# Hypergeometric motives

June 25 to June 29, 2012

at the

International Centre for Theoretical Physics, Trieste, Italy

organized by

Henri Cohen and Fernando Rodriguez Villegas

This workshop, sponsored by AIM, NSF, 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 ICTP 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 $F(\frac12,\frac12, 1; t)$ 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.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for funding to participate in this workshop has passed.