$$\rho(X_p^{\mathrm{al}})\geq\begin{cases}\rho(X^{\mathrm{al}})&\text{if }E\text{ is CM or }m\text{ is even,}\\\rho(X^{\mathrm{al}})+d&\text{if }E\text{ is totally real and }m\text{ is odd,}\end{cases}$$
Equality occurs infinitely often (density $1$ after some finite extension).
If $E$ is totally real and $m$ is odd, infinitely many good ordinary prime pairs $(p,q)$ satisfy $\rho(X_p^{\mathrm{al}})=\rho(X_q^{\mathrm{al}})=\rho(X^{\mathrm{al}})+d$ and
$$\operatorname{disc}\operatorname{Pic}(X_p^{\mathrm{al}})\not\equiv\operatorname{disc}\operatorname{Pic}(X_q^{\mathrm{al}})\bmod(\Bbb{Q}^{\times})^2$$
The Kloosterman—van Luijk method works, if it is aware of $E$.
NEEDS APPROVAL: [s22-m05] Recommendation: Recall T=T(X)_Q from slide 15; use algebraic divisor classes in its definition. SETTLED: the complement is algebraic Pic, not topological line bundles.
Source: Charles 2014, introduction.
Retained correction: T is the orthogonal complement of algebraic Pic in rational H^2, not topological line bundles. Source: Charles 2014, introduction. The repaired body was already present before this pass.
NEEDS APPROVAL: [s22-m06] Recommendation: Keep geometric discriminants and the conjunction "totally real and m odd". SETTLED by the theorem: both geometric base changes and AND are necessary corrections to I15:L533-541.
Source: Charles 2014, Thm. 1 and Prop. 18.
Content anchors: the conjunction is in "Computing rho(X^{al})" (slide 16); the geometric discriminants are in "Two primes at the minimum" (slide 25). This single ID covers both parts.
NEEDS APPROVAL: [s22-m07] Recommendation: Keep the quoted candidate wording, formulas and arrangement on this slide.
Exact candidate text: "Computing $\rho(X^{\mathrm{al}})$" | "$T=T(X)_{\Bbb{Q}}$; $E=\operatorname{End}_{\mathrm{Hdg}}(T)$; $d=[E:\Bbb{Q}]$; $m=\dim_E T$" | "Theorem (Charles)" | "$$\rho(X_p^{\mathrm{al}})\geq\begin{cases}\rho(X^{\mathrm{al}})&\text{if }E\text{ is CM or }m\text{ is even,}\\\rho(X^{\mathrm{al}})+d&\text{if }E\text{ is totally real and }m\text{ is odd.}\end{cases}$$" | "Equality occurs infinitely often (density $1$ after some finite extension)." | "Further, assume that we are in the second case, then exist infinitely many pairs $(p,q)$ such that the equality holds and" | "$$\operatorname{disc}\operatorname{Pic}(X_p^{\mathrm{al}})\not\equiv\operatorname{disc}\operatorname{Pic}(X_q^{\mathrm{al}})\bmod(\Bbb{Q}^{\times})^2$$"
Speaker notes proposed: "Charles computes the minimum and proves its attainment. Density one is over a suitable finite extension, not necessarily over the original field." | "His original characteristic bound supplied the then-known Tate theorem. The proof with modern finite-field Tate gives the all-good-primes statement. The pair discriminants are geometric; apply Artin-Tate after extending the residue field to define every divisor class." | "The primes are good; the pairs can be chosen ordinary, with both ranks equal to $\rho(X^{\mathrm{al}})+d$. The minimum $\eta$ is zero in the first case and $d$ in the second."
SETTLED mathematical source: I15:L523-550; Charles 2014, Thm. 1 and Prop. 18. Exact teaching arrangement still needs approval.
Allocation after the move: this single ID still covers the complete original candidate. Charles's minimum and equality are on slide 16; its original prime-pair sentence and discriminants are on slide 25. The implicit second-case reference is flagged by s22-m90.
NEEDS APPROVAL: [s22-m04] Recommendation: Keep the source sentence "Further, assume that we are in the second case, then exist infinitely many pairs (p,q) such that the equality holds and" before the geometric discriminant display; say ordinary in the notes.
Source: I15:L539-543; Charles 2014, Prop. 18.
NEEDS APPROVAL: [s22-m90] The historical sentence "Further, assume that we are in the second case" refers to Charles's theorem in "Computing rho(X^{al})": E is totally real and m is odd. "The equality" means both reduction ranks equal rho(X^{al})+d. The active sentence and notes now name that theorem explicitly. Historical candidate quotations in s22-m04 and s22-m07 retain their original wording.
AUTHOR'S CALL: [s22-m91] Keep "Computing $\rho(X^{\mathrm{al}})$", or choose "Charles's specialization theorem" or "The minimum rank under reduction"?