Hypergeometric Groups Research Articles

Thinness of some hypergeometric groups in

Abstract We show the thinness of $7$ of the $40$ hypergeometric groups having a maximally unipotent monodromy in $\mathrm{Sp}(6)$ .

Thin Monodromy in $$\textrm{O}(5)$$

R\'esum\'eThis article studies the orthogonal hypergeometric groups of degree five. We establish the thinness of 12 out of the 19 hypergeometric groups of type O(3, 2) from [4, Table 6]. Some of these examples are associated with Calabi-Yau 4-folds. We also establish the thinness of 9 out of the 17 hypergeometric groups of type O(4, 1) from [13], where the thinness of 7 other cases was already proven. The O(4, 1) type groups were predicted to be all thin and our result leaves just one case open.

K3 surfaces, Picard numbers and Siegel disks

If a K3 surface admits an automorphism with a Siegel disk, then its Picard number is an even integer between 0 and 18. Conversely, using the method of hypergeometric groups, we are able to construct K3 surface automorphisms with Siegel disks that realize all possible Picard numbers. The constructions involve extensive computer searches for appropriate Salem numbers and computations of algebraic numbers arising from holomorphic Lefschetz-type fixed point formulas and related Grothendieck residues.

Arithmeticity of some hypergeometric groups

We show that the hypergeometric groups associated to the pairs of the parameters (0,112,512,712,1112),(12,13,13,23,23); and (0,0,13,23), (12,12,14,34) are arithmetic.

Hypergeometric groups and dynamics on K3 surfaces

A hypergeometric group is a matrix group modeled on the monodromy group of a generalized hypergeometric differential equation. This article presents a fruitful interaction between the theory of hypergeometric groups and dynamics on K3 surfaces by showing that a certain class of hypergeometric groups and related lattices lead to a lot of K3 surface automorphisms of positive entropy, especially such automorphisms with Siegel disks.

Symplectic hypergeometric groups of degree six

Our computations show that there is a total of 40 pairs of degree six coprime polynomials f,g where f(x)=(x−1)6, g is a product of cyclotomic polynomials, g(0)=1 and f,g form a primitive pair. The aim of this article is to determine whether the corresponding 40 symplectic hypergeometric groups with a maximally unipotent monodromy follow the same dichotomy between arithmeticity and thinness that holds for the 14 symplectic hypergeometric groups corresponding to the pairs of degree four polynomials f,g where f(x)=(x−1)4 and g is as described above. As a result we prove that at least 18 of these 40 groups are arithmetic in Sp6.In addition, we extend our search to all degree six symplectic hypergeometric groups and find that there is a total of 458 pairs of polynomials (up to scalar shifts) corresponding to such groups. For 211 of them, the absolute values of the leading coefficients of the difference polynomials f−g are at most 2 and the arithmeticity of the corresponding groups follows from Singh and Venkataramana, while the arithmeticity of one more hypergeometric group follows from Detinko, Flannery and Hulpke.In this article, we show the arithmeticity of 163 of the remaining 246 hypergeometric groups.

On orthogonal hypergeometric groups of degree five

A computation shows that there are 77 77 (up to scalar shifts) possible pairs of integer coefficient polynomials of degree five having roots of unity as their roots and satisfying the conditions of Beukers and Heckman, so that the Zariski closures of the associated monodromy groups are either finite or the orthogonal groups of non-degenerate and non-positive quadratic forms. Following the criterion of Beukers and Heckman, it is easy to check that only 4 4 of these pairs correspond to finite monodromy groups, and only 17 17 pairs correspond to monodromy groups, for which the Zariski closure has real rank one. There are 56 56 pairs remaining, for which the Zariski closures of the associated monodromy groups have real rank two. It follows from Venkataramana that 11 11 of these 56 56 pairs correspond to arithmetic monodromy groups, and the arithmeticity of 2 2 other cases follows from Singh. In this article, we show that 23 23 of the remaining 43 43 rank two cases correspond to arithmetic groups.

Commensurability and Arithmetic Equivalence for Orthogonal Hypergeometric Monodromy Groups

ABSTRACTInvariants are computed of quadratic forms associated with orthogonal hypergeometric groups of degree five. Some commensurabilities are then determined between these groups, and it is established that some thin groups cannot be conjugated to each other.

Hypergeometric groups of orthogonal type

We obtain an infinite family of orthogonal hypergeometric groups which are higher rank arithmetic groups. We also list cases of arithmetic monodromy when the real Zariski closure of the hypergeometric group is O (2,3).

Orthogonal Hypergeometric Groups with a Maximally Unipotent Monodromy

Similar to the symplectic cases, there is a family of 14 orthogonal hypergeometric groups with a maximally unipotent monodromy (cf. Table 1). We show that 2 of the 14 orthogonal hypergeometric groups associated to the pairs of parameters (0, 0, 0, 0, 0), and (0, 0, 0, 0, 0), are arithmetic. We also give a table (cf. Table 2) which lists the quadratic forms Q preserved by these 14 hypergeometric groups, and their two linearly independent Q-orthogonal isotropic vectors in ; it shows in particular that the orthogonal groups of these quadratic forms have -rank two.

Hyperbolic monodromy groups for the hypergeometric equation and Cartan involutions

We give a criterion which ensures that a group generated by Cartan involutions in the automorph group of a rational quadratic form of signature (n-1,1) is "thin", namely it is of infinite index in the latter. It is based on a graph defined on the integral Cartan root vectors, as well as Vinberg's theory of hyperbolic reflection groups. The criterion is shown to be robust for showing that many hyperbolic hypergeometric groups for _nF_{n-1} are thin.

Arithmeticity of certain symplectic hypergeometric groups

We give a sufficient condition on a pair of (primitive) polynomials that the associated hypergeometric group (monodromy group of the corresponding hypergeometric differential equation) is an arithmetic subgroup of the integral symplectic group.

On the bilinear structure associated to Bezoutians

This paper is partly a survey of known results on quadratic forms that are hard to find in the literature. Our main focus is a twisted form of a construction due to Bezout. This skew Bezoutian is a symplectic (resp. quadratic) space associated to a pair of reciprocal (or skew reciprocal) coprime polynomials of same degree. The isometry group of this space turns out to contain a certain associated hypergeometric group. Using the skew Bezoutian we construct explicit isometries of bilinear spaces with given invariants (such as the characteristic polynomial or Jordan form and, in the quadratic case, the spinor norm).

A negatively curved Kähler threefold not covered by the ball

We construct examples of three-dimensional compact Kahler manifolds with negative curvature, not covered by the ball. Our manifolds are obtained as a natural generalization of the two-dimensional examples discovered by Mostow and Siu, using their description in terms of monodromy covers of hypergeometric functions. Each example is obtained from a hypergeometric monodromy group in PU(3,1) that is not discrete but has finite local monodromy. We describe a manifold on which the group acts discretely, and check that it has a compact quotient with the above features. The examples are locally described as branched covers of the ball, with totally geodesic branch locus.

Invariant of the hypergeometric group associated to the quantum cohomology of the projective space

We present a simple method to calculate the Stokes matrix for the quantum cohomology of the projective spaces CP k−1 in terms of certain hypergeometric group. We present also an algebraic variety whose fibre integrals are solutions to the given hypergeometric equation.

A nonadaptive version of Ulam's problem with one lie

In this paper we use separating systems to construct a nonadaptive hypergeometric group testing procedure that identifies one defect in a set of cardinality n when at most one sample test can be erroneous, and no sample of cardinality greater than k can be tested.

Hypergeometric groups

SynopsisWe introduce and begin an investigation of the family of hypergeometric groups. We determine which hypergeometric groups of Euclidean type are infinite.

Construction on non-adaptive hypergeometric group testing desings for identifying at most two defectives

In the literature a systematic method of obtaining a group testing design is not available at present. Weideman and Raghavarao (1987a, b) gave methods for the construction of non - adaptive hypergeometric group testing designs for identifying at most two defectives by using a dual method. In the present investigation we have developed a method of construction of group testing designs from (i) Hypercubic Designs for t ≡ 3 (mod 6) and (ii) Balanced Incomplete Block Designs for t ≡ 1 (mod 6) and t ≡ 3 (mod 6). These constructions are accomplished by the use of dual designs. The designs so constructed satisfy specified properties and attained an optimal bound as discussed by Weidman and Raghavarao (1987a,b). Here it is also shown that the condition for pairwise disjoint sets of BIBD for t ≡ 1 (mod 6) given by Weideman and Raghavarao (1987b) is not true for all such designs.

Some optimum non-adaptive hypergeometric group testing designs for identifying two defectives

A systematic study of non-adaptive hypergeometric group testing designs for identifying two defectives is presented. Bounds are given for the number of items that can be tested in a given number of tests. Constructions of some classes of designs are presented.

Multiple Hypergeometric Functions and Simple Lie Groups ${\bf {\text{SL}}}$ and ${\bf {\text{Sp}}}$

With any multiple hypergeometric series one can associate a Lie group called its group of symmetries. This article is devoted to those series which group of symmetries contains a subgroup isomorphic to the special linear group ${\bf {\text{SL}}}$ or to the sympletic group ${\bf {\text{Sp}}}$. The work owes much to the group-theoretic analysis of the Lauricella series $F_D$ in Miller [J. Math. Phys.,13 (1972), pp. 1393–1399], Symmetry and the Separation of Variables, Addison-Wesley, Reading, MA, 1977.