# Hypergeometric Motives

David P. Roberts
Fernando Rodriguez Villegas

##  1 Introduction

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

$F(t)=\frac{1}{\pi}\int_{0}^{1}\frac{1}{\sqrt{x(1-x)(1-tx)}}\,dx$ (1.1)

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; they have remained relevant and are
connected to a great deal of the mathematics of the last 200 years.

Indeed $F$ is not an elementary function. Its Maclaurin expansion

$F(t)=\sum_{k=0}^{\infty}\binom{2k}{k}^{2}\,\left(\frac{t}{16}\right)^{k}$
(1.2)

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 [Katz-RLS, p.3], Riemann was lucky. His analysis only works
because any rank two differential equation on
${\mathbb{P}}^{1}({\mathbb{C}})-\\{0,1,\infty\\}$ is 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 $\pi F$
as a period of the family of elliptic curves defined by

$E_{t}:\quad y^{2}=x(1-x)(x-t).$ (1.3)

This fact implies as well that $F$ satisfies an order two linear differential
equation, ultimately because $H^{1}(E_{t},{\mathbb{Q}})$ is two-dimensional.
††David P. Roberts is a professor at the University of Minnesota Morris. His
e-mail address is<EMAIL_ADDRESS>and his research is supported by
grant DMS-1601350 from the National Science Foundation. ††Fernando Rodriguez
Villegas is a senior research scientist at the Abdus Salam International
Centre for Theoretical Physics. His email address is<EMAIL_ADDRESS>
Shifting now to more arithmetic topics, if we fix a rational number $t\neq
0,1$ then for almost all primes $p$ the number $a_{p}$ defined by

$|E_{t}({\mathbb{F}}_{p})|=p+1-a_{p}$ (1.4)

is of fundamental importance. With these $a_{p}$ as the main ingredients, one
builds an $L$-function

$L(E_{t},s)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}.$ (1.5)

Much of the importance of the $a_{p}$ is seen through this $L$-function. For
example, the famous Birch-Swinnerton-Dyer conjecture says that the group
$E_{t}({\mathbb{Q}})$ modulo its torsion is isomorphic to ${\mathbb{Z}}^{r}$,
where $r$ is the order of vanishing of $L(E_{t},s)$ at $s=1$. A critical
advance is the result of Wiles et al. that the function

$f(z)=\sum_{n=1}^{\infty}a_{n}e^{2\pi iz}$ (1.6)

on the upper half plane is a modular form. In particular, this result implies
that $L(E_{t},s)$ is at least well-defined at $s=1$.

The equations displayed so far represent a standard general paradigm in
arithmetic geometry. One can start with any variety $X$ over ${\mathbb{Q}}$,
not just the varieties (1.3). There are fully developed theories of periods
and point counts, and in principle one can produce analogs of the period
formulas (1.1)-(1.2) and the point count formula (1.4). Interacting now with
deep but widely-believed conjectures, one can break the cohomology of $X$ into
irreducible motives, study $L$-functions like (1.5), and try to find
corresponding automorphic forms like (1.6).

This survey is an informal invitation to hypergeometric motives, hereafter
abbreviated HGMs; see Section 4 for their definition. We write them as
$H(Q,t)$, with a rational function $Q\in{\mathbb{Q}}(T)$ satisfying certain
conditions being the family parameter and $t\in{\mathbb{Q}}-\\{0,1\\}$ the
specialization parameter. The introductory family of examples is

$H\left({(T+1)^{2}}/{(T-1)^{2}},t\right)=H^{1}(E_{t},{\mathbb{Q}}).$ (1.7)

Rigidity makes HGMs much more tractable than general motives: periods, point
counts, and other invariants are given by explicit formulas in the parameters
$(Q,t)$. Our broader goal in this survey is to use HGMs to gain insight into
the general theory of motives; we illustrate all topics with explicit examples
throughout.

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, $h=(h^{w,0},\dots,h^{0,w})$. A theme
here is that HGMs form quite a broad class of irreducible motives, as very
general $h$ arise. Sections 10-15 are arithmetic in nature, with the focus
being on generalizations of (1.4), (1.6), and especially (1.5). Watkins has
written a very useful hypergeometric motives package [Wat] in Magma and
throughout this article we indicate how to use it by including small snippets
of Magma code. Together these snippets are enough to let Magma beginners
numerically compute with $L$-functions $L(H(Q,t),s)$ using the free online
Magma calculator.

##  2 Hypergeometric functions

We begin by generalizing (1.1)-(1.2) and explaining how this generalization
leads to family parameters.

### Integrals and series.

Let $\alpha=(\alpha_{1},\dots,\alpha_{n})$, and
$\beta=(\beta_{1},\dots,\beta_{n})$ be vectors of complex numbers with
$\mbox{Re}(\beta_{j})>\mbox{Re}(\alpha_{j})>0$ and $\beta_{n}=1$. For $|t|<1$
define, making use of the standard Gamma function
$\Gamma(s)=\int_{0}^{\infty}e^{-x}x^{s-1}dx$,

$\displaystyle
F(\alpha,\beta,t)=\prod_{i=1}^{n}\frac{\Gamma(\beta_{i})}{\Gamma(\alpha_{i})\Gamma(\beta_{i}-\alpha_{i})}\cdot$
$\displaystyle\int_{0}^{1}\cdots\int_{0}^{1}\frac{\prod_{i=1}^{n-1}\left(x_{i}^{\alpha_{i}-1}(1-x_{i})^{\beta_{i}-\alpha_{i}-1}dx_{i}\right)}{(1-tx_{1}\cdots
x_{n-1})^{\alpha_{n}}}.$

Via $\Gamma(1)=1$ and $\Gamma(1/2)=\sqrt{\pi}$, (1.1) is the special case
$\alpha=(1/2,1/2)$ and $\beta=(1,1)$.

Expand the denominator of the integrand of (2) via the binomial theorem and
use Euler’s beta integral to evaluate the individual terms. Written in terms
of Pochhammer symbols $(u)_{k}=u(+1)\cdots(u+k-1)$, the result is

$F(\alpha,\beta,t)={}_{n}F_{n-1}(\alpha,\beta,t):=\sum_{k=0}^{\infty}\frac{(\alpha_{1})_{k}\cdots(\alpha_{n})_{k}}{(\beta_{1})_{k}\dots(\beta_{n})_{k}}t^{k}.$
(2.2)

In other words, the integral (2) is an alternative definition of the standard
hypergeometric power series (2.2). The case $\alpha=(1/2,1/2)$ and
$\beta=(1,1)$ simplifies to (1.2).

### Monodromy.

An excellent general reference for hypergeometric functions is [BH], and we
now give a summary sufficient for this survey. The function
$F(\alpha,\beta,t)$ is in the kernel of an $n^{\rm th}$ order differential
operator $D(\alpha,\beta)$ with singularities only at $0$, $1$ and $\infty$.
This means in particular that $F(\alpha,\beta,t)$, initially defined on the
unit disk, extends to a “multivalued function” on the thrice-punctured
projective line
${\mathbb{P}}^{1}({\mathbb{C}})-\\{0,1,\infty)={\mathbb{C}}-\\{0,1\\}$. With
respect to a given basis, this multivaluedness is codified by a representation
$\rho$ of the fundamental group $\pi_{1}({\mathbb{C}}-\\{0,1\\},1/2)$ into
$GL_{n}({\mathbb{C}})$. The fundamental group is free on $g_{0}$ and $g_{1}$,
with these elements coming from counterclockwise circular paths of radius
$1/2$ about $0$ and $1$ respectively. To emphasize the equal status of
$\infty$ and $0$, it is better to present this group as generated by
$g_{\infty}$, $g_{1}$, and $g_{0}$, subject to the relation
$g_{\infty}g_{1}g_{0}=1$. The assumption $\beta_{n}=1$ was only imposed to
present the classical viewpoint (2)-(2.2) cleanly; we henceforth drop it.

A useful fact due to Levelt is the explicit description of the matrices
$h_{\tau}=\rho(g_{\tau})\in GL_{n}({\mathbb{C}})$ with respect to a certain
well-chosen basis. Define polynomials

$\displaystyle q_{\infty}$ $\displaystyle:=(T-e^{2\pi
i\alpha_{1}})\cdots(T-e^{2\pi i\alpha_{n}}),$ $\displaystyle q_{0}$
$\displaystyle:=(T-e^{-2\pi i\beta_{1}})\cdots(T-e^{-2\pi i\beta_{n}}).$

Then $h_{\infty}$ and $h_{0}$ are companion matrices of $q_{\infty}$ and
$q_{0}$, while $h_{1}$ is determined by $h_{\infty}h_{1}h_{0}=I$. The matrix
$h_{1}$ differs minimally from the identity in that $h_{1}-I$ has rank $1$. We
will henceforth consider only cases where no $\alpha_{i}-\beta_{j}$ is an
integer. This ensures that the $h_{\tau}$ generate an irreducible subgroup
$\Gamma$ of $GL_{n}({\mathbb{C}})$. Moreover the representation is rigid, in
the following sense: suppose $h_{\infty}^{\prime}$, $h^{\prime}_{1}$, and
$h_{0}^{\prime}$ are conjugate to $h_{\infty}$, $h_{1}$, and $h_{0}$
respectively. Then there is a single matrix $c$ such that
$h_{\tau}=ch_{\tau}^{\prime}c^{-1}$ for all three $\tau$.

### Family parameters.

The parameters $(\alpha,\beta)$ contain information which is irrelevant for
the sequel. First, the individual $\alpha_{j}$ and $\beta_{j}$ are important
only modulo integers. Second, the orderings of the $\alpha_{j}$ and
$\beta_{j}$ do not matter. To remove these irrelevancies, we will regard the
degree $n$ rational function $Q=q_{\infty}/q_{0}$ as the primary index in the
sequel, calling it the family parameter. A bonus of this shift in emphasis is
that an important field $E\subset{\mathbb{C}}$ is made evident, the field
generated by the coefficients of $q_{\infty}$ and $q_{0}$. By construction,
all three $h_{\tau}$ lie in $GL_{n}(E)$.

The cases which naturally have underlying motives are exactly the ones with
all $\alpha_{j}$ and $\beta_{j}$ rational, so that $E$ is some cyclotomic
field. In this survey we will substantially simplify by restricting to cases
with $E={\mathbb{Q}}$. With this simplification, there are two natural ways to
present $Q$ as follows. Write $\Psi_{m}=T^{m}-1$ and consider its
factorization into irreducible polynomials, $\Psi_{m}=\prod_{d|m}\Phi_{d}$. So
the factors are cyclotomic polynomials
$\Phi_{d}=\prod_{j\in({\mathbb{Z}}/d)^{\times}}(T-e^{2\pi ij/d})$ and thus
have degree the Euler totient $\phi(d)=|({\mathbb{Z}}/d)^{\times}|$.

In our introductory example, the ways are

$Q\;\;\;=\;\;\;\frac{(T^{2}-1)^{2}}{(T-1)^{4}}=\frac{\Psi^{2}_{2}}{\Psi_{1}^{4}}\;\;\;=\;\;\;\frac{(T+1)^{2}}{(T-1)^{2}}=\frac{\Phi^{2}_{2}}{\Phi_{1}^{2}}.$
(2.3)

In general, the second way is just the canonical factorization into
irreducibles, while the first way is the unique “unreduction” to products of
$\Psi_{m}$ in which no factor appears in both a numerator and denominator.

To enter a family parameter $Q$ into Magma, one can use either of these two
ways, as in the equivalent commands

Q:=HypergeometricData([*-2,-2,1,1,1,1*]); Q:=HypergeometricData([1,1],[2,2]);
(2.4)

In the first method, one inputs just the gamma vector
$\gamma=[\gamma_{1},\dots,\gamma_{l}]$ formed by subscripts on the $\Psi$’s,
using signs to distinguish between numerator and denominator. In the second
method, one inputs just the subscripts of the denominator and then numerator
$\Phi$’s, these being called the cyclotomic parameters. When working with
underlying varieties, the gamma vectors are so useful that we often simply
write $H(\gamma,t)$ rather than $H(Q,t)$. After the transition to motives, the
cyclotomic presentation is generally more convenient. To simplify slightly, we
henceforth require that $\gcd(\gamma_{1},\dots,\gamma_{l})=1$.

Note that initialization commands like (2.4) do nothing by themselves; in this
survey Magma will first start returning useful information in Sections 5 and
11. Note also that Magma requires a semicolon at the end of all commands, as
in (2.4). We often omit these semicolons in the sequel.

### Orthogonal vs. symplectic.

The number $Q(0)=\det(h_{1})$ is either $-1$ or $1$ under our restriction
$E={\mathbb{Q}}$. This dichotomy is strongly felt throughout this survey. It
can also be expressed in terms of the fundamental bilinear form
$\left(\cdot,\cdot\right)$ on ${\mathbb{Q}}^{n}$ preserved by the monodromy
group $\Gamma=\langle h_{\infty},h_{0}\rangle$; see [BH, §4], [RV-bez, §3.5].
In the orthogonal case, $h_{1}$ is conjugate to $\mbox{diag}(-1,1,\dots,1)$
and $\left(\cdot,\cdot\right)$ is symmetric. In the symplectic case, $h_{1}$
is conjugate to $\binom{1\;1}{0\;1}\oplus\mbox{diag}(1,\dots,1)$, and
$\left(\cdot,\cdot\right)$ is antisymmetric.

##  3 Source varieties

We now describe varieties which give rise to hypergeometric motives.

### Euler varieties.

We have already generalized (1.1) to (2) and (1.2) to (2.2). Assuming briefly
$\beta_{n}=1$ again, a natural generalization of (1.3) is to

$y^{m}=\prod_{j=1}^{n-1}x_{j}^{a_{j}}(1-x_{j})^{b_{j}}(1-tx_{1}\cdots
x_{n-1})^{a_{n}}.$ (3.1)

Here $m$ is the least common denominator of the $\alpha_{j}$ and $\beta_{j}$,
and the exponents are integers $0\leq a_{j},b_{j}<m$ such that

$a_{j}\equiv-m\alpha_{j}\bmod m,\qquad b_{j}\equiv
m(\alpha_{j}-\beta_{j})\bmod m,$

for $j=1,\ldots,n-1$ and $a_{n}\equiv m\alpha_{n}\bmod m$. The equations
(2)-(2.2) show that a specified scalar multiple of
${}_{n}F_{n-1}(\alpha,\beta,t)$ arises as a period of this variety. However
the varieties (3.1) depend on how the parameters are paired:
$(\alpha_{1},\beta_{1})$, …, $(\alpha_{n},\beta_{n})$. This dependence
complicates the arithmetic of these varieties, so we will use an alternative
collection of varieties to define hypergeometric motives.

### Canonical varieties.

The alternative varieties appear under the term “circuits” in [GKZ] and are
studied at greater length in [BCM]. For a gamma vector $\gamma$ and a complex
number $t$, define $X^{\rm bcm}_{\gamma,t}\subset{\mathbb{P}}^{l-1}$ by two
homogeneous equations,

$\displaystyle\sum_{j=1}^{l}y_{j}$ $\displaystyle=0,$
$\displaystyle\prod_{\gamma_{j}>0}y_{j}^{\gamma_{j}}=u\prod_{\gamma_{i}<0}y_{j}^{-\gamma_{j}}.$
(3.2)

Here and in the sequel, we systematically use the abbreviation
$u=t\prod_{j}\gamma_{j}^{\gamma_{j}}$. The canonical variety is by definition
the open subvariety $X_{\gamma,t}$ on which all the homogeneous coordinates
$y_{j}$ are nonzero. The point $(\gamma_{1}:\cdots:\gamma_{l})$ is an ordinary
double point on $X_{\gamma,1}$ and otherwise all the $X_{\gamma,t}$ are
smooth. Because of this double point, we exclude the case $t=1$ from
consideration until Section 9.

### Toric models.

From a dimension-count viewpoint, the BCM equations (3.2) for canonical
varieties are inefficient. They start with the $l=\kappa+3$ variables $y_{i}$
and use two equations and projectivization to get the desired
$\kappa$-dimensional variety $X_{\gamma,t}$. The toric models from [GKZ] start
instead with $d=\kappa+1$ variables $x_{i}$ and present $X_{\gamma,t}$ by just
one equation.

To obtain a toric model from a gamma vector $\gamma$, one proceeds as
illustrated by Table 3.1. First, for each new variable $x_{i}$ choose a row
vector $m_{i*}$ in ${\mathbb{Z}}^{l}$ which is orthogonal to the given
$l$-vector $\gamma$. These row vectors are required to be such that
${\mathbb{Z}}^{l}/\langle m_{i*}\rangle$ is torsion-free. Second, choose a row
vector $k\in{\mathbb{Z}}^{l}$ which satisfies $\gamma\cdot k=1$. The toric
model is then

$\sum_{i=1}^{l}u^{k_{i}}\prod_{j=1}^{d}x_{i}^{m_{ij}}=0.$ (3.3)

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

$x_{1}^{2}+ux_{1}x_{2}^{2}+u+x_{1}^{3}x_{2}=0,$ (3.4)

with $u=-2^{6}3^{3}t/5^{5}$. In general, the variety $X_{\gamma,t}$ is the
subvariety of the torus ${\mathbb{G}}_{m}^{d}$ given by the equation (3.3).

${{{{{\begin{array}[]{r|cccc|}\cline{2-5}\cr&\gamma_{1}&\gamma_{2}&\gamma_{3}&\gamma_{4}\\\
\cline{2-5}\cr x_{1}:&m_{11}&m_{12}&m_{13}&m_{14}\\\
x_{2}:&m_{21}&m_{22}&m_{23}&m_{24}\\\ \cline{2-5}\cr
u:&k_{1}&k_{2}&k_{3}&k_{4}\\\
\cline{2-5}\cr\end{array}=\begin{array}[]{|cccc|}\cr\hrule
height=0.4pt}-5&-2&3&4\\\ \cr\hrule height=0.4pt}2&1&0&3\\\ 0&2&0&1\\\
\cr\hrule height=0.4pt}0&1&1&0\\\ \cr\hrule height=0.4pt}\end{array}}$ Table
3.1:  Derivation of the equation (3.4) for $X_{[-5,-2,3,4],t}$

The relation between the BCM equation for $X_{\gamma,t}$ and a toric model for
$X_{\gamma,t}$ is very simple:

$y_{j}=u^{k_{j}}\prod_{i=1}^{d}x_{i}^{m_{ij}}.$ (3.5)

When one uses (3.5) to write (3.2) in terms of the $x_{i}$, the second
equation is identically satisfied while the first becomes (3.3). Conversely,
any point $(y_{1}:\dots:y_{l})$ comes from a unique $(x_{1},\dots,x_{d})$
because of the torsion-free condition.

### Polytopes.

A toric model gives a polytope $\Delta\subset{\mathbb{R}}^{d}$ which is an aid
to understanding the $X_{\gamma,t}$. The case $d=2$ is readily visualizable
and Figure 3.1 continues our example. In general, one interprets the column
vectors $m_{*j}$ of the chosen matrix as points in ${\mathbb{Z}}^{d}$ and
$\Delta$ is their convex hull. Let $\Delta_{j}$ be the convex hull of all the
points except the $j^{\rm th}$ one. Normalize volume so that the standard
$d$-dimensional simplex has volume $1$, and thus $[0,1]^{d}$ has volume $d!$.
Then the volume of $\Delta_{j}$ is $|\gamma_{j}|$. The $\Delta_{j}$ with
$\gamma_{j}>0$ form one triangulation of $\Delta$, while the $\Delta_{j}$ with
$\gamma_{j}<0$ form another. The total volume of $\Delta$ is the important
number $\mbox{vol}(\gamma)=\frac{1}{2}\sum_{j=1}^{l}|\gamma_{j}|$

$-2$$-5$$4$$3$ Figure 3.1:  The triangulations
$\Delta=\Delta_{1}\cup\Delta_{2}$ and $\Delta=\Delta_{3}\cup\Delta_{4}$ of the
polytope $\Delta$ for the family with $\gamma=[-5,-2,3,4]$. The points are at
the column vectors $\binom{m_{1j}}{m_{2j}}$ of Table 3.1, and $\gamma_{j}$ is
printed in the opposite triangle.

The common topology of the $X_{\gamma,t}$ with $t\neq 1$ is reflected in the
combinatorics of $\Delta$. In the case of $d=2$, the genus $g$ of
$X_{\gamma,t}$ is the number of lattice points on the interior, while the
number of punctures $k$ is the number of lattice points on the boundary.
Pick’s theorem then says that the Euler characteristic $\chi=2-2g-k$ of
$X_{\gamma,t}$ is $-\mbox{vol}(\gamma)$. In the example of Figure 3.1,
$(g,k,\chi)=(2,5,-7)$. For larger ambient dimension $d$, the situation is of
course much more complicated, but always $\chi=(-1)^{d-1}\mbox{vol}(\gamma)$.

### Compactifications.

In algebraic geometry, one normally wants to compactify a given open variety
such as $X_{\gamma,t}$ and there are typically many natural ways of doing it.

We already saw the compactification $X_{\gamma,t}^{\rm bcm}$. It is a
hypersurface of degree $\mbox{vol}(\gamma)$ in the projective space
${\mathbb{P}}^{d}$ defined by the first equation of (3.2). On the other hand,
for any choice of matrix $m$ with all entries nonnegative, homogenization of
(3.3) gives a alternative compactification
$\overline{X}_{\gamma,t}\subset{\mathbb{P}}^{d}$.

In our continuing example $\gamma=[-5,-2,3,4]$, the plane curve
$X_{\gamma,t}^{\rm bcm}$ has degree seven. In contrast, the plane curve
$\overline{X}_{\gamma,t}$ has degree just four, this number arising as the
maximum column sum of the matrix $m$ in Table 3.1. Smooth curves in these
degrees have genera $15$ and $3$ respectively. For $t\neq 1$, $X_{\gamma,t}$
has genus $2$ so $X_{\gamma,t}^{\rm bcm}$ must have bad singularities while
$\overline{X}_{\gamma,t}$ has just a single node.

Another compactification $X_{\gamma,t}^{\rm BCM}$ is a major focus of [BCM].
It is typically not smooth, but only has quotient singularities. These
singularities are mild in the sense that $X_{\gamma,t}^{\rm BCM}$ looks smooth
from the viewpoint of rational cohomology, and may be ignored when discussing
motives as in the next section.

##  4 HGMs from cohomology

Here we define HGMs and explain how their behavior is simpler than other
similar motives.

### Motivic formalism.

Let $K$ and $E$ be subfields of ${\mathbb{C}}$; the case of principal interest
to us is $K=E={\mathbb{Q}}$. Minimally modifying Grothendieck’s original
conditional definitions, André unconditionally defined a category
$\mathcal{M}(K,E)$ of pure motives over $K$ with coefficients in $E$ [And].
The formal structures of this category can best be understood in terms of a
huge proreductive algebraic group ${\mathbb{G}}_{K}$ over ${\mathbb{Q}}$, the
absolute motivic Galois group of $K$. Then $\mathcal{M}(K,E)$ is exactly the
category of representations of ${\mathbb{G}}_{K}$ on finite-dimensional $E$
vector spaces.

When taking cohomology, we are always implicitly working with the complex
points of a variety. For a smooth projective variety $X$ over $K$ and an
integer $w$, the singular cohomology space $M=H^{w}(X,E)$ is an object of
$\mathcal{M}(K,E)$. The image $G_{M}$ of ${\mathbb{G}}_{K}$ in the general
linear group of $M$ is by definition the motivic Galois group of $M$. The
purpose of $G_{M}$, as the rest of this survey will make clear, is to group-
theoretically coordinate very concrete structures on the vector space
$H^{w}(X,E)$.

Two copies of the multiplicative group ${\mathbb{G}}_{m}=GL_{1}$ play
important roles in the formalism of motives. One is a normal subgroup and the
other a quotient:
${\mathbb{G}}_{m}\subset{\mathbb{G}}_{K}\twoheadrightarrow{\mathbb{G}}_{m}$. A
rank $n$ motive $M$ is said to be of weight $w$ if the representation
restricted to the subgroup ${\mathbb{G}}_{m}$ consists of $n$ copies the
representation $r\mapsto r^{w}$. The motives $H^{w}(X,E)$ all have weight $w$.
The representation of ${\mathbb{G}}_{K}$ on the rank one motive
$E(-1):=H^{2}({\mathbb{P}}^{1},E)$ corresponds to the representation $t\mapsto
t$ of the quotient group ${\mathbb{G}}_{m}$. The motive corresponding to the
representation $t\mapsto t^{j}$ is denoted $E(-j)$. The motives
$M(j):=M\otimes E(j)$ are called the Tate twists of $M$.

Each category of pure motives $\mathcal{M}(K,E)$ is contained in a larger
category $\mathcal{M}\mathcal{M}(K,E)$ of mixed motives, where an irreducible
motive has a canonical weight filtration with subquotients in
$\mathcal{M}(K,E)$. We will mention mixed motives at several junctures, but
our focus is sharply on pure motives.

### Definition of HGMs.

Let $\gamma$ be a gamma vector of length $\kappa+3$ with $r$ negative entries
and let $t\in{\mathbb{Q}}^{\times}-\\{1\\}$. The hypergeometric motive
$H(\gamma,t)$ is defined from the cohomology of the affine variety
$X_{\gamma,t}$ [RV-Mixed]. We start with the compactly supported middle
cohomology space $H_{c}^{\kappa}(X_{\gamma,t},{\mathbb{Q}})$ and begin by
cutting out a subquotient $H^{\prime}(\gamma,t)$ in two steps.

First, we eliminate the contribution of the ambient $d$-dimensional torus to
obtain the primitive subspace $PH_{c}^{\kappa}(X_{\gamma,t},{\mathbb{Q}})$.
Second, we take any smooth compactification $\bar{X}$ of $X_{\gamma,t}$, or
one with at worst mild singularities as mentioned above, and consider the
image $H^{\prime}(\gamma,t)$ of $PH_{c}^{\kappa}(X_{\gamma,t},{\mathbb{Q}})$
under the natural map to $H^{\kappa}(\bar{X},{\mathbb{Q}})$. As a quotient of
$PH_{c}^{\kappa}(X_{\gamma,t},{\mathbb{Q}})$, the space $H^{\prime}(\gamma,t)$
is independent of the choice of compactification.

For example, for the compactification $X^{BCM}_{\gamma,t}$ there is a
decomposition of its middle cohomology,
$H^{\kappa}(X^{BCM}_{\gamma,t},{\mathbb{Q}})=H^{\prime}(\gamma,t)\oplus T$. It
is described at the level of point counts in [BCM, Thm 1.5]. Here $T$ is zero
if $\kappa$ is odd and the sum of $\binom{\kappa+1}{r-1}$ copies of
${\mathbb{Q}}(-\kappa/2)$ if $\kappa$ is even.

Finally, we define the hypergeometric motive
$H(\gamma,t)\in\mathcal{M}({\mathbb{Q}},{\mathbb{Q}})$ as the Hodge-normalized
Tate twist $H^{\prime}(\gamma,t)(j)$, as discussed in the next section. So
$H(\gamma,t)$ has weight $w=\kappa-2j$ with $j\in{\mathbb{Z}}_{\geq 0}$
specified there.

More conceptually, $PH_{c}^{\kappa}(X_{\gamma,t},{\mathbb{Q}})$ is a mixed
motive of rank $\operatorname{vol}(\gamma)-1$ and the pure motive
$H^{\prime}(\gamma,t)$ is its weight $\kappa$ quotient. The passage from
$PH_{c}^{\kappa}(X_{\gamma,t},{\mathbb{Q}})$ to $H^{\prime}(\gamma,t)$ is
closely related to the reduction of fractions as in (2.3). In particular,
$H^{\prime}(\gamma,t)$ has rank $n=\deg(Q)$.

It is worth stressing that the full mixed motive
$PH_{c}^{\kappa}(X_{\gamma,t},{\mathbb{Q}})$ is itself of great interest, with
its lower weight parts playing an important role in deeper studies of
hypergeometric motives.

### Motivic Galois groups of HGMs.

For $t\in{\mathbb{C}}^{\times}-\\{1\\}$, one likewise gets a motive
$M=H(\gamma,t)\in\mathcal{M}({\mathbb{Q}}(t),{\mathbb{Q}})$. lf $t$ is
transcendental then the motivic Galois group of $M$ can be cleanly expressed
in terms of the monodromy group $\Gamma$ of Section 2, as follows. If $w=0$,
then $\Gamma$ is finite and $G_{M}=\Gamma$. If $w>0$, then $\Gamma$ is
infinite and $G_{M}$ is the smallest algebraic group containing both $\Gamma$
and scalars; more explicitly, $G_{M}$ is the conformal symplectic group
$\mbox{CSp}_{n}$ in the symplectic case of odd $w$, and a conformal orthogonal
group $\mbox{CO}_{n}$ in the orthogonal case of even $w$. In the case that $t$
is algebraic, including our main case that $t$ is rational, the same
identification of $G_{M}$ holds almost always.

### Related motives.

The toric model viewpoint is part of the program in [GKZ] to approach
algebraic geometry by emphasizing the number $\kappa+\epsilon$ of terms in
polynomials defining $\kappa$-dimensional varieties. By scaling to normalize
coefficients, such varieties come in $(\epsilon-2)$-dimensional families.

For $\epsilon=2$, the Newton polytope $\Delta$ is a simplex. An abelian group
$A$ of order $\mbox{vol}(\Delta)$ and some exponent $m$ acts on the single
associated complex variety $X$. The essential cases here are the Fermat
varieties in ${\mathbb{P}}^{\kappa+1}$, defined by

$x_{1}^{m}+\cdots+x_{\kappa+2}^{m}=0.$ (4.1)

The group $A$ comes from scaling the variables by $m^{\rm th}$ roots of unity
and has order $m^{\kappa+1}$ and exponent $m$. Writing $K={\mathbb{Q}}(e^{2\pi
i/m})$, the action decomposes $H^{\kappa}(X,K)$ into one-dimensional motives
in $\mathcal{M}(K,K)$. This setting of $\epsilon=2$ was the focus of several
influential papers of Weil from around 1950, and the rank one motives
appearing are Jacobi motives.

The case $\epsilon=3$ corresponds to general hypergeometric motives where the
$\alpha_{j}$ and $\beta_{j}$ can be arbitrary rational numbers. The group $A$
now has order a divisor of $\mbox{vol}(\Delta)$ and some exponent $m$. For
example, for $m=\kappa+2\geq 3$ one can add the term $ux_{1}\cdots x_{m}$ to
(4.1); then $A$ is reduced to having order $m^{\kappa}$ but still has exponent
$m$. The action of $A$ again decomposes $H^{\kappa}(X,K)$ in
$\mathcal{M}(K,K)$ and the summands include general hypergeometric motives.
Our torsion-free requirement for exponent matrices is equivalent to requiring
that the column vectors affine span ${\mathbb{Z}}^{d}$; in turn, this means
that our HGMs constitute exactly the case $|A|=1$.

Much of what we are describing in this article both has simpler analogs for
Jacobi motives and extends to general hypergeometric motives. Indeed [BH] and
[Katz-ESDE] are in the latter setting. However the associated $L$-functions
correspond to motives that have been descended to
$\mathcal{M}({\mathbb{Q}},{\mathbb{Q}})$ and have rank $\phi(m)n$. Because of
the factor $\phi(m)$, inclusion of these other settings would only modestly
increase the collection of computationally accessible $L$-functions. Also the
resulting motives in $\mathcal{M}({\mathbb{Q}},{\mathbb{Q}})$ have motivic
Galois groups which are more complicated than the $\mbox{CSp}_{n}$ and
$\mbox{CO}_{n}$ arising ubiquitously in our setting of $H(\gamma,t)$.

##  5 Hodge numbers

One of the very first things one wants to know about a motive is its Hodge
numbers. Fortunately, this desire is easily satisfied for HGMs by an appealing
procedure.

### Background.

For a smooth projective variety $X$ over $K\subseteq{\mathbb{C}}$, there is a
decomposition of complex vector spaces
$H^{w}(X,{\mathbb{C}})=\bigoplus_{p=0}^{w}H^{p,w-p}$, with complex conjugation
on coefficients switching $H^{p,q}$ and $H^{q,p}$. The Hodge numbers
$h^{p,q}:=\dim(H^{p,q})$ therefore satisfy Hodge symmetry $h^{p,q}=h^{q,p}$
and sum to the Betti number $b_{w}:=\dim(H^{w}(X,{\mathbb{C}}))$. Classical
examples are given in (6.1)-(6.2) below.

Likewise, the rank of a weight $w$ motive $M\in\mathcal{M}(K,E)$ is decomposed
into Hodge numbers $h^{p,w-p}$. The decomposition has a simple group-theoretic
reformulation: ${\mathbb{G}}_{K}({\mathbb{R}})$ contains a subgroup
${\mathbb{C}}^{\times}$ which acts on $H^{p,q}$ by $z^{p}\overline{z}^{q}$. If
either $K$ or $E$ is in ${\mathbb{R}}$, as will generally be the case for us,
then Hodge symmetry continues to hold.

If a motive $M$ has Hodge numbers $\underline{h}^{p,q}$ then the Hodge numbers
of its Tate twist $M(j)$ are ${h}^{p-j,q-j}=\underline{h}^{p,q}$. The Hodge-
normalization of a pure weight motive is the Tate twist for which all the
nonzero Hodge numbers are in the vector $h=(h^{w,0},\dots,h^{0,w})$ and at
least one of the outer ones is nonzero.

### Zigzag procedure.

The procedure we are about to describe is equivalent to a formula conjectured
by Corti and Golyshev [CG] and proved by different methods in Fedorov [Fed]
and [RV-Mixed]. The procedure is completely combinatorial and only depends on
the interlacing pattern of the roots of $q_{\infty}$ and $q_{0}$ in the unit
circle.

$\frac{1}{4}$$\frac{1}{3}$$\frac{1}{2}$$\frac{2}{3}$$\frac{3}{4}$$\frac{1}{5}$$\frac{2}{5}$$\frac{3}{5}$$\frac{4}{5}$$1$$1$$3$$1$$1$$3$$1$$\alpha_{j}$’s:$\beta_{j}$’s:
Figure 5.1:  The zigzag procedure with input the family parameter
$Q={\Phi_{2}\Phi_{3}\Phi_{4}}/{\Phi_{1}\Phi_{5}}$ and output the Hodge vector
$h=(1,3,1)$

To pass from a family parameter $Q=q_{\infty}/q_{0}$ to its Hodge vector $h$
one proceeds as illustrated by Figure 5.1. One orders the parameters
$\alpha_{j}$ and $\beta_{j}$, viewed as elements in say $(0,1]$; for more
immediate readability, we associate the colors red and blue to $\infty$ and
$0$ respectively. One draws a point at $(0,0)$ corresponding to the smallest
parameter in a Cartesian plane. One then proceeds in uniform steps from left
to right, drawing a point for each parameter and then moving diagonally
upwards after red points and diagonally downwards after blue points. One
focuses on one color or the other, counting the number of points on horizontal
lines. The numbers obtained form the Hodge vector $h$. The red and blue dots
yield the same Hodge vector but contain more information. They may be used to
describe the limiting mixed Hodge structure at $t=\infty$ and $0$
respectively.

### The completely intertwined case.

Complete intertwining of the $\alpha_{j}$ and $\beta_{j}$ gives the extreme
where the resulting Hodge vector is just $(n)$. Beukers and Heckman [BH]
proved that complete intertwining is exactly the condition one needs for the
monodromy group $\langle h_{\infty},h_{0}\rangle$ to be finite. They also
established the complete list of such pairs $(\alpha,\beta)$. Actually they,
like Schwarz who famously treated the $n=2$ case more than a century earlier,
worked without our standing assumption $E={\mathbb{Q}}$. Then one needs to
require complete intertwining of all the natural conjugates of
$(\alpha,\beta)$ and the list obtained is longer.

In our setting of $E={\mathbb{Q}}$, the corresponding $\gamma$-vectors are of
odd lengths $3$ to $9$. There are infinite collections of length $3$ and $5$
given by coprime positive integers $a,b$:

$\displaystyle(i)$ $\displaystyle\quad[-(a+b),a,b],$ (5.1) $\displaystyle(ii)$
$\displaystyle\quad[-2(a+b),-a,2a,b,a+b],$ $\displaystyle(iii)$
$\displaystyle\quad[-2a,-2b,a,b,a+b].$

Here and always when discussing classification, we omit consideration of
$-\gamma$ whenever $\gamma$ is listed. In case $(i)$, the canonical variety
consists of just $a+b$ points. Removing a variable, the BCM presentation takes
the form

$X_{a,b,t}:y^{a}(1-y)^{b}-\frac{a^{a}b^{b}}{(a+b)^{a+b}}t=0.$

For $b=1$ this presentation is already trinomial; in general, one has to make
a non-trivial change of variables to pass to the trinomial presentation of
$X_{a,b,t}$ given by a toric model.

Beyond the closely related collections $(i)$ \- $(iii)$, there are only
finitely many further $\gamma$, all related to Weyl groups. [BH, Table 8.3]
says that, modulo the quadratic twisting operation $Q(T)\mapsto Q(-T)$, there
are just one, five, five, and fifteen respectively for the groups $W(F_{4})$,
$W(E_{6})$, $W(E_{7})$, and $W(E_{8})$. One of the $W(E_{6})$ cases is
discussed in Section 7 and the remaining $W(E_{n})$ cases are similarly
treated in [Rob-polys].

### The completely separated case.

Complete separation of the $\alpha_{j}$ and $\beta_{j}$ gives the extreme
where the resulting Hodge vector is $(1,1,\dots,1,1)$. The subcase where
$q_{0}=(T-1)^{n}$ has the simplifying feature that $h_{0}$ consists of a
single Jordan block. Families in this subcase have received particular
attention in the literature; the condition is sometimes verbalized as MUM, for
maximal unipotent monodromy.

Classification of families in the completely separated case is easier than in
the completely intertwined case. It becomes trivial in the MUM subcase because
$q_{\infty}$ is arbitrary except for the fact that it contains no factors of
$(T-1)$. Accordingly, the number $c_{n}$ of rank $n$ families in the MUM
subcase is given by a generating function

$\displaystyle\sum_{n=0}^{\infty}c_{n}x^{n}$ $\displaystyle=$
$\displaystyle\prod_{k=2}^{\infty}\frac{1}{1-x^{\phi(k)}}$ $\displaystyle=$
$\displaystyle 1+x+4x^{2}+4x^{3}+14x^{4}+14x^{5}+\cdots$

Always $c_{2j}=c_{2j+1}$ as under the MUM restriction multiplying by
$(T+1)/(T-1)$ gives a bijection on parameters. Arithmetic information about
the list underlying $c_{4}=14$ is in [RV]. For general $n$, the case
$q_{\infty}=1+T+\cdots+T^{n}$ is the “mirror dual” of the Dwork case discussed
after (4.1), and so motives of this family have been given special attention
in the physics literature.

### Signature and the Magma implementation.

A motive defined over a subfield of ${\mathbb{R}}$ has a signature $\sigma$,
which is the trace of complex conjugation. For odd weight motives, it is
always zero. For even weight HGMs $H(Q,t)$, it depends only on $Q$ and the
interval $(-\infty,0)$, $(0,1)$, or $(1,\infty)$ in which $t$ lies. Magma’s
command `HodgeStructure` returns both the Hodge vector and the signature in
coded form. To see just the Hodge vector clearly, one can implement `Q` as in
(2.4) and get the Hodge vector from say

`HodgeVector(HodgeStructure(Q,2));`

For example, from the gamma vector $[-21,1,2,3,4,5,6]$ one gets the Hodge
vector $(1,2,12,2,1)$.

##  6 Projective Hypersurfaces

Here we realize some HGMs in the cohomology of the most classical varieties of
all, smooth hypersurfaces in projective space.

### Hodge numbers.

Let $X\subset{\mathbb{P}}^{\kappa+1}$ be a smooth hypersurface of degree
$\delta$. Let $PH^{\kappa}(X,{\mathbb{Q}})$ be the primitive part of its
middle cohomology, meaning the part that does not come from the ambient
projective space. If $\kappa$ is odd, then this primitive part is all of
$H^{\kappa}(X,{\mathbb{Q}})$. If $\kappa$ is even, then the complementary
piece that we are discarding is ${\mathbb{Q}}(-\kappa/2)$.

Hirzebruch gave a formula for the Hodge numbers of
$PH^{\kappa}(X,{\mathbb{Q}})$ as a function of $\kappa$ and $\delta$. For
example, the sum of the Hodge numbers and first Hodge number are respectively

$\displaystyle b_{\kappa}$
$\displaystyle=\frac{(\delta-1)^{\kappa+2}+(-1)^{\kappa}(\delta-1)}{\delta},$
$\displaystyle h^{\kappa,0}$ $\displaystyle=\binom{\delta-1}{\kappa+1}.$ (6.1)

These special cases and Hodge symmetry are sufficient to give Hodge vectors
when $\kappa\leq 3$:

${{\begin{array}[]{c|ccc}\delta&\mbox{Curves in
${\mathbb{P}}^{2}$}&\mbox{Surfaces in ${\mathbb{P}}^{3}$}&\mbox{Threefolds in
${\mathbb{P}}^{4}$}\\\ \cr\hrule height=0.4pt}3&(1,1)&(0,6,0)&(0,5,5,0)\\\
4&(3,3)&(1,19,1)&(0,30,30,0)\\\ 5&(6,6)&(4,44,4)&(1,101,101,1)\\\
6&(10,10)&(10,85,10)&\,(5,255,255,5).\end{array}}$ (6.2)

For $\kappa=1$, either part of (6.1) reduces to the genus formula for smooth
plane curves, $g=(\delta-1)(\delta-2)/2$.

### One example for every $(\delta,\kappa)$.

Let $\delta=e+1\geq 3$ be a desired degree and let $\kappa$ be a desired
dimension. Define

$\gamma=[1,-e,e^{2},\ldots,(-e)^{\kappa},(-e)^{\kappa+1}-1,\frac{(-e)^{\kappa+2}+e}{e+1}].$
(6.3)

The toric procedure illustrated by Table 3.1 yields the completed canonical
variety:

$\displaystyle X_{t}:\ \quad
ux_{\kappa+1}x_{1}^{e}+\sum_{i=2}^{\kappa+2}x_{i-1}x_{i}^{e}+x_{\kappa+2}^{e+1}=0.$
(6.4)

The necessary orthogonality relations on each variable’s exponents are
illustrated by the case of cubic fourfolds where
$\gamma=[1,-2,4,-8,16,-33,22]$. Then (6.4) becomes

$X_{t}:ux_{5}x_{1}^{2}+x_{1}x_{2}^{2}+x_{2}x_{3}^{2}+x_{3}x_{4}^{2}+x_{4}x_{5}^{2}+x_{5}x_{6}^{2}+x_{6}^{3}=0.$

For $x_{5}$, the relation is that $m_{5*}=(1,0,0,0,2,1,0)$ is orthogonal to
$\gamma$. In general, partial derivatives of (6.4) are very simple since the
row vectors $m_{i*}$ have just two nonzero entries, except for the case
$i=\kappa+1$ and its three nonzero entries. It is then a pleasant exercise to
check via the Jacobian criterion that $X_{t}$ is smooth for
$t\in{\mathbb{C}}^{\times}-\\{1\\}$.

The degree of the rational function $Q$ determined by $\gamma$ can be computed
uniformly in $(\delta,\kappa)$ as the cancellations to be analyzed are very
structured. This degree agrees with the Betti number $b_{\kappa}$ from (6.1).
Thus $H(\gamma,t)$ is the full primitive middle cohomology of $X_{t}$, while a
priori it might have been a proper subspace. The zigzag procedure for
computing Hodge numbers must agree in the end with the Hirzebruch formula. The
reader might want to check the above case of cubic fourfolds, where
Hirzebruch’s full formula gives $(0,1,20,1,0)$.

### All examples for a given $(\delta,\kappa)$.

An interesting problem is to find all $\gamma$ which give projective smooth
$\kappa$-folds of degree $\delta$. For small parameters, this problem can be
solved by direct computation. For example, consider $(\delta,\kappa)=(3,4)$,
thus cubic fourfolds. In this case, one has the standardization
$[-33,-8,-2,1,4,16,22]$ of the above example, and then exactly ten more:

${{\\!\\!\begin{array}[]{ll}\,[-48,-15,-12,5,16,24,30],&[-36,-9,-4,3,8,18,20],\\\
\,[-48,-12,-3,1,6,24,32],&[-33,-16,-4,2,8,11,32],\\\
\,[-48,-12,-3,6,16,17,24],&[-33,-10,-7,5,11,14,20],\\\
\,[-36,-16,-9,3,8,18,32],&[-33,-4,-1,2,8,11,17],\\\
\,[-36,-9,-8,4,15,16,18],&[-21,-20,-16,7,8,10,32].\end{array}\\!\\!}}$

##  7 Dimension reduction

An HGM $H(\gamma,t)$ is defined in terms of a $\kappa$-dimensional variety but
its Hodge vector $(h^{w,0},\dots,h^{0,w})$ raises the question of whether it
also comes from a variety of dimension $w=\kappa-2j$. The exterior zeros for
low degree projective hypersurfaces as illustrated in (6.2) raise the same
question. The generalized Hodge conjecture says that this dimension reduction
is always possible. We illustrate here some of the appealing geometry that
arises from reducing dimension.

### Reduction to points.

When $w=0$ the reduction to dimension zero is possible in all cases. For
example, $\gamma=[-12,-3,1,6,8]$ corresponds to entry 45 on the Beukers-
Heckman list [BH, Table 8.3]. Formula (3.3) then gives a family $X_{t}$ of
cubic surfaces. An equation whose roots correspond to the famous twenty-seven
lines on $X_{t}$ is

$\displaystyle 2^{4}tx^{3}(x^{2}-3)^{12}-3^{9}(x^{3}-3x^{2}+x+1)^{8}(x-2)=0.$
(7.1)

The Galois group of this polynomial $g(t,x)$ for generic
$t\in{\mathbb{Q}}^{\times}-\\{1\\}$ is $W(E_{6})$. It has $51840=2^{7}3^{4}5$
elements and is also the monodromy group $\Gamma=\langle
h_{\infty},h_{0}\rangle$.

### Reduction via splicing.

Suppose $\gamma$ can be written as the concatenation of two lists each summing
to zero. Then one can use a general splicing technique from [BCM, §6] to
reduce the dimension by two. This technique is behind the scenes even of our
introduction: in the family of examples there, the canonical varieties for
$[-2,-2,1,1,1,1]$ are three-dimensional, although the more familiar source
varieties are just the Legendre curves (1.3).

For an example complicated enough to be representative of the general case,
take $\gamma=[-12,-3,-2,1,1,1,6,8]$ so that the canonical variety has
dimension $\kappa=5$. The Hodge vector is just $(3,3)$, so one would like to
realize $H(\gamma,t)$ in the cohomology of a curve.

Splicing is possible because both $[-12,-3,1,6,8]$ and $[-2,1,1]$ sum to zero.
No further splicing is possible, but fortunately we have just treated the
first sublist by other means. Splicing corresponds to taking a fiber product
over the $t$-line which in turn corresponds to just multiplying rational
functions. In our case, solving (7.1) for $t$ to get the first factor, the
dimension-reduced variety is given by

$\frac{3^{9}(x-2)\left(x^{3}-3x^{2}+x+1\right)^{8}}{2^{4}x^{3}\left(x^{2}-3\right)^{12}}\cdot\frac{2y-1}{y^{2}}=t.$
(7.2)

The variable $y$ from $[-2,1,1]$ enters only quadratically and so (7.2)
defines a double cover of the $x$-line. Taking the discriminant with respect
to $y$ and removing unneeded square factors presents this hyperelliptic curve
in standard form:

$z^{2}=-3(x-2)g(t,x).$

As the right side has degree 28, this curve has genus $13$.

In both the new examples of this section, the middle cohomology of the
dimension-reduced varieties contains not only the desired motives, with Hodge
vectors $(6)$ and $(3,3)$ respectively, but also parasitical motives, with
Hodge vectors $(21)$ and $(10,10)$. In this regard, they are less attractive
than the original canonical varieties. HGMs provide many illustrations like
these two of the motivic principle that a motive $M$ comes from many varieties
$X$, and often no single $X$ should be viewed as the best source.

##  8 Distribution of Hodge vectors

In this section, we explain one of the great features of HGMs: they represent
many Hodge vectors.

### Completeness in ranks $\leq 19$.

By direct computation starting from all family parameters $Q$ in degrees $\leq
19$, we have verified the following fact. Let $h=(h^{w,0},\cdots,h^{0,w})$ be
a vector of positive integers satisfying $h^{q,p}=h^{p,q}$ for all $p+q=w$ and
let $n=\sum_{i=0}^{w}h^{p,w-p}$. Then if $n\leq 19$ there exists an HGM with
Hodge vector $h$.

### Many families per Hodge vector in ranks $\leq 100$.

In ranks $20$ to $23$, the only vectors not realized by a family of HGMs are

$\begin{array}[]{rc}20:&(6,1,1,1,2,1,1,1,6),\\\
22:&(6,1,1,1,1,2,1,1,1,1,6),\\\ 22:&(4,1,2,1,1,1,2,1,1,1,2,1,4),\\\
23:&(1,21,1).\end{array}$

Table 8.1 gives a fuller sense of the situation for $n=24$, where there are
about $460,000,000$ family parameters. It gives the extremes of the list of
$4096$ possible Hodge vectors $h$, sorted by how many families realize $h$.

${\begin{array}[]{c|r}h&\\#\\\ \cr\hrule height=0.4pt}$(9, 1, 1, 2, 1, 1,
9)$&0\\\ $(7, 1, 1, 1, 1, 2, 1, 1, 1, 1, 7)$&0\\\ $(1, 6, 1, 1, 1, 1, 2, 1, 1,
1, 1, 6, 1)$&0\\\ $(4, 1, 3, 1, 1, 1, 2, 1, 1, 1, 3, 1, 4)$&0\\\ $(5, 1, 2, 1,
1, 1, 2, 1, 1, 1, 2, 1, 5)$&0\\\ $(6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1,
6)$&0\\\ $(4, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 4)$&0\\\ $(4, 1, 2, 1, 1,
1, 1, 2, 1, 1, 1, 1, 2, 1, 4)$&0\\\ $(6, 2, 1, 1, 1, 2, 1, 1, 1, 2, 6)$&$2$\\\
$(8, 1, 1, 1, 2, 1, 1, 1, 8)$&$4$\\\ $(1, 22, 1)$&$ 4$\\\ $(8, 1, 1, 4, 1, 1,
8)$&$ 6$\\\ $\vdots$&$\vdots$\\\ $(1, 5, 6, 6, 5, 1)$&$ 7637828$\\\ $(1, 2, 4,
5, 5, 4, 2, 1)$&$ 7982874$\\\ $(2, 4, 6, 6, 4, 2)$&$ 9504072$\\\ $(1, 4, 7, 7,
4, 1)$&$ 9905208$\end{array}$ Table 8.1:  Hodge vectors with total $24$ and
their number of hypergeometric realizations

The ratio of the numbers just reported say that the number of family
parameters per Hodge vector in degree $24$ is about $113,000$. This ratio
increases to a maximum at $n=58$ where it is about four million. It then
decreases to zero, with some approximate sample values being two million for
$n=100$ but only $0.00001$ for $n=300$. These numbers are computed via
generating functions, similar to (5) but more complicated.

### Perspective.

Section 6 offers some perspective on the general inverse problem of finding an
irreducible motive $M\in\mathcal{M}({\mathbb{Q}},{\mathbb{Q}})$ with a given
Hodge vector. From (6.1)-(6.2), one sees that the Hodge vectors coming from
hypersurfaces are very sparse. When one looks at broader standard classes of
varieties, such as complete intersections in projective spaces, more Hodge
vectors arise, but they all have the same rough form: bunched in the middle.
Ad hoc techniques, such as reducing Hodge numbers by imposing singularities,
give many more Hodge vectors. But for many $h$, it does not seem easy to find
a corresponding motive and then prove irreducibility in this geometric way.
For example, imposing $k$ ordinary double points on a sextic surface reduces
the Hodge vector to $(10,85-k,10)$. However the family of sextic surfaces is
only $68$-dimensional, and so it would it seem to be difficult to get down to
e.g. $(10,1,10)$. There does not seem to be even a conjectural expectation of
which Hodge vectors arise from irreducible motives in
$\mathcal{M}({\mathbb{Q}},{\mathbb{Q}})$.

### The cases $(1,b,1)$.

One could go into much more detail about the families behind any given Hodge
vector. Here we say a little more about the cases $(1,b,1)$, which are
particularly interesting for several reasons. The $\gamma$ giving Hodge
vectors of the form $(1,b,1)$ typically have canonical dimension
$\kappa=\dim(X_{\gamma,t})$ greater than two, posing instances of the
dimension reduction problem. If $b\leq 19$, then the moduli theory of $K3$
surfaces says that there is at least one family $Y_{t}$ of $K3$ surfaces also
realizing $H(\gamma,t)$. Finding such a family is a challenge.

Cases with $b\geq 20$ present a greater challenge, as they cannot be realized
by K3 surfaces. There are seventy-two parameters giving $(1,20,1)$. None of
the eleven listed in Section 6 can be spliced, underscoring the difficulty of
dimension reduction. One of the four gamma vectors giving $(1,22,1)$ has
canonical dimension eight, namely $[-60,-5,-4,-3,-2,8,9,10,12,15,20]$. The
other three have canonical dimension ten:

$\begin{array}[]{l}\,[-66,-11,-6,-5,-4,-4,1,2,8,12,18,22,33],\\\
\,[-60,-15,-9,-6,-4,-2,3,5,8,12,18,20,30],\\\
\,[-33,-10,-6,-4,-4,-1,2,2,5,\;\,8,11,12,18].\end{array}$

In all four cases, there are many ways to splice, but no path to a surface.

##  9 Special and semi HGMs

We have so far been excluding the singular specialization point $t=1$ from
consideration. Now we explain how it yields a particularly interesting motive
$H(Q,1)\in\mathcal{M}({\mathbb{Q}},{\mathbb{Q}})$. We also explain how other
interesting motives arise when the family parameter $Q$ is reflexive, in the
sense of satisfying $Q(-T)=Q(T)^{-1}$.

### Interior zeros.

A Hodge-normalized motive $M\\!\in\\!\mathcal{M}({\mathbb{Q}},{\mathbb{Q}})$
of weight $w$ has Hodge vector $h=(h^{w,0},\dots,h^{0,w})$ with
$h^{w,0}=h^{0,w}>0$. But for the Hodge vectors explicitly considered so far,
the remaining numbers $h^{p,w-p}$ are also positive. There is a reason for
this restriction: Griffiths transversality says that any collection of motives
moving in a family with irreducible monodromy group has Hodge vector with no
interior zeros. Special and semi HGMs do not move in families, and they
include cases with interior zeros.

### Special HGMs.

The way to account for the double point on the canonical variety
$X_{\gamma,1}$ is to first of all take inertial invariants with respect to the
monodromy operator $h_{1}$. In the orthogonal case, this already give the
right motive $H(Q,1)$. Its Hodge vector differs from the generic Hodge vector
only in that $h^{w/2,w/2}$ is decreased by $1$. In the symplectic case, the
motive of inertial invariants is mixed, and quotienting out by its submotive
of weight $w-1$ and rank $1$ gives $H(Q,1)$. Its Hodge vector now comes from
the generic one by decreasing the two central Hodge numbers by $1$. These
drops obviously can cause interior zeros, as in
$(10,1,10)\rightarrow(10,0,10)$ or $(1,1,1,1,1,1)\rightarrow(1,1,0,0,1,1)$.

### Semi HGMs.

For a reflexive parameter $Q$ and any $t\in{\mathbb{Q}}^{\times}$, the motives
$H(Q,t)$ and $H(Q,t^{-1})$ are quadratic twists of one another. The interest
in reflexive parameters is that nongeneric behavior is thereby forced at
$t=\pm 1$. The motive $H(Q,(-1)^{n})$ is a direct sum of two motives in
$\mathcal{M}({\mathbb{Q}},{\mathbb{Q}})$ of roughly equal rank. We call the
summands semi HGMs and their Hodge vectors can have many interior zeros. For
example, the summands of $H(\Phi_{2}^{16}/\Phi_{1}^{16},1)$ are studied in
[Rob] and the two Hodge vectors are

$\begin{array}[]{c}\,(1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1),\\\
\,(1,0,1,0,1,0,0,0,0,1,0,1,0,1).\end{array}$ (9.1)

There is a similar decomposition of $H(Q,-(-1)^{n})$, but only after viewing
it in $\mathcal{M}({\mathbb{Q}}(i),{\mathbb{Q}}(i))$.

##  10 Point counts

We now turn to arithmetic. The point counts of this section form the principal
raw material from which the $L$-functions studied in the remaining sections
are built.

### Background.

Let $X$ be a smooth projective variety over ${\mathbb{Q}}$. Then for all
primes $p$ outside a finite set $S$, the equations defining $X$ have good
reduction and so define a smooth projective variety over ${\mathbb{F}}_{p}$.
For any power $q=p^{e}$, one has the finite set of solutions
$X({\mathbb{F}}_{q})$ to the defining equations. The key invariants that need
to be input into the motivic formalism are the cardinalities
$|X({\mathbb{F}}_{q})|$, and famous results of Grothendieck, Deligne, and
others provide the tools.

The vector spaces $H^{k}(X,{\mathbb{Q}})$ do not see that $X$ is defined over
${\mathbb{Q}}$. The arithmetic origin of $X$ yields extra structure as
follows. For any prime $\ell$, one can extend coefficients to obtain vector
spaces $H^{k}(X,{\mathbb{Q}}_{\ell})$ over the field ${\mathbb{Q}}_{\ell}$ of
$\ell$-adic numbers. Then the group
$\mathrm{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$ acts on
$H^{k}(X,{\mathbb{Q}}_{\ell})$.

For every prime $p$ the group
$\mathrm{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$ contains Frobenius
elements $\mathrm{Fr}_{p}$, well-defined up to ambiguities that will disappear
from our considerations. For any power $q=p^{e}$ of a prime $p\not\in S$, and
any $\ell\neq p$, one has the trace of the operator
$\mathrm{Fr}_{q}=\mathrm{Fr}_{p}^{e}$ acting on
$H^{k}(X,{\mathbb{Q}}_{\ell})$. These $\ell$-adic numbers are in fact rational
and independent of $\ell$. We emphasize the independence of $\ell$ by denoting
them $\operatorname{Tr}(\mathrm{Fr}_{q}|H^{k}(X,{\mathbb{Q}}))$. The
connection with point counts is the Lefschetz trace formula:
$|X({\mathbb{F}}_{q})|=\sum_{k}(-1)^{k}\operatorname{Tr}(\mathrm{Fr}_{q}|H^{k}(X,{\mathbb{Q}}))$.
The left side for fixed $p$ and varying $e$ determines the summands on the
right side in principle because the complex eigenvalues of $\mathrm{Fr}_{p}$
on weight $k$ cohomology have absolute value $p^{k/2}$.

Much of this transfers formally to the motivic setting. Thus for a motive $M$
and a prime $\ell$, there is an action of
$\mathrm{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$ on the corresponding
$\ell$-adic vector space $M_{\ell}$. This action has image in
$G_{M}({\mathbb{Q}}_{\ell})$. Indeed the Tate conjecture predicts that the
${\mathbb{Q}}_{\ell}$-Zariski closure of the image of
$\mathrm{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$ is all of
$G_{M}({\mathbb{Q}}_{\ell})$.

One technical problem with André’s category
$\mathcal{M}({\mathbb{Q}},{\mathbb{Q}})$ is that the projectors used to define
motives are not known to come from algebraic cycles. As a consequence, for a
general $M\in\mathcal{M}({\mathbb{Q}},{\mathbb{Q}})$ the above compatibility
of Frobenius traces is not known. However this problem does not arise for
hypergeometric motives, because they are essentially the entire middle
cohomology of varieties. Accordingly one has well-defined rational numbers
$\operatorname{Tr}(\mathrm{Fr}_{q}|H(\gamma,t))$. There are similar technical
problems at the primes $p\in S$, but they do not affect our computations and
we will ignore them.

### Wild, tame, and good primes.

Returning now to very concrete considerations, we sort primes for a parameter
$(\gamma,t)$ as follows. A prime $p$ is wild if it divides a $\gamma_{j}$. For
$t\neq 1$, a prime $p$ is tame if it is not wild but it divides either the
numerator of $t$, the denominator of $t$, or the numerator of $t-1$; these
last three conditions say that $t$ is $p$-adically close to the special points
$0$, $1$, and $\infty$ respectively. For $t=1$, no primes are tame. We say
that a prime is bad if it is either wild or tame, and all other primes are
good.

### Split powers of a good prime $p$.

A power $q$ of a good prime $p$ is split for $\gamma$ if $q\equiv 1\bmod m$,
where $m$ is the least common multiple of the $\gamma_{j}$. One then has a
collection of Jacobi sums indexed by characters $\chi$ of
${\mathbb{F}}^{\times}_{q}$:

$J(\gamma,\chi):=\prod_{j=1}^{n}g(\omega^{\alpha_{j}(q-1)}\chi,\psi)\overline{g(\omega^{\beta_{j}(q-1)}\chi,\psi)}.$

Here $\psi:{\mathbb{F}}_{q}\rightarrow{\mathbb{C}}^{\times}$ is any nonzero
additive character,
$\omega:{\mathbb{F}}_{q}^{\times}\rightarrow{\mathbb{C}}^{\times}$ is any
generator of the group of multiplicative characters, $(\alpha,\beta)$
underlies $\gamma$ as in Section 2, and
$g(\rho,\psi)=\sum_{t\in{\mathbb{F}}_{q}^{\times}}\rho(t)\psi(t)$ is the
standard Gauss sum. The desired quantity is then given by a sum due to Katz
[Katz-ESDE, p. 258]. Renormalizing to fit our conventions, it is

$\operatorname{Tr}(\mathrm{Fr}_{q}|H(\gamma,t))=\frac{q^{\phi_{0}}}{1-q}\sum_{\chi}\frac{J(\gamma,\chi)}{J(\gamma,1)}\,\chi(t).$
(10.1)

Here $\phi_{0}$ is the vertical coordinate of a lowest point on the zigzag
diagram of $\gamma$, e.g. $\phi_{0}=-1$ in Figure 5.1.

### General powers of a good prime $p$.

The Gross-Koblitz formula lets one replace the above Gauss sums by values of
the $p$-adic gamma function. This is both a computational improvement and
extends the formula to all powers of any good prime. With this method, the
desired integers $\operatorname{Tr}(\mathrm{Fr}_{q}|H(\gamma,t))$ are first
approximated $p$-adically. Errors are under control and exact values are
determined from sufficiently good approximations. See [BCM] for a closely
related approach to the essential numbers
$\operatorname{Tr}(\mathrm{Fr}_{q}|H(\gamma,t))$ and references to earlier
contributions.

##  11 Frobenius polynomials

Frobenius polynomials are a concise way of packaging the point counts of the
preceding section. They play the leading role in the formula for $L$-functions
of the next section. After saying what they are, this section explains several
reasons why they are useful, even before one gets to $L$-functions.

### Capturing point counts.

Consider the numbers
$c_{p,e}=\operatorname{Tr}(\mathrm{Fr}^{e}_{p}|M)\in{\mathbb{Q}}$ for a fixed
motive $M\in\mathcal{M}({\mathbb{Q}},{\mathbb{Q}})$ of rank $n$, a fixed good
prime $p$, and varying $e$. They can be captured in a single degree $n$
polynomial $F_{p}(M,x)=\det(1-\mathrm{Fr}_{p}x|M)$. The relation, which comes
from summing the geometric series belonging to each of the $n$ eigenvalues, is

$\exp\left(\sum_{e=1}^{\infty}\frac{c_{p,e}}{e}x^{e}\right)=\frac{1}{F_{p}(M,x)}.$
(11.1)

Write

$F_{p}(M,x)=1+a_{p,1}x+\dots+a_{p,n-1}x^{n-1}+a_{p,n}x^{n}.$

Then the $c_{p,e}$ for $e\leq k$ determine $a_{p,k}$. Thus the $c_{p,e}$ for
$e\leq n$ determine $F_{p}(M,x)$. But, even better, Poincaré duality on a
source variety ultimately implies that one has
$a_{p,e}=\epsilon(p)a_{p,n-e}p^{(n-2e)w/2}$ for a sign $\epsilon(p)$. For
HGMs, this sign is known and in fact always $1$ when $w$ is odd. So
$F_{p}(M,x)$ can be computed using only $c_{p,e}$ for $e\leq\lfloor
n/2\rfloor$.

### Relation with Hodge vectors.

Indexing by weight $w$, consider as examples the rank six family parameters

$\displaystyle Q_{0}$
$\displaystyle=\frac{\Phi_{3}\Phi_{12}}{\Phi_{1}\Phi_{2}\Phi_{8}},$
$\displaystyle Q_{1}$
$\displaystyle=\frac{\Phi_{3}\Phi_{12}}{\Phi_{1}^{2}\Phi_{8}},$ $\displaystyle
Q_{5}$ $\displaystyle=\frac{\Phi^{3}_{3}}{\Phi_{1}^{6}}.$ (11.2)

The first two are the families from Section 7, with Hodge vectors respectively
$(6)$ and $(3,3)$; the last one has Hodge vector $(1,1,1,1,1,1)$. Specializing
at a randomly chosen common point gives motives $M_{6,w}=H(Q_{w},3/2)$.

After the required initialization of a variable $x$ by
`_<x>:=PolynomialRing(Integers())`, and after inputting `Q`$w$ as in (2.4),
Magma quickly gives some Frobenius polynomials via e.g.
`EulerFactor(Q0,3/2,5)`:

$\displaystyle F_{5}(M_{6,0},x)$ $\displaystyle=1-x-x^{5}+x^{6},$
$\displaystyle F_{7}(M_{6,0},x)$ $\displaystyle=1\hskip 5.0pt\hskip
5.0pt\hskip 5.0pt\hskip 5.0pt\hskip 5.0pt\hskip 5.0pt\hskip 5.0pt-x^{6},$
$\displaystyle F_{5}(M_{6,1},x)$ $\displaystyle=1+\hskip 5.0ptx+\hskip
5.0pt6x^{2}+16x^{3}+\cdots,$ $\displaystyle F_{7}(M_{6,1},x)$
$\displaystyle=1-2x+12x^{2}-28x^{3}+\cdots,$ $\displaystyle F_{5}(M_{6,5},x)$
$\displaystyle=1-\hskip
5.0pt9x+5\\!\cdot\\!156x^{2}-5^{3}\\!\cdot\\!2556x^{3}+\cdots,$ $\displaystyle
F_{7}(M_{6,5},x)$
$\displaystyle=1+12x+7\\!\cdot\\!888x^{2}+7^{3}\\!\cdot\\!1816x^{3}+\cdots.$

These displays illustrate a basic motivic principle: as weight increases,
motives of a given rank $n$ become more complicated. A more refined principle
involves Hodge numbers and can be expressed by forming a weakly increasing
vector $(s_{1},\dots,s_{n})=(0,\dots,w)$, where an entry $i$ appears
$h^{i,w-i}$ times. Then the Newton-over-Hodge inequality is
$\mbox{ord}_{p}(a_{p,k})\geq\sum_{j=1}^{k}s_{j}.$ For $k=1$, …, $6$, these
lower bounds from the Hodge vector $(1,1,1,1,1,1)$ controlling $M_{6,5}$ are
$(0,1,3,6,10,15)$. For $(3,0,0,0,0,3)$ the bounds $(0,0,0,5,10,15)$ would be
smaller, leaving more possibilities for Frobenius polynomials. In this sense,
spread out Hodge vectors correspond to more complicated motives.

### Congruences.

Reduced to ${\mathbb{F}}_{\ell}$, the numbers $a_{p,k}$ for $p\neq\ell$ depend
only on the mod $\ell$ Galois representation belonging to $M$. In our
examples, suppose one kills $\ell=2$ in (11.2) by replacing all
$\Phi_{2^{j}a}$ by $\Phi_{a}^{\phi(2^{j})}$. Then $Q_{0}$ and $Q_{1}$ both
become $Q_{5}$. This agreement implies that
$F_{p}(M_{6,w},x)\in{\mathbb{F}}_{\ell}[x]$ is independent of $w$. This
independence can be seen for the primes $5$ and $7$ in the displayed Frobenius
polynomials. The analogous congruences hold for any $\ell$, when one changes
our Tate twist convention to make the weight of $H(Q,t)$ the number of
integers among the $\alpha_{j}$ and $\beta_{j}$, minus one. This web of
congruences, like the web corresponding to splicing considered in Section 7,
makes it clear that HGMs constitute a natural collection of motives.

### Finite Galois groups.

Frobenius polynomials render Galois-theoretic aspects of the situation very
concrete. As a warm-up, consider $Q_{0}$ as a representative of the relatively
familiar case of ordinary Galois theory. Here the $\ell$-adic representations
all come from a single representation
$\mathrm{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}}){\rightarrow}W(E_{6})\subset
GL_{6}({\mathbb{Q}})$. Let $\lambda_{p}$ be the partition of $27$ obtained by
taking the degrees of the irreducible factors of $g(3/2,x)$ from (7.1). Then
the twenty-five possibilities for the pair $(\lambda_{p},F_{p})$ correspond to
the twenty-five conjugacy classes in the finite group $W(E_{6})$. If one can
collect enough classes, then one can conclude that the image $G$ is all of
$W(E_{6})$. In our example $t=3/2$, the above primes $5$ and $7$ give
$(5^{5}1^{2},1-x-x^{5}+x^{6})$ and $(6^{4}3,1-x^{6})$ respectively. In ATLAS
notation, these are the classes $5A$ and $6I$. They do not quite suffice to
prove $G=W(E_{6})$. But the prime $11$ gives the class $12C$ and since no
maximal subgroup contains elements from $5A$, $6I$, and $12C$, indeed
$G=W(E_{6})$.

The Chebotarev density theorem says that each pair appears proportionally to
the number of elements in its conjugacy class. For example, the classes $5A$,
$6I$ and $12C$ occur with frequency $1/10$, $1/12$, and $1/12$ respectively.

### Infinite Galois groups.

The cases $Q_{1}$ and $Q_{5}$ are beyond classical Galois theory as the
motivic Galois groups have positive dimension. But the situation remains quite
similar. Consider for example odd weight motives of rank $n=2r$ so that $G$ is
in the conformal symplectic group $\mbox{CSp}_{n}$. The Weyl group of
$\mbox{CSp}_{n}$ is the hyperoctahedral group $W(C_{r})$ of signed permutation
matrices, with order $2^{r}r!$. A separable $F_{p}(M,x)$, being conformally
palindromic, has Galois group within $W(C_{r})$. If it has Galois group all of
$W(C_{r})$ then $G$ necessarily contains a certain twisted maximal torus.
Suppose a second prime $p^{\prime}$ satisfies the same condition and moreover
the joint Galois group of $F_{p}(M,x)F_{p^{\prime}}(M,x)$ is all of
$W(C_{r})\times W(C_{r})$. Then $G$ contains two maximal tori which are
sufficiently different to force $G=\mbox{CSp}_{n}$, by the classification of
subgroups containing a maximal torus.

To analyze a given motive, the necessary computations can be done using
Magma’s `GaloisGroup` command. The order of the Galois group of
$F_{p}(M_{6,1},x)$ is $16$, $16$, $4$, $48$, $48$ for $p=5$, $7$, $11$, $13$,
$17$, and the pair $(p,p^{\prime})=(13,17)$ satisfies the criterion. For
$5\leq p<100$, all $F_{p}(M_{6,5},x)$ have Galois group $W(C_{3})$ except
$p=13$. Excluding 13, all $\binom{22}{2}=231$ pairs $(p,p^{\prime})$ satisfy
the criterion. In general, it becomes easier to establish genericity as the
weight increases, a reflection of the growth in complexity discussed above.

Applying this two-prime technique to the special and semi HGMs of Section 9
suggests that almost always their motivic Galois groups are as big as
possible. In particular, the exotic Hodge vectors with interior zeros arising
there indeed come from irreducible motives. Details in the case (9.1) are
given in [Rob].

The Chebotarev density theorem extends to the full motivic setting if all the
$L$-functions described below have their expected analytic properties. Readers
wishing to see a glimpse of this theory can compute hundreds of
$a_{p,1}/p^{w/2}$ for $M_{6,w}$ for $w=1$ or $w=5$. By all appearances, the
data matches the Sato-Tate measure $\mu$, meaning the pushforward of Haar
measure on the compact group $\mbox{Sp}_{6}$ to $[-6,6]$ via the defining
character. One would have to compute thousands of $a_{p,1}/p^{w/2}$ before one
could confidently distinguish this measure from the Gaussian measure of mean
$0$ and standard deviation $1$.

##  12 $L$-functions

We now finally define $L$-functions and illustrate how everything works by
some numeric computations.

### Local invariants.

Let $M\in\mathcal{M}({\mathbb{Q}},{\mathbb{Q}})$ be a motive of rank $n$ and
weight $w$, having bad reduction within a finite set $S$ of primes. We have
discussed two types of local invariants associated to $M$. Corresponding to
the place $\infty$ of ${\mathbb{Q}}$ is the Hodge vector
$h=(h^{w,0},\dots,h^{0,w})$ with total $n$, and also a signature $\sigma$.
Corresponding to a prime $p\not\in S$ is the degree $n$ Frobenius polynomial
$F_{p}(M,x)$. For primes $p\in S$, there is also a Frobenius polynomial
$F_{p}(M,x)$, now of degree $\leq n$, and moreover a conductor exponent
$c_{p}\geq n-\mbox{deg}(F_{p}(M,x))$, both to be discussed shortly. The
conductor of $M$, which can be viewed as quantifying the severity of its bad
reduction, is the integer $N=\prod_{p\in S}p^{c_{p}}$.

### Formal products.

The local invariants can be combined into a holomorphic function in the right
half-plane $\mbox{Re}(s)>\frac{w}{2}+1$, called the completed $L$-function of
$M$:

$\Lambda(M,s)=N^{s/2}\Gamma_{h,\sigma}(s)\prod_{p}\frac{1}{F_{p}(M,p^{-s})}.$
(12.1)

The product over primes alone is the $L$-function $L(M,s)$, while the
remaining factors give the completion. The infinity factor is given by an
explicit formula:

$\Gamma_{h,\sigma}(s)=\Gamma_{\mathbb{R}}(s-\frac{w}{2})^{h_{+}}\Gamma_{\mathbb{R}}(s-\frac{w}{2}+1)^{h_{-}}\prod_{p<q}\Gamma_{\mathbb{C}}(s-p)^{h^{p,q}}\\!\\!.$
(12.2)

Here $\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2)$ and
$\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)$. The factors involving
$h_{\pm}=(h^{w/2,w/2}\pm(-1)^{w/2}\sigma)/2$ only appear when $w$ is even; in
the common case that $\sigma=0$, they can be replaced by
$\Gamma_{\mathbb{C}}(s-\frac{w}{2})^{h^{w/2,w/2}/2}$, by the duplication
formula.

Both the $L$-function and the completing factor are multiplicative in $M$ so
that $\Lambda(M_{1}\oplus M_{2},s)=\Lambda(M_{1},s)\Lambda(M_{2},s)$. Another
simple aspect of the formalism is that Tate twists correspond to shifts:
$\Lambda(M(j),s)=\Lambda(M,s+j)$.

### Expected analytic properties.

The $L$-function $L({\mathbb{Q}},s)$ of the unital motive ${\mathbb{Q}}$ is
just the Riemann zeta function $\zeta(s)=\prod_{p}(1-p^{-s})^{-1}$, and the
completing factor is $\Gamma_{\mathbb{R}}(s)$. Riemann established that
$\Lambda({\mathbb{Q}},s)$ has a meromorphic continuation to the whole
$s$-plane, with poles only at $0$ and $1$; moreover he proved that
$\Lambda({\mathbb{Q}},1-s)=\Lambda({\mathbb{Q}},s)$. The product
$\Lambda(M,s)$ is expected to have similar analytic properties. First, for $M$
irreducible and not of the form ${\mathbb{Q}}(j)$, there should be an analytic
continuation to the entire $s$-plane, bounded in vertical strips. Second,
always

$\Lambda(M,w+1-s)=\epsilon\Lambda(M,s),$ (12.3)

for some sign $\epsilon$. For comparison with Section 14, note that most
everything said in the last three sections generalizes to motives in
$\mathcal{M}({\mathbb{Q}},E)$, with Frobenius polynomials being in $E[x]$.
However (12.3) takes the more complicated form
$\Lambda(\overline{M},w+1-s)=\epsilon\Lambda(M,s)$, with $\overline{M}$ the
complex conjugate motive and $\epsilon$ only on the unit circle.

### Determining invariants at bad primes.

One approach to the conductor exponents $c_{p}$ and Frobenius polynomials
$F_{p}(M,x)$ associated to bad primes $p$ is to compute them directly by
studying the bad reduction of an underlying variety. For an HGM $H(Q,t)$,
Magma takes this approach for primes which are tame for $(Q,t)$, as sketched
in Section 13.

A very different approach uses the fact that the list of possible
$(c_{p},F_{p}(M,x))$ for a given prime $p$ is finite, and the product (12.1)
has the conjectured analytic properties for at most one member of the product
list. The current state of HGMs for the wild primes of $Q$ mixes the two
approaches: we first greatly reduce the length of the lists by using proved
and conjectured general facts. Then we search within the much smaller product
list for the right quantities.

Our view is that numerical computations such as those that follow in this
section and Section 15 admit only one plausible interpretation: the bad
factors have been properly identified and the analytic properties indeed hold.
However rigorous confirmation does not seem to be in sight at the moment,
despite the progress described in Section 14.

### A rank four example.

For $Q$ of degree $\leq 6$ and $t=1$, Watkins numerically identified all the
bad quantities, so that the corresponding $L$-functions are immediately
accessible on Magma. For example, take the family parameter to be
$Q=\Phi_{2}^{2}\Phi_{12}/\Phi_{18}$, implemented as always by modifying (2.4).
At the specialization point $t=1$, the Hodge vector is $(1,1,1,1)$. The
corresponding $L$-function, set up so that calculations are done with 10
digits of precision, is

`L := LSeries(Q,1:Precision:=10);`

The bad information stored in Magma is revealed by `EulerFactor(L,`$p$`)` and
`Conductor(L)` to be $F_{2}(M,x)=1+2x$, $F_{3}(M,x)=1$, and $N=2^{6}3^{9}$.
The sign $\epsilon$ is calculated numerically, with `Sign(L)` returning
$-1.000000000$. So the order of vanishing of $L(M,s)$ at the central point
$s=2$ should be odd. This order is apparently three since

`Evaluate(L,2:Derivative:=1);`

returns zero to ten decimal places, but the same command with $1$ replaced by
$3$ returns 51.72756346.

### A rank six example.

More typically, Magma does not know $F_{p}(M,x)$ and $c_{p}$ for wild primes
$p$ and one needs to input this information. As an example, take
$M=H(\Phi_{3}^{4}/\Phi_{1}^{8},1)$ with Hodge vector $(1,1,1,0,0,1,1,1)$. The
only prime bad for the data is $p=3$. A good first guess is that $F_{3}(M,x)$
is just the constant $1$. A short search over some possible $c_{3}$ is
implemented after redefining `Q` by

`[CFENew(LSeries(Q,1:Precision:=10),`

`BadPrimes:=[<3,c,1>]): c in [6..10]];`

The returned number for $c=9$ is $0.0000000000$, while the numbers for the
other $c$ are all at least $0.1$. This information strongly suggests that
indeed $F_{3}(M,x)=1$ and $c_{3}=9$. After setting up `L` with `[<3,9,1>]`,
analytic calculations can be done as before. For example, here the order of
central vanishing is apparently 2. In the miraculous command `CFENew`, `CFE`
stands for the Magma command `CheckFunctionalEquation`, implemented by Tim
Dokchitser using his [Dok]; `New` reflects subsequent improvements by Watkins.

##  13 Bad primes

Fix a hypergeometric motive $M=H(Q,t)$ and a prime $p$. We now sketch how
Magma computes the local invariants when $p$ is tame for $(Q,t)$, and describe
some conjectural basic features for the case when $p$ is wild for $(Q,t)$.

### Tame primes.

When $p$ is tame for $(Q,t)$, the conductor exponent $c_{p}$ is the
codimension of the invariants of a power of a Levelt matrix $h_{\tau}$ from
Section 2. When $\mbox{ord}_{p}(t-1)\geq 1$, the simple shape of $h_{1}$ gives
a completely explicit formula: $c_{p}=1$ except in the orthogonal case with
$\mbox{ord}_{p}(t-1)$ even, where $c_{p}=0$. When $\mbox{ord}_{p}(t-1)\leq 0$,

$c_{p}=\mbox{rank}(h_{\tau}^{|k|}-I).$ (13.1)

Here $k=\mbox{ord}_{p}(t)$, $\tau=\infty$ if $k$ is negative, and $\tau=0$ if
$k$ is positive. So there is separate periodic behavior for $k<0$ and $k>0$,
as illustrated by the top part of Figure 13.1. The example of this table comes
from the case $(a,b)=(3,5)$ of (5.1), so the conductor there is very simply
computed as the discriminant of the octic algebra
${\mathbb{Q}}[x]/(5x^{8}+8tx^{5}+3t^{3})$.

Because ramification is at worst tame, the degree of $F_{p}(M,x)$ is
$n-c_{p}$. When $\mbox{ord}_{p}(t-1)$ is positive, $F_{p}(M,x)$ is computed by
slightly modifying the formulas for point counts sketched in Section 10. In
the other cases, $F_{p}(M,x)$ comes from Jacobi motives as mentioned around
(4.1), extracted from how the family $X_{Q,t}$ degenerates at the relevant
cusp $\tau\in\\{0,\infty\\}$.

### Wild primes.

To simplify the overview, we just exclude the case where
$\mbox{ord}_{p}(t-1)\geq 1$. Write specialization points as $t=vp^{k}$ with
$k=\mbox{ord}_{p}(t)$. The bottom part of Figure 13.1 shows right away that
the situation is complicated.

Figure 13.1:  Pairs $(k,c_{p})$ where $k=\mbox{ord}_{p}(t)$ and
$c_{p}=\mbox{ord}_{p}(\mbox{Conductor}(H([-8,3,5],t)))$, compared with the
graph of the corresponding $\sigma$. Top: The tame cases $p>5$. Bottom: The
wild case $p=2$.

A function $\sigma$ is graphed in both parts of Figure 13.1 and its general
definition goes as follows. For $d$ a positive integer, write

$s(d)=\left\\{\begin{array}[]{ll}1&\mbox{if $\gcd(d,p)=1$,}\\\
1+\mbox{ord}_{p}(d)+\frac{1}{p-1}&\mbox{else.}\end{array}\right.$

Let

$\displaystyle\sigma_{\infty}$
$\displaystyle=\sum_{i=1}^{n}s(\mbox{denom}(\alpha_{i})),$
$\displaystyle\sigma_{0}$
$\displaystyle=\sum_{i=1}^{n}s(\mbox{denom}(\beta_{i})).$

Define $k_{\rm
crit}=\sigma_{\infty}-\sigma_{0}=-\sum_{j}\gamma_{j}\mbox{ord}_{p}(\gamma_{j})$
and transition points $k_{\infty}=\min(k_{\rm crit},0)$ and $k_{0}=\max(k_{\rm
crit},0)$. Then

$\sigma(k)=\left\\{\begin{array}[]{ll}\sigma_{\infty}&\mbox{if $k\leq
k_{\infty}$},\\\ \max(\sigma_{\infty},\sigma_{0})-|k|&\mbox{if $k_{\infty}\leq
k\leq k_{0}$},\\\ \sigma_{0}&\mbox{if $k\geq k_{0}$}.\\\ \end{array}\right.$

In the tame case, $\sigma$ is just the constant function $n$. In general,
there are plateaus corresponding to the cusps $\infty$ and $0$, and then a
ramp of length $|k_{\rm crit}|$ between them.

We conjecture that

$c_{p}\leq\sigma(k)-\mbox{degree}(F_{p}(M,x)),$ (13.2)

with equality if $k$ and $p$ are relatively prime. The second statement is
proved in [LNV] in the general trinomial setting of (5.1). All of (13.2) has
been computationally verified in many instances. As one passes from one family
to another via mod $\ell$ congruences as in Section 11, wild ramification at
$p$ does not change. This fact and other theoretical stabilities give us
confidence in (13.2). To make Magma more fully automatic, a key step would be
to define a more complicated function $\sigma(k,v)$ with
$c_{p}\leq\sigma(k,v)-\mbox{degree}(F_{p}(M,x))$, and equality under broad
circumstances.

At present, we understand a factor $f_{p}(M,x)$ of the Frobenius polynomial
$F_{p}(M,x)$ as follows. For $k\neq k_{\rm crit}$, $f_{p}(M,x)$ comes from
modifying the tame formulas; in particular its degree is given by replacing
$k$ by $k-k_{\rm crit}$ in (13.1). If $k=k_{\rm crit}$, corresponding to being
at the bottom of the ramp, we use an erasing principle explained to us by
Katz. Here one simply ignores all $\alpha_{j}$ and $\beta_{j}$ that have
denominator divisible by $p$. Let $n_{\infty}$ and $n_{0}$ be respectively the
number of $\alpha_{j}$’s and $\beta_{j}$’s remaining. Then $n_{\infty}-n_{0}$
is a multiple of $p-1$, so that the formulas described in Section 10 still
make sense, as the choice of an auxiliary additive character on
${\mathbb{F}}_{p}$ again does not matter. The resulting $f_{p}(M,x)$ has
degree $\max(n_{\infty},n_{0})$. We conjecture that the complementary factor
$F_{p}(M,x)/f_{p}(M,x)$ is $1$ whenever $p$ and $k$ are relatively prime. In
practice, when $p\mid k$ it is usually $1$ also, but not always.

##  14 Automorphy

One of the most exciting aspects of the theory of motives is its conjectured
extremely tight connection to automorphic representations of adelic groups
through the Langlands program.

### Background.

Let ${\mathbb{A}}$ be the adele ring of ${\mathbb{Q}}$; it is a restricted
product of all the completions ${\mathbb{Q}}_{p}$, including
${\mathbb{Q}}_{\infty}={\mathbb{R}}$. A cuspidal automorphic representation of
$GL_{n}({\mathbb{A}})$ has an $L$-function known to have an analytic
continuation and functional equation. The main conjecture is that, after
incorporating Tate twists to make normalizations match, the set of
$L$-functions coming from irreducible rank $n$ motives in
$\mathcal{M}({\mathbb{Q}},{\mathbb{C}})$ is exactly the subset of automorphic
$L$-functions for which the infinity factor has the form (12.2).

### The case $n=2$.

For a motive $M\in\mathcal{M}({\mathbb{Q}},{\mathbb{C}})$ with nonvanishing
Hodge numbers $h^{w,0}=h^{0,w}=1$ and conductor $N$, one can switch to
classical language. The desired automorphic representation is entirely given
by a power series in $q=e^{2\pi iz}$ as in (1.6), but now this newform on
$\Gamma_{0}(N)$ has weight $w+1$.

To exhibit some matches between motivic and automorphic $L$-functions,
consider the four reflexive parameters $Q$ yielding motives $H(Q,1)$ with
Hodge vector $(1,1,0,0,1,1)$:

${{\begin{array}[]{c|rrrr|rrrr}Q&a_{5}&a_{7}&N^{\prime}&&b_{5}&b_{7}&N^{\prime\prime}&\\\
\cr\hrule
height=0.4pt}\Phi_{2}^{6}/\Phi_{1}^{3}&-2&24&8&\eta_{2}^{4}\eta_{4}^{4}&74&24&8&\\\
\Phi_{2}^{4}\Phi_{3}/\Phi_{1}^{4}\Phi_{6}&-18&8&12&&54&-88&4&\eta_{2}^{12}\\\
\Phi_{2}^{2}\Phi_{3}^{2}/\Phi_{1}^{2}\Phi_{6}^{2}&-6&-16&18&&-66&176&6&\\\
\Phi_{3}^{3}/\Phi_{6}^{3}&-16&-12&72&&-16&12&72&\end{array}}$

Magma computes automatically with these reducible motives, reporting their
conductors to be $N=64$, $48$, $108$, and $5184$. However these computations
do not see the decompositions $H(Q,1)=M^{\prime}(-1)\oplus M^{\prime\prime}$
analogous to (9.1), where now $M^{\prime}$ and $M^{\prime\prime}$ respectively
have Hodge vectors $(1,0,0,1)$ and $(1,0,0,0,0,1)$. In the Frobenius
polynomial

$F_{p}(M(Q,1),x)=(1-pa_{p}x+p^{5}x^{2})(1-b_{p}x+p^{5}x^{2}),$

the $pa_{p}$ belonging to $M^{\prime}(-1)$ can be distinguished from the
$b_{p}$ belonging to $M^{\prime\prime}$ whenever the latter is not a multiple
of $p$. The reader might enjoy searching in the LMFDB’s complete lists [lmfdb]
of modular forms to see that the $a_{p}$ and $b_{p}$ for $p=5$ and $p=7$ let
one identify the relevant forms and in particular determine the above-
displayed factorizations $N=N^{\prime}N^{\prime\prime}$. Part of the further
information given by the LMFDB is that two of the forms are expressible using
the Dedekind eta function $\eta_{1}$, via
$\eta_{d}=q^{d/24}\prod_{j=1}^{\infty}(1-q^{dj})$.

### Higher rank.

For a given motive $M\in\mathcal{M}({\mathbb{Q}},{\mathbb{C}})$ with larger
rank $n$, one can usually replace $GL_{n}({\mathbb{A}})$ by the adelic points
of a smaller group determined by the motivic Galois group $G$ of $M$. In
favorable cases, the representation sought again corresponds to a holomorphic
form. For rank three orthogonal motives, classical modular forms are again
relevant, but a symmetric square is now involved. In rank four, Hilbert
modular forms are needed for orthogonal motives and Siegel modular forms are
needed for symplectic motives. Numerical and sometimes proved matches have
been found in these three settings. For example, [DPVZ] treats some
interesting rank four orthogonal cases.

Generally speaking, the Hodge numbers of central concern earlier in this
survey continue to play a large role. In particular, motives for which all
$h^{p,q}$ are $0$ or $1$ have theoretical advantages, and their motivic
$L$-functions at least have a meromorphic continuation with the right
functional equation [PT].

##  15 Numerical computations

We promised in the introduction that we would equip the reader to numerically
explore a large collection of motivic $L$-functions. We conclude this survey
by giving sample computations in the context of two important topics, always
assuming that the expected analytic continuation and functional equation
indeed hold. In both topics, we let $c=\frac{1}{2}+\frac{w}{2}$ be the center
of the functional equation. The conductors $N$ in our examples are small for
their Hodge vectors $h$, allowing us to keep runtimes short and/or work to
high precision.

### Special values.

If $M$ is a motive in $\mathcal{M}({\mathbb{Q}},{\mathbb{Q}})$ then the
numbers $L(M,k)$ for integers $k\leq c$ are mostly forced to be $0$, because
of poles in the infinity factor (12.2) and the functional equation. However,
when $L(M,k)$ is nonzero it is expected to be arithmetically significant [Del-
PL]. The arithmetic interpretation involves a determinant of periods like (2).
To see the significance without entering into periods, one can look at the
ratio $r_{d}=L(M\otimes\chi_{d},k)/L(M,k)$, for $d$ a positive quadratic
discriminant. Then the periods cancel out so that $r_{d}$ should be rational.

For a sample computation, take $M=H(\Phi_{2}^{5}/\Phi_{1}^{5},2^{10})$ and use
(2.4) and `L:=LSeries(Q,1024)` to define its $L$-function, as usual. While $2$
is wild for the family, it is unramified in $M$ because because the exponent
$10$ is at the bottom of the ramp of Section 13. The erasing procedure from
the end of that section applies, yielding

$F_{2}(M,x)=(1-4x)(1+5x+10x^{2}+80x^{3}+256x^{4}).$

Since $t-1=1023=3\cdot 11\cdot 31$ is squarefree, it is the conductor, by the
recipe before (13.1). Magma gets all the bad factors right automatically. As a
confirmation, `CFENew(L)` quickly returns $0$ to the default $30$ digits.

`Evaluate(L,2)` gives $0.4278180899\cdots$. Twisting by a $d$ with
$\gcd(d,1023)=1$ makes the conductor go up by a factor of $d^{5}$ and
precision needs to be reduced.

`Evaluate(LSeries(Q,1024:`

`QuadraticTwist:=5,Precision:=10),2);`

takes six minutes to give its answer of $35.04685793$. This ratio and then two
others are apparently

$\displaystyle r_{5}$ $\displaystyle=\frac{2^{11}}{5^{2}},$ $\displaystyle
r_{8}$ $\displaystyle={2^{6}\,5},$ $\displaystyle r_{13}$
$\displaystyle=\frac{2^{10}\,251}{13^{2}}.$

The two $L$-functions appearing in $r_{d}$ are completely different
analytically, and so the apparent fact that quotients are rational is very
remarkable.

Readers wanting to work out their own examples might want to begin with $M$
having odd weight. Then if $L(M,c)\neq 0$, one has conjecturally rational
quotients $r_{d}$ for $k=c$. The lateral argument $k=c-j$ fits into the theory
only in the rare case that the $2j$ most central entries of the Hodge vector
are $0$. In the even weight case, one needs to have $h_{+}=0$ to make
$k=c-\frac{1}{2}$ fit into the theory, as in our example.

### Critical zeros.

For a weight $w$ motive $M$, all the zeros of the completed $L$-function
$\Lambda(M,s)$ lie in the critical strip $c-\frac{1}{2}\leq\mbox{Re}(s)\leq
c+\frac{1}{2}$. The Riemann hypothesis for $M$ then predicts that all the
zeros lie on the critical line $\mbox{Re}(s)=c$. We now show by examples that
numerical identification of low-lying zeros is possible in modestly high rank.

For the examples, take $M_{10,w}=H(Q_{w},1)$ with
$Q_{10}={\Phi_{4}\Phi_{2}^{9}}/{\Phi^{11}_{1}}$ and
$Q_{7}={\Phi_{4}^{4}\Phi_{2}^{4}}/{\Phi_{8}^{2}\Phi_{1}^{4}}$. So $M_{10,10}$
is orthogonal with Hodge vector $(1,1,1,1,1,0,1,1,1,1,1)$ while $M_{10,7}$ is
symplectic with Hodge vector $(1,1,2,1,1,2,1,1)$.

The only bad prime in each case is $2$. A search says that
$F_{2}(M_{10,10},x)=1+32x$ and $c_{2}=11$. For $M_{10,7}$, $k=k_{\rm crit}=0$
so erasing applies, yielding $1+4x+96x^{2}+512x^{3}+16384x^{4}$ as a factor of
$F_{2}(M_{10,7},x)$. A short search says that this factor is all of
$F_{2}(M_{10,7},x)$ and $c_{2}=18$.

In general, the Hardy Z-function of a motive $M$ is

$Z(M,t)=\epsilon^{1/2}\frac{N^{s/2}\Gamma_{h,\sigma}(s)}{|N^{s/2}\Gamma_{h,\sigma}(s)|}L(M,s),$

with $s=c+it$. It is a real-valued function of the real variable $t$, even or
odd depending on whether the sign $\epsilon$ is $1$ or $-1$.

Figure 15.1:  Graphs of $Z(M_{10,10},t)$ and $Z(M_{10,7},t)$

Figure 15.1 was computed via many calls to `Evaluate` at points of the form
$c+it$. The signs in the two cases are $1$ and $-1$, and the orders of central
vanishing are the minimum possible, $0$ and $1$. On both plots, all local
maxima are above the axis and all local minima are beneath the axis. Zeros off
the critical line would likely cause a disruption of this pattern; thus the
plots not only identify zeros on the critical line, but suggest a lack of
zeros off the critical line.

## References


2.5cm2.5cm2.5cm2.5cm

# Cosmic Gamma Ray Bursts

A. Janiuk1<EMAIL_ADDRESS>B. James1 K. Sapountzis1
1Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników
32/46, 02-668 Warsaw, Poland

(Key words: accretion, accretion disks – black hole physics –
Magnetohydrodynamics (MHD))

Gamma ray bursts (GRBs) are astronomical phenomena detected at highest
energies. The gamma ray photons carry energies on the order of mega-
electronovolts and arrive to us from the point-like sources that are uniformly
distributed on the sky. A typical burst has a form of a pulse that lasts for
about a minute. As the Earth’s atmosphere is not transparent to the very high
energy radiation, the bursts are detected by means of the telescopes onboard
satellites that are placed on the orbit (Gehrels et al., 2004). The total
energetics of GRB events, which is given by the integrated energy flux by the
detector unit area, implies that we are witnessing very powerful explosions,
where an enormously great power is released within a short time. There is only
one way to obtain such huge energies in cosmos: the disruption of a star
(Paczynski, 1986).

##  1 Introduction

Multi-wavelength and multi-messenger observations show that black holes are
the central engines responsible of the most violent astrophysical events such
as, for instance, active nuclei of galaxies, X-ray binaries, core-collapse
supernovae or gamma-ray bursts. This central engine is subject to strong
gravity, strong electromagnetic fields and rotation. The governing physical
laws of such engines are well known (General Relativistic
MagnetoHydroDynamics, hereafter GRMHD) but are nonlinear, time-dependent and
multidimensional. Thereby, it is necessary to develop a numerical approach to
simulate their evolution and observational appearance where a first-principles
theory can not be achieved.

In order to produce a gamma ray burst, the black hole is launching a jet of
plasma. In these jets, plasma is expanding with ultra-relativistic velocities.
Particles are accelerated in the strong magnetic field and produce high energy
radiation. The jet is powered by the Blandford-Znajek mechanism which can
extract energy from a rotating black hole Blandford and Znajek (1977). This
process requires a magnetized accretion disk with a strong poloidal magnetic
field generated around a spinning black hole. The gamma-ray emission produced
in the jet at large distances is not uniform and short time-scale variability
suggests that it originates close to the black hole.

##  2 Two classes of gamma ray bursts

Two disctinct classes of Gamma Ray bursts have been identified and are known
to constitute statistically distinct populations of sources (Kouveliotou et
al., 1993). The long bursts cluster around few tens of seconds of gamma ray
emission, while the short bursts are typically lasting a fraction of a second
only. Also, their characteristic spectral energy distributions have distinct
characteristics (see Table 1).

Type | Mean Duration | Peak in Energy Spectrum | Origin
---|---|---|---
Long | 25 [s] |  $\log(E_{\rm peak})=2.2$ [keV] | Collapse of massive star
Short | 0.7 [s] |  $\log(E_{\rm peak})=2.7$ [keV] | Merger of two compact stars

Table 1: Properties of short and long GRBs

###  2.1 Long gamma ray bursts

Already from 1990’s we know that the GRBs originate mostly in the distant
galaxies, and many of them are associated with supernovae. They have to be in
fact special supernova types (only 10 per cent of them meet the criteria),
because the core of collapsing star needs to form a black hole, surrounded by
a disk composed from the remnant matter from the stellar envelope. It is the
accretion of matter onto a rotating black hole that is able to provide energy
large enough to account for the observed properties of GRB phenomena. This
process is relatively long (several tens – hundreds seconds).

###  2.2 Short gamma ray bursts

Another mechanism of producing a shorter GRB is the coalescence of two neutron
stars. A transient structure is then formed, and collapses to a black hole.
The surrounding remnants of dense matter form a disk composed of elementary
particles and neutrinos. The process of disk accretion, mediated by magnetic
fields provides power to extract the rotational energy of the black hole and
launch a relativistic jet (Janiuk and Yuan, 2010). This collimated outflow is
where the gamma rays are produced.

##  3 Central engine

###  3.1 Progenitors

The collapsar scenario is able explain the long-duration GRBs, while the short
GRBs are associated with the mergers of compact objects. In the long GRB case,
the energetics of the explosion is consistent with the gravitational mass of
the progenitor star: $E=GM_{\star}^{2}/R\approx 10^{54}$ [erg]. The jet has to
have a highly relativistc speed, with a bulk lorentz factor of $\Gamma\sim
100$, in order to be able to break through the stellar envelope. Also, the
duration, time variability, and the following afterglow emission of these GRBs
at lower energies are consistent with the collapsing massive stars that reside
in star-forming host galaxies. In the short GRB case, the compact binary
megergers are almost equally energetic, while the duation of the burst is set
by the viscous timescale of an accretion disk formed from the tidally-
disrupted remnant, $t_{vis}\approx\alpha_{\rm vis}^{-1}(R_{\rm
disk}^{3}/GM_{NS})^{1/2}(H/R_{\rm disk})^{-2}$, and is about 0.5 [s] for
typical parameters of such system. The highly relativistic speeds of the jets
are also supported by the observation of non-thermal energy spectra, which
otherwise would not be possible because of a lare optical depht due to the
electron-positon pair production (see Thompson (1994)). Recently, the
discovery of the binary neutron star (NS-NS) mergers in the gravitational wave
observation made by LIGO (GW170817 and GW190425), as well as the detection of
associated electromagnetic counterparts, provided a direct proof of the NS-NS
system being sources of short GRBs (see review by Janiuk and Sapountzis
(2018)). Schematic view of a central engine of a GRB, in a unified aproach, is
depicted in Fig. 1.

Figure 1: Schematic idea of GRB. The ralativistic jet is ejected (Lorentz
factor about 100). Its origin is the “central engine” where the black hole
sits. The jet emits gamma ray radiation, collimated towards the observer.

###  3.2 Numerical modeling of the engine and jet structure

We compute the structure and evolution of black hole accretion disks using the
numerical simulations, governed by the equations of general relativistic
magnetohydrodynamics (GRMHD). In particular, such disks and outflows can be
found at the base of relativistic jets in the extragalactic sources, like
blazars, or gamma ray bursts. Long-lasting, detailed computations are
essential to properly determine the physics of these jets, and confront the
theoretical models with observables available from astrophysical observatories
in space and from ground-based detectors.

Our numerical scheme works in a conservative manner, by solving a set of non-
linear equations at each time-step, to advance the conserved quantities from
one time step to the next. The efficiency of computations is enhanced thanks
to the code parallelisation. We use the Message Passing Interface (MPI)
techniques to distribute the computational grid over the threads. Such
calculations are computationally demanding, and also required special fine-
tuning of the code algorithm.

Our calculations are started with a steady state model of the flow, based on
the analytical, equilibrium solution driven by the main physical parameters of
the black hole accretion disk (namely, BH mass, its spin, and size and density
of the accreting torus). Properties of the pressure equilibrium torus around a
black hole, supported by a constant specific angular momentum over radius,
were found by Fishbone and Moncrief (1976) and Chakrabarti (1985). In the
latter model, the angular momentum distribution is chosen to have a power law
distribution, which differs from the default disk model of Fishbone and
Moncrief (1976), where the angular momentum is assumed to be constant in the
disk. This model will allow us to create a initial torus with a large amount
of poloidal magnetic flux.

Starting from the Chakrabarti solution, implemented as the initial condition
for our simulations, we use a dynamical scheme, and we follow the flow
evolution. This is achieved by solving numerically the continuity and
momentum-energy conservation equations in GR MHD framework:

$(\rho u_{\mu})_{;\nu}=0;\hskip
28.45274ptT^{\mu\nu}=T^{\mu\nu}_{gas}+T^{\mu\nu}_{EM}=0$ (1)

$T^{\mu\nu}_{gas}=\rho
hu^{\mu}u^{\nu}+pg^{\mu\nu}=(\rho+u+p)u^{\mu}u^{\nu}+pg^{\mu\nu}$ (2)
$T^{\mu\nu}_{EM}=b^{2}u^{\mu}u^{\nu}+\frac{1}{2}b^{2}g^{\mu\nu}-b^{\mu}b^{\nu};b^{\mu}=u^{*}_{\nu}F^{\mu\nu}$
(3)

Here we account for both the gas and electromagnetic components of the stress
tensor, $u^{\mu}$ is the four-velocity of gas, $u$ is internal energy, $\rho$
is density, $p$ denotes pressure, and $b^{\mu}$ is the magnetic four-vector.
Here, $F$ is the Faraday tensor, and in the force-free approximation the
Lorentz force vanishes, $E_{\nu}=u^{\nu}F^{\mu\nu}=0$.

The model system of equations is supplemented with an equation of state, in
the polytropic form

$p_{\rm g}=K\rho^{\gamma}$ (4)

where $p_{\rm g}$ is the gas pressure, $\rho$ is density, $K$ is constant
specific entropy, and the polytropic index $\gamma=4/3$ is typically used in
the context of gamma ray bursts. The plasma $\beta-$parameter is defined as
the ratio of the fluid’s thermal to the magnetic pressure, $\beta\equiv p_{\rm
g}/p_{\rm mag}$. We normalize the magnetic field in the torus to have
$\beta=(\gamma-1)u_{\rm max}/(b^{2}_{\rm max}/2)$, where $u_{max}$ is the
internal energy at the torus pressure maximum radius, $r_{\rm max}$.

Resulting evolved structure of the GRB central engine (disk in the upper
panel, and jet in the bottom panel) is shown in Figure 2. The snapshot is
taken at time $t=2000~{}[t_{\rm g}]$, where $t_{\rm g}=GM/c^{3}$ denotes
geometric time unit, and scales with black hole mass. For GRB engine, typical
mass of central black hole is in the range from $M\sim 3$ up to $M\sim 30$
Solar masses. Note that in the evolved state we are able to obtain the
physically motivated structure of the accretion torus in the equatorial plane
of the rotating black hole, and low-density polar funnels, with dominant
magnetic field. These funnels carry both thermal and Poynting energy along the
jet axis. The ultimate bulk Lorentz factor in the jet, achieved at ’infinity’
(outside the computational domain) will possibly reach the order of magnitude
of this total energetics parameter, so $\Gamma\sim$ few hundreds.

Figure 2: Top: Distribution of mass density and configuration of magnetic
field lines in the region close to the black hole, at the base of gamma ray
burst. Bottom: Structure of jet launched from rotating black hole, in terms of
its total energetics (thermal plus Poynting energy). The results from
axisymmetric numerical simulation are shown, for black hole spin $a=J/M^{2}$ =
0.7. The snapshot is taken at time $t=2000~{}[t_{\rm g}]$.

+

##  4 Discussion

Figure 3: Time variability of the jet in terms of its total energetics
(thermal plus Poynting energy), as measured during the simulation at a chosen
point in the jet (located at an angle $\theta=5^{\circ}$ from the jet axis).
The result of numerical 2-dimensional simulation is shown, for black hole spin
$a=J/M^{2}=0.7$, and magnetic field normalized in torus with $\beta(r_{\rm
max})=60$. The dashed lines show the characteristic timescale of the MRI
instability, measured as the maximum growth rate timescale (Gammie 2004) which
correlates with the pulse width.

Our numerical simulations, performed by the computational astrophysics group
at the Center for Theoretical physics, PAS, provide working tools to model the
central engine of both long and short GRBs. The unified approach is possible
because we scale the size and timescale of the system by converting the
geometric units to physical ones, as adequate in the GR MHD modeling. We also
notice that the variability timescales, modeled in our simulations as the time
variability of the jet energetics (see Fig. 3), are governed by the timescale
of the magneto-rotational instability in the accretion disk. In order to
properly govern the regime of long-duration GRB, with a sustained magnetic
turbulence in the engine, full 3-dimensional simulations are needed. This is
the work in progress carried by our group (Fig. 4).

Figure 4: Structure of the accretion flow and large scale magnetic field in
the jet. Figure shows the result of 3-dimensional numerical simulation.
Exemplary field line is depicted as launched from the black hole horizon, with
dominant toroidal field component.

Such models, with a good spatial resolution to solve physical process close to
the black hole, need large computational resources. We ran already several
series of calculations on the OKEANOS supercomputer of the Warsaw ICM
supercomputing center, and in the Prometheus supercomputer within the PL-Grid.
The purpose is to study the polarimetric signatures of strong magnetic fields
near event horizon of a black hole in the Galaxy center (Moscibrodzka et al.,
2020, in preparation).

## Acknowledgement

The authors acknowledge the financial support by the grant No.
2016/23/B/ST9/03114 and 2019/35/B/ST9/04000 from the Polish National Science
Center. We have used the computational resources of the Warsaw ICM through
grant Gb79-9, and the PL-Grid through the grant grb3.

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