arXiv subjects
Publications and source records attributed to Sandip Singh.
We determine the existence of cocompact lattices in groups of the form $\V\rtimes\SL_2(\R)$, where $\V$ is a finite dimensional real representation of $\SL_2(\R)$. It turns out that the answer depends on the parity of $\dim(\V)$ when the representation is irreducible.
We show that the hypergeometric groups corresponding to the seven pairs of the parameters $\alpha$, $\beta$ where $\alpha$ = (0, 0, 0, 0, 0, 0) and $\beta$ is any of the parameters (1/2, 1/2, 1/2, 1/2, 1/2, 1/2), (1/2, 1/2, 1/2, 1/2, 1/3, 2/3), (1/2, 1/2, 1/2, 1/2, 1/4, 3/4), (1/2, 1/2, 1/2, 1/2, 1/6, 5/6), (1/2, 1/2, 1/3, 2/3, 1/3, 2/3), (1/2, 1/2, 1/3, 2/3, 1/4, 3/4), (1/2, 1/2, 1/5, 2/5, 3/5, 4/5) are thin.
We show that the hypergeometric groups associated to the pairs of the parameters $\left(0,0,\frac{1}{3}, \frac{2}{3}\right)$, $\left(\frac{1}{2},\frac{1}{2},\frac{1}{4},\frac{3}{4}\right)$; and $\left(0,\frac{1}{12}, \frac{5}{12},\frac{7}{12},\frac{11}{12}\right), \left(\frac{1}{2},\frac{1}{3},\frac{1}{3},\frac{2}{3},\frac{2}{3}\right)$ are arithmetic.
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 $\mathrm{Sp}(6)$. In addition, we extend our search to all degree six symplectic hypergeometric groups. We 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 $160$ of the remaining $246$ hypergeometric groups.
A computation shows that there are 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 [1], 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 [1], it is easy to check that only 4 of these pairs correspond to finite monodromy groups and only 17 pairs correspond to monodromy groups, for which, the Zariski closures have real rank one. There are remaining 56 pairs, for which, the Zariski closures of the associated monodromy groups have real rank two. It follows from Venkataramana [16] that 11 of these 56 pairs correspond to arithmetic monodromy groups and the arithmeticity of 2 other cases follows from Singh [11]. In this article, we show that 23 of the remaining 43 rank two cases correspond to arithmetic groups.
We compute invariants of quadratic forms associated to orthogonal hypergeometric groups of degree five. This allows us to determine some commensurabilities between these groups, as well as to say when some thin groups cannot be conjugate to each other.
The article [14] gives a list of 51 symplectic hypergeometric monodromy groups corresponding to primitive pairs of degree four polynomials, which are products of cyclotomic polynomials, and for which, the absolute value of the leading coefficient of the difference polynomial is greater than 2. It follows from [12] and [14] that 12 of the 51 monodromy groups are arithmetic (cf. Table 1); and the thinness of 13 of the remaining 39 monodromy groups follows from [3] (cf. Table 2). In this article, we show that 15 of the remaining 26 monodromy groups are arithmetic (cf. Table 3).
Similar to the symplectic cases, there is a family of fourteen orthogonal hypergeometric groups with a maximally unipotent monodromy (cf. Table 1.1). We show that two of the fourteen orthogonal hypergeometric groups associated to the pairs of parameters $(0, 0, 0, 0, 0)$, $(\frac{1}{6}, \frac{1}{6}, \frac{5}{6}, \frac{5}{6}, \frac{1}{2})$; and $(0, 0, 0, 0, 0)$, $(\frac{1}{4}, \frac{1}{4}, \frac{3}{4}, \frac{3}{4}, \frac{1}{2})$ are arithmetic. We also give a table (cf. Table 2.1) which lists the quadratic forms $\mathrm{Q}$ preserved by these fourteen hypergeometric groups, and their two linearly independent $\mathrm{Q}$- orthogonal isotropic vectors in $\mathbb{Q}^5$; it shows in particular that the orthogonal groups of these quadratic forms have $\mathbb{Q}$- rank two.
In [12], we show that 3 of the 14 hypergeometric monodromy groups associated to Calabi-Yau threefolds, are arithmetic. Brav-Thomas (in [3]) show that 7 of the remaining 11, are thin. In this article, we settle the arithmeticity problem for the 14 monodromy groups, by showing that, the remaining 4 are arithmetic.
Let W be an irreducible finitely generated Coxeter group. The geometric representation of W in GL(V) provides a discrete embedding in the orthogonal group of the Tits form (the associated bilinear form of the Coxeter group). If the Tits form of the Coxeter group is non-positive and non-degenerate, the Coxeter group does not contain any finite index subgroup isomorphic to an irreducible lattice in a semisimple group of R-rank greater or equal to 2.
We give a sufficient condition on a pair of (primitive) integral polynomials that the associated hypergeometric group (monodromy group of the corresponding hypergeometric differential equation) is an arithmetic subgroup of the integral symplectic group.