Finite Monodromy Groups Research Articles

On the semi‐stability degree for abelian varieties

For an abelian variety A $A$ over a number field we study bounds depending only on the dimension of A $A$ for the minimal degree d ( A ) $d(A)$ of a field extension over which A $A$ acquires semi-stable reduction. We first compute d ( A ) $d(A)$ in terms of the cardinalities of the finite monodromy groups of A $A$ which leads to a bound on d ( A ) $d(A)$ in terms of the classical Minkowski bound. We then show this bound is tight up to its 2-part by considering p $p$ -adic coverings of the local points of a universal abelian scheme.

Variétés abéliennes CM et grosse monodromie finie sauvage

In this paper we study the wild part of the finite monodromy groups of abelian varieties over number fields. We solve Grunwald problems for groups of the form Z/pZ≀Sn over number fields to build CM abelian varieties with maximal wild finite monodromy in the odd prime case. For the even prime case we prove a new bound on the 2-part of the order of the finite monodromy group for CM abelian varieties and build varieties that reach it.

Degenerating families with finite monodromy groups

A degenerating family of Riemann surfaces over a Riemann surface gives us a monodromy representation, which is a homomorphism from the fundamental group of a punctured surface to the mapping class group. We show that, given such a homomorphism, if its image is finite, then there exists an (isotrivial) degenerating family of Riemann surfaces whose monodromy representation coincides with it. Moreover, we discuss the special sections of such a degenerating family.

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.

On a characterization of finite vector bundles as vector bundles admitting a flat connection with finite monodromy group

We prove that a holomorphic vector bundle E E over a compact connected Kähler manifold admits a flat connection, with a finite group as its monodromy, if and only if there are two distinct polynomials f f and g g , with nonnegative integral coefficients, such that the vector bundle f ( E ) f(E) is isomorphic to g ( E ) g(E) . An analogous result is proved for vector bundles over connected smooth quasi-projective varieties, of arbitrary dimension, admitting a flat connection with finite monodromy group. When the base space is a connected projective variety, or a connected smooth quasi-projective curve, the above characterization of vector bundles admitting a flat connection with finite monodromy group was established by M. V. Nori.

On a Certain Class of Generalized Hypergeometric Functions with Finite Monodromy Groups

On a Certain Class of Generalized Hypergeometric Functions with Finite Monodromy Groups