Rigid Borrowing and Spectral Reconstruction D: Split Complex Multiplication Criterion for Genus-Two Curves—Quotient-Curve Isogeny Verification and a Machine-Checkable Pipeline for CM Principal-Type Decision
Year: 2026
Type: preprint
Abstract: The complex multiplication (CM) criterion for Jacobians of genus-two curves isa core machine-checkable problem in arithmetic geometry, but the direct decisionpath relies on symbolic computation with Igusa class polynomials or period matricesand has a high entry threshold. This paper establishes a conditional verificationpipeline for split CM: for curves whose Jacobians are isogenous to a product ofelliptic curves, the decision reduces to the elliptic-curve CM principal-type test forthe two quotient curves. The pipeline takes as its entry the structural criterionfor genus-two (2,2)-splitting—a Jacobian splits if and only if the curve admits anon-hyperelliptic involution, equivalently a double cover of a genus-one curve; thequotient by the hyperelliptic involution is a genus-zero curve and does not constituteevidence of splitting. On this structural basis, the pipeline is organized into threechannels: automorphism-sign verification and order-two classification, prime-byprime consistency verification of isogeny on a registered verification set, and CMprincipal-type testing on a Heegner discriminant set (the split branch is given bya norm representation, and the inert branch by ap = 0 together with Kroneckersymbol −1). Throughout, we introduce a registered verification set S and evidencelevels L1 to L4: L1 is prime-by-prime consistency over a finite prime set, L2 isstability under prime enlargement, L3 is independent benchmark checking, andL4 is a constructive isogeny proof. A positive conclusion, when L2 and L3 areincomplete, is recorded only as L1-level supportive positivity; the split algebraiccondition is taken strictly as End0(J(C)) being isomorphic to a product of twoimaginary quadratic fields. When the two quotient curves are isogenous, End0 isa central simple algebra and does not satisfy this condition, so the output mustbe annotated separately. A non-split negative conclusion given by the structuralcriterion does not depend on the prime set or the Heegner set, but its scope isstrictly limited to the split case and does not exclude primitive CM. The pipelinecompletes machine-checked certification on registered samples and records a no-goasset: isogenous splitting of the Jacobian does not imply CM; the quotient curvehas j = 64/9, which is not an algebraic integer, and the j-integrality theorem forCM elliptic curves excludes CM for the quotient curve. Primitive CM and exactsymbolic implementation of Igusa invariants are outside the scope of the pipeline.
Authors: zhou, changzheng, zhou, ziqing
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225145369