The asymptotic Fermat’s Last Theorem for five-sixths of real quadratic fields
The asymptotic Fermat’s Last Theorem for five-sixths of real quadratic fields
复制标题
六分之五实二次域的渐近费马大定理
作者：
Nuno Freitas;S. Siksek
让$K$成为一个完全真实的领域。关于$K上的渐近Fermat最后定理，我们指的是存在一个常数$B_(K)$，使得对于任何素数指数$p>；B_(K)$，Fermat方程$$egin{eqnarray}a^{p}+b^{p}+c^{p}=0，quad a，b，Cin Kend{eqnarray}$$的唯一解是满足$ABC=0$的平凡解。借助于模性、水平降低和惯性像比较，我们给出了一个算法上可测试的判据，如果满足$K$，则蕴含着$K$上的渐近Fermat最后定理。利用解析数论的技巧，我们证明了当$dgeqslant 2$的子集在无平方正整数中具有密度${extstyle frac{5}{6}}$时，满足我们的判据：$K=mathbb{q}(sqrt{d})$.如果我们假设一个标准的‘Eichler-Shimura’猜想，我们可以将这个密度提高到$1$。
Let $K$ be a totally real field. By the asymptotic Fermat’s Last Theorem over$K$ we mean the statement that there is a constant $B_{K}$ such that for any prime exponent $p>B_{K}$, the only solutions to the Fermat equation $$egin{eqnarray}a^{p}+b^{p}+c^{p}=0,quad a,b,cin Kend{eqnarray}$$ are the trivial ones satisfying $abc=0$. With the help of modularity, level lowering and image-of-inertia comparisons, we give an algorithmically testable criterion which, if satisfied by $K$, implies the asymptotic Fermat’s Last Theorem over $K$. Using techniques from analytic number theory, we show that our criterion is satisfied by $K=mathbb{Q}(sqrt{d})$ for a subset of $dgeqslant 2$ having density ${ extstyle frac{5}{6}}$ among the squarefree positive integers. We can improve this density to $1$ if we assume a standard ‘Eichler–Shimura’ conjecture.