Home | Math Dept Here you will find information about the Summer 2003 REU Program: Rational and Integer Points on Elliptic Curves that took place between May 12 and June 20, 2003.

## Two new sections were added to this web page: Projects with information about the different projects (available for exploration or the final results), and LaTeX resources with some basic links to introductory material on LaTeX.

## • Projects: here you will find a list of projects --some of which are mandatory-- for you to try.

• Counting curves: here are some histograms showing the distribution of the number of elliptic curves over Z/(p Z) for p prime between 5 and 293. This note (ps, pdf) explains why the histograms are symmetric.

• Counting points for different positive characteristics and experimental evidence for the Birch and Swinnerton-Dyer conjecture: here (ps, pdf) is a short note describing very simple experiments in this direction.

• Torsion groups: here (ps, pdf) you can find all groups that appear as torsion groups of elliptic curves, including a curves realizing them and some partial frequency data.

• Competition! Idea: find an integer point (x,y) on an elliptic curve y**2 = x**3 + a*x**2 + b * x +c with a, b, c integers so that R=sqrt(|x|/|discriminant(a,b,c)|) is as large as possible. For example, the point (-1,1) is on y**2=x**3-2*x and we have R=sqrt(|-1|/|discriminant(0,-2,0)|) = sqrt(1/32) ~ 0.18.

Some better points are:

(x,y) a b c R by
(28186307315582916794821057511400951
0272983487003358,*)
0 132 -935552736240624 3453575350.92 N. Elkies
(23330479,112690010143) -7 9 -14 55.5 Anon.
(17454560,72922784957) -9 4 9 30.1 Anon.
(47884,10477737) -4 1 5 13.6 M. Woodbury
(5853886516781223,4478849284284020423
07918)
0 0 -1641843 8.97 N. Elkies
(4677933,10117668338) -12 -13 4 8.08 Anon.
(3307172,6014313923) 14 -4 -7 6.20 Anon.
(1057027,1086752792) 8 -9 -8 5.65 Anon.

Do you know of other big points? Let us know!.

• Cryptography. This is an interesting overview on applications of elliptic curves to cryptography.

## Noam Elkies' mathematics web page: lots of interesting stuff on computational number theory. In particular, check the material related to Hall's conjecture. Rational Points on Elliptic Curves: Hisanori Mishima's page with lots of numerical results. ECMNET Project with lots of information on factorization using Lenstra's Elliptic Curve algorithm. W. Stein's pages with (among other things) interesting information on elliptic curves. Check here for some (elementary) readings.

## A (preliminary version of) a tutorial (ps, pdf) on using python for doing computations on elliptic curves. Code to add to a .emacs file to make emacs python-aware. Python module to operate with rational numbers. Download and save in the directory where you keep your python stuff. Another very similar implementation written by Jim Carlson, called Rational. Jim Carlson's Computations in Number Theory Using Python. Python's web site. Contains lots of documentation as well as versions of python for different operating systems. Dive into python: an on-line book on python.

## This is a medium sized introduction to LaTeX. These notes produced by the AMS concentrate on typing Mathematics into LaTex. The TUG web site contains lots of information (including references to tutorials and books) on TeX and its family. Here are two sample files: art1.tex is a simple file to show the basic structure of a latex file, while art2.tex is the file of a medium sized paper where you can see more latex features. art1.ps and art2.ps are the "printer ready" version of the previous files. Modify your .emacs file: add, as a single line, the statement `(require 'tex-site)` and remember to re-start emacs. 3 step program to compile and view or print the file "art.tex": `latex2e art` This step will generate a file called "art.dvi". This file has yet to be converted so that it can be seen on the screen or sent to the printer. This step may have to be repeated two (and rarely three times): check the last few lines of the output for more information. `xdvi art` This will open a window on your computer and will display the formatted article. `dvips art` This step will generate the file "art.ps" that is ready to be sent to the printer, usually with the command `lpr art.ps`

## Campus map and building locator. University of Utah homepage. Mathematics undergraduate colloquium.

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