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A hypergeometric motive $H(A,B,t)$ in $M(\Q,\Q)$ is described by a hypergeometric family $H(A,B)$ in $M(\Q(t),t)$ along with a specialization point $t \in \Q-\{0,1\}$. Here, $A$ and $B$ are disjoint lists of positive integers and are the defining parameters of the hypergeometric family. The value $t$ along with $A$ and $B$ are the defining parameters of the hypergeometric motive $H(A,B,t)$.

The list $\alpha$ is obtained from $A$ by replacing each $a \in A$ with the list of all elements of $\Q \cap (0,1]$ with denominator $a$, then sorting into ascending order; the list $\beta$ is obtained from $B$ by the same procedure. Both $\alpha$ and $\beta$ have length equal to the degree of the family.

The list $\gamma$ of nonzero integers in ascending order is characterized by the property $$\frac{\prod_{g \in \gamma, g>0} (x^g-1)}{\prod_{g \in \gamma,g<0} (x^{-g}-1)} = \frac{\prod_{a \in \alpha} (x - e^{2 \pi i a})}{\prod_{b \in \beta} (x - e^{2\pi i b})} = \frac{\prod_{a \in A} \Phi_a(x)}{\prod_{b \in B} \Phi_b(x)}$$ where $\Phi_n(x)$ denotes the $n$-th cyclotomic polynomial.

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• Review status: beta
• Last edited by Alex J. Best on 2019-09-25 16:15:43
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