Arizona Winter School 1999
Course Descriptions and Notes

John Cremona: Symbolic computation and number theory

• Cremona's project was about “Rational points on conics.” There is a description of the project, and some suggested readings for the project.
• Cremona's web site has a number of preprints and and slides related to his talk at the Winter School, and to his student project. It also contains a lots of information about symbolic computation packages, other software, and mathematics.

Jean-Louis Colliot-Thélène, assisted by David Harari and Alexei Skorobogatov: The Hasse principle

• A detailed description of Colliot-Thélène's project on “The local-global principle” is available in dvi, ps, and pdf formats.
• A list of references for Colliot-Thélène's talks: dvi, ps, pdf
• Slides for Barbu's talk on the Brauer group: dvi, ps, pdf

Barry Mazur: Visualizing elements of the Shafarevich-Tate group

• There are 4 documents available for Mazur's project on “Visualizing elements in the Shafarevich-Tate group”:
  1. A bibliography (dvi, ps, pdf)
  2. Mazur's comments on the topic of “visualizing elements in the Shafarevich-Tate group,” (dvi, ps, pdf)
  3. The paper “Visualizing elements in the Shafarevich-Tate group,” by Cremona and Mazur (to appear in the Journal of Experimental Math) (dvi, ps, pdf)
  4. The paper “Visualizing elements of order three in the Shafarevich-Tate group,” by Mazur (to appear in the Journal of Asian Mathematics). (dvi, ps, pdf).
• The Mazur student project led to a preprint by William Stein entitled “Some Abelian Varieties with Visible Shafarevich-Tate Groups.” (dvi, ps, pdf). Stein's web site also has many interesting tables, preprints, and program related to modular forms and modular Jacobians.

Bill McCallum, assisted by Pavlos Tzermias and Joseph Wetherell: The method of Coleman and Chabauty

• Pavlos Tzermias wrote up his Winter School Lectures as “The Manin-Mumford Conjecture: A Brief Survey” (dvi, ps, pdf)
• Wetherell's project was on “An application of the method of Coleman and Chabauty.” Here is a description of the project; it refers to Wetherell's thesis, which is available in dvi, ps, and pdf formats.
• Cathy O'Neil's thesis (dvi, ps, pdf) on Jacobians of Curves of Genus One was the basis of Tomas Klenke's talk.
• Several U of A graduates students worked with Bill McCallum on constructing the Jacobian of a genus 1 curve. Their web site has the details. Slides for David Marshall's talk on this topic are available in dvi, ps, and pdf formats.

Bjorn Poonen: Mordell-Weil groups, Selmer groups, and Shafarevich-Tate groups

• Poonen's lectures at the Winter School were on background material, but two (or more) of the papers on his web site are relelvant to the topic: “Algebraic families of nonzero elements of Shafarevich-Tate groups” (with Colliot-Thélène) (pdf), and “Computational aspects of curves of genus at least 2” (ps, pdf)
• Poonen's project was on “The Mordell-Weil group of a genus 2 Jacobian.” Here is a description of the project.

Karl Rubin: Shafarevich-Tate groups of CM and modular elliptic curves

• Here are three papers related to Rubin's lectures at the Winter School:
  • “Elliptic curves with complex multiplication” (dvi, ps, pdf)
  • “Euler systems and modular elliptic curves” (dvi, ps, pdf)
  • “General Euler systems” (large: 840K dvi file, 9MB pdf file!) (dvi, ps, pdf)
• Karl's web site has other preprints that may be of interest.
• Rubin's project was on “Tate-Shafarevich groups of twists on an elliptic curve.” Here is a description of the project.