A basic problem of diophantine analysis is to investigate the asymptotics as T of (1.2) N(T, V)= {m V(Z): Ilmll T} where we denote by V(A), for any ring A, the set of A-points of V. Hence I1" is some… (More)

(1) zd = { i √ d 2 if d ≡ 0 (mod 4), −1+i √ d 2 if d ≡ 3 (mod 4). The j-function has the remarkable property that j(zd) is an algebraic integer of degree h(−d), the class number of K = Q(zd). In… (More)

190 NOTICES OF THE AMS VOLUME 44, NUMBER 2 Introduction It may be a challenging problem to describe the integer solutions to a polynomial equation in several variables. Which integers, for example,… (More)

It is a basic problem to determine the dimension of the space of cusp forms of a given type. For classical holomorphic forms of integral weight larger than one, the dimension is well understood by… (More)

Assuming the 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… (More)

τ(n)e(nz). The two equations were proven for τ(n) by Mordell, using what are now known as the Hecke operators. The inequality was proven by Deligne as a consequence of his proof of the Weil… (More)

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… (More)

The law of quadratic reciprocity is a gem from number theory. In this article we show that it has a natural interpretation that can be generalized to an arbitrary finite group. Our treatment relies… (More)

After a review of the quadratic case, a general problem about the existence of number fields of a fixed degree with extremely large class numbers is formulated. This problem is solved for abelian… (More)

We define traces associated to a weakly holomorphic modular form f of arbitrary negative even integral weight and show that these traces appear as coefficients of certain weakly holomorphic forms of… (More)