Picard lattice, over finite fields

$P_2(t) = \det(t - \operatorname{Frob}_q \mid H^2_{\mathrm{et}}(X_p^{\mathrm{al}},\Bbb{Q}_\ell))$; $q=p^n$, $\ell\neq p$.
$q^{-22}P_2(qt)$ monic; roots $\zeta_i$, $|\zeta_i| = 1$.

$$\begin{gathered} q^{-22}P_2(qt) = h(t)\prod_i\Phi_{k_i}(t)^{\gamma_i} \\ \Phi_k\text{ the }k\text{-th cyclotomic polynomial};\quad h\text{ has no cyclotomic factor} \end{gathered}$$

Example: $X := Z(y^4 - x^3z + yz^3 + zw^3 + w^4) \subset \mathbf{P}^3$, $p = 89$.

$$p^{-22}P_2(pt) = (t-1)^{1+4}(t+1)(t^4+1)h(t), \qquad \deg h = 12$$

$$H^2:=H^2_{\mathrm{et}}(X_{89}^{\mathrm{al}},\Bbb{Q}_\ell(1))=P_{\Phi_1}\oplus P_{\Phi_2}\oplus P_{\Phi_8}\oplus P_h$$

$$\dim (H^2)^{\operatorname{Frob}_{89}^8=1}=(1+4)+1+4=10$$

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.

$P_{\Phi_k}=\ker\Phi_k(\operatorname{Frob}_{89})$ and $P_h=\ker h(\operatorname{Frob}_{89})$ on this $H^2$. Their dimensions are $5,1,4,12$.

Since $1,2,8$ divide $8$, the first three summands give $10$ classes fixed by $\operatorname{Frob}_{89}^8$. 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"?