Restriction to H^2,0 embeds E into End _ℂ(H^2,0)=ℂ. Thus the division algebra is commutative, and finite dimensionality over ℚ makes it a number field. E=ℚ means no real or complex multiplication.

Zarhin proves simplicity and fieldhood in "Hodge groups of K3 surfaces", J. reine angew. Math. 341 (1983), Theorem 1.6(a) and proof 1.6.1, p. 207; the classification is Theorem 1.5.1, p. 206. Van Geemen explains the Hodge structures in "Real multiplication on K3 surfaces and Kuga Satake varieties", Michigan Math. J. 56 (2008), 375-399, Sections 1.3, 1.5, 1.7-1.8 and 2.1; Lemma 3.2 gives m≥3 in the totally real case.

The twist divides Frobenius eigenvalues by the residue-field size. Choose a positive power lying in connected monodromy; each totally real embedding then gives an SO_m block. Odd m forces a fixed vector in each block. Before taking the power these give roots of unity, hence new divisor classes by Tate.

These new cyclotomic roots are removed from h. Commutation with E does not force every eigenvalue orbit to have size d.

Zarhin classifies the Hodge endomorphism field. Charles computes the minimum increase of the geometric Picard rank under specialization in "On the Picard number of K3 surfaces over number fields", Algebra & Number Theory 8 (2014), 1-17, Theorem 1, p. 3; Proposition 15 and Lemma 16, pp. 8-9, give the fixed-space argument. The Tate theorem holds in every residue characteristic; see Ito-Ito-Koshikawa, arXiv:1809.09604.

NEEDS APPROVAL: [s14-m04] Recommendation: Keep the qualified SO-block statement and the odd-m condition on slide 15, before Charles.
GPT 6 astra: Use embedding blocks after a suitable Frobenius power; reject fixed root-orbit sizes.
GPT 5.6 sol: Use the same qualified block decomposition; reject fixed root-orbit sizes.
Source: Zarhin 1983, Thms. 1.5.1, 1.6(a), 2.2.1; van Geemen 2008, Lem. 3.2; Charles 2014, Prop. 15 and Lem. 16.

NEEDS APPROVAL: [s18-m90] Current host: Endomorphisms of the transcendental Hodge structure (slide 15). The existing SO setup defines $V:=T(1)\otimes\Bbb{Q}_\ell$ using $T$ from Pic inside H^2. No decision or comment status is closed here.

NEEDS APPROVAL: [s14-m06] Recommendation: Keep the quoted candidate wording, formulas and arrangement on this slide. Author comments s15-c01--s15-c04 are incorporated; other arrangement decisions retain this stable ID.
Exact candidate text: "Endomorphisms of the transcendental Hodge structure" | "$$T:=T(X)_{\Bbb{Q}}=c_1(\operatorname{Pic}(X^{\mathrm{al}}))_{\Bbb{Q}}^{\perp}\subset H^2(X_{\Bbb{C}},\Bbb{Q})$$" | "$$E:=\operatorname{End}_{\mathrm{Hdg}}(T)=\{a\in\operatorname{End}_{\Bbb{Q}}(T):a_{\Bbb{C}}(T^{i,j})\subset T^{i,j}\}$$" | "$T$ minimal rational sub-Hodge structure of $H^2$ with $H^{2,0}\subset T_{\Bbb{C}}$ $\Rightarrow$ $(0\neq\alpha\in E\Rightarrow\alpha(H^{2,0})=H^{2,0}\Rightarrow\operatorname{im}\alpha=T\Rightarrow\alpha^{-1}\in E)$" | "Theorem (Zarhin)" | "$E$: a totally real field or a totally imaginary quadratic extension of one, i.e., a CM field" | "$d:=[E:\Bbb{Q}]$, $m:=\dim_E T$; $dm=22-\rho(X^{\mathrm{al}})$" | "$E$ totally real $\Rightarrow m\geq3$ [van Geemen]" | "$E$ totally real; $V:=T(1)\otimes\Bbb{Q}_\ell$; $g=\operatorname{Frob}_p^a$ in connected monodromy" | "$$V\otimes\Bbb{Q}_\ell^{\mathrm{al}}=\bigoplus_{\sigma:E\hookrightarrow\Bbb{Q}_\ell^{\mathrm{al}}}V_\sigma,\quad\dim V_\sigma=m,\quad g|V_\sigma\in SO(V_\sigma)$$" | "$$m\text{ odd}\Rightarrow\dim\ker(g-1)\geq d\Rightarrow\rho(X_p^{\mathrm{al}})\geq\rho(X^{\mathrm{al}})+d$$"
Speaker notes: "We are still working with a projective K3 surface $X/k$, with $k\subset\Bbb{C}$ a number field. For reduction, $p$ is a finite place of good reduction and $\ell$ differs from its residue characteristic." | "$T$ is the smallest rational sub-Hodge structure of $H^2(X_{\Bbb{C}},\Bbb{Q})$ whose complexification contains $H^{2,0}(X_{\Bbb{C}})$. That line has complex dimension one. Endomorphisms preserve the Hodge decomposition; their kernels and images are rational sub-Hodge structures." | "If $\alpha$ kills $H^{2,0}$, minimality gives $\ker\alpha=T$, hence $\alpha=0$. Otherwise its image contains that line, so minimality gives $\operatorname{im}\alpha=T$. Finite dimension gives $\ker\alpha=0$; the inverse also preserves the Hodge decomposition." | "Restriction to $H^{2,0}$ embeds $E$ into $\operatorname{End}_{\Bbb{C}}(H^{2,0})=\Bbb{C}$. Thus the division algebra is commutative, and finite dimensionality over $\Bbb{Q}$ makes it a number field. $E=\Bbb{Q}$ means no real or complex multiplication." | "Zarhin proves simplicity and fieldhood in "Hodge groups of K3 surfaces", J. reine angew. Math. 341 (1983), Theorem 1.6(a) and proof 1.6.1, p. 207; the classification is Theorem 1.5.1, p. 206. Van Geemen explains the Hodge structures in "Real multiplication on K3 surfaces and Kuga Satake varieties", Michigan Math. J. 56 (2008), 375-399, Sections 1.3, 1.5, 1.7-1.8 and 2.1; Lemma 3.2 gives $m\geq3$ in the totally real case." | "The twist divides Frobenius eigenvalues by the residue-field size. Choose a positive power lying in connected monodromy; each totally real embedding then gives an $SO_m$ block. Odd $m$ forces a fixed vector in each block. Before taking the power these give roots of unity, hence new divisor classes by Tate." | "These new cyclotomic roots are removed from $h$. Commutation with $E$ does not force every eigenvalue orbit to have size $d$." | "Zarhin classifies the Hodge endomorphism field. Charles computes the minimum increase of the geometric Picard rank under specialization in "On the Picard number of K3 surfaces over number fields", Algebra & Number Theory 8 (2014), 1-17, Theorem 1, p. 3; Proposition 15 and Lemma 16, pp. 8-9, give the fixed-space argument. The Tate theorem holds in every residue characteristic; see Ito-Ito-Koshikawa, arXiv:1809.09604."
Checked sources: I15:L523-527; Zarhin, Hodge groups of K3 surfaces (1983), Thms. 1.4.1 (p. 205), 1.5.1 (p. 206), 1.6(a) and proof 1.6.1 (p. 207); van Geemen, Real multiplication on K3 surfaces and Kuga Satake varieties, Secs. 1.3, 1.5, 1.7-1.8, 2.1, 2.4, Thm. 2.8 and Lem. 3.2; Charles, On the Picard number of K3 surfaces over number fields (2014), Thm. 1 (p. 3), Prop. 15 (pp. 8-9) and Lem. 16 (p. 9).
Current allocation: the complete original candidate is reunited on slide 15 before Charles on slide 16. Definitions, invertibility, Zarhin, dimensions, the Frobenius/SO block and all original notes are retained. The direct V definition is covered by s18-m90. Historical quotations retain their original wording and IDs.

APPLIED (2026-09-14): [s14-m09] The existing invertibility line is CORRECT with its stated minimality, but the first implication uses the kernel argument. Clarified the ambient H^2. Exact line: $T$ minimal rational sub-Hodge structure of $H^2$ with $H^{2,0}\subset T_{\Bbb{C}}$ $\Rightarrow$ $(0\neq\alpha\in E\Rightarrow\alpha(H^{2,0})=H^{2,0}\Rightarrow\operatorname{im}\alpha=T\Rightarrow\alpha^{-1}\in E)$. If alpha kills H^{2,0}, its kernel is a rational sub-Hodge structure containing that line after complexification; minimality gives alpha=0. Otherwise the image contains H^{2,0}, hence equals T; finite dimension gives ker alpha=0 and the inverse is Hodge. Restriction to the one-dimensional H^{2,0} embeds E into C, proving commutativity. Source: Zarhin, Thm. 1.6(a), proof 1.6.1, p. 207; van Geemen, Secs. 1.3, 1.5, 1.7-1.8. Full check: artifacts/orch/comments-E.md, s15-c01. The stable mark ID is retained.

SETTLED (2026-09-14): [s14-m11] Zarhin owns the classification of E as totally real or CM; Charles owns the specialization cases and eta. Source: Zarhin, Thms. 1.5.1 and 1.6(a); Charles, Thm. 1 and Prop. 15. This supersedes the earlier attribution question; no new verdict is requested.

Computing ρ(X^al)

T=T(X)_ℚ; E=End _Hdg(T); d=[E:ℚ]; m=dim _E T

Theorem (Charles)

ρ(X_p^al)≥{ρ(X^al)if E is CM or m is even,ρ(X^al)+dif E is totally real and m is odd,

Equality occurs infinitely often (density 1 after some finite extension).

If E is totally real and m is odd, infinitely many good ordinary prime pairs (p,q) satisfy ρ(X_p^al)=ρ(X_q^al)=ρ(X^al)+d and

disc Pic (X_p^al)≢disc Pic (X_q^al) mod (ℚ^×)²

Corollary (Charles)

The Kloosterman—van Luijk method works, if it is aware of E.

NEEDS APPROVAL: [s22-m05] Recommendation: Recall T=T(X)_Q from slide 15; use algebraic divisor classes in its definition. SETTLED: the complement is algebraic Pic, not topological line bundles.
Source: Charles 2014, introduction.
Retained correction: T is the orthogonal complement of algebraic Pic in rational H^2, not topological line bundles. Source: Charles 2014, introduction. The repaired body was already present before this pass.

NEEDS APPROVAL: [s22-m06] Recommendation: Keep geometric discriminants and the conjunction "totally real and m odd". SETTLED by the theorem: both geometric base changes and AND are necessary corrections to I15:L533-541.
Source: Charles 2014, Thm. 1 and Prop. 18.
Content anchors: the conjunction is in "Computing rho(X^{al})" (slide 16); the geometric discriminants are in "Two primes at the minimum" (slide 25). This single ID covers both parts.

NEEDS APPROVAL: [s22-m07] Recommendation: Keep the quoted candidate wording, formulas and arrangement on this slide.
Exact candidate text: "Computing $\rho(X^{\mathrm{al}})$" | "$T=T(X)_{\Bbb{Q}}$; $E=\operatorname{End}_{\mathrm{Hdg}}(T)$; $d=[E:\Bbb{Q}]$; $m=\dim_E T$" | "Theorem (Charles)" | "$$\rho(X_p^{\mathrm{al}})\geq\begin{cases}\rho(X^{\mathrm{al}})&\text{if }E\text{ is CM or }m\text{ is even,}\\\rho(X^{\mathrm{al}})+d&\text{if }E\text{ is totally real and }m\text{ is odd.}\end{cases}$$" | "Equality occurs infinitely often (density $1$ after some finite extension)." | "Further, assume that we are in the second case, then exist infinitely many pairs $(p,q)$ such that the equality holds and" | "$$\operatorname{disc}\operatorname{Pic}(X_p^{\mathrm{al}})\not\equiv\operatorname{disc}\operatorname{Pic}(X_q^{\mathrm{al}})\bmod(\Bbb{Q}^{\times})^2$$"
Speaker notes proposed: "Charles computes the minimum and proves its attainment. Density one is over a suitable finite extension, not necessarily over the original field." | "His original characteristic bound supplied the then-known Tate theorem. The proof with modern finite-field Tate gives the all-good-primes statement. The pair discriminants are geometric; apply Artin-Tate after extending the residue field to define every divisor class." | "The primes are good; the pairs can be chosen ordinary, with both ranks equal to $\rho(X^{\mathrm{al}})+d$. The minimum $\eta$ is zero in the first case and $d$ in the second."
SETTLED mathematical source: I15:L523-550; Charles 2014, Thm. 1 and Prop. 18. Exact teaching arrangement still needs approval.
Allocation after the move: this single ID still covers the complete original candidate. Charles's minimum and equality are on slide 16; its original prime-pair sentence and discriminants are on slide 25. The implicit second-case reference is flagged by s22-m90.

NEEDS APPROVAL: [s22-m04] Recommendation: Keep the source sentence "Further, assume that we are in the second case, then exist infinitely many pairs (p,q) such that the equality holds and" before the geometric discriminant display; say ordinary in the notes.
Source: I15:L539-543; Charles 2014, Prop. 18.

NEEDS APPROVAL: [s22-m90] The historical sentence "Further, assume that we are in the second case" refers to Charles's theorem in "Computing rho(X^{al})": E is totally real and m is odd. "The equality" means both reduction ranks equal rho(X^{al})+d. The active sentence and notes now name that theorem explicitly. Historical candidate quotations in s22-m04 and s22-m07 retain their original wording.

AUTHOR'S CALL: [s22-m91] Keep "Computing $\rho(X^{\mathrm{al}})$", or choose "Charles's specialization theorem" or "The minimum rank under reduction"?

A real multiplication example

Elsenhans-Jahnel
X: minimal resolution of

w²=(−y²/8+yz−z²)(7x²/8+5xz+7z²)(2x²+3xy+y²)

6 lines; 15=(62) nodes; 15 exceptional (−2)-curves
H,E_ij: 16 independent classes

ρ(X^al)=16
RM: E=ℚ(√(2)); dim _E T=(22−16)/2=3
η=2; ρ(X_p^al)≥18 at every good prime

Rank-18 pair at p=17, q=23, unequal square classes, certified RM ⇒ ρ(X^al)≤16
Without certified real multiplication, one could only prove ρ(X^al)≤17.

This is X^(2,1) in Elsenhans-Jahnel 2014. The three quadratics split over ℚ(√(2)) into six lines; no three meet. H is the pullback of a general line. With the fifteen exceptional curves its Gram matrix is diag (2,−2,...,−2), determinant −65536.

The proof of Theorem 6.6 uses rank-eighteen reductions at seventeen and twenty-three with unequal geometric discriminant square classes. The RM field is proved, not numerically guessed. Point counts at the good primes p=17 and q=23 give both geometric ranks 18; Artin-Tate gives the unequal square classes after extending the residue fields to define all divisor classes. Specialization gives ρ(X^al)≤18; the unequal square classes exclude 18, so ρ(X^al)≤17. Certified quadratic RM makes 2 divide 22−ρ(X^al), so ρ(X^al) is even and at most 16. The sixteen independent classes H,E_ij give ρ(X^al)≥16, hence equality. Point counts alone do not prove characteristic-zero rank 16.

Further integral generators satisfy 2D_i=H+∑_j≠iE_ij. For w²=∏_i l_i, the quintic ∏_j≠il_j−l_i⁵=0 splits into w=±l_i³; a split component has class D_i+2H. The full characteristic-zero lattice has index thirty-two over the displayed sublattice and discriminant −64. These generators add no rational rank and do not identify the two new reduction classes.

At eighty-three there are two additional divisor-class directions. We have not identified explicit curves representing them.

References: Elsenhans-Jahnel, "Examples of K3 surfaces with real multiplication" (2014), arXiv:1402.4555v2, Theorems 5.12 and 6.6.

AUTHOR'S CALL: [s24-m01] Recommendation: Keep the certified-RM application after the known sixteen classes on slide 27. The authorized running order keeps the certified bound on slide 26, before the example on slide 27. The equation, nodes, rank and field remain unchanged.
GPT 6 astra: KEEP the interpretation between Charles and the RM example.
GPT 5.6 sol: KEEP the interpretation; the earlier absorption proposal is withdrawn.
Source: Elsenhans-Jahnel 2014, proof of Thm. 6.6; Charles 2014, Prop. 23.
The authorized running order keeps the complete certified bound on slide 26 before this example on slide 27; the earlier KEEP/DROP placement alternative is superseded.

AUTHOR'S CALL: [s24-m02] See s24-m01 for this identical recommendation and its evidence.

NEEDS APPROVAL: [s24-m03] Recommendation: Keep the equation and the short geometry, rank, RM and certified-bound bullets shown in the candidate.
Source: Elsenhans-Jahnel 2014, Thms. 5.12 and 6.6.

AUTHOR'S CALL: [s24-m04] Recommendation: Show the equation and credit first; reveal geometry at 0, rank and RM at 1, and the certified bound at 2.
GPT 6 astra: Use geometry, then rank/RM, then the method application.
GPT 5.6 sol: Use three groups: equation/source, geometry/rank, then RM and the certified conclusion.
Source: Elsenhans-Jahnel 2014, Thm. 6.6.

APPROVED (2026-09-14): [s24-m05] title "A real multiplication example"; correct the intended word "multiplication" from the author's typo. Author, verbatim: "Regarding Slide : "A surface where that happens", The title should be "A real multiplaction example", we should credit Elsenhans and Jahnel, We should explain that 15 = 6 choose 2. I think Elsenhans--Jahnel even tell us the shape of the extra cycles. I do not understand the questions about that slide"

APPROVED (2026-09-14): [s24-m06] credit Elsenhans-Jahnel 2014, Theorems 5.12 and 6.6; this is X^(2,1). Explain 15 = 6 choose 2 nodes and 15 exceptional (-2)-curves. Author, verbatim: "Regarding Slide : "A surface where that happens", The title should be "A real multiplaction example", we should credit Elsenhans and Jahnel, We should explain that 15 = 6 choose 2. I think Elsenhans--Jahnel even tell us the shape of the extra cycles. I do not understand the questions about that slide"

NEEDS APPROVAL: [s24-m07] Recommendation: Keep the split-quintic integral generators in notes; leave the two reduction-only representatives explicitly unverified. No clear answer for explicit representatives of the two new classes at 83 was found.
GPT 6 astra: The split quintics complete the same rank-16 integral lattice; the reduction-only shapes are unverified.
GPT 5.6 sol: Give no description of the two additional reduction classes; their representatives are unverified.
Source: Elsenhans-Jahnel, period integration, Rem. 4.6; 2-adic point counting, Lem. 3.11.
UNVERIFIED: explicit representatives in Pic(X_83^{al})_Q / sp(Pic(X^{al})_Q). Checked EJ-RM Theorem 6.6 and family definition; period integration Remark 4.6; 2-adic point counting Lemma 3.11, equation (11). These passages supply no representatives for the two quotient directions. The split component has class D_i+2H; D_i is an integral saturation generator. The author can supply another exact locator.

DECIDED, SUPERSEDED (2026-09-14): [s24-m08] Superseded by the explicit approved Elsenhans-Jahnel credit in s24-m06. Retain this ID and its history; no separate credit decision remains.

NEEDS APPROVAL: [s24-m09] Recommendation: Keep the quoted candidate wording, formulas and arrangement on this slide.
Exact candidate text: "A real multiplication example" | "Elsenhans-Jahnel" | "$X$: minimal resolution of" | "$$w^2=(-y^2/8+yz-z^2)(7x^2/8+5xz+7z^2)(2x^2+3xy+y^2)$$" | "$6$ lines; $15=\binom{6}{2}$ nodes; $15$ exceptional $(-2)$-curves" | "$H,E_{ij}$: $16$ independent classes" | "$\rho(X^{\mathrm{al}})=16$" | "RM: $E=\Bbb{Q}(\sqrt{2})$; $\dim_E T=(22-16)/2=3$" | "$\eta=2$; $\rho(X_p^{\mathrm{al}})\geq18$ at every good prime" | "Rank-$18$ pair, unequal square classes, certified RM $\Rightarrow$ $\rho(X^{\mathrm{al}})\leq16$"
Speaker notes proposed: "This is $X^{(2,1)}$ in Elsenhans-Jahnel 2014. The three quadratics split over $\Bbb{Q}(\sqrt2)$ into six lines; no three meet. $H$ is the pullback of a general line. With the fifteen exceptional curves its Gram matrix is $\operatorname{diag}(2,-2,\ldots,-2)$, determinant $-65536$." | "The proof of Theorem 6.6 uses rank-eighteen reductions at seventeen and twenty-three with unequal geometric discriminant square classes. The RM field is proved, not numerically guessed." | "Further integral generators satisfy $2D_i=H+\sum_{j\neq i}E_{ij}$. For $w^2=\prod_i l_i$, the quintic $\prod_{j\neq i}l_j-l_i^5=0$ splits into $w=\pm l_i^3$; a split component has class $D_i+2H$. The full characteristic-zero lattice has index thirty-two over the displayed sublattice and discriminant $-64$. These generators add no rational rank and do not identify the two new reduction classes." | "At eighty-three there are two additional divisor-class directions. We have not identified explicit curves representing them."
SETTLED mathematical source: Saard PDF, physical p. 35; Elsenhans-Jahnel 2014, Thms. 5.12 and 6.6; period integration, Rem. 4.6; 2-adic point counting, Lem. 3.11. Exact teaching arrangement still needs approval.
SETTLED author instructions, s24-m05 and s24-m06: title "A real multiplication example"; credit Elsenhans and Jahnel; explain "15 = 6 choose 2". Approval is requested for the additional content and arrangement.
Allocation after the move: the complete original example and its unchanged reveals are on slide 27, after the certified bound on slide 26.

AUTHOR'S CALL: [s24-m90] Within s24-m01, keep the opening placement recommendation, the later placement note, or both? Their wording differs; both are preserved pending your choice.

AUTHOR'S CALL: [s23-m01] Recommendation: KEEP the interpretation on slide 26; Charles's minimum is on slide 16, the prime pairs are on slide 25, and the RM example follows on slide 27. The earlier DROP alternative is superseded by the authorized running order; the certified bound stays on slide 26.
GPT 6 astra: KEEP the interpretation between Charles and the RM example.
GPT 5.6 sol: KEEP the interpretation; the earlier absorption proposal is withdrawn.
Source: Charles 2014, Thm. 1, Props. 18 and 23.

AUTHOR'S CALL: [s23-m02] Recommendation: Keep the certified quadratic-RM box, including both rank-18 reductions and unequal geometric square classes.
GPT 6 astra: Show the elementary 18 -> 17 -> 16 argument.
GPT 5.6 sol: Show the degree-sensitive subtraction, then its quadratic instance.
Source: Charles 2014, Prop. 23.

NEEDS APPROVAL: [s23-m03] Recommendation: Keep "eta: forced minimum; an individual reduction can exceed it" in the notes. SETTLED: Charles proves the minimum and its attainment.
Source: Charles 2014, Thm. 1.

APPROVED (2026-09-14): [s23-m04] matching lower bound on slide 12, via s12-m02; symplectic order-5 action and a polarization, Garbagnati-Sarti, Proposition 1.1. Author, verbatim: "is it symplectic? if so we should reference the paper"

NEEDS APPROVAL: [s23-m05] Recommendation: Keep the two cases with their actual two-prime bounds; reserve the broad discussion for the notes.
Source: Charles 2014, Remark 19.

AUTHOR'S CALL: [s23-m06] Recommendation: Show the parity case and rank-17 example first; reveal nontrivial RM at 0 and the certified bound at 1.
Source: Charles 2014, Thm. 1 and Prop. 23.

AUTHOR'S CALL: [s23-m07] See s23-m01 for this identical recommendation and its evidence.

NEEDS APPROVAL: [s23-m08] Recommendation: Keep "Charles 2014" visible; cite van Geemen 2008, Lem. 3.2, after Zarhin 1983, in the notes.
Source: reference-years, entries 23, 43, 46.

NEEDS APPROVAL: [s23-m09] See s23-m03 for this identical recommendation and its evidence.

NEEDS APPROVAL: [s23-m10] Recommendation: Keep the quoted candidate wording, formulas and arrangement on this slide.
Exact candidate text: "When every prime overshoots" | "$r:=\rho(X^{\mathrm{al}})$; $d:=[E:\Bbb{Q}]$; $m:=\dim_E T$" | "$E=\Bbb{Q}$, $m$ odd $\Rightarrow$ $\eta=1$; van Luijk succeeds [Charles] $$\text{Order-5 example:}\quad17\leq\rho(X^{\mathrm{al}})<18$$" | "$E$ totally real, $E\neq\Bbb{Q}$, $m$ odd $\Rightarrow$ $\eta=d\geq2$ $$\min_p\rho(X_p^{\mathrm{al}})=r+d;\qquad\text{two-prime upper bound: }r+d-1$$" | "Certified quadratic RM" | "$$F\hookrightarrow E,\quad[F:\Bbb{Q}]=2;\qquad\rho(X_p^{\mathrm{al}})=\rho(X_q^{\mathrm{al}})=18$$" | "$$\operatorname{disc}\operatorname{Pic}(X_p^{\mathrm{al}})\not\equiv\operatorname{disc}\operatorname{Pic}(X_q^{\mathrm{al}})\bmod(\Bbb{Q}^{\times})^2$$" | "$$\rho(X^{\mathrm{al}})\leq17,\quad\rho(X^{\mathrm{al}})\text{ even}\quad\Rightarrow\quad\rho(X^{\mathrm{al}})\leq16$$"
Speaker notes proposed: "$\eta$ is the forced minimum; an individual reduction can exceed it. At the minimum, the usual two-prime discriminant comparison leaves a gap of $d-1$ when $d>1$. This is a limitation of that criterion." | "The order-five example has lower bound seventeen from its symplectic action and a polarization, by Garbagnati-Sarti 2007, Proposition 1.1. Its two reductions supply the matching upper bound." | "For certified quadratic RM, $2$ divides $22-\rho$, so $\rho$ is even. The unequal rank-eighteen discriminants exclude eighteen, hence give at most sixteen. This is the elementary quadratic case of Charles, Proposition 23." | "A projective Kummer surface has transcendental dimension at most five. Nontrivial totally real multiplication requires $dm\geq2\cdot3=6$ by van Geemen 2008, Lemma 3.2, after Zarhin 1983. Kummer surfaces avoid this obstruction even when their determinant character is nontrivial."
SETTLED mathematical source: Charles 2014, Thm. 1, Remark 19 and Prop. 23; Garbagnati-Sarti 2007, Prop. 1.1. Exact teaching arrangement still needs approval.

AUTHOR'S CALL: [s22-m01] See s23-m01 for this identical recommendation and its evidence.

AUTHOR'S CALL: [s22-m02] See s23-m01 for this identical recommendation and its evidence.

NEEDS APPROVAL: [s22-m03] See s23-m01 for this identical recommendation and its evidence.

Infinitely many rational curves

So far we have been trying to improve the inequality ρ(X^al)≤ρ(X_p^al).
Can we use the inequality to our advantage?

Theorem (Li-Liedtke)

If there are infinitely many p primes such that

ρ(X^al)<ρ(X_p^al) and ρ(X_p^al)≠22,

then X^al contains infinitely many rational curves.

Theorem (Joshi-Rajan; Bogomolov-Zarhin)

The set {p:ρ(X_p^al)≠22} has positive density (density 1 after finite extension).

Corollary (Li-Liedtke)

ρ(X^al) odd ⇒ infinitely many integral rational curves on X^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.

Jumping Picard ranks

η(X^al):=min _{p good}(ρ(X_p^al)−ρ(X^al))

Consider

Π_jump(X):={p good:ρ(X_p^al)>ρ(X^al)+η(X^al)}

Is this set infinite? What is its density?

What about

X/ℚ: γ(X,B):=(#{p≤B:p∈Π_jump(X)})/(#{p≤B:p prime}) as B→∞ ?

NEEDS APPROVAL: [s14-m01] Recommendation: Keep the displayed minimum definition before Pi_jump on slide 18; recall sharpness from slide 16.
Source: Charles 2014, Thm. 1.

NEEDS APPROVAL: [s14-m02] Recommendation: Keep the source questions and displays on slide 18.
Source: V:L559-576.

NEEDS APPROVAL: [s14-m07] Recommendation: Keep the quoted candidate wording, formulas and arrangement on this slide.
Exact candidate text: "Jumping Picard ranks" | "$$\eta(X^{\mathrm{al}}):=\min_{p\text{ good}}\bigl(\rho(X_p^{\mathrm{al}})-\rho(X^{\mathrm{al}})\bigr)$$" | "Consider" | "$$\Pi_{\mathrm{jump}}(X):=\{p\text{ good}:\rho(X_p^{\mathrm{al}})>\rho(X^{\mathrm{al}})+\eta(X^{\mathrm{al}})\}$$" | "Is this set infinite? What is its density?" | "What about" | "$$X/\Bbb{Q}:\quad\gamma(X,B):=\frac{\#\{p\leq B:p\in\Pi_{\mathrm{jump}}(X)\}}{\#\{p\leq B:p\text{ prime}\}}\quad\text{as }B\rightarrow\infty\quad ?$$"
Speaker notes proposed: "$\eta$ is the minimum excess. Odd characteristic-zero rank forces an increase but does not imply $\eta=1$. Which primes exceed the minimum? Is that set infinite? What is its density?" | "Charles will compute this minimum and prove its attainment. The counting function here uses rational primes; over a number field, count places by norm." | "Charles's theorem identifies the forced minimum: zero in the CM or even-dimensional case, and the endomorphism-field degree in the totally real odd-dimensional case."
SETTLED mathematical source: V:L559-576; Charles 2014, Thm. 1. Exact teaching arrangement still needs approval.
Allocation after the move: this candidate is on slide 18. The active Charles note points back to slide 16; the earlier finding is retained in s14-m90; Charles is now on slide 16.

PROPOSED (2026-09-14): [s14-m10] Recommendation: Charles is on slide 16, before the separate eta definition on slide 18, as authorized by the reorder instruction. The author said "Maybe have that in slide 16, and then eta definition is natraul". Current allocation: ORDER-astra.md section 4 and artifacts/plan/lecture1.md. The historical quoted suggestion is retained.

NEEDS APPROVAL: [s14-m90] The historical speaker note said "Charles will compute this minimum and prove its attainment." That future reference is stale after the move: Charles computes the minimum on slide 16. Its case formula identifies eta as 0 or d. Applied s17-c01: the active note now points back to Charles on slide 16; the quoted sentence is retained as history.

Product of elliptic curves

A=E₁×E₂; E_i/ℚ
X=Kum (E₁×E₂)

rk NS (E₁×E₂)=rk End (E₁×E₂)^†=2+rk Hom (E₁,E₂)

ρ(X^al)=18+rk Hom (E₁^al,E₂^al)

A	ρ(X^al)	γ(X,B), predicted	What is known
square of CM	20	1/2	1/2+o(1), CM theory
square of non-CM	19	∼c_X/√(B)	infinitely many [Elkies]
CM times CM	18	1/4	1/4+o(1), CM theory
CM times non-CM	18	∼c_X/√(B)	infinitely many [Charles]
non-CM times non-CM	18	∼c_X/√(B)	infinitely many [Charles]

What happens for K3 surfaces in general?

NS, End and Hom are over ℚ^al; † is Rosati for the product polarization.

Recall "Pic plays the role of End (A)": ρ(X^al)=16+ρ(A^al). At common good primes of odd residue characteristic, ρ(X_p^al)=16+ρ(A_p^al).

Use the definitions in "Jumping Picard ranks" also for A, with ρ(A^al)=rk (Pic (A^al)/Pic ⁰(A^al)). Compare A and X on the same common good odd primes. The two rank identities give equal rank excesses and equal minima over this prime set, hence Π_jump(X)=Π_jump(A) on this set.

All isogenies are geometric. For A^al∼E², a jump occurs exactly when E has supersingular reduction. For geometrically non-isogenous factors, a jump occurs exactly when E_1,p^al∼E_2,p^al.

The product rows have geometrically non-isogenous factors. For two CM factors the CM fields are distinct. The square-root rates are conjectural; infinitude is unconditional.

For a fixed non-CM square and sufficiently large B, c(log log B)log B/B<γ(X,B)<Clog B/B^1/4. The lower bound assumes GRH for real Dirichlet characters; the upper bound is unconditional. The constants depend on the fixed curve. Elkies 1991, Theorems A and B; the upper-bound proof uses Kaneko.

For a CM square, the good unramified jump primes are exactly the primes inert in the CM field. Their density is one half.

References: Elkies, "The existence of infinitely many supersingular primes for every elliptic curve over Q", Inventiones Mathematicae 89 (1987), 561-567. Charles, "Frobenius distribution for pairs of elliptic curves and exceptional isogenies", Duke Mathematical Journal 167 (2018), 2039-2072, Theorem 1.1; arXiv:1411.2914. The quantitative bounds use Elkies, "Distribution of supersingular primes", Asterisque 198-200 (1991), 127-132, Theorems A and B.

NEEDS APPROVAL: [s16-m02] Recommendation: Keep the combined transfer, geometric product formula and comparison table as this candidate synthesis.
GPT 6 astra: The table is a synthesis of source frames and needs explicit approval.
GPT 5.6 sol: The table is a synthesis of source frames and needs explicit approval.
Source: I15:L298-352, L512-518; V:L577-596.
SYNTHESIS, not a transcription: this combines I15:L298-352 and L512-518 with V:L577-596 into the current four-column table. The report describing five columns refers to an older version. Recommendation remains to keep this single four-column candidate; no structural approval is inferred.
Allocation after the move: the original combined subject spans the dictionary on slide 19 and product/frequencies on slide 20; no content approval is inferred from the move.

AUTHOR'S CALL: [s16-m90] Keep the four-column product table as shown, or cut rows or columns? Rows: square of CM (20, 1/2, 1/2 + o(1) CM theory); square of non-CM (19, ~ c_X/sqrt(B), infinitely many [Elkies]); CM times CM (18, 1/4, 1/4 + o(1) CM theory); CM times non-CM (18, ~ c_X/sqrt(B), infinitely many [Charles]); non-CM times non-CM (18, ~ c_X/sqrt(B), infinitely many [Charles]). This table is a synthesis of source frames, not a transcription, which is why it needs your call.

NEEDS APPROVAL: [s16-m03] Recommendation: Keep geometric Hom and End; read the product rows as geometrically non-isogenous factors.
Source: I15:L298-326; Skorobogatov-Zarhin, Sec. 1, eq. (10).

NEEDS APPROVAL: [s16-m04] Recommendation: Keep Charles in both product rows and Elkies for infinitude; put publication years, the 1991 bounds and their GRH qualification in notes.
GPT 6 astra: Correct the mixed row; omit the original-field density-zero footer unless its separate source is supplied.
GPT 5.6 sol: Correct the mixed row; distinguish infinitude from an unproved asymptotic.
Source: Charles 2018, Thm. 1.1; Elkies 1991, Thms. A-B.

NEEDS APPROVAL: [s16-m06] Recommendation: Keep the quoted candidate wording, formulas and arrangement on this slide.
Exact candidate text: "Product of elliptic curves" | "$X=\operatorname{Km}(A)$; $A/\Bbb{Q}$ an abelian surface" | "$\rho(A^{\mathrm{al}}):=\operatorname{rk}(\operatorname{Pic}(A^{\mathrm{al}})/\operatorname{Pic}^0(A^{\mathrm{al}}))$" | "$\rho(X^{\mathrm{al}})=16+\rho(A^{\mathrm{al}})$" | "$\rho(X_p^{\mathrm{al}})=16+\rho(A_p^{\mathrm{al}})$; $p>2$ good" | "$\eta(X^{\mathrm{al}})=\eta(A^{\mathrm{al}})=\rho(A^{\mathrm{al}})\bmod2$" | "$\Pi_{\mathrm{jump}}(X)=\Pi_{\mathrm{jump}}(A)$" | "Fix a polarization on $A$; $\dagger$ the Rosati involution" | "$$(\operatorname{Pic}(A^{\mathrm{al}})/\operatorname{Pic}^0(A^{\mathrm{al}}))_{\Bbb{Q}}\simeq\{\phi\in\operatorname{End}(A^{\mathrm{al}})_{\Bbb{Q}}:\phi^\dagger=\phi\}$$" | "$A=E_1\times E_2$; $E_i/\Bbb{Q}$" | "$$\rho(X^{\mathrm{al}})=18+\operatorname{rk}\operatorname{Hom}(E_1^{\mathrm{al}},E_2^{\mathrm{al}})$$" | "$X$" | "$\rho(X^{\mathrm{al}})$" | "$\gamma(X,B)$, predicted" | "What is known" | "square of CM" | "20" | "$1/2$" | "$1/2+o(1)$, CM theory" | "square of non-CM" | "19" | "$\sim c_X/\sqrt{B}$" | "infinitely many [Elkies]" | "CM times CM" | "18" | "$1/4$" | "$1/4+o(1)$, CM theory" | "CM times non-CM" | "18" | "$\sim c_X/\sqrt{B}$" | "infinitely many [Charles]" | "non-CM times non-CM" | "18" | "$\sim c_X/\sqrt{B}$" | "infinitely many [Charles]" | "Product rows: geometrically non-isogenous factors" | "Non-CM rates: Lang-Trotter heuristics; per-prime scale $1/\sqrt{p}$" | "Remark" | "$p\in\Pi_{\mathrm{jump}}(X)$ depends uniquely on the pair $(a_{E_1}(p),a_{E_2}(p))$."
Speaker notes proposed: "The product rows have geometrically non-isogenous factors. For two CM factors the CM fields are distinct. The square-root rates are conjectural; infinitude is unconditional." | "For a fixed non-CM square and sufficiently large $B$, $c(\log\log B)\log B/B<\gamma(X,B)<C\log B/B^{1/4}$. The lower bound assumes GRH for real Dirichlet characters; the upper bound is unconditional. The constants depend on the fixed curve. Elkies 1991, Theorems A and B; the upper-bound proof uses Kaneko." | "For a CM square, the good unramified jump primes are exactly the primes inert in the CM field. Their density is one half."
SETTLED mathematical source: I15:L298-352, L512-518; V:L577-596; Charles 2018, Thm. 1.1; Elkies 1991, Thms. A-B. Exact teaching arrangement still needs approval.
Allocation after the move: this single ID still covers the complete original candidate, with its dictionary and Rosati display on slide 19 and product formula, table and trace-pair remark on slide 20.

NEEDS APPROVAL: [s17-m01] Recommendation: Keep the two rank equivalences and three geometric criteria on slide 19, after the Kummer dictionary; omit the closing question. No verified current answer to the simple-surface frequency question is supplied. Its research status is unverified.
GPT 6 astra: Remove the closing question; the geometric criterion completes the example.
GPT 5.6 sol: Keep a boxed question, "What happens in this case?"; do not claim its present research status is known.
Source: V:L597-623.

AUTHOR'S CALL: [s17-m02] Recommendation: Reveal the square, non-isogenous product and End = Z cases at 0/1/2.
GPT 6 astra: Reveal one complete geometric case at a time.
GPT 5.6 sol: Reveal one complete geometric case at a time.
Source: V:L597-623.

AUTHOR'S CALL: [s17-m03] See s17-m01 for this identical recommendation and its evidence.

NEEDS APPROVAL: [s17-m04] Recommendation: Keep the quoted candidate wording, formulas and arrangement on this slide.
Exact candidate text: "Jumping Picard ranks for Kummer surfaces" | "$\rho(A_p^{\mathrm{al}})\geq4\Longleftrightarrow A_p^{\mathrm{al}}\sim E^2$, $E$ an elliptic curve" | "$\rho(A_p^{\mathrm{al}})=6\Longleftrightarrow A_p^{\mathrm{al}}\sim E^2$, $E$ a supersingular elliptic curve" | "If $A^{\mathrm{al}}\sim E^2$, then $p\in\Pi_{\mathrm{jump}}(A)$ iff $p$ is supersingular for $E$." | "If $A^{\mathrm{al}}\sim E_1\times E_2$ with $E_1^{\mathrm{al}}\not\sim E_2^{\mathrm{al}}$, then $p\in\Pi_{\mathrm{jump}}(A)$ iff $E_{1,p}^{\mathrm{al}}\sim E_{2,p}^{\mathrm{al}}$." | "If $\operatorname{End}(A^{\mathrm{al}})=\Z$, then $p\in\Pi_{\mathrm{jump}}(A)$ iff $A_p^{\mathrm{al}}\sim E^2$."
Speaker notes proposed: "All isogenies are geometric. For factors defined after a finite extension, choose a place above $p$; the geometric criterion is independent of that choice. Take common good primes of odd residue characteristic." | "When $\operatorname{End}(A^{\mathrm{al}})=\Z$, the abelian Picard number is one and the Kummer Picard number is seventeen. Here $\eta=1$, so a jump means $\rho(A_p^{\mathrm{al}})>2$. The later $\eta=0$ theorem does not answer its frequency question."
SETTLED mathematical source: V:L597-623; C22:L779-790. Exact teaching arrangement still needs approval.
Allocation after the move: the original two equivalences and three criteria are on slide 19, preceded by the dictionary moved from "Product of elliptic curves". This ID continues to cover only its original criteria content.

NEEDS APPROVAL: [s16-m01] Recommendation: Keep the geometric rationalized Pic/Pic^0 quotient, with a fixed polarization.
Source: V:L590-591; Milne, Abelian Varieties, Prop. 17.2.

NEEDS APPROVAL: [s16-m05] Recommendation: Keep the mathematical convention "For an abelian surface, rho is the rank of Pic/Pic^0." SETTLED: the quotient convention. NEEDS APPROVAL: the combined slide structure.
Source: V:L590; author-approved quotient on slide 8.

NEEDS APPROVAL: [s17-m90] The moved dictionary uses eta(A) and Pi_jump(A) before explicitly extending the definitions. Use the definitions on slide 18 with rho(A)=rank(Pic(A)/Pic^0(A)); the +16 identities here give the transfer. ORDER-astra.md V05 requests this spoken bridge. The existing slide text is preserved.

AUTHOR'S CALL: [s17-m91] The rank quotient, Kummer rank transfer and Rosati display recall the Pic/End slide (8), now with geometric base changes and a fixed polarization. Keep these formulations here, on slide 8, or in both places? Both sets are preserved pending your choice. The two reduction-rank equivalences and three geometric cases have different hypotheses and are retained.

AUTHOR'S CALL: [s17-m92] Keep rho(A) := rk(Pic(A)/Pic^0(A)), rho(X^al) = 16 + rho(A^al), and the Rosati display (Pic(A^al)/Pic^0(A^al))_Q = {phi in End(A^al)_Q : phi^dagger = phi} on this slide, on slide 8, or on both? Both copies are preserved pending your choice.

Numerical experiments for ρ(X^al)=2

ρ(X)=ρ(X^al)=2 and E=ℚ or CM

gamma(X,B) for three rank two examples

No obvious trend …

We can explain the 1/2 observed in even rank

d_X:=Δ_H²(X)Δ_Pic(X) modulo squares; d_X∈ℤ∖{0}

Theorem (C-Elsenhans-Jahnel)

p good, p∤2d_X⇒det (Frob _p∣T(1)⊗ℚ_ℓ)=((d_X)/(p))=−1⇒ρ(X_p^al)≥ρ(X^al)+2

Corollary

If η(X^al)=0, then

d_X nonsquare ⇒ L=ℚ(√(d_X)), [L:ℚ]=2
p good, inert in L ⇒p∈Π_jump(X), up to finitely many primes
liminf _B→∞γ(X,B)≥1/2
E=ℚ ⇒ infinitely many integral rational curves on X^al

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

Δ_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 η=0 makes the rank increase a jump. For E=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=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" | "$$d_X=-1\cdot5\cdot151\cdot22490817357414371041\cdot387308497430\allowbreak 149337233666\allowbreak 358807996260\allowbreak 780875056740\allowbreak 850984213276\allowbreak 970343278935\allowbreak 342068889706\allowbreak 146733313789$$"

Discriminant of a K3 surface

τ:Gal (k^al/k)⟶O(V:=T(1)⊗ℚ_ℓ)

det τ=1⟺im τ⊂SO(V)

det τ≠1 ⇒ nontrivial quadratic character

ρ(X^al) even; φ:=Frob _p|V

det φ=−1⇒ρ(X_p^al)≥ρ(X^al)+2

D_X:=Δ_H²(X)∈ℚ^×/(ℚ^×)²: determinant-character square class

D_X∈ℤ∖{0} a representative; p good, p∤2D_X

Theorem (Deligne; Suh)

The functional equation of Frobenius on H²(X) has the plus sign iff D_X is square mod p.

ε_p=det (−Frob _p∣H²_et(X^al,ℚ_ℓ(1)))=((D_X)/(p))

D_X represents the quadratic extension cut out by the determinant on H²(1). It is neither the Picard intersection discriminant nor an unspecified equation discriminant.

Dimension twenty-two gives det (−Frob )=det (Frob ). C-Elsenhans-Jahnel, Proposition 2.1, attributes the projective sign statement to Deligne; Suh treats the proper nonprojective extension.

References: Costa-Elsenhans-Jahnel, "On the distribution of the Picard ranks of the reductions of a K3 surface" (2020), arXiv:1610.07823v3, Proposition 2.1, including the attribution to Deligne.

The Tate twist makes the pairing orthogonal. The determinant detects a quotient of order two, not all components of monodromy.

Forced excess can survive determinant one: quadratic RM with dim _E T=3 still forces two Tate classes.

For Kummer surfaces the universal SO assertion is false. Serre, Lectures on N_X(p), Section 8.5.6.4, removes one polarization from H²(A)(1). Removing all divisor classes gives determinant equal to the algebraic determinant. For A=(y²=x³−x)², complex conjugation on the CM field gives a nontrivial character.

For X=Kum (A), the determinant on V equals the determinant on (Pic (A^al)/Pic ⁰(A^al))_{ℚ_ℓ}. It can be nontrivial.

Remove the reciprocal pairs other than ±1. Determinant minus one makes the multiplicity of minus one odd. Even dimension then makes the multiplicity of plus one odd.

The two eigenvalues become one over a finite residue extension. Tate identifies the new geometric classes. In odd dimension minus one is forced but plus one need not be.

NEEDS APPROVAL: [s20-m01] Recommendation: Use "Theorem (Deligne; C-Elsenhans-Jahnel 2020)"; keep Suh in the notes.
GPT 6 astra: Credit Deligne visibly for the projective sign theorem; Suh belongs in broader notes.
GPT 5.6 sol: Keep the split credit Deligne-Suh for the sign and C-Elsenhans-Jahnel for the character.
Source: CEJ 2020, Prop. 2.1.

NEEDS APPROVAL: [s20-m02] Recommendation: Define D_X as the cohomological determinant square class before its residue symbol; choose a nonzero integer representative.
Source: CEJ 2020, Def. 2.4, Thm. 2.15.

AUTHOR'S CALL: [s20-m04] Recommendation: Keep the cohomological D_X and the explicit Galois-fixed Picard premise on slide 23.
GPT 6 astra: Credit Deligne visibly for the projective sign theorem; Suh belongs in broader notes.
GPT 5.6 sol: Keep the split credit Deligne-Suh for the sign and C-Elsenhans-Jahnel for the character.
Source: CEJ 2020, Def. 2.4 and Thm. 2.15.

NEEDS APPROVAL: [s20-m05] Recommendation: Keep the quoted candidate wording, formulas and arrangement on this slide.
Exact candidate text: "Discriminant of a K3 surface" | "$X/\Bbb{Q}$ quartic K3" | "$D_X:=\Delta_{H^2}(X)\in\Bbb{Q}^{\times}/(\Bbb{Q}^{\times})^2$: determinant-character square class" | "$D_X\in\Z\setminus\{0\}$ a representative; $p$ good, $p\nmid2D_X$" | "Theorem (Deligne; C-Elsenhans-Jahnel)" | "The functional equation of the Frobenius action on $H^2(X)$ has the plus sign if and only if $D_X$ is square mod $p$." | "$$\varepsilon_p=\det(-\operatorname{Frob}_p\mid H^2_{\mathrm{et}}(X^{\mathrm{al}},\Bbb{Q}_\ell(1)))=\left(\frac{D_X}{p}\right)$$" | "$\operatorname{Gal}(\Bbb{Q}^{\mathrm{al}}/\Bbb{Q})$ fixes $\operatorname{Pic}(X^{\mathrm{al}})$ $\Rightarrow$ $\Delta_{\operatorname{Pic}}(X)=1$" | "Theorem (C-Elsenhans-Jahnel)" | "$$\rho(X^{\mathrm{al}})=2r,\quad\left(\frac{D_X}{p}\right)=-1\quad\Rightarrow\quad\rho(X_p^{\mathrm{al}})\geq2r+2$$"
Speaker notes proposed: "$D_X$ represents the quadratic extension cut out by the determinant on $H^2(1)$. It is neither the Picard intersection discriminant nor an unspecified equation discriminant." | "Dimension twenty-two gives $\det(-\operatorname{Frob})=\det(\operatorname{Frob})$. C-Elsenhans-Jahnel, Proposition 2.1, attributes the projective sign statement to Deligne; Suh treats the proper nonprojective extension." | "$\Delta_{\operatorname{Pic}}$ is the square class of the Picard representation determinant. Galois fixing every geometric class makes it one; an integral descent equality is unnecessary."
SETTLED mathematical source: I15:L626-646; C-Elsenhans-Jahnel 2020, Prop. 2.1, Def. 2.4 and Thm. 2.15. Exact teaching arrangement still needs approval.

AUTHOR'S CALL: [s20-m90] Keep the prose plus-sign criterion, the determinant/Legendre-symbol display, or both? The display also identifies the determinant; both formulations are preserved pending your choice.

NEEDS APPROVAL: [s18-m02] Recommendation: Keep "Forced excess can survive determinant one" spoken on slide 21.
GPT 6 astra: Keep the forced-excess qualification in notes.
GPT 5.6 sol: Keep the forced-excess qualification in notes.
Source: Charles 2014, Prop. 15.

APPROVED (2026-09-14): [s18-m03] replace the quoted closing sentence with "An easy way to explain some jumps: O vs SO." The author calls it slide 15; the exact sentence is on slide 21 (historical slide 18). Apply at the text location; preserve the author's quoted numbers. 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)"

AUTHOR'S CALL: [s18-m04] Recommendation: Show the representation first, then the determinant-one and nontrivial-character alternatives, then the approved closing line.
Source: CEJ 2020, Prop. 2.13.

NEEDS APPROVAL: [s18-m05] Recommendation: Keep the quoted candidate wording, formulas and arrangement on this slide.
Exact candidate text: "O or SO?" | "$V:=T_\ell(1)$; cup-product pairing" | "$$\tau:\operatorname{Gal}(k^{\mathrm{al}}/k)\longrightarrow O(V)$$" | "$\det\tau=1\Longleftrightarrow\operatorname{im}\tau\subset SO(V)$" | "$\det\tau\neq1$ $\Rightarrow$ nontrivial quadratic character" | "An easy way to explain some jumps: $O$ vs $SO$."
Speaker notes proposed: "The Tate twist makes the pairing orthogonal. The determinant detects a quotient of order two, not all components of monodromy." | "Forced excess can survive determinant one: quadratic RM with $\dim_E T=3$ still forces two Tate classes." | "For Kummer surfaces the universal SO assertion is false. Serre, Lectures on N_X(p), Section 8.5.6.4, removes one polarization from $H^2(A)(1)$. Removing all divisor classes gives determinant equal to the algebraic determinant. For $A=(y^2=x^3-x)^2$, complex conjugation on the CM field gives a nontrivial character."
SETTLED mathematical source: C-Elsenhans-Jahnel 2020, Prop. 2.13; Serre, Sec. 8.5.6.4. Exact teaching arrangement still needs approval.
SETTLED author instruction, s18-m03: "An easy way to explain some jumps: O vs SO." Approval is requested for the additional content and arrangement.

NEEDS APPROVAL: [s15-m04] Recommendation: Keep "An easy way to explain some jumps: O vs SO." on slide 21 only.
GPT 6 astra: Keep the approved line at its actual location on the O/SO frame.
GPT 5.6 sol: Keep the O/SO frame and its approved transition; the earlier report proposed a second pointer.
Source: author decision on s18-m03.

NEEDS APPROVAL: [s15-m06] Recommendation: Keep the qualified Kummer determinant explanation in the notes.
GPT 6 astra: The universal Kummer-to-SO claim is false; a swapping lift need not have a fixed spectrum.
GPT 5.6 sol: The final jury agrees the universal claim is false; its earlier fixed-spectrum argument is superseded.
Source: Serre, Lectures on N_X(p), Sec. 8.5.6.4; CEJ 2020, Ex. 2.36.

AUTHOR'S CALL: [s19-m01] Recommendation: Keep slide 22 as the four-step proof of the two new Tate classes.
GPT 6 astra: KEEP the elementary determinant proof.
GPT 5.6 sol: KEEP the elementary determinant proof.
Source: CEJ 2020, Prop. 2.13.

NEEDS APPROVAL: [s19-m02] Recommendation: Keep the even-rank hypothesis in the setup and the bound rho(X_p^al) >= rho(X^al)+2 in the final step.
GPT 6 astra: Put the even-rank hypothesis in the setup.
GPT 5.6 sol: Say "In this even-rank case" in the final step.
Source: CEJ 2020, Prop. 2.13.

NEEDS APPROVAL: [s19-m03] Recommendation: Keep "Orthogonality $\Rightarrow$ lambda and lambda^{-1}, with equal multiplicities."
GPT 6 astra: State reciprocal pairing explicitly; absolute value one alone is insufficient.
GPT 5.6 sol: Rely on the preceding orthogonal representation; omit the repeated word.
Source: CEJ 2020, Prop. 2.13.

AUTHOR'S CALL: [s19-m04] Recommendation: Show reciprocal pairing first; reveal minus one, plus one and Tate at 0/1/2.
Source: CEJ 2020, Prop. 2.13.

AUTHOR'S CALL: [s19-m05] See s19-m01 for this identical recommendation and its evidence.

NEEDS APPROVAL: [s19-m06] Recommendation: Keep the quoted candidate wording, formulas and arrangement on this slide.
Exact candidate text: "What $\det=-1$ costs you" | "$\rho(X^{\mathrm{al}})$ even; $\varphi:=\operatorname{Frob}_p|T_\ell(1)$; $\det\varphi=-1$" | "Orthogonality $\Rightarrow$ $\lambda$ and $\lambda^{-1}$, with equal multiplicities." | "Other pairs: determinant $+1$; multiplicity of $-1$ odd." | "$\dim T_\ell(1)$ even $\Rightarrow$ multiplicity of $+1$ odd." | "Tate: $+1,-1$ give two new geometric divisor classes. $$\rho(X_p^{\mathrm{al}})\geq\rho(X^{\mathrm{al}})+2$$"
Speaker notes proposed: "Remove the reciprocal pairs other than $\pm1$. Determinant minus one makes the multiplicity of minus one odd. Even dimension then makes the multiplicity of plus one odd." | "The two eigenvalues become one over a finite residue extension. Tate identifies the new geometric classes. In odd dimension minus one is forced but plus one need not be."
SETTLED mathematical source: C-Elsenhans-Jahnel 2020, Prop. 2.13. Exact teaching arrangement still needs approval.

AUTHOR'S CALL: [s18-m01] Recommendation: Keep the separate four-step determinant proof on slide 22.
GPT 6 astra: KEEP the proof; it explains the increase by two.
GPT 5.6 sol: KEEP the proof; the earlier drop recommendation is superseded.
Source: CEJ 2020, Prop. 2.13.

AUTHOR'S CALL: [s19-m90] Which proof-placement recommendation should remain: s18-m01 or s19-m01? Their wording differs; both are preserved pending your choice.

Lecture 1