Weyl Sums for Quadratic Roots
W. Duke, J. B. Friedlander and H. Iwaniec
1
Introduction
The topic of exponential sums entered rmly into the mainstream of number
theory with the work [We] of H. Weyl. Although today very general sums
e (F (x) ,
e(z ) = exp (
CYCLE INTEGRALS OF THE J-FUNCTION AND MOCK MODULAR
FORMS
W. DUKE, O. IMAMOGLU, AND A . TOTH
Abstract. In this paper we construct certain mock modular forms of weight 1/2 whose Fourier
coecients are given in terms of cycle integrals of the modular j -funct
REAL QUADRATIC ANALOGUES OF TRACES OF SINGULAR
INVARIANTS
W. DUKE, O. IMAMOGLU, AND A . TOTH
1. Introduction
There are numerous connections between quadratic elds and modular forms. One of the most
beautiful is provided by the theory of singular invariant
RATIONAL PERIOD FUNCTIONS AND CYCLE INTEGRALS
W. DUKE, O. IMAMOGLU, AND A . TOTH
1. Introduction
In this paper, which is an outgrowth of [3], we will give some applications of the theory of
weakly holomorphic modular forms to rational period functions for
INTEGRAL TRACES OF SINGULAR VALUES OF WEAK MAASS FORMS
W. DUKE AND PAUL JENKINS
A BSTRACT. We dene traces associated to a weakly holomorphic modular form f of arbitrary negative even integral weight and show that these traces appear as coefcients of
certa
The zeros of the Weierstrass function and
hypergeometric series
W. Duke and O. Imamoglu
A BSTRACT. We express the zeros of the Weierstass -function in
terms of generalized hypergeometric functions. As an application
of our main result we prove the transce
Some Entries in Ramanujans Notebooks
W. Duke
1. Introduction
Some of Ramanujans original discoveries about hypergeometric
functions and their relation to modular integrals, especially Eisenstein series of negative weight, are still not very well understoo
ON THE ZEROS AND COEFFICIENTS OF CERTAIN WEAKLY
HOLOMORPHIC MODULAR FORMS
W. DUKE AND PAUL JENKINS
To J-P. Serre on the occasion of his eightieth birthday.
1. Introduction
For this paper we assume familiarity with the basics of the theory of modular forms
A combinatorial problem related to Mahlers
measure
W. Duke
A BSTRACT. We give a generalization of a result of Myerson on the
asymptotic behavior of norms of certain Gaussian periods. The
proof exploits properties of the Mahler measure of a trinomial.
1. I
On a Formula of Bloch
W. Duke and O. Imamoglu
To Jean-Marc Deshouillers, on his sixtieth birthday.
A BSTRACT. We give a new proof of a formula of Bloch for a special value of a certain Eisenstein series of weight one with an additive character.
1. Introdu
AN INTRODUCTION TO THE LINNIK PROBLEMS
W. Duke
UCLA
Abstract. This paper is a slightly enlarged version of a series of lectures on the Linnik problems
given at the SMSNATO ASI 2005 Summer School on Equidistribution in Number Theory.
Key words: Linnik prob
Journal de Th orie des Nombres
e
de Bordeaux 00 (XXXX), 000000
Special values of multiple gamma functions
par W. D UKE et O. I MAMO GLU
R E SUM E . Nous donnons une formule de type Chowla-Selberg
qui relie une g n ralisation de leta-fonction a GL(n) avec
Modular functions and the uniform
distribution of CM points
W. Duke
1. Introduction
The classical j -function is dened for z in the upper half plane H
by
j (z ) =
1 + 240
q
n=1
n=1 (1
2iz
m|n m
q n )24
3n3
q
= q 1 + 744 + 196884q + ,
where q = e(z ) = e .
Continued Fractions and Modular Functions
W. Duke
The mathematical universe is inhabited not only by important
species but also by interesting individuals. C.L. Siegel1
1. Introduction
It is widely recognized that the work of Ramanujan deeply inuenced the
Quadratic Reciprocity in a Finite Group
William Duke
Kimberly Hopkins
Dedicated to the memory of Abe Hillman
1
INTRODUCTION
The law of quadratic reciprocity is a gem from number theory. In this article
we show that it has a natural interpretation that can
On Ternary Quadratic Forms
W. Duke
Department of Mathematics, University of California, Los Angeles, CA 98888.
Dedicated to the memory of Arnold E. Ross
1
Introduction.
Let q (x) = q (x1 , x2 , x3 ) be a positive denite ternary quadratic form with integra
Lattice Points in Cones and Dirichlet Series
W. Duke and O. Imamolu
g
Department of Mathematics, University of California, Los Angeles, CA 98888.
Department of Mathematics, University of California, Santa Barbara, CA
93106
Abstract
Hecke proved the meromo
Number elds with large class groups
W. Duke
Department of Mathematics, University of California, Los Angeles, CA 98888.
Abstract
After a review of the quadratic case, a general problem about the existence of number elds of a xed degree with extremely larg
Extreme values of Artin L-functions and class
numbers
W. Duke
Department of Mathematics, University of California, Los Angeles, CA 98888.
Abstract
Assuming the GRH and Artin conjecture for Artin L-functions, we prove
that there exists a totally real numbe
Almost all reductions of an elliptic curve have a large
exponent
W. Duke
UCLA Mathematics Department
Box 951555
Los Angeles, CA 90095-1555
duke@math.ucla.edu
Abstract
Let E be an elliptic curve dened over Q. Suppose that f (x) is any positive function
ten
Rational points on the sphere
W. Duke
Department of Mathematics, University of California, Los Angeles, CA 98888.
Dedicated to the memory of Robert Rankin
Abstract
Using only basic tools from the theory of modular forms, the rational
points of bounded hei
The Splitting of Primes in Division Fields of
Elliptic Curves
o
W.Duke and A. Tth
z
Dedicated to the memory of Petr Ciek
Introduction
Given a Galois extension L/K of number elds with Galois group G, a fundamental problem is to describe the (unramied) prim
REGULARIZED INNER PRODUCTS OF MODULAR FUNCTIONS
W. DUKE, O. IMAMOGLU, AND A . TOTH
Dedicated to the memory of Marvin Knopp
Abstract. In this note we give an explicit basis for the harmonic weak forms of weight two.
We also show that their holomorphic coec
MOCK-MODULAR FORMS OF WEIGHT ONE
W. DUKE AND Y. LI
Abstract. The object of this paper is to initiate a study of the Fourier coecients of
a weight one mock-modular form and relate them to the complex Galois representations
associated to the forms shadow, w