No abstract

Define B(n) to be the largest height of a polynomial in Z[x] dividing x n − 1. We formulate a number of conjectures related to the value of B(n) when n is of a prescribed form. Additionally, we prove a lower bound for B(p a q b ) where p, q are distinct primes.

ABSTRACT. There are a variety of characterizations of Saito-Kurokawa lifts from elliptic modular forms to Siegel modular forms of degree 2. In addition to giving a survey of known characterizations, we apply a recent result of Weissauer to provide a number of new and simpler characterizations of Saito-Kurokawa lifts.

We address the problem of evaluating an L-function when only a small number of its Dirichlet coefficients are known. We use the approximate functional equation in a new way and find that it is possible to evaluate the L-function more precisely than one would expect from the standard approach. The method, however, requires considerably more computational effort to achieve a given accuracy than would be needed if more Dirichlet coefficients were available.

L-functions can be viewed axiomatically, such as in the formulation due to Selberg, or they can be seen as arising from cuspidal automorphic representations of GL(n), as first described by Langlands. Conjecturally, these two descriptions of L-functions are the same, but it is not even clear that these are describing the same set of objects. We propose a collection of axioms that bridges the gap between the very general analytic axioms due to Selberg and the very particular and representation-theoretic construction due to Langlands. Along the way we prove theorems about L-functions that satisfy our axioms and state conjectures that arise naturally from our axioms.