Summer 2004 Number Theory Seminar

This seminar has concluded for the summer.



Tuesdays and Thursdays, 11 am to noon, 341 Altgeld Hall



University of Illinois Urbana-Champaign

Organizer:

  • Tim Kilbourn (tkilbour@math.uiuc.edu)

    Description:

    We will work through chapters 1-3 and 10 of Henri Darmon's book Rational Points on Modular Elliptic Curves. These chapters present the proofs of the theorems of Gross, Zagier, and Kolyvagin concerning special cases of the Birch and Swinnerton-Dyer conjecture.

    This book can be ordered from the AMS bookstore, or downloaded from the author's website.

    Schedule of Talks:

  • Monday, June 7: Tim Kilbourn:
    Group cohomology.

    References:
    [1] J. Silverman,
    The Arithmetic of Elliptic Curves,
    Springer, Graduate Texts in Mathematics, 106, (1986), Appendix B, p.330-335.
    [2] M. F. Atiyah and C. T. C. Wall,
    Cohomology of Groups,
    in Algebraic Number Theory, J. W. S. Cassels and A. Froehlich, eds., Academic Press, (1967) p.94-115.

  • Thursday, June 10: Scott Ahlgren:
    The Mordell-Weil Theorem (Chapter 1, sections 1 and 2).

  • Tuesday, June 15: Barry Walker:
    The Birch and Swinnerton-Dyer Conjecture (Chapter 1, sections 3 and 4).

  • Thursday, June 17: Tim Kilbourn:
    Review of modular forms (Chapter 2, sections 1 through 4).

  • Tuesday, June 22: No meeting.

  • Thursday, June 24: No meeting.

  • Tuesday, June 29: Sam Kadziela:
    Eichler-Shimura Theory and Shimura-Taniyama-Weil (Chapter 2, sections 5 and 6).

    Reference:
    [3] A. W. Knapp,
    Elliptic Curves,
    Princeton University Press, Mathematical Notes, 40, (1992).

  • Thursday, July 1: Sam Kadziela:
    Examples of Eichler-Shimura Theory (Knapp, Chapter 11).

  • Tuesday, July 6: Nadia Masri:
    Modular symbols (Chapter 2, section 7).

    References:
    [4] J.E. Cremona,
    Algorithms for Modular Elliptic Curves,
    online, (1997), Chapter 2.
    [5] William A. Stein,
    An introduction to computing modular forms using modular symbols.
    (postscript file, from Stein's homepage)

  • Thursday, July 8: Nadia Masri:
    Examples of modular symbols (Cremona, appendix to Chapter 2).

  • Tuesday, July 13: Jennifer Paulhus:
    Complex Multiplication (Chapter 3, Section 1).

    References:
    [6] J.P. Serre,
    Complex Multiplication,
    in Algebraic Number Theory, J. W. S. Cassels and A. Froehlich, eds., Academic Press, (1967) p.292-296.

  • Thursday, July 15: Tim Kilbourn:
    Heegner points (Chapter 3, Section 2-4).

  • Tuesday, July 20: Tim Kilbourn:
    Heegner systems (Chapter 3, Section 5-8).

    Last updated: July 20th, 2004

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