不可约二元三次代数与数域上的广义超椭圆方程
Irreducible binary cubics and the generalized superelliptic equation over number fields
摘要：
对于一类不可约二元三次型$F(x,y) \in \mathbb Z[x,y]$，Bennett和Dahmen证明了广义超椭圆方程$F(x,y)=z^l$在$x,y \in \mathbb Z$互质、$z \in \mathbb Z$和指数$l \in \mathbb Z_{\geq 4} $中至多有1/2个解.他们的证明使用了某些mod $l$伽罗瓦表示的模块性和Ribet的水平降低定理。本文的目的是处理同样的问题，二元三次系数在$\mathcal O_K$，一个任意数域$K$的整数环，使用到现在为止，有据可查的模块性结构。
英文摘要：
For a large class (heuristically most) of irreducible binary cubic forms $F(x,y) \in \mathbb Z[x,y]$, Bennett and Dahmen proved that the generalized superelliptic equation $F(x,y)=z^l$ has at most finitely many solutions in $x,y \in \mathbb Z$ coprime, $z \in \mathbb Z$ and exponent $l \in \mathbb Z_{\geq 4} $. Their proof uses, among other ingredients, modularity of certain mod $l$ Galois representations and Ribet's level lowering theorem. The aim of this paper is to treat the same problem for binary cubics with coefficients in $\mathcal O_K$, the ring of integers of an arbitrary number field $K$, using by now well-documented modularity conjectures.
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