Hypergeometric motives

Abstract: 

The Riemann zeta-function is the most well-known example of a class of objects known as LL-functions. Historically, it and the other degree 1 LL-functions, those of Dirichlet, were the first to be studied. Degree 2 LL-functions, specifically those of modular forms and elliptic curves, cropped up in the work of Wiles on Fermat's last theorem among other places, while LL-functions of number fields (Dedekind zeta-functions) are examples of higher degree LL-functions. We describe a new subclass of LL-functions (sometimes only conjectural), namely those corresponding to hypergeometric motives. An example of this are the degree 4 L-function associated to the parametrised tt-family of quintic 3-folds x51+x25+x53+x54+x55=5tx1x2x3x4x5x15+x52+x35+x45+x55=5tx1x2x3x4x5 of interest to physicists, and incidentally the modularity of similar generalized Fermat hypersurfaces played a key role in the proof of the Sato-Tate conjecture. We describe what hypergeometric motives are by analogising hypergeometric equations over the complex field to over the p-adic CpCp, and how a trace formula of Katz can be used to compute Euler factors of the LL-functions at sufficiently good primes. We also describe how to compute with hypergeometric motives with programs such as MAGMA.