Introduction

The 1999 Graduate Summer School of the Institute for Advanced Study/Park City Mathematics Institute was held in Park City, Utah from June 20 to July 10, 1999. The topic was arithmetic algebraic geometry. Joe Buhler, Fernando Gouvêa, Ralph Greenberg, Ken Ribet, and Alice Silverberg each gave a lecture course, and there were two special lectures by John Tate.

In general terms, arithmetic algebraic geometry is the use of methods from algebraic geometry to study problems in number theory. This is both a very active and very broad area. In deciding which topics to choose for the lectures, we were heavily (but not exclusively) influenced by the recent spectacular work of Wiles on modular elliptic curves and Fermat’s Last Theorem.

Many problems in number theory involve finding solutions of polynomial equations (or systems of polynomial equations) in integers, in rational numbers, or in other number fields. In studying such questions one is quickly led to try to understand the collection of all number fields, or equivalently, to understand the absolute Galois group G_{Q} = Gal(Q̄/Q) of the rational number field Q.

How can one try to understand G_{Q}? One of the most productive ways has been by studying its continuous representations (over the complex numbers, or over p-adic or finite fields). For example, the one-dimensional representations of G_{Q} are described by the Kronecker-Weber theorem, which says that there is a natural bijection between the set of representations of G_{Q} into C^{×} and the set of Dirichlet characters. More generally, if F is a number field then the one-dimensional representations of G_{F} are described by class field theory, and have been understood since the early 20th century.

The situation with higher-dimensional representations is much more difficult, and a great deal remains to be understood, even for two-dimensional representations. It is generally expected that all “well-behaved” representations should “come from geometry” in a certain precise sense.

What does this mean? First of all, one can use algebraic geometry to produce examples of higher dimensional representations of G_{Q}. For example, if E is an elliptic curve defined over Q, then the torsion points on E with coordinates in Q̄ form a group isomorphic to (Q/Z)^{2}. For every prime p, G_{Q} acts linearly on the p-part of this group, and thus we get a representation ρ_{E,p} of G_{Q} into GL_{2}(Q_{p}). In a similar but more complicated way, certain modular forms f (cusp forms which are eigenfunctions of the Hecke operators) give rise to two-dimensional representations ρ_{f,𝔭} of G_{Q} into GL_{2}(K_{f,𝔭}) where K_{f} is a number field determined by f and K_{f,𝔭} is its completion at a prime 𝔭.

The lectures of Buhler give an introduction to these topics, including elliptic curves, modular forms, and their L-functions and Galois representations.

The lectures of Silverberg discuss some of the current frontiers of research in arithmetic algebraic geometry by focusing on important open problems that are guiding the development of the field. For example: what is known, and what is not known, about torsion points on elliptic curves and abelian varieties? What is known, and what is conjectured, about ranks of elliptic curves? What is the Birch and Swinnerton-Dyer conjecture, and how much of it has been proved? What does it mean for an elliptic curve to be modular?

One formulation of the modularity conjecture of Shimura and Taniyama^{1} states that for every elliptic curve E defined over Q there is a cusp form f with K_{f} = Q such that ρ_{E,p} is isomorphic to ρ_{f,p} (in which case we say that E is modular).

Beginning in the early 1970’s Serre developed conjectures which (among other things) give a “first approximation” to the Shimura-Taniyama conjecture. Namely, Serre conjectured (the “weak” conjecture) that if ρ̄ is an irreducible representation of G_{Q} into GL_{2}(F̄_{p}), and det(ρ̄(τ)) = −1 where τ is a complex conjugation, then ρ̄ is modular in the sense that there is a cusp form f such that for some prime 𝔭 of K_{f} dividing p, the reduction of ρ_{f,𝔭} modulo 𝔭 is isomorphic to ρ̄ (after a suitable embedding of the residue field at 𝔭 into F̄_{p}). Serre also gave a recipe (the “strong” conjecture) for the minimal weight, level, and character of such an f in terms of ρ̄.

Not much progress has been made in proving Serre’s weak conjecture. A notable exception is when F = F_{3}, in which case work of Langlands and Tunnell shows that such a form f does exist. This case is crucial for Wiles’ program.

On the other hand, when one such f exists, a great deal of progress has been made in showing that an f of the prescribed weight, level and character exists. The lectures of Ribet (written for this volume jointly by Ribet and William Stein) introduce Serre’s conjectures and give some details of the methods and results which enable one to go from Serre’s weak conjecture to the strong conjecture.

Suppose now that one has a representation ρ : G_{Q} → GL_{2}(Q_{p}) which one wants to show is modular, and let ρ̄ : G_{Q} → GL_{2}(F_{p}) be the reduction of ρ modulo p. In the 1980’s Mazur introduced his deformation theory of Galois representations to study deformations of ρ̄, i.e., two-dimensional representations of G_{Q} which reduce to ρ̄. With appropriate restrictions on the deformations, Mazur showed that there is a “universal deformation ring” R, and a “universal deformation” ρ_{R} : G_{Q} → GL_{2}(R), which parametrizes all deformations in the sense that every representation of G_{Q} into GL_{2}(S) (of the appropriate type) whose reduction is ρ̄ is the composition of ρ_{R} with a unique homomorphism R → S.

Deformation theory is the subject of the lectures of Gouvêa. The goal is to get information about all deformations by studying the universal deformation ring. For example, if Serre’s conjecture holds for ρ̄, then one can also define a “universal modular deformation” 𝕋 which parametrizes all deformations of ρ̄ which are modular (i.e., isomorphic to some ρ_{f,𝔭}). Then (since R is universal) there is a homomorphism R → 𝕋, and if (after placing appropriate restrictions on the deformation problem) one can show that this map is an isomorphism, then one can conclude that “every deformation is modular”, including the original ρ. This was Wiles’ successful strategy.

In a different direction, not so directly related to the modularity problem, are Greenberg’s lectures on Iwasawa theory and elliptic curves. One can attach an L-function L(E, s) to an elliptic curve E over Q. This is a function of a complex variable s, initially defined by an Euler product in a right half-plane. It follows from the fact that E is modular that L(E, s) has an analytic continuation to all of C and satisfies a functional equation relating L(E, s) and L(E, 2 − s). The Birch and Swinnerton-Dyer conjecture predicts the order of vanishing of L(E, s) at s = 1, as well as the value of the first non-vanishing derivative of L(E, s) at s = 1, in terms of important invariants of E. One can think of L(E, s) as a generalization of Dirichlet L-functions, and the Birch and Swinnerton-Dyer conjecture as a generalization of classical analytic class number formulas.

1The senior organizer would like to thank the junior organizer and his colleagues Breuil, Diamond, and Taylor for completing their work on the program begun by Wiles just in time to announce a complete proof of this conjecture in Park City.

Iwasawa theory was introduced by Iwasawa beginning in the late 1950’s as a tool to study ideal class groups of cyclotomic fields. He observed that the group algebra \underleftarrow{lim} Z_{p}[Gal(Q(μ_{p^{n}})/Q)] is in many ways easier to work with than the individual group rings Z_{p}[Gal(Q(μ_{p^{n}})/Q)]. In particular, he found that the ideal class groups of the fields Q(μ_{p^{n}}) for n ≥ 1 can be much better understood if one studies them all at once rather than for each n separately. In the early 1970’s, Mazur used Iwasawa theory to study elliptic curves, and since then it has been extended by Greenberg and others to study a large class of problems.

Iwasawa theory for elliptic curves has two sides, analytic and algebraic. On the analytic side one can attach to E an analytic p-adic L-function, defined by interpolating values of classical L-functions. On the algebraic side one can study the arithmetic of E over the infinite abelian extension Q(μ_{p^{∞}}) of Q, and define an algebraic p-adic L-function which encodes some of the arithmetic of E. If one can show that these two p-adic L-functions are the same (the “main conjecture”), then one has begun to find the link between L(E, s) and arithmetic invariants of E which was predicted by Birch and Swinnerton-Dyer. Greenberg’s lectures deal with the algebraic side of the theory.

Galois cohomology is an essential tool used throughout arithmetic algebraic geometry, and Tate kindly agreed to give two introductory lectures which are included in this volume.

Also in this volume are the lectures which were delivered at the Mentoring Program for Women in Mathematics portion of PCMI at the Institute for Advanced Study in May 1999. The three lecturers were Winnie Li, Noriko Yui, and Alice Silverberg. For details about that program see the introduction to Li’s lectures.

Finally we would like to express our thanks to all the lecturers for their lectures and their written contributions to this volume. Their efforts to keep the Park City lectures accessible to as many of the participants as possible were successful and greatly appreciated. We would also like to thank all the teaching assistants, the audiences, and Catherine Jordan and the IAS/PCMI staff for making the program a success.

Karl Rubin Brian Conrad