Classification of Rigid Irregular G2G_2-Connections

Using the Katz-Arinkin algorithm we give a classification of irreducible rigid irregular connections on a punctured $\mathbb{P}^1_{\mathbb{C}}$ having differential Galois group $G_2$, the exceptional simple algebraic group, and slopes having numerator 1. In addition to hypergeometric systems and their Kummer pull-backs we construct families of $G_2$-connections which are not of these types

Rigid G2-Representations and motives of Type G2

We prove an effective Hilbert Irreducibility result for residual realizations of a family of motives with motivic Galois group G2

Non-Schlesinger Isomonodromic Deformations of Fuchsian Systems and Middle Convolution

The paper is devoted to non-Schlesinger isomonodromic deformations for resonant Fuchsian systems. There are very few explicit examples of such deformations in the literature. In this paper we construct a new example of the non-Schlesinger isomonodromic deformation for a resonant Fuchsian system of order 5 by using middle convolution for a resonant Fuchsian system of order 2. Moreover, it is known that middle convolution is an operation that preserves Schlesinger's deformation equations for non-resonant Fuchsian systems. In this paper we show that Bolibruch's non-Schlesinger deformations of resonant Fuchsian systems are, in general, not preserved by middle convolution

On globally nilpotent differential equations

In a previous work of the authors, a middle convolution operation on the category of Fuchsian differential systems was introduced. In this note we show that the middle convolution of Fuchsian systems preserves the property of global nilpotence. This leads to a globally nilpotent Fuchsian system of rank two which does not belong to the known classes of globally nilpotent rank two systems. Moreover, we give a globally nilpotent Fuchsian system of rank seven whose differential Galois group is isomorphic to the exceptional simple algebraic group of type $G_2.

Comparison of integral structures on spaces of modular forms of weight two, and computation of spaces of forms mod 2 of weight one, with appendices by Jean-Francois Mestre and Gabor Wiese

Two integral structures on the Q-vector space of modular forms of weight two on X_0(N) are compared at primes p exactly dividing N. When p=2 and N is divisible by a prime that is 3 mod 4, this comparison leads to an algorithm for computing the space of weight one forms mod 2 on X_0(N/2). For p arbitrary and N>4 prime to p, a way to compute the Hecke algebra of mod p modular forms of weight one on Gamma_1(N) is presented, using forms of weight p, and, for p=2, parabolic group cohomology with mod 2 coefficients. Appendix A is a letter from Mestre to Serre, of October 1987, where he reports on computations of weight one forms mod 2 of prime level. Appendix B reports on an implementation for p=2 in Magma, using Stein's modular symbols package, with which Mestre's computations are redone and slightly extended.Comment: 39 pages, Late

Non-integrability of density perturbations in the FRW universe

We investigate the evolution equation of linear density perturbations in the Friedmann-Robertson-Walker universe with matter, radiation and the cosmological constant. The concept of solvability by quadratures is defined and used to prove that there are no "closed form" solutions except for the known Chernin, Heath, Meszaros and simple degenerate ones. The analysis is performed applying Kovacic's algorithm. The possibility of the existence of other, more general solutions involving special functions is also investigated.Comment: 13 pages. The latest version with added references, and a relevant new paragraph in section I