## Workshop at ICTP June 2012

I am co-organizing a two-week activity this June at ICTP. The general theme of the school is that of computational algebra and number theory. The second week the school will include a more specific AIM workshop on hypergeometric motives. This workshop is a natural continuation of that in Benasque in 2009. I will be posting in this blog some comments on the matter. We will have a more formal site to organize the activities of the workshop shortly.

One of the main goals of the AIM workshop is to be able to explicitly compute the $L$-function of hypergeometric motives. This means computing numerically enough of the ingredients of the $L$-function in detail so as to, for example, check that it satisfies the expected functional equation. In fact, testing the functional equation is a good way to find a few of the ingredients, say the conductor, that might be missing. Once we have a large computable set of higher degree $L$-functions (i.e., with Euler factors of degree bigger than two say) we can begin exploring numerically various conjectures about their zeros, special values, etc.

So, what is a hypergeometric motive? The best place to start is the Legendre family of elliptic curves

$E_t: y^2=x(x-1)(x-t),$

Deuring computed the number of points of this family over the finite field $\mathbb F_p$ and showed that modulo $p$ is given in terms of the polynomial

$(-1)^{(p-1)/2}\sum_{n=0}^{(p-1)/2} \binom{2n}{n}^2t^n.$

Igusa later pointed out that this polynomial satisfies the same hypergeometric differential equation as a period of $E_t$ over the complex numbers.

On one hand we have for $t \in \mathbb Q$ a two dimensional vector space $H^1(E_t,\mathbb Q_l)$ with an action of Galois and on the other the two dimensional space of solutions to the hypergeometric differential equation. These are two incarnations of an abstract object: a “motive”.

Now the point is that we can start with *any* hypergeometric equation with appropriate parameters and it will yield a family of motives parameterized by $t$. (Because we have a rigid system the hypergeometric equation is determined by very simple discrete data; in other words, there is no moduli.) For each rational value of $t$ we get an associated $L$-function. These are what we want to be able to calculate.

The key fact is that though it is not entirely clear how to describe the motive (for example, how do we come up with the Legendre family of elliptic curves starting from the differential equation?) this description is not strictly necessary! Indeed, there is a formula like that of Deuring’s above that gives the trace of Frobenius on the motive. This finite field analogue of the hypergeometic series (due to N. Katz) is quite computable and it actually gives the trace as a $p$-adic number and not just modulo $p$. With this we can compute the Euler factors of our $L$-series at least for all but finitely many primes $p$ up to some bound.

The challenge that remains is how to compute the rest of the data of the $L$-function. Namely, i) the Euler factors for the remaining primes, ii) the Gamma factors, iii) the conductor and iv) the sign of the functional equation.

To complete this program some further description of the motive would in fact be helpful. But there isn’t necessarily a unique or canonical way to exhibit the motive. Different ways of presenting it bring their own advantages. To my mind a motive is somewhat like what genes are in biology. The same gene can appear in the DNA of very different life forms.

An example we worked out with Henri Cohen is a good illustration of this point. The motive has weight zero and corresponds to a Galois extension of $L/\mathbb Q(t)$ with Galois group the Weyl group of $F_4$ of order $1152$. The $L$-function of the motive is the Artin $L$-function attached to the reflection representation (of dimension $4$). As it turned out the motive also sits naturally in $H^2$ of an affine cubic surface $S$ with an Eckardt point. This surface has $24$ lines that are permuted by the Galois group. This description gives us an explicit degree $24$ extension $K/\mathbb Q(t)$ with Galois closure $L/\mathbb Q (t)$. From here it is not too hard to give the associated Artin $L$-function directly.