Charting the Land of Elliptic CurvesWilliam Stein

Benjamin Peirce Asst. Prof.

March 2002

Computers in Mathematics Computers are increasingly used in mathematics in

many ways, some sensible and others not. The computations I will talk about today are

completely precise. They are not about drawing pretty pictures, but about seeing exactly how certain mathematical objects (elliptic curves) behave.

Over the last few decades computations of the type I will describe today have repeatedly had a major influence on the direction of research in number theory. A famous mathematician, Bryan Birch, once said to me: "It is always best to prove true theorems."

What Is An Elliptic Curve?

An elliptic curve is a cubic curve where a and b are integers and

The conductor of E is an integer divisible only by primes that divide disc(E): The powers of the primes that divide the conductor

encode information about what the graph of E looks like modulo those primes.

There are only finitely many "essentially different" elliptic curves with a given conductor.

Why Are Elliptic Curves So Interesting?

The set has a natural group structure.

Wiles and Taylor (at Harvard) proved that purported counterexamples to Fermat's Last Theorem give rise to elliptic curves that can't exist.

Elliptic curves (modulo p) of great practical importance in cryptography (work of Lenstra, Elkies, etc.).

The Graph Of An Elliptic Curve

Rational Points(2,3)(0,1)(-1,0)(0,-1)(2,-3)

point at infinity

Adding Two Points

(0,1) + (2,3) = (-1,0)

Two More Graphs

Some Points(1,0),(0,2),

(25/16, -3/64),(352225/576, 209039023/13824)

... infinitely many ...

Tables of Elliptic Curves1 Antwerp IV (1970s: Birch, Swinnerton-Dyer, et al.):

all 749 curves of conductor up to 200 (modulo errors)

2 Cremona (1980s-now):all 78198 curves of conductor < 12000.

3 Brumer-McGuinness (1989-90): 310716 curves of prime conductor <(not all curves of prime conductor < )

4 Watkins-Stein (in progress):44 million curves of conductor < . (Not all!)

5 Stein (2000-now): Abelian varieties: higher-dimensional analogues of curves.

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What is in These Tables?Answer: Cubic equations and extensive data about each listed elliptic curve

Most of the standard arithmetic invariants of each curve. This data gives very strong corroboration for the famous conjecture of Birch and Swinnerton-Dyer, which ties these invariants together. (There is no known provably-correct algorithm to compute all the invariants appearing in the conjecture, but we usually succeed in practice.)

If there is a "homomorphism" from E onto F, we say that E and F are isogenous (isogeny is an equivalence relation). The curves are divided up into isogeny classes, and the structure of the isogenies is given.

Gave evidence for the Shimura-Taniyama conjecture (before it was proved by Taylor, Wiles, Breuil, Conrad, and Diamond).

The Antwerp (Belgium) Tables Table of all elliptic curves of conductor up to 200. Created around 1972 by Swinnerton-Dyer, Birch,

Davenport, V₫lu, Tingley, & Stephens. Also used methods of Serre, Tate, and Deligne. Published in Springer LNM 476

Beginning of the table:

Birch and Swinnerton-Dyer

From Antwerp...

John Cremona's Tables

1992, 97: Published extensive data about every single elliptic curves of conductor up to 1000 in a hefty book.

Used Algol68 and the ICL3980 computer (batch jobs), which limited portability; later used C++.

Subsequently extended table to conductor up to 12000 (data available on the web).

Inspiring story of me photocopying the whole book at Arizona Winter School...

The Brumer-McGuinness Tables

1989-1990 using Macintosh II computer. Table of 310716 curves of prime conductor <

(some curves were undoubtedly missed...) They systematically enumerated equations of elliptic

curves and threw out those curves whose conductor is bigger than or composite.

Computed points on these curves and were surprised to find that 40% of their "even" curves have infinitely many rational points. ("Conventional wisdom: asymptotically 0% of all even curves have rank > 0.")

Brumer

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The Stein-Watkins Database

Now: Database of (probably most) curves with |discriminant| < and conductor < , along with extensive data about each curve.

Contain about 44 million curves (which contains at least 80% of Cremona's data).

Would take years to create with a single standard computer, so computation is being done at Penn State, Berkeley, NSA, and soon on MECCAH, the Mathematics Extreme Computation Cluster at Harvard, which is a gift of The Friends and the Clark/Tozier fund.

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Higher Dimensional Analogues(Abelian Varieties)

Elliptic curves are modular, which means they live in Jacobians of modular curves.

Most of the Jacobian of a modular curve consists of higher-dimensional analogues of elliptic curves called abelian varieties.

I have created a database about most abelian varieties of level < 4000. (Maybe give live demonstration via internet.)

I intend to greatly extend this database using MECCAH.

What is MECCAH?Mathematics Extreme Computation Cluster At Harvard

Inside Outside

Six dual-processor Athlon MP 2000+ rackmounted computers with at least 2GB memory each, running Linux and linked together as a single computer via MOSIX. (Currently under construction.)

From users' point of view, the 6 computers appear as a single computer with 12 processors.

MOSIX supports job-level parallelization: Users do not have to rewrite their code in order to take

full advantage of the cluster; they simply run several jobs at once.

MOSIX doesn't support fine-grain parallelization, e.g., multiplying a huge matrix quickly using lots of nodes of a network. Thus MOSIX isn't good for weather forecasters.

Why MOSIX?

A Top Output Under MOSIX

Primes and dragon3 run on node 0, and mathematica and two copiesof primes are running on node 2.(Log in to MECCAH and type "mtop". Run some jobs, etc.)

Thanks! Any Questions?