#
Hypergeometric motives

June 25 to June 29, 2012
at the

American Institute of Mathematics,
Palo Alto, California

organized by

Henri Cohen and Fernando Rodriguez Villegas

## Original Announcement

This workshop and ICTP will focus
on the L-functions of arithmetic
geometry whose Euler factors are generically of degree higher than
two. It will consist of a two-pronged approach combining theory and
computations. This workshop
will take place within the framework of a larger two-week ICPT activity.
Specifically, we will (mostly) concentrate on the L-functions of
hypergeometric motives. These are certain one-parameter families of
motives, which in one incarnation correspond to the classical
hypergeometric differential equations with rational parameters. A
prototypical example is the equation satisfied by the Gauss
hypergeometric function $F(1/2, 1/2, 1; t)$ whose associated L-function
is that of the Legendre elliptic curve $y^2= x(x-1)(x-t)$.
Another example is the basic period of the Dwork pencil of quintic
threefolds
\[
x_1+\cdots+x_5 - 5\psi x_1 \cdots x_5 = 0
\]
that plays a prominent role in mirror symmetry.

The approach to computing the L-function of these motives does not
require the direct counting of points of varieties over finite fields
nor the calculation of a corresponding automorphic form. Instead it
uses a p-adic formula for the trace of Frobenius, which is a finite
version of a hypergeometric function.

This approach has already proven to be quite efficient. However, some
issues need yet to be resolved in order to tackle a broader class of
cases (both higher conductors and higher degree of the Euler
factors). On the theoretical side these include:

- The precise general description of the Euler factors for primes of
bad reduction.
- An a priori calculation of the conductor, or at least an upper
bound on the power to which a prime might appear.
- A description of the parameters of an associated automorphic form
for a given motive (and possibly its calculation in some cases).
- A detailed proof of modularity in some specific cases.

On the computational side some issues to address are:
- The sharpening of the implementation of the p-adic formula for
the trace of Frobenius. This includes the computation of the
p-adic gamma function.
- The implementation of a broad and robust test for modularity.
- The computation of (higher degree) L-functions knowing only
their gamma factors and conductors, assuming integrality of
coefficients and the Ramanujan bound, and checking the
correspondence of these L-functions with those coming from
geometry.

## Material from the workshop

A list of participants.
The workshop schedule.

A report on the workshop activities.