Differential equations and Hecke operators

Differential equations and intertwining operators

We show that if every module W for a vertex operator algebra V = ∐ n∈Z V(n) satisfies the condition dimW/C1(W ) < ∞, where C1(W ) is the subspace of W spanned by elements of the form u−1w for u ∈ V+ = ∐ n>0 V(n) and w ∈W , then matrix elements of products and iterates of intertwining operators satisfy certain systems of differential equations. Moreover, for prescribed singular points, there exi...

Hecke Operators

Integral Operators and Delay Differential Equations

The monodromy operator of a linear delay differential equation with periodic coefficients is formulated as an integral operator. The kernel of this operator includes a factor formed from the fundamental solution of the linear delay differential equation. Although the properties of the fundamental solutions are known, in general there is no closed form for the fundamental solution. This paper de...

Hecke Operators on Cohomology

Hecke operators play an important role in the theory of automorphic forms, and automorphic forms are closely linked to various cohomology groups. This paper is mostly a survey of Hecke operators acting on certain types of cohomology groups. The class of cohomology on which Hecke operators are introduced includes the group cohomology of discrete subgroups of a semisimple Lie group, the de Rham c...

Modular Symbols and Hecke Operators

We survey techniques to compute the action of the Hecke operators on the cohomology of arithmetic groups. These techniques can be seen as generalizations in different directions of the classical modular symbol algorithm, due to Manin and Ash-Rudolph. Most of the work is contained in papers of the author and the author with Mark McConnell. Some results are unpublished work of Mark McConnell and ...