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Hodge classes on certain hyperelliptic prymians

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 $P$.