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# Hypergeometric Motives

Communicated by *Notices* Associate Editor William McCallum

## 1. Introduction

It must have been frustrating in the early days of calculus that an integral like

appeared not to be expressible in terms of known functions. This type of integral arises in computing the movement of the ideal pendulum or the length of an arc of an ellipse for example; it has remained relevant and is connected to a great deal of the mathematics of the last 200 years.

Indeed

is an example of a hypergeometric series. It satisfies a linear differential equation of order two of the type brilliantly analyzed by Riemann. As mentioned by Katz Kat96, p.3, Riemann was lucky. His analysis only works because any order two differential equation on *rigid* in the sense that the local behavior of solutions around the missing points uniquely determines their global behavior.

Taking a more geometric perspective, 1.1 is presenting the function

This fact implies as well that

Shifting now to more arithmetic topics, if we fix a rational number

for almost all primes

Much of the importance of the *et al.* that the function

on the upper half plane is a modular form.

The discussion so far represents a standard general paradigm in arithmetic geometry. One starts with a smooth projective variety

This paper is an informal invitation to *hypergeometric motives*, hereafter called HGMs. They are very concrete so that their definition which we give in Section 4 can be understood with just the minimal background on motives that we provide there. We write an HGM as *gamma vector* *specialization point*

Sections 2–9 are geometric in nature. The main focus is on varieties generalizing 1.3 and the discrete aspect of periods like 1.1–1.2, as captured in vectors of Hodge numbers,

Sections 10–15 are arithmetic in nature, with the focus being on generalizations of 1.4, 1.6, and especially on the

Rigidity makes HGMs much more tractable than general motives: periods, Frobenius traces, and other invariants are given by explicit formulas in the parameters

## 2. Family Parameters

We begin by generalizing 1.1–1.2 using classical parameters. We then describe how in the cases we are interested in the same parameter data can be encoded in two other formats: the

### Integrals and series

Let

Via

Expanding the denominator of the integrand of 2.1 via the binomial theorem and using Euler’s beta integral to evaluate the individual terms, one obtains

where

### Monodromy

An excellent general reference for hypergeometric functions is BH89, and we now give a brief summary. The function

A useful fact due to Levelt is the explicit description of the matrices

Then

### Other formats

A way to encode the classical parameters

We call *family parameter*; its degree is the common degree of

for non-zero integers

As an illustration, the two formats for the parameter data in our introductory example are

where

To enter a family parameter

In the first method, one inputs just the gamma vector. In the second method, one inputs just the subscripts of the denominator and then numerator

### Three Magma notes

Initialization commands like 2.4 do not return output; in this survey, Magma will first start returning useful information in Sections 5 and 11. Also essential to know is that Magma requires a semicolon at the end of all commands, as in 2.4. We omit these semicolons in our in-line text when giving explicit Magma commands in the sequel. Finally, Magma conventions are reversed from the classical conventions, as

## 3. Source Varieties

There are different choices of varieties that can be used to define hypergeometric motives. Here we use certain affine varieties *canonical*. They appear under the term “circuits” in GKZ94 and are studied at greater length by Beukers, Cohen, and Mellit in BCM15.

### The canonical variety

For a gamma vector

Here and in the sequel, we systematically use the abbreviation

### Toric models

The BCM equations 3.1 are appealing since they define

First, for

So in the example of Table 3.1, the resulting equation is

with

### Toric models as parameterizations

The relation between the BCM equations 3.1 and a toric model for the same variety

When one uses 3.4 to write 3.1 in terms of the

### Polytopes

A toric model determines a polytope

The common topology of the

### Compactifications

In algebraic geometry, one normally wants to compactify a given open variety such as

We already saw the compactification

In our continuing example

Another compactification

## 4. HGMs from Cohomology

The HGMs

### Motives on an intuitive level

Let

“Motive” in this paper officially means “object in the category

As an example, suppose that