401-4119-64L  Reading Course: Topics in Fermat Curves

SemesterAutumn Semester 2014
LecturersJ. Fresán
Periodicitynon-recurring course
Language of instructionEnglish


AbstractFor the last half century, Fermat curves and hypersurfaces have been touchstones for far-reaching conjectures in algebraic geometry and number theory. For instance, they provided crucial support to the Weil conjectures on the number points of varieties over finite fields, or to Deligne and Beilinson's conjectures relating special values of L-functions to periods and regulators.
ObjectiveAfter a couple of introductory sessions, participants will contribute to the reading seminar with 45 minutes talks.
Content(1) Fermat curves over finite fields. The Weil conjectures. Jacobi sums as Hecke characters

(2) Periods of Fermat curves. The Chowla-Selberg formula. Transcendence of beta and gamma values

(3) The Beilinson conjectures for Fermat curves
LiteratureB. H. Gross, "On the periods of abelian integrals and a formula of Chowla and Selberg". With an appendix by David E. Rohrlich. Invent. Math. 45 (1978), 193-211.

K. Ireland, M. Rossen, A classical introduction to modern number theory, Grad. Texts in Math. 84, Springer-Verlag, New-York, 1990. Chapter 11

S. Lang, Introduction to algebraic and abelian functions. Second edition. Grad.Texts in Math. 89, Springer-Verlag, 1982, Chapters II and V

N. Otsubo, "On the regulator of Fermat motives and generalized hypergeometric functions", J. reine angew. Math. 660 (2011), 27-82, available at Link.

A. Weil, "Number of solutions of equations in finite fields", Bull. Amer. Math. Soc. 55 (1949), 497-508, available at Link
Prerequisites / NoticeAny interested student is encouraged to participate!

The level of the reading course will be increasing, depending on the audience. Only basic knowledge of number theory is needed for part (1). Part (2) requires some familiarity with algebraic geometry of curves and Riemann surfaces. The relevant notions from cohomology will be recalled. Part (3) will be a bit more advanced, but possible to follow after (1) and (2).

For further information or if you want to give a talk, please contact Link.