论文标题
非卵形椭圆曲线和过度椭圆形雅各比人II
Non-isogenous elliptic curves and hyperelliptic jacobians II
论文作者
论文摘要
让$ k $成为一个特征的字段，不同于$ 2 $，$ \ bar {k} $它的代数关闭。令$ n \ ge 3 $为一个奇数。令$ f（x）$和$ h（x）$为$ n $多项式，系数为$ k $，而没有重复的根。让我们考虑一下属$（n-1）/2 $ hyperelliptic曲线$ c_f：y^2 = f（x）$和$ c_h：y^2 = h（x）$，以及他们的jacobians $ j（c_f）$ j（c_f）$和$ j（c_h）$，它们是$（n-1）/2 $ d $ - d $ dimemential Abelian abelian abelian abelian abelian abelian abelian abelian abelian abelian pecnecties $ k $ k。 假设其中一个多项式是不可约的，而另一个则完全划分了$ k $。我们证明，如果$ j（c_f）$和$ j（c_h）$在$ \ bar {k} $上是等不的，那么有一个（奇数的）prime $ \ ell $划分$ n $，以至于$ j（c_f）$ j（c_f）$和$ j（c_h）$ cupfield的$ j $ phote的$ phote $ phot $ the eell y eell field $ the Ell y y y $ the Ell y eell osomorphic fore field $ the Ell $ the Ell osomorphic，
Let $K$ be a field of characteristic different from $2$, $\bar{K}$ its algebraic closure. Let $n \ge 3$ be an odd integer. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider genus $(n-1)/2$ hyperelliptic curves $C_f: y^2=f(x)$ and $C_h: y^2=h(x)$, and their jacobians $J(C_f)$ and $J(C_h)$, which are $(n-1)/2$-dimensional abelian varieties defined over $K$. Suppose that one of the polynomials is irreducible and the other splits completely over $K$. We prove that if $J(C_f)$ and $J(C_h)$ are isogenous over $\bar{K}$ then there is an (odd) prime $\ell$ dividing $n$ such that the endomorphism algebras of both $J(C_f)$ and $J(C_h)$ contain a subfield that is isomorphic to the field of $\ell$th roots of $1$.
