Jeremy Teitelbaum

Banach space representations and Iwasawa theory

We develop a duality theory between the continuous representations of a compactp-adic Lie groupG in Banach spaces over a givenp-adic fieldK and certain compact modules over the completed group ringoK[[G]]. We then introduce a “finiteness” condition for Banach space representations called admissibility. It will be shown that under this duality admissibility corresponds to finite generation over the ringK[[G]]: =K ⊗oK[[G]]. Since this latter ring is noetherian it follows that the admissible representations ofG form an abelian category. We conclude by analyzing the irreducibility properties of the continuous principal series of the groupG: = GL2(ℤp).

Lines on Calabi-Yau complete intersections, mirror symmetry, and Picard-Fuchs equations

A relation between the number of rational curves of fixed degree on Calabi Yau threefolds and the Picard Fuchs equations, which was suggested as part of the study of mirror symmetry, is verified in the case of complete intersection of two cubics and lines.

Euclid's algorithm and the lanczos method over finite fields

This paper shows that there is a close relationship between the Euclidean algorithm for polynomials and the Lanczos method for solving sparse linear systems, especially when working over finite fields. It uses this relationship to account rigorously for the appearance of self-orthogonal vectors arising in the course of the Lanczos algorithm. It presents an improved Lanczos method which overcomes problems with self-orthogonality and compares this improved algorithm with the Euclidean algorithm.