Elliptic Genera of Level N and Elliptic Cohomology

Elliptic genera of level N have been defined by F. Hirzebruch,generalising the earlier notion of elliptic genus due to S.Ochanine. We show that there are corresponding elliptic cohomologytheories which are naturally associated to such genera and thatthese are obtained from the level 1 case by algebraic extensionof the coefficient rings from level 1 to level N modular forms.     

Functional Equations and Weierstrass Transforms

The purpose of this article is to illustrate the utility of the Weierstrass transform in the study of functional equations (and systems) of the form 1 $${\mathop \sum^N\limits_{k=0}}\alpha_{k}f(x+r_{k})=f_{0}(x)\ \ \ \, x\in\ {\rm R}.$$ One may think of α₀, α₁,…, α_N as given complex numbers, r₀, r₁,…, r_N as given real numbers, ?₀: ? → C as a given function and ? as the unknown.     

A Bruen chain forq=19

The one-to-one correspondence between the class of two-dimensional translation planes of orderq ² and the collection of spreads ofPG(3,q) has long provided a natural context for describing new planes. The method often used for constructing interesting spreads is to start with a regular spread, corresponding to a desarguesian plane, and then replace some nice subset of lines by another partial spread covering the same set of points. Indeed the first approach was replacing the lines of a regulus by the lines of its opposite regulus, or doing this process for a set of disjoint reguli. Nontrivial generalizations of this idea include thechains of Bruen and thenests of Baker and Ebert. In this paper we construct a replaceable subset of a regular spread ofPG (3, 19) which is the union of 11 reguli double covering the lines in their union, hence is a chain in the terminology of Bruen or a 11-nest in the Baker-Ebert terminology.