Geodesic continued fractions and LLL

We discuss a proposal for a continued fraction-like algorithm to determine simultaneous rational approximations to $d$ real numbers $\alpha_1,\ldots,\alpha_d$. It combines an algorithm of Hermite and Lagarias with ideas from LLL-reduction. We dynamically LLL-reduce a quadratic form with parameter $t$ as $t\downarrow0$. The new idea in this paper is that checking the LLL-conditions consists of solving linear equations in $t$

Dwork crystals I

We present an elementary elaboration of Dwork's idea of explicit $p$-adic limit formulas for zeta functions of toric hypersurfaces.Comment: This version is identical with the earlier one, we only change the numeration of theorems so that it agrees with the version published by IMR

Duality relations for hypergeometric series

We explicitly give the relations between the hypergeometric solutions of the general hypergeometric equation and their duals, as well as similar relations for q-hypergeometric equations. They form a family of very general identities for hypergeometric series. Although they were foreseen already by N. M. Bailey in the 1930's on analytic grounds, we give a purely algebraic treatment based on general principles in general differential and difference modules.Comment: 16 page

pp-Linear schemes for sequences modulo prp^r

We construct finite $p$-automata for the computation of interesting combinatorial sequences modulo $p^r$. They are presented in the form of so-called $p$-linear schemes.Comment: 6 page

Gauss congruences for rational functions in several variables

We investigate necessary as well as sufficient conditions under which the Laurent series coefficients $f_{\boldsymbol{n}}$ associated to a multivariate rational function satisfy Gauss congruences, that is $f_{\boldsymbol{m}p^r} \equiv f_{\boldsymbol{m}p^{r-1}}$ modulo $p^r$. For instance, we show that these congruences hold for certain determinants of logarithmic derivatives. As an application, we completely classify rational functions $P/Q$ satisfying the Gauss congruences in the case that $Q$ is linear in each variable.Comment: 20 page

Finite hypergeometric functions

Finite hypergeometric functions are complex valued functions on finite fields which are the analogue of the classical analytic hypergeometric functions. From the work of N.M.Katz it follows that their values are traces of Frobenius on certain l-adic sheafs. More concretely, in many instances their values can be used to give formulas for pointcounts of F_q-rational points on certain varieties. In this paper we work out the case of one-variable functions whose monodromy in the analytic case can be defined over the rational integers.Comment: 26 pages, 2 figure

Dwork crystals II

We give a generalization of $p$-adic congruences for truncated period functions, that were originally discovered for a class of hypergeometric functions by Bernard Dwork.Comment: This version is identical with the previous one, we only changed the numbering of theorems to coincide with the version published by IMR