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Let vecα, vecβ∈(C)n be n-tuples
(or multisets) of complex numbers.
For arithmetic applications we will eventually take them to be rationals,
and for purposes of monodromy will largely need only to consider them
modulo 1.
Consider the generalised hypergeometric differential equation
z(θ + α1) ... (θ + αn)F(z)
=(θ + β1 - 1) ... (θ + βn - 1)F(z), qquadθ=z(d/dz),
whose only singularities are regular at 0, 1, and ∞.
For simplicity of exposition, we assume that the β's
are distinct modulo 1,
when a basis of solutions around z=0 is given by
z1 - βi()nFn - 1
biggl((α1 - βi + 1, ..., αn - βi + 1atop
β1 - βi + 1, ..v.., βn - βi + 1)biggm|z biggr)
for i=1 ... n, and the ith term βi - βi + 1 is suppressed.
The generalised hypergeometric function ()nFn - 1 is given by
()nFn - 1
biggl((a1, ..., anatop b1, ..., bn - 1)biggm|z biggr)=
∑k=0^∞((a1)k ... (an)k/(b1)k ... (bn - 1)k)
(zk/k!),
where the Pochhammer symbol is given by (x)k=(x)(x + 1) ... (x + k - 1);
the k! in the denominator of the above display can
thus be thought of as (1)k, which was the suppressed term.
Note that shifting all the α and β by some fixed amount
keeps the ()nFn - 1 expression the same, while only modifying
the z1 - βi term.
Also, switching α and β can be envisaged in terms of
the map z -> 1/z that swaps 0 and ∞.
A theorem of Pochhammer says that the above differential equation has (n - 1)
independent holomorphic solutions around z=1. Let G denote
the fundamental group of the thrice punctured Riemann sphere, and
Vvecα, vecβ the solution space around a base point.
We have a monodromy representation
M:G -> GLn(Vvecα, vecβ).
Writing g0, g1, g_∞∈G for loops about 0,1,∞,
we find that M(g0) has eigenvalues e - 2π iβj
and M(g_∞) has eigenvalues e2π iαj,
implying that we are mainly concerned
with vecα and vecβ only modulo 1.
Indeed, one can note that if we take F=()nFn - 1(vec a, vec b|z)
and vec x∈(Z)n, vec y∈(Z)n - 1, then generically
()nFn - 1(vec a + vec x, vec b + vec y|z) is a linear combination
of rational functions times derivatives of F (this is a contiguity relation).
Meanwhile, the above fact from Pochhammer implies that M(g1)
must have (n - 1) eigenvalues equal to 1 (all with independent eigenvectors),
and so this element is a pseudo-reflection.
It turns out that if H⊆GLn((C)) is generated by A and B
with AB - 1 a pseudo-reflection, the H-action on (C)n is
irreducible if and only if A and B have disjoint sets of eigenvalues.
This is equivalent to all the αi - βj being nonintegral. Moreover,
in his 1961 Amsterdam thesis, Levelt showed that, given any eigenvalues,
there are (up to conjugacy) unique A and B realising these eigenvalues
with AB - 1 a pseudo-reflection.
(Much of the above comes from notes of Beukers.)
For arithmetic purposes, one usually also desires that the
eigenvalues be roots of unity and the sets of them be Galois-invariant.
Thus we can specify hypergeometric data H by (say) two products
of cyclotomic polynomials, these products being coprime and of equal degree.
Given such an H, Rodriguez-Villegas conjectures the existence
of a family of pure motives (defined over Q), for which the trace
of Frobenius at good primes is given by a hypergeometric sum defined
by Katz [Kat90] (see also [Kat96]).
For each rational t != 0, 1, there should be a motive Ht
whose L-function satisfies a functional equation of a prescribed type,
with the Euler factors at good primes given in terms of Gauss sums
(the bad Euler factors are less understood, and depend on deformation theory).
One can also relate such motives to more traditional objects in many cases.
For instance, there is one hypergeometric datum in degree 1, which can
be specified by α=[(1/2)] and β=[0], these being rationals
corresponding to the second and first cyclotomic polynomials respectively.
The L-function here corresponds to
the quadratic field (Q)(Sqrt(t(t - 1))).
In degree 2 there are 13 such data, of which 3 are of weight 0 (see below)
and give Artin representations of number fields, while
the other 10 are of weight 1, and yield elliptic curves
(explicitly calculated by Cohen). An example in higher degree
is α=[(1/5), (2/5), (3/5), (4/5)]
and β=[0, 0, 0, 0], corresponding respectively to the
5th cyclotomic polynomial and the 4th power of the
first cyclotomic polynomial, and this is associated to the Calabi-Yau
quintic 3-fold given by x15 + x25 + x35 + x45 + x55=5tx1x2x3x4x5.
The weight w of a hypergeometric motive can be defined in terms of how much
the α and β interlace (considered as roots of unity).
In particular, if they are completely interlacing, then the weight is 0,
and the resulting motive corresponds to an Artin representation.
Write D(x)=#{α : α≤x} - #{β : β≤x}.
Then w + 1=maxx D(x) - minx D(x), so that the above 3-fold has weight 3
(from the four β's at 0). This weight controls how large the
coefficients of the Euler factors will be.
The trace at q of a hypergeometric motive (for the parameter t)
is given in terms of Gauss sums gq over (F)q.
Associated to hypergeometric data is a GammaArray,
and one defines Gq(r)=∏v gq( - rv)γv,
and also the MValue by M=∏v vvγv.
For primes p with vp(Mt)=0 the hypergeometric trace is then given by
Uq(t)=(1/1 - q)biggl(∑r=0q - 2ωp(Mt)rQq(r)biggr),
where ωp is the Teichmüller character and
Qq(r)=( - 1)m0qD + m0 - mrGq(r) where mr is the multiplicity
of (r/q - 1) in the β and D is a scaling parameter that
involves the Hodge structure (one expression is m0=w + 1 - 2D).
One uses p-adic Γ-functions to expedite the computation
of the above Gauss sums (indeed, the above gives the hypergeometric trace
as a p-adic number, which one recognises as an integer via sufficiently
high precision). The Euler factor is given by the standard recipe
Ep(T)=exp( - ∑n Upn(t)Tn/n), and this is a
polynomial that satisfies a local functional equation.
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