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.