## Instructor Information

Instructor:William Stein (Benjamin Peirce Assistant Professor of Mathematics)

Office:Science Center 515 (by the balcony)

Office Hours:Monday 2-3PM, Friday 3-4PM

Phone:617-495-1790 (office), 617-308-0144 (mobile)

Email:[email protected]

Web page:http://modular.fas.harvard.edu/edu/Fall2003/252/## Course Objectives

Andrew Wiles proved Fermat's Last Theorem by showing that most elliptic curves

y^{2}=x^{3}+ax+bare "modular". In this course we will study modular elliptic curves along with their higher-dimensional generalizations, which are called "modular abelian varieties". I willnotassume that you already know about modular forms and abelian varieties.

My main goal for this course is to describe how to construct an abelian variety from a modular form.I will then state the Birch and Swinnerton-Dyer Conjecture for modular abelian varieties and explain some of the main theorems and computational evidence for it, thus taking you to the forefront of current research. I will also discuss methods for computing with modular abelian varieties.## Evaluation

There will be

homework(60%) and afinal project(40%), but no exams.

Homeworkwill consist of weekly problem sets that will be graded.- The
final projectwill be a paper about something related to the course. This paper could be a further exploration of examples or a proof of something not explained in the lectures.## Textbooks

The primary textbooks for the course are

Seminar on Fermat’s Last Theorem(ed. Murty) andArithmetic Geometry(ed. Cornell and Silverman), but we will look at many other articles and books. See the references section of the web site for comments, reviews, and scans of some of the relevant material.## Tables and Software

Tables:You'll certainly want to look at the modular forms database and, in particular, at the modular forms explorer.Computations:For doing computations with modular forms and modular symbols, use the MAGMA computer algebra system. This is not a free program, but since you are in my course I can get you a free copy for Windows, Mac OS X, or Linux. I can also get you an account on the MECCAH cluster, which has MAGMA installed on it.## Detailed Outline

I will fill in the references for the topics listed below as the semester progresses. Each bullet represents approximately one lecture.

## 1. Modular Curves

- The
modular groupSL_{2}(Z): action on upper half plane, generators, relations, fundamental domain (Chapter VII of Serre'sA course in arithmetic)Congruence subgroups: Gamma_{0}(N), Gamma_{1}(N), a simple method to find generators (Section III.2 of Lang'sIntroduction to modular forms); modular curvesoverC: Quotient of the upper half plane by a congruence subgroup; compactification (Section 7.1 of Diamond-Im, Section III.1 of Lang'sIntroduction to modular forms)Moduli-theoretic interpretation: Isomorphism classes of elliptic curves with levelNstructure (Section 7.2 of Diamond-Im);natural mapsbetween modular curves andgenusformulasHomologyof modular curves andmodular symbols; Manin symbols: finite presentation for modular symbol (Cremona,Algorithms for modular elliptic curves;Manin,Parabolic points and zeta functions of modular curves)- Modular curves are
defined over Q; equations for modular curves: Low genus, canonical embedding, Galbraith's thesis## 2. Abelian Varieties

Complex tori: morphisms, isogenies, endomorphismsHermitianandRiemann formsPoincare reducibility theoremoverCandsemisimplicityof endomorphism algebraAlgebraic groupsandgroup schemesTheta functionsand theLefschetz embedding theoremNeron-Severiandpicard group;polarizationsof abelian varietiesComputing intersectionsof complex tori- Abelian varieties
over number fields## 3. Jacobian Varieties

Definition;Abel-Jacobi theory(Mumford,Curves and Their Jacobians, chapter 3)Autodualityof Jacobians and thetheta divisorJacobians of modular curves,modular symbols, thecuspidal subgroup## 4. Modular Forms

Definitions;q-expansions,cusp formsandEisenstein seriesDirichlet charactersand thePetersson inner productMultiplicity oneDimensionformulas## 5. Hecke Algebras

- Hecke operators as
correspondencesand onq-expansions- Hecke operators on
modular and Manin symbolsStructureof Hecke algebras; Sturm boundNewformsand how to enumerate them## 6. Attaching an Abelian Variety to a Modular Form

Attachingan abelian variety to a newform- The
endomorphism ringofAoverQand the algebraic closureModularity conjectures:Which abelian varieties are modular? Serre's conjectures.## 7. The Birch and Swinnerton-Dyer Conjecture

attached to modular forms and abelian varietiesL-functions; Hecke's theoremthat modularL-functions extend to the whole complex plane- The
Mordell-Weil group,heightsand theregulator(sketch of proof of Mordell-Weil theorem)- The
Neron model, real volume,Manin constant, andcomputingL(1)/OmegaTamagawa numbersand component groups- The
Shafarevich-Tate groupandvisibility- Statement of the
Birch and Swinnerton-Dyer conjecture; results and examplesHeegner pointsand theGross-Zagier theoremEuler systemsofKolyvaginandKato