某些超椭圆Prymians的Hodge类
Hodge classes on certain hyperelliptic prymians
摘要：
设$n=2g+2$为正偶数，$f(x)$为$n$次无重根复多项式，$C_f: y^2=f(x)$为复数域$\C$上相应亏格的$g$超椭圆曲线.设$(g-1)$维复阿贝尔簇$P$是$C_f$的Prym簇，它对应于$C_f$的一个非分歧双覆盖.设存在$\C$的一个子域$K$，使得$f(x)$位于$K[x]$中，在$K$上不可约，且它的伽罗瓦群是全对称群.假设$g>2$，我们证明了$End(P)$是整数环$Z$或$Z$的两个副本的直和，此外，在这两种情况下的霍奇组$P$是“尽可能大”。特别是，霍奇猜想对PDMATHAMT的所有自积都成立。
英文摘要：
Let $n=2g+2$ be a positive even integer, $f(x)$ a degree $n$ complex polynomial without multiple roots and $C_f: y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve over the field $\C$ of complex numbers. Let a $(g-1)$-dimensional complex abelian variety $P$ be a Prym variety of $C_f$ that corresponds to a unramified double cover of $C_f$. Suppose that there exists a subfield $K$ of $\C$ such that $f(x)$ lies in $K[x]$, is irreducible over $K$ and its Galois group is the full symmetric group. Assuming that $g>2$, we prove that $End(P)$ is either the ring of integers $Z$ or the direct sum of two copies of $Z$; in addition, in both cases the Hodge group of $P$ is "as large as possible". In particular, the Hodge conjecture holds true for all self-products of $P$.
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