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.

**Knowl status:**

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

**Referred to by:**

- hgm.bezout_matrix
- hgm.defining_parameter_ppart
- hgm.defining_parameter_primetoppart
- hgm.degree
- hgm.imprimitivity_index
- mot.hodgevector
- lmfdb/hypergm/main.py (line 652)
- lmfdb/hypergm/main.py (line 658)
- lmfdb/hypergm/templates/hgm-show-motive.html (line 5)
- lmfdb/hypergm/templates/hgm_family.html (line 5)
- lmfdb/motives/templates/hypergeometric-index.html (line 5)
- lmfdb/motives/templates/motive-index.html (line 5)

**History:**(expand/hide all)

- 2019-09-25 16:15:43 by Alex J. Best
- 2019-08-22 16:18:06 by Kiran S. Kedlaya
- 2019-08-22 11:54:02 by Kiran S. Kedlaya
- 2019-08-22 11:35:20 by Kiran S. Kedlaya
- 2017-11-05 23:55:37 by John Jones

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