Elliptic (p,q)-difference modules

Drinfeld Modules and Elliptic Sheaves

Elliptic Hypergeometric Solutions to Elliptic Difference Equations⋆

It is shown how to define difference equations on particular lattices {xn}, n ∈ Z, made of values of an elliptic function at a sequence of arguments in arithmetic progression (elliptic lattice). Solutions to special difference equations have remarkable simple interpolatory expansions. Only linear difference equations of first order are considered here.

Rank-one Drinfeld Modules on Elliptic Curves

The sgn-normalized rank-one Drinfeld modules 4> associated with all elliptic curves E over ¥q for 4 < q < 13 are computed in explicit form. (Such 4> for q < 4 were computed previously.) These computations verify a conjecture of Dormán on the norm of j{) = aq+l and also suggest some interesting new properties of . We prove Dorman's conjecture in the ramified case. We also prove the formula...

Linear Elliptic Difference Inequalities with Random Coefficients

We prove various pointwise estimates for solutions of linear elliptic difference inequalities with random coefficients. These estimates include discrete versions of the maximum principle of Aleksandrov and Harnack inequalities and Holder estimates of Krylov and Safonov for elliptic differential operators with bounded coefficients.

Difference-elliptic Operators and Root Systems

Recently a new technique in the harmonic analysis on symmetric spaces was suggested based on certain remarkable representations of affine and double affine Hecke algebras in terms of Dunkl and Demazure operators instead of Lie groups and Lie algebras. In the classic case (see [O,H,C5]) it resulted (among other applications) in a new theory of radial part of Laplace operators and their deformati...