Charting the Land of Elliptic Curves

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Charting the Land of Elliptic Curves
William Stein Benjamin Peirce Asst. Prof.
March 2002
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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."

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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.

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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.).

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The Graph Of An Elliptic Curve
Rational Points (2,3) (0,1) (-1,0) (0,-1) (2,-3) p
oint at infinity
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Adding Two Points
(0,1) (2,3) (-1,0)
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Two More Graphs
Some Points (1,0), (0,2), (25/16,
-3/64), (352225/576, 209039023/13824) ...
infinitely many ...
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Tables of Elliptic Curves

• Antwerp IV (1970s Birch, Swinnerton-Dyer, et
  al.) all 749 curves of conductor up to 200
  (modulo errors)
• Cremona (1980s-now)all 78198 curves of
  conductor lt 12000.
• Brumer-McGuinness (1989-90) 310716 curves of
  prime conductor lt(not all curves of prime
  conductor lt )
• Watkins-Stein (in progress)44 million curves of
  conductor lt . (Not all!)
• 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).

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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
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From Antwerp...
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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...

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The Brumer-McGuinness Tables

• 1989-1990 using Macintosh II computer.
• Table of 310716 curves of prime conductor lt
  (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 gt 0.")

Brumer
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The Stein-Watkins Database

• Now Database of (probably most) curves with
  discriminant lt and conductor lt ,
  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 lt 4000. (Maybe give live
  demonstration via internet.)
• I intend to greatly extend this database using
  MECCAH.

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What is MECCAH?Mathematics Extreme Computation
Cluster At Harvard
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.)
Inside
Outside
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Why MOSIX?

• 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.

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A Top Output Under MOSIX
Primes and dragon3 run on node 0, and mathematica
and two copies of primes are running on node
2. (Log in to MECCAH and type "mtop". Run some
jobs, etc.)
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Thanks! Any Questions?