Kitaoka’s Conjecture (finiteness of fields with universal ternary classical forms)

Establish that there are only finitely many totally real number fields K for which there exists a positive definite classical ternary quadratic form over OK that is universal (represents every totally positive algebraic integer of K).

Background

Universal quadratic forms over totally real number fields are central objects linking arithmetic and quadratic form theory. A form is universal if it represents all totally positive integers in the field. Classical forms additionally have even off-diagonal coefficients.

Kitaoka’s Conjecture asserts a finiteness property for fields admitting universal ternary classical forms. Significant progress has been made: e.g., classification over real quadratic fields and nonexistence over biquadratic fields, and this paper resolves the case where 2 is unramified (odd discriminant). However, the full finiteness conjecture across all totally real fields remains open.