AI 中文总结
研究从平坦二维环面到二维球面的一族恰当双调和映射的零度猜想，通过揭示谱几何与算术几何联系，构造椭圆曲线同构转化问题，确定莫德尔 - 韦伊群，证实猜想并得出该族映射零度为5的结论。
AI 中文摘要
我们证明了蒙塔尔多、奥尼丘克和拉托关于从平坦二维环面到二维球面的一族恰当双调和映射的零度的猜想。证明揭示了谱几何与算术几何之间意想不到的联系。我们表明，混合傅里叶特征值的消失会在明确界定的仿射四次曲线上产生一个有理点。通过构造与有理数域上椭圆曲线的显式多项式同构，该问题简化为确定莫德尔 - 韦伊群。这给出了谱曲线上有理点的完整描述，并表明没有一个满足混合傅里叶模式所需的正性条件。因此，混合特征值从不消失，证实了蒙塔尔多 - 奥尼丘克 - 拉托猜想，并证明该族中每个映射的零度等于5。
英文摘要
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 $5$.
