Endomorphisms of the transcendental Hodge structure

$$E:=\operatorname{End}_{\mathrm{Hdg}}(T)=\{a\in\operatorname{End}_{\Bbb{Q}}(T):a_{\Bbb{C}}(T^{i,j})\subset T^{i,j}\}$$

$T$ minimal rational sub-Hodge structure of $H^2$ with $H^{2,0}\subset T_{\Bbb{C}}$

$$0\neq\alpha\in E\Rightarrow\alpha(H^{2,0})=H^{2,0}\Rightarrow\operatorname{im}\alpha=T\Rightarrow\alpha^{-1}\in E$$

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; $V:=T(1)\otimes\Bbb{Q}_\ell$; $g=\operatorname{Frob}_p^a$ in connected monodromy

$$V\otimes\Bbb{Q}_\ell^{\mathrm{al}}=\bigoplus_{\sigma:E\hookrightarrow\Bbb{Q}_\ell^{\mathrm{al}}}V_\sigma,\quad\dim V_\sigma=m,\quad g|V_\sigma\in SO(V_\sigma)$$

$$m\text{ odd}\Rightarrow\dim\ker(g-1)\geq d\Rightarrow\rho(X_p^{\mathrm{al}})\geq\rho(X^{\mathrm{al}})+d$$

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.

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.