Product of elliptic curves

$A=E_1\times E_2$; $E_i/\Bbb{Q}$
$X=\operatorname{Kum}(E_1\times E_2)$

$$\operatorname{rk}\operatorname{NS}(E_1\times E_2)=\operatorname{rk}\operatorname{End}(E_1\times E_2)^\dagger=2+\operatorname{rk}\operatorname{Hom}(E_1,E_2)$$

$$\rho(X^{\mathrm{al}})=18+\operatorname{rk}\operatorname{Hom}(E_1^{\mathrm{al}},E_2^{\mathrm{al}})$$

$A$	$\rho(X^{\mathrm{al}})$	$\gamma(X,B)$, predicted	What is known
square of CM	20	$1/2$	$1/2+o(1)$, CM theory
square of non-CM	19	$\sim c_X/\sqrt{B}$	infinitely many [Elkies]
CM times CM	18	$1/4$	$1/4+o(1)$, CM theory
CM times non-CM	18	$\sim c_X/\sqrt{B}$	infinitely many [Charles]
non-CM times non-CM	18	$\sim c_X/\sqrt{B}$	infinitely many [Charles]

What happens for K3 surfaces in general?

NS, End and Hom are over $\Bbb{Q}^{\mathrm{al}}$; $\dagger$ is Rosati for the product polarization.

Recall "Pic plays the role of $\operatorname{End}(A)$": $\rho(X^{\mathrm{al}})=16+\rho(A^{\mathrm{al}})$. At common good primes of odd residue characteristic, $\rho(X_p^{\mathrm{al}})=16+\rho(A_p^{\mathrm{al}})$.

Use the definitions in "Jumping Picard ranks" also for $A$, with $\rho(A^{\mathrm{al}})=\operatorname{rk}(\operatorname{Pic}(A^{\mathrm{al}})/\operatorname{Pic}^0(A^{\mathrm{al}}))$. Compare $A$ and $X$ on the same common good odd primes. The two rank identities give equal rank excesses and equal minima over this prime set, hence $\Pi_{\mathrm{jump}}(X)=\Pi_{\mathrm{jump}}(A)$ on this set.

All isogenies are geometric. For $A^{\mathrm{al}}\sim E^2$, a jump occurs exactly when $E$ has supersingular reduction. For geometrically non-isogenous factors, a jump occurs exactly when $E_{1,p}^{\mathrm{al}}\sim E_{2,p}^{\mathrm{al}}$.

The product rows have geometrically non-isogenous factors. For two CM factors the CM fields are distinct. The square-root rates are conjectural; infinitude is unconditional.

For a fixed non-CM square and sufficiently large $B$, $c(\log\log B)\log B/B<\gamma(X,B)<C\log B/B^{1/4}$. The lower bound assumes GRH for real Dirichlet characters; the upper bound is unconditional. The constants depend on the fixed curve. Elkies 1991, Theorems A and B; the upper-bound proof uses Kaneko.

For a CM square, the good unramified jump primes are exactly the primes inert in the CM field. Their density is one half.

References: Elkies, "The existence of infinitely many supersingular primes for every elliptic curve over Q", Inventiones Mathematicae 89 (1987), 561-567. Charles, "Frobenius distribution for pairs of elliptic curves and exceptional isogenies", Duke Mathematical Journal 167 (2018), 2039-2072, Theorem 1.1; arXiv:1411.2914. The quantitative bounds use Elkies, "Distribution of supersingular primes", Asterisque 198-200 (1991), 127-132, Theorems A and B.

NEEDS APPROVAL: [s16-m02] Recommendation: Keep the combined transfer, geometric product formula and comparison table as this candidate synthesis.
GPT 6 astra: The table is a synthesis of source frames and needs explicit approval.
GPT 5.6 sol: The table is a synthesis of source frames and needs explicit approval.
Source: I15:L298-352, L512-518; V:L577-596.
SYNTHESIS, not a transcription: this combines I15:L298-352 and L512-518 with V:L577-596 into the current four-column table. The report describing five columns refers to an older version. Recommendation remains to keep this single four-column candidate; no structural approval is inferred.
Allocation after the move: the original combined subject spans the dictionary on slide 19 and product/frequencies on slide 20; no content approval is inferred from the move.

AUTHOR'S CALL: [s16-m90] Keep the four-column product table as shown, or cut rows or columns? Rows: square of CM (20, 1/2, 1/2 + o(1) CM theory); square of non-CM (19, ~ c_X/sqrt(B), infinitely many [Elkies]); CM times CM (18, 1/4, 1/4 + o(1) CM theory); CM times non-CM (18, ~ c_X/sqrt(B), infinitely many [Charles]); non-CM times non-CM (18, ~ c_X/sqrt(B), infinitely many [Charles]). This table is a synthesis of source frames, not a transcription, which is why it needs your call.

NEEDS APPROVAL: [s16-m03] Recommendation: Keep geometric Hom and End; read the product rows as geometrically non-isogenous factors.
Source: I15:L298-326; Skorobogatov-Zarhin, Sec. 1, eq. (10).

NEEDS APPROVAL: [s16-m04] Recommendation: Keep Charles in both product rows and Elkies for infinitude; put publication years, the 1991 bounds and their GRH qualification in notes.
GPT 6 astra: Correct the mixed row; omit the original-field density-zero footer unless its separate source is supplied.
GPT 5.6 sol: Correct the mixed row; distinguish infinitude from an unproved asymptotic.
Source: Charles 2018, Thm. 1.1; Elkies 1991, Thms. A-B.

NEEDS APPROVAL: [s16-m06] Recommendation: Keep the quoted candidate wording, formulas and arrangement on this slide.
Exact candidate text: "Product of elliptic curves" | "$X=\operatorname{Km}(A)$; $A/\Bbb{Q}$ an abelian surface" | "$\rho(A^{\mathrm{al}}):=\operatorname{rk}(\operatorname{Pic}(A^{\mathrm{al}})/\operatorname{Pic}^0(A^{\mathrm{al}}))$" | "$\rho(X^{\mathrm{al}})=16+\rho(A^{\mathrm{al}})$" | "$\rho(X_p^{\mathrm{al}})=16+\rho(A_p^{\mathrm{al}})$; $p>2$ good" | "$\eta(X^{\mathrm{al}})=\eta(A^{\mathrm{al}})=\rho(A^{\mathrm{al}})\bmod2$" | "$\Pi_{\mathrm{jump}}(X)=\Pi_{\mathrm{jump}}(A)$" | "Fix a polarization on $A$; $\dagger$ the Rosati involution" | "$$(\operatorname{Pic}(A^{\mathrm{al}})/\operatorname{Pic}^0(A^{\mathrm{al}}))_{\Bbb{Q}}\simeq\{\phi\in\operatorname{End}(A^{\mathrm{al}})_{\Bbb{Q}}:\phi^\dagger=\phi\}$$" | "$A=E_1\times E_2$; $E_i/\Bbb{Q}$" | "$$\rho(X^{\mathrm{al}})=18+\operatorname{rk}\operatorname{Hom}(E_1^{\mathrm{al}},E_2^{\mathrm{al}})$$" | "$X$" | "$\rho(X^{\mathrm{al}})$" | "$\gamma(X,B)$, predicted" | "What is known" | "square of CM" | "20" | "$1/2$" | "$1/2+o(1)$, CM theory" | "square of non-CM" | "19" | "$\sim c_X/\sqrt{B}$" | "infinitely many [Elkies]" | "CM times CM" | "18" | "$1/4$" | "$1/4+o(1)$, CM theory" | "CM times non-CM" | "18" | "$\sim c_X/\sqrt{B}$" | "infinitely many [Charles]" | "non-CM times non-CM" | "18" | "$\sim c_X/\sqrt{B}$" | "infinitely many [Charles]" | "Product rows: geometrically non-isogenous factors" | "Non-CM rates: Lang-Trotter heuristics; per-prime scale $1/\sqrt{p}$" | "Remark" | "$p\in\Pi_{\mathrm{jump}}(X)$ depends uniquely on the pair $(a_{E_1}(p),a_{E_2}(p))$."
Speaker notes proposed: "The product rows have geometrically non-isogenous factors. For two CM factors the CM fields are distinct. The square-root rates are conjectural; infinitude is unconditional." | "For a fixed non-CM square and sufficiently large $B$, $c(\log\log B)\log B/B<\gamma(X,B)<C\log B/B^{1/4}$. The lower bound assumes GRH for real Dirichlet characters; the upper bound is unconditional. The constants depend on the fixed curve. Elkies 1991, Theorems A and B; the upper-bound proof uses Kaneko." | "For a CM square, the good unramified jump primes are exactly the primes inert in the CM field. Their density is one half."
SETTLED mathematical source: I15:L298-352, L512-518; V:L577-596; Charles 2018, Thm. 1.1; Elkies 1991, Thms. A-B. Exact teaching arrangement still needs approval.
Allocation after the move: this single ID still covers the complete original candidate, with its dictionary and Rosati display on slide 19 and product formula, table and trace-pair remark on slide 20.

NEEDS APPROVAL: [s17-m01] Recommendation: Keep the two rank equivalences and three geometric criteria on slide 19, after the Kummer dictionary; omit the closing question. No verified current answer to the simple-surface frequency question is supplied. Its research status is unverified.
GPT 6 astra: Remove the closing question; the geometric criterion completes the example.
GPT 5.6 sol: Keep a boxed question, "What happens in this case?"; do not claim its present research status is known.
Source: V:L597-623.

AUTHOR'S CALL: [s17-m02] Recommendation: Reveal the square, non-isogenous product and End = Z cases at 0/1/2.
GPT 6 astra: Reveal one complete geometric case at a time.
GPT 5.6 sol: Reveal one complete geometric case at a time.
Source: V:L597-623.

AUTHOR'S CALL: [s17-m03] See s17-m01 for this identical recommendation and its evidence.

NEEDS APPROVAL: [s17-m04] Recommendation: Keep the quoted candidate wording, formulas and arrangement on this slide.
Exact candidate text: "Jumping Picard ranks for Kummer surfaces" | "$\rho(A_p^{\mathrm{al}})\geq4\Longleftrightarrow A_p^{\mathrm{al}}\sim E^2$, $E$ an elliptic curve" | "$\rho(A_p^{\mathrm{al}})=6\Longleftrightarrow A_p^{\mathrm{al}}\sim E^2$, $E$ a supersingular elliptic curve" | "If $A^{\mathrm{al}}\sim E^2$, then $p\in\Pi_{\mathrm{jump}}(A)$ iff $p$ is supersingular for $E$." | "If $A^{\mathrm{al}}\sim E_1\times E_2$ with $E_1^{\mathrm{al}}\not\sim E_2^{\mathrm{al}}$, then $p\in\Pi_{\mathrm{jump}}(A)$ iff $E_{1,p}^{\mathrm{al}}\sim E_{2,p}^{\mathrm{al}}$." | "If $\operatorname{End}(A^{\mathrm{al}})=\Z$, then $p\in\Pi_{\mathrm{jump}}(A)$ iff $A_p^{\mathrm{al}}\sim E^2$."
Speaker notes proposed: "All isogenies are geometric. For factors defined after a finite extension, choose a place above $p$; the geometric criterion is independent of that choice. Take common good primes of odd residue characteristic." | "When $\operatorname{End}(A^{\mathrm{al}})=\Z$, the abelian Picard number is one and the Kummer Picard number is seventeen. Here $\eta=1$, so a jump means $\rho(A_p^{\mathrm{al}})>2$. The later $\eta=0$ theorem does not answer its frequency question."
SETTLED mathematical source: V:L597-623; C22:L779-790. Exact teaching arrangement still needs approval.
Allocation after the move: the original two equivalences and three criteria are on slide 19, preceded by the dictionary moved from "Product of elliptic curves". This ID continues to cover only its original criteria content.

NEEDS APPROVAL: [s16-m01] Recommendation: Keep the geometric rationalized Pic/Pic^0 quotient, with a fixed polarization.
Source: V:L590-591; Milne, Abelian Varieties, Prop. 17.2.

NEEDS APPROVAL: [s16-m05] Recommendation: Keep the mathematical convention "For an abelian surface, rho is the rank of Pic/Pic^0." SETTLED: the quotient convention. NEEDS APPROVAL: the combined slide structure.
Source: V:L590; author-approved quotient on slide 8.

NEEDS APPROVAL: [s17-m90] The moved dictionary uses eta(A) and Pi_jump(A) before explicitly extending the definitions. Use the definitions on slide 18 with rho(A)=rank(Pic(A)/Pic^0(A)); the +16 identities here give the transfer. ORDER-astra.md V05 requests this spoken bridge. The existing slide text is preserved.

AUTHOR'S CALL: [s17-m91] The rank quotient, Kummer rank transfer and Rosati display recall the Pic/End slide (8), now with geometric base changes and a fixed polarization. Keep these formulations here, on slide 8, or in both places? Both sets are preserved pending your choice. The two reduction-rank equivalences and three geometric cases have different hypotheses and are retained.

AUTHOR'S CALL: [s17-m92] Keep rho(A) := rk(Pic(A)/Pic^0(A)), rho(X^al) = 16 + rho(A^al), and the Rosati display (Pic(A^al)/Pic^0(A^al))_Q = {phi in End(A^al)_Q : phi^dagger = phi} on this slide, on slide 8, or on both? Both copies are preserved pending your choice.