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Let $H(A,B)$ be a hypergeometric family of degree $d$ and weight $w$ with defining parameters $A=[a_1,\ldots,a_m)$ and $B=(b_1,\ldots,b_n)$. The Hodge vector of $H(A,B)$ depends on how the indices $a_j$ and $b_k$ and intertwine on the circle $\R/\Z$. At the extreme of complete intertwining, the Hodge vector is $(h^{0,0}) = (d)$. At the extreme of complete separation, $d = w+1$ and the Hodge vector is $(h^{0,w},\dots,h^{w,0}) = (1,\dots,1)$.

For $t \in \Q^\times - \{1\}$ the motive $H(A,B,t)$ has the same Hodge vector. For $t=1$ the motive $H(A,B,t)$ has degree $d-1$ or $d-2$, according to whether $w$ is even or odd. If $w$ is even, the middle Hodge number $h^{w/2,w/2}$ drops by $1$. If $w$ is odd, the two middle Hodge numbers drop by $1$.

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  • Review status: beta
  • Last edited by John Jones on 2017-11-06 16:04:35
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