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 at $p=17$, $q=23$, unequal square classes, certified RM $\Rightarrow$ $\rho(X^{\mathrm{al}})\leq16$
Without certified real multiplication, one could only prove $\rho(X^{\mathrm{al}})\leq17$.

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. 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 $\rho(X^{\mathrm{al}})\leq18$; the unequal square classes exclude $18$, so $\rho(X^{\mathrm{al}})\leq17$. Certified quadratic RM makes $2$ divide $22-\rho(X^{\mathrm{al}})$, so $\rho(X^{\mathrm{al}})$ is even and at most $16$. The sixteen independent classes $H,E_{ij}$ give $\rho(X^{\mathrm{al}})\geq16$, hence equality. Point counts alone do not prove characteristic-zero rank $16$.

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.

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.