Arithmeticity of certain Symplectic Monodromy Groups

Abstract: Monodromy groups of hypergeometric differential equations are defined as image of the fundamental group $G$ of Riemann sphere minus three points namely 0, 1 and the point at infinity, under some certain representation of $G$ inside the general linear group $GL_n$. By a theorem of Levelt (1961), the monodromy groups are (up to conjugation in $GL_n$) the subgroups of $GL_n$ generated by the companion matrices of two degree $n$ polynomials $f$ and $g$ with complex coefficients and having no common roots.

If we start with $f$, $g$ two integer coefficient polynomials of degree $n$ (an even integer) which satisfy some "conditions" with $f(0)=g(0)=1$, then the associated monodromy group preserves a non-degenerate integral symplectic form, that is, the monodromy group is contained in the integral symplectic group of the associated symplectic form.
 
In this talk, we will describe a sufficient condition on a pair of the polynomials that the associated monodromy group is an arithmetic subgroup (a subgroup of finite index) of the integral symplectic group.