The Nullity of a Family of Proper Biharmonic Maps via Elliptic Curves

Abstract

We prove a conjecture of Montaldo, Oniciuc and Ratto concerning the nullity of a family of proper biharmonic maps from the flat two-torus to the round two-sphere. The proof reveals an unexpected connection between spectral geometry and arithmetic geometry. We show that the vanishing of a mixed Fourier eigenvalue produces a rational point on an explicitly defined affine quartic. By constructing an explicit polynomial isomorphism with an elliptic curve over $\Q$, the problem is reduced to the determination of a Mordell--Weil group. This yields a complete description of the rational points on the spectral curve and shows that none satisfies the positivity conditions required for a mixed Fourier mode. As a consequence, the mixed eigenvalues never vanish, confirming the Montaldo--Oniciuc--Ratto conjecture and proving that the nullity of every map in the family is equal to $5$.

Disclosure

“hat curve. Section 5 pulls the rational points back to the quartic, and Section 6 completes the proof of Theorem 1.1. The elementary algebraic identities and the SageMath verifications can be found in the appendix. Acknowledgements: I used Deepl to improve the English. 2. Reduction to an algebraic curve In this ection we provide the reduction of the problem to the study of an algebraic curve. The maps considered by Montaldo, Oniciuc and Ratto are defined”

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2026-08-23T18:59:29.962915Z

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