Hypergeometric Motives ?
A good introduction
David Roberts & Fernando Rodriguez Villegas, Hypergeometric Motives,
https://arxiv.org/abs/2109.00027
Frits Beukers, Henri Cohen & Anton Mellit, Finite Hypergeometric Functions,
https://arxiv.org/abs/1505.02900
For the purpose of this tutorial, HGMs are a nice source of motivic L-functions (sometimes
conjecturally!), related to point counting on families of algebraic varieties of the form
n
Y
i=1
xγi
i = t,
n
X
i=1
xi = 0,
n
Y
i=1
xi 6= 0 ,
where (γi) ∈ Zn
and t ∈ Q∗
specifies a variety in the family. One can write periods in terms of
classical hypergeometric functions nFn−1(α, β; t) and count points in terms of Jacobi sums.
One recovers in this way L-functions attached to Artin representations, curves over number fields,
Siegel modular forms, etc. Example: the Legendre family of elliptic curves,
Et : y2 = x(x − 1)(x − t); note that t = 0, 1 correspond to singular points.
Atelier 2022 (13/01/2022) – p. 2/12