Certificate escape for CM j-invariants in the complex rational function field

Definitions and mathematical statement

Let $K={\mathbb{C}}(Z)$ be the rational-function field in one indeterminate $Z$, not the complex field expanded by a predicate for the integers. For $a,b\in{\mathbb{C}}$ with $4a^3+27b^2\ne0$, let \[ E_{a,b}:Y^2=X^3+aX+b,\qquad j(E_{a,b})=1728\frac{4a^3}{4a^3+27b^2}. \] Write $\operatorname{End}_{{\mathbb{C}}}(E)$ for the endomorphism ring of the elliptic curve with its identity. In characteristic zero its underlying abelian group has rank one or two. Complex multiplication means $\operatorname{End}_{{\mathbb{C}}}(E)\ne{\mathbb{Z}}$, where ${\mathbb{Z}}$ denotes the scalar multiplication endomorphisms. Put \[ S_{\mathrm{CM}}=\{j(E)\in{\mathbb{C}}:E/{\mathbb{C}}\text{ has complex multiplication}\}. \] These are the singular moduli. The source uses the criterion \[ |\operatorname{End}(E)/2\operatorname{End}(E)|>2 \quad\Longleftrightarrow\quad \operatorname{End}(E)\ne{\mathbb{Z}}. \] The rank-one/rank-two fact for complex elliptic curves is essential; freeness and finite generation alone would not imply this for an arbitrary ring or module.

Question 23 (Pheidas). The source reports that the displayed criterion is Diophantine and that $S_{\mathrm{CM}}$, viewed as a subset of $K$, can be existentially defined in $K$. What arithmetic or logical consequences can be obtained from this definable set? The question does not specify a desired consequence. In particular, a Diophantine interpretation of integer arithmetic would require additional definable data and operations, not just a unary predicate. The claimed existential formula and its permitted parameters are not supplied, so existential definability is retained as a source assertion rather than an explicit construction in this draft.

Reported certificate obstructions. If the endomorphism order of a CM elliptic curve has discriminant $D<0$, the aid reports that every non-scalar $\alpha\in\operatorname{End}(E)\setminus{\mathbb{Z}}$ satisfies \[ \deg(\alpha)\geq\frac{|D|}{4}. \] Here degree is the degree of the elliptic-curve morphism, distinct from rational-function witness degree below. Consequently, for every fixed bound $N$, singular moduli admitting a non-scalar endomorphism of degree at most $N$ form a finite set. No converse based only on this inequality is asserted.

More generally, fix a Diophantine formula $\Phi(t)$ over $K$, with finitely many fixed coefficients and a fixed finite tuple of existential witnesses, such that \[ \{c\in{\mathbb{C}}:K\models\Phi(c)\}=S_{\mathrm{CM}}. \] For a rational function $h=A(Z)/B(Z)$ in reduced form, $B\ne0$, use $\deg_Z(h)=\max\{\deg A,\deg B\}$, with constants, including zero, of degree zero. For each integer $d\geq0$, let \[ S_{\Phi,d}=\{c\in S_{\mathrm{CM}}:\Phi(c)\text{ has witnesses in }K \text{ each of }Z\text{-degree at most }d\}. \] The supplied aid reports \[ |S_{\Phi,d}|<\infty\quad\text{for every fixed }\Phi,d, \qquad S_{\mathrm{CM}}=\bigcup_{d\geq0}S_{\Phi,d}. \] Thus, for a fixed formula, no uniform finite witness-degree bound covers the entire CM locus. The formula, its coefficients and its witness presentation must remain fixed as $c$ varies. This statement is conditional on an exact Diophantine definition on constants and is not a non-definability theorem. It does not rule out unbounded certificates, additional definable relations, or an eventual interpretation of arithmetic. The record supplies no such interpretation or decision-theoretic consequence.

Short English statement

Pheidas asks what can be deduced from the reported existential definition of CM $j$-invariants in ${\mathbb{C}}(Z)$. The aid states that bounded-degree endomorphisms, and bounded rational-function witnesses for any fixed definition, account for only finitely many singular moduli. This requires unbounded certificates and leaves the requested arithmetic applications unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $20002360$, catalogue entry, aim-logic-notes.json, notes[135], workshop Extensions of Hilbert's tenth problem, Question 23 (Pheidas): https://aimath.org/WWN/hilberts10th/hilberts10th.pdf.
• Supplied research summary on CM discriminants, endomorphism-degree bounds and bounded rational-function witness strata. No separate authoritative citation or explicit Diophantine formula is supplied.
• Attempt 1 is labelled partial_result, problem status at run "open", rigor "proved", low novelty confidence; the registry currently says partially_solved. These labels concern a machine-generated research aid, not peer-reviewed verification; independently verify before citation. The supplied statement-verification field reports inspection of page 5 of the workshop PDF; this draft has not independently visited it.