What the characteristic polynomial gives you

Tate conjecture (proved)

$X_p/\Bbb{F}_q$ an abelian surface or a K3 surface; $q=p^n$, $\ell\neq p$.
For abelian surfaces: $\rho:=\operatorname{rk}(\operatorname{Pic}/\operatorname{Pic}^0)$.

$\rho(X_p) = \operatorname{ord}_{t = q} P_2(t)$
$\rho(X_p^{\mathrm{al}}) = \sum_{\zeta} \operatorname{ord}_{t = q\zeta} P_2(t)$, where $\zeta$ runs over all roots of unity.
For K3 surfaces $\Rightarrow$ $\rho(X_p^{\mathrm{al}})$ is even.

Artin-Tate for K3 surfaces over $\Bbb{F}_q$

$$\boxed{ \lim_{t\to q}\frac{P_2(t)}{(t-q)^\rho} =(-1)^{\rho-1}q^{21-\rho}\#\operatorname{Br}(X_p)\,\operatorname{disc}(\operatorname{Pic}(X_p)) }$$

$\#\operatorname{Br}(X_p)\in\Bbb{Q}^{\times 2}$ and $P_2(t)$ give $\operatorname{disc}(\operatorname{Pic}(X_p)) \bmod \Bbb{Q}^{\times 2}$

$P_2(t)$ is the characteristic polynomial introduced in "Picard lattice, over finite fields".

$X_p$ is over $\Bbb{F}_q$. The multiplicity at $q$ is the arithmetic rank; summing over $q\zeta$ gives the geometric rank.

Several papers proved the Tate conjecture. The Kuga-Satake construction plays a crucial role.

In the K3 formula, $\rho$ is the arithmetic rank $\rho(X_p)$. Artin-Tate gives the discriminant of $\operatorname{Pic}(X_p)$ over $\Bbb{F}_q$.

The Brauer order is a square, so the polynomial and the known sign and power of $q$ give the discriminant modulo squares.

For the geometric discriminant, pass first to an extension defining every divisor class and use that extension's Frobenius polynomial, as in "Let's apply it to a K3 surface with a $\Bbb{Z}/5$ automorphism".

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): [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.