William Duke

Continued fractions and modular functions

It is widely recognized that the work of Ramanujan deeply influenced the direction of modern number theory. This influence resonates clearly in the “Ramanujan conjectures.” Here I will explore another part of his work whose position within number theory seems to be less well understood, even though it is more elementary, namely that related to continued fractions. I will concentrate on the special values of continued fractions that represent modular functions, especially the Rogers-Ramanujan continued fraction. These give analogues of the simple continued fraction expansions of units in real quadratic fields. My primary motivation is to furnish a coherent treatment of this topic, around which an air of mystery seems to linger. Another is to provide an inviting and non-standard introduction to the classical theory of modular functions. This is largely an expository paper; most of the ideas I discuss are well known. Yet it is hoped that the elaboration given here combines these ideas in a novel way. Although this paper is not intended to be comprehensive, its later sections contain more material than is likely needed to gain a clear impression of the main themes, which the first six sections should provide. These will take the general reader through a proof of the first main result, Theorem 1, introducing the needed concepts along the way. The sections that follow these assume a somewhat greater background in number theory.

Extreme Values of Artin L-Functions and Class Numbers

Assuming the generalized Riemann hypothesis (GRH) and Artin conjecture for Artin L-functions, we prove that there exists a totally real number field of any fixed degree (>1) with an arbitrarily large discriminant whose normal closure has the full symmetric group as Galois group and whose class number is essentially as large as possible. One ingredient is an unconditional construction of totally real fields with small regulators. Another is the existence of Artin L-functions with large special values. Assuming the GRH and Artin conjecture it is shown that there exist an Artin L-functions with arbitrarily large conductor whose value at s = 1 is extremal and whose associated Galois representation has a fixed image, which is an arbitrary nontrivial finite irreducible subgroup of GL(n, ℂ) with property GalT.

The Splitting of Primes in Division Fields of Elliptic Curves

We give a global description of the Frobenius for the division fields of an elliptic curve E that is strictly analogous to the cyclotomic case. This is then applied to determine the splitting of a prime p in a subfield of such a division field. These subfields include a large class of nonsolvable quintic extensions and our application provides an arithmetic counterpart to Kleins “solution” of quintic equations using elliptic functions. A central role is played by the discriminant of the ring of endomorphisms of the reduced curve modulo p.

The graphic nature of Gaussian periods

Recent work has shown that the study of supercharacters on abelian groups provides a natural framework within which to study certain exponential sums of interest in number theory. Our aim here is to initiate the study of Gaussian periods from this novel perspective. Among other things, our approach reveals that these classical objects display dazzling visual patterns of great complexity and remarkable subtlety.

Quadratic Reciprocity in a Finite Group

A key role in our story is played by group characters. Recall that a character X of a finite Abelian group G is a homomorphism from G into C*, the multiplicative group of nonzero complex numbers. The set of all distinct characters forms a group under pointwise multiplication that is isomorphic to G. Later we will need the notion of a character defined on an arbitrary finite group G, which is the trace of a finitedimensional representation of G. A character x of the group (Z/nZ)* of reduced residue classes modulo a positive integer n gives rise to a Dirichlet character modulo n, also denoted by X, which is the function on the integers defined by

Rational Points on the Sphere

Using only basic tools from the theory of modular forms, the rational points of bounded height on the sphere are counted and shown to be uniformly distributed. The more difficult case of points with a given height is also treated.

Infinite products of cyclotomic polynomials

We study analytic properties of certain infinite products of cyclotomic polynomials that generalize some introduced by Mahler. We characterize those that have the unit circle as a natural boundary and use associated Dirichlet series to obtain their asymptotic behavior near roots of unity.

Some entries in Ramanujan's notebooks

Some of Ramanujan’s original discoveries about hypergeometric functions and their relation to modular integrals, especially Eisenstein series of negative weight, are still not very well understood. These discoveries take the form of identities that he recorded, without proof, as entries in his notebooks.∗ In the following sections I will introduce some of these entries, discuss their status, give new proofs of several of them and also provide new results of a similar nature.

AN INTRODUCTION TO THE LINNIK PROBLEMS

This paper is a slightly enlarged version of a series of lectures on the Linnik problems given at the SMS-NATO ASI 2005 Summer School on Equidistribution in Number Theory.