Let's apply it to a K3 surface with a $\Bbb{Z}/5$ automorphism

$$X : x^3 z + 3x^2 y^2 + 5xw^3 + y^3 w + 3yz^3 - 5z^2 w^2 = 0 \ \subset \ \mathbf{P}^3$$

$p$	$\rho(X_p^{\mathrm{al}})$	$\operatorname{disc}\operatorname{Pic}(X_p^{\mathrm{al}})\bmod\Bbb{Q}^{\times2}$
11	18	$-55$
13	18	$-85$

Theorem (Artin-Tate)

$$P_2(t) \leadsto \operatorname{disc}\operatorname{Pic}(X_p) \bmod (\Bbb{Q}^{\times})^2.$$

$55/85 = 11/17 \notin (\Bbb{Q}^{\times})^2$, hence $\rho(X^{\mathrm{al}}) \leq 17$.
Symplectic order-5 action $\Rightarrow$ $\rho(X^{\mathrm{al}}) \geq 17$ [Garbagnati-Sarti].

The prime-field ranks are $1$ at $11$ and $5$ at $13$. Extend to $\Bbb{F}_{11^{30}}$ and $\Bbb{F}_{13^4}$, respectively $\Rightarrow$ $\rho(X_{\Bbb{F}_{p^r}})=\rho(X_p^{\mathrm{al}})=18$ 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 $\rho(X^{\mathrm{al}})=17$.

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

Take $\sigma(x:y:z:w)=(x:\zeta y:\zeta^2z:\zeta^4w)$, with $\zeta$ of order $5$. Every monomial has weight $2$ mod $5$, and $\det(\sigma)=\zeta^2$. The residue 2-form is fixed because $\det(\sigma)/\zeta^2=1$.

There are $16$ coinvariant classes and an invariant polarization, giving $\rho\geq17$.

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.