Irreducibility of A-hypergeometric systems

We give an elementary proof of the Gel'fand-Kapranov-Zelevinsky theorem that non-resonant A-hypergeometric systems are irreducible. We also provide a proof of a converse statement In this second version we have removed the condition of saturatedness in Theorems 1.2 and 1.3. In Theorem 1.3 it is replaced by the condition of Cohen-Macaulayness of the toric ideal

Irrationality of some p-adic L-values

We give a proof of the irrationality of the $p$-adic zeta-values $\zeta_p(k)$ for $p=2,3$ and $k=2,3$. Such results were recently obtained by F.Calegari as an application of overconvergent $p$-adic modular forms. In this paper we present an approach using classical continued fractions discovered by Stieltjes. In addition we show irrationality of some other $p$-adic $L$-series values, and values of the $p$-adic Hurwitz zeta-function

On B.Dwork's accessory parameter problem

Let P ε C [z] be a monic quadratic polynomial with non-zero discriminant and P(0) ≠ 0. Let λ ε C. Consider the linear differential equation zP(z) d2u/dz2 + (zP(z)) ,du/dz + (z-λ)u=0. Note that this is the general shape of a Fuchsian differential equation on P1 with singularities in four points, including ∞, having local exponents 0,0 at the nite points and 1; 1 at ∞. By scaling z if necessary we can assume that P has the form P(z) =z2+az-

Zeeman's monotonicity conjecture

In this paper we prove a conjecture of Zeeman about the monotonicity of the rotation number of a family of dieomorphisms φ of the first quadrant Q of R

Fields of definition of finite hypergeometric functions

Finite hypergeometric functions are functions of a finite field ${\bf F}_q$ to ${\bf C}$. They arise as Fourier expansions of certain twisted exponential sums and were introduced independently by John Greene and Nick Katz in the 1980's. They have many properties in common with their analytic counterparts, the hypergeometric functions. One restriction in the definition of finite hypergeometric functions is that the hypergeometric parameters must be rational numbers whose denominators divide $q-1$. In this note we use the symmetry in the hypergeometric parameters and an extension of the exponential sums to circumvent this problem as much as posssible.Comment: 8 page

Log Fano varieties over function fields of curves

Consider a smooth log Fano variety over the function field of a curve. Suppose that the boundary has positive normal bundle. Choose an integral model over the curve. Then integral points are Zariski dense, after removing an explicit finite set of points on the base curve.Comment: 18 page