Arizona Winter School 1999
Course Descriptions and Notes

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

  • “The local-global principle” project description: dvi, ps, pdf
  • References for talks: dvi, ps, pdf
  • Slides for Barbu's talk on the Brauer group: dvi, ps, pdf

• SHORTTITLE  Barry Mazur: Visualizing elements of the Shafarevich-Tate group

  • “Visualizing elements in the Shafarevich-Tate group” project:
    • Bibliography: dvi, ps, pdf
    • Mazur's comments on “visualizing elements in the Shafarevich-Tate group,”: dvi, ps, pdf
    • “Visualizing elements in the Shafarevich-Tate group,” by Cremona and Mazur (to appear in the Journal of Experimental Math): dvi, ps, pdf
    • “Visualizing elements of order three in the Shafarevich-Tate group,” (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.

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

  • Pavlos Tzermias' lectures on “The Manin-Mumford Conjecture: A Brief Survey”: dvi, ps, pdf
  • Wetherell's project on “An application of the method of Coleman and Chabauty.”: description. It refers to Wetherell's thesis: dvi, ps, pdf
  • 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: dvi, ps, pdf

• SHORTTITLE  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
    • “Computational aspects of curves of genus at least 2”: ps, pdf
  • “The Mordell-Weil group of a genus 2 Jacobian”: project description

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

  • Three papers related to Rubin's lectures:
    • “Elliptic curves with complex multiplication”: dvi, ps, pdf
    • “Euler systems and modular elliptic curves”: dvi, ps, pdf
    • “General Euler systems”: dvi, ps, pdf
  • Karl's web site has other preprints that may be of interest.
  • “Tate-Shafarevich groups of twists on an elliptic curve”: project description

• SHORTTITLE  Professional development: John Cremona: Symbolic computation and number theory

  • “Rational points on conics.”: project description, suggested readings
  • 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.