北京雁栖湖应用数学研究院

杨武岳

Satyabrat Sahoo

2026年03月19日 13:30 至 14:45

A3-4-301

Zoom 435 529 7909 (BIMSA)

In 1994, Wiles proved the famous Fermat’s Last Theorem by establishing the modularity of elliptic curves over $\mathbb{Q}$, showing that the equation $x^p + y^p + z^p = 0$ with primes $p\le 3$ has no non-trivial primitive integer solutions. A similar study over totally real number fields $\mathbb{K}$ was studied by Freitas and Siksek in 2015, and showed that an asymptotic version of this result holds for certain classes of $\mathbb{K}$. In this talk, we investigate the asymptotic solutions of the generalized Fermat equation $Ax^p + By^p + Cz^p = 0$ over $\mathbb{K}$, where $A, B, C ∈ \mathcal{O}_{\mathbb{K}} \backslash \{0\}$. Using the modular method, we first show that for certain classes of fields $\mathbb{K}$, the equation $Ax^p + By^p + Cz^p = 0$ has no asymptotic solutions $(a, b, c) ∈ \mathcal{O}_{\mathbb{K}^3}$ with $2|abc$. Under additional assumptions on $A, B, C$, we further prove that this equation has no asymptotic solution in $\mathbb{K}^3$. Finally, we provide several purely local criteria on K such that $Ax^p + By^p + Cz^p = 0$ has no asymptotic solutions in $\mathbb{K}^3$.