Don Zagier

Values of Zeta Functions and Their Applications

Zeta functions of various sorts are all-pervasive objects in modern number theory, and an ever-recurring theme is the role played by their special values at integral arguments, which are linked in mysterious ways to the underlying geometry and often seem to dictate the most important properties of the objects to which the zeta functions are associated. It is this latter property to which the word “applications” in the title refers. In this article we will give a highly idiosyncratic and prejudiced tour of a number of these “applications,” making no attempt to be systematic, but only to give a feel for some of the ways in which special values of zeta functions interrelate with other interesting mathematical questions. The prototypical zeta function is “Riemann’s” (math) and the prototypical result on special values is the theorem that ζ(k) = rational number × π k (k > 0 even), (1) which Euler proved in 1735 and of which we will give a short proof in Section 1. (The “applications” in this case are the role which the rational numbers occurring on the right-hand side of this formula play in the theory of cyclotomic fields, in the construction of p-adic zeta functions, and in the investigation of Fermât’s Last Theorem.)

Volumes of hyperbolic three-manifolds

BY“hyperbolic 3-manifold” we will mean an orientable complete hyperbolic 3-manifold M of finite volume. By Mostow rigidity the volume of M is a topological invariant, indeed a homotopy invariant, of the manifold M. There is in fact a purely topological definition of this invariant, due to Gromov. The set of all possible volumes of hyperbolic 3-manifolds is known to be a well-ordered subset of the real numbers and is of considerable interest (for number theoretic aspects see, for instance, [2], [13]) but remarkably little is known about it: the smallest element is not known even approximately, and it is not known whether any element of this set is rational or whether any element is irrational. For more details see Thurston’s Notes [73. In this paper we prove a result which, among other things, gives some metric or analytic information about the set of hyperbolic volumes. Given a hyperbolic 3-manifold M with h cusps, one can form the manifold MK = M(P1.41. . , Ph4J obtained by doing a (pi, q,)-Dehn surgery on the i-th cusp, where (pi, qi) is a coprime pair of integers, or the symbol 03 if the cusp is left unsurgered. This notation is well defined only after choosing a basis mi, di for the homology HI (Q, where Z is a torus cross section of the i-th cusp. Then (pi, q,)-Dehn surgery means: cut off the i-th cusp and paste in a solid torus to kill Pimi+qiei.

Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 57 Chapter 1 : The Intersection Behaviour of the Curves T N . . . . . . 60 1.1. Special Points . . . . . . . . . . . . . . . . . . . . . 60 1.2. Modules in Imaginary Quadratic Fields . . . . . . . . . . 68 1.3. The Transversal Intersections of the Curves T N . . . . . . . 74 1.4. Contributions from the Cusps . . . . . . . . . . . . . . 78 1.5. Self-Intersections . . . . . . . . . . . . . . . . . . . . 82 Chapter 2: Modular Forms Whose Fourier Coefficients Involve Class Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.1. The Modular Form ~oo(z ) . . . . . . . . . . . . . . . . 88 2.2. The Eisenstein Series of Weight 3 . . . . . . . . . . . . . 91 2.3. A Theta-Series Attached to an Indefinite Quadratic Form . . 96 2.4. Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . I00 Chapter 3: Modular Forms with Intersection Numbers as Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.1. Modular Forms of Nebentypus and the Homology of the Hilbert Modular Surface . . . . . . . . . . . . . . . . . . . . t03 3.2. The Relationship to the Doi-Naganuma Mapping . . . . . 108 References . . . . . . . . . . . . . . . . . . . . . . . . . . 111

A Kronecker limit formula for real quadratic fields

Then ((s, A) is (after analytic continuation) a meromorphic function of s with a simple pole at s = 1 as its only singularity. Moreover, the residue of ((s, A) at s = 1 is independent of the ideal class A chosen; this fact, discovered by Dirichlet (for the case of quadratic fields) is at the basis of the analytic determination of the class number of K. If we consider the Laurent expansion of ((s, A) at s = 1, however, say

A generalized Jacobi theta function and quasimodular forms

In this note we give a direct proof using the theory of modular forms of a beautiful fact explained in the preceding paper by Robbert Dijkgraaf [1, Theorem 2 and Corollary]. Let \( {\tilde M_*}({\Gamma _1}) \) denote the graded ring of quasi-modular forms on the full modular group Γ= PSL(2, ℤ). This is the ring generated by G2, G4, G6, and graded by assigning to each G k the weight where \( {G_k} = - \frac{{{B_k}}}{{2k}} + \sum\limits_{n = 1}^\infty {\left( {{{\sum\limits_{d|n} d }^{k - 1}}} \right)} {q^n}\left( {k \geqslant 2,{B_k} = kth Bernoulli number} \right) \) are the classical Eisenstein series, all of which except G 2 are modular.

Derivation and double shuffle relations for multiple zeta values

Derivation and extended double shuffle (EDS) relations for multiple zeta values (MZVs) are proved. Related algebraic structures of MZVs, as well as a ‘linearized’ version of EDS relations are also studied.

Period functions for Maass wave forms. I

Recall that a Maass wave form on the full modular group Gamma=PSL(2,Z) is a smooth gamma-invariant function u from the upper half-plane H = {x+iy | y>0} to C which is small as y \to \infty and satisfies Delta u = lambda u for some lambda \in C, where Delta = y^2(d^2/dx^2 + d^2/dy^2) is the hyperbolic Laplacian. These functions give a basis for L_2 on the modular surface Gamma\H, with the usual trigonometric waveforms on the torus R^2/Z^2, which are also (for this surface) both the Fourier building blocks for L_2 and eigenfunctions of the Laplacian. Although therefore very basic objects, Maass forms nevertheless still remain mysteriously elusive fifty years after their discovery; in particular, no explicit construction exists for any of these functions for the full modular group. The basic information about them (e.g. their existence and the density of the eigenvalues) comes mostly from the Selberg trace formula: the rest is conjectural with support from extensive numerical computations.

Modular forms and differential operators

In 1956, Rankin described which polynomials in the derivatives of modular forms are again modular forms, and in 1977, H Cohen defined for eachn ≥ 0 a bilinear operation which assigns to two modular formsf andg of weightk andl a modular form [f, g]n of weightk +l + 2n. In the present paper we study these “Rankin-Cohen brackets” from two points of view. On the one hand we give various explanations of their modularity and various algebraic relations among them by relating the modular form theory to the theories of theta series, of Jacobi forms, and of pseudodifferential operators. In a different direction, we study the abstract algebraic structure (“RC algebra”) consisting of a graded vector space together with a collection of bilinear operations [,]n of degree + 2n satisfying all of the axioms of the Rankin-Cohen brackets. Under certain hypotheses, these turn out to be equivalent to commutative graded algebras together with a derivationS of degree 2 and an element Φ of degree 4, up to the equivalence relation (∂,Φ) ~ (∂ - ϕE, Φ - ϕ2 + ∂(ϕ)) where ϕ is an element of degree 2 andE is the Fuler operator (= multiplication by the degree).

The Dilogarithm Function

1. Special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2. Functional equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3. The Bloch-Wigner function D(z) and its generalizations . . . . . . . . . . . . 10 4. Volumes of hyperbolic 3-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5. . . . and values of Dedekind zeta functions . . . . . . . . . . . . . . . . . . . . . . . . . 16

Periods of modular forms and Jacobi theta functions

co where L ( f s) denotes the L-series o f f ( = analytic continuation of ~ = 1 a:( l ) l s ) . The Eichler-Shimura-Manin theory tells us that the maple--, r: is an injection from the space Sg of cusp forms of weight k on F to the space of polynomials of degree _-< k 2 and that the product of the nth and ruth coefficients of r: is an algebraic multiple of the Petersson scalar product ( f , f ) i f f is a Hecke eigenform and n and m have opposite parity. More precisely, for each integer l > 1 the polynomial in two variables