Elliptic Curves II

4 hours of lectures per week.

Content: The course is a continuation of the spring course: "Elliptic curves over Q and C " and will treat modular forms and elliptic curves.

Prerequisites: Elliptic Curves, Chap. I-IV.

Literature:

  • Anthony W. Knapp: Elliptic Curves, Princeton University Press
  • Jean-Pierre Serre: A Course in Arithmetic, GTM Springer Verlag
  • Lars V. Ahlfors: Complex Analysis, McGraw-Hill
  • F. R. Gantmacher: Matrix Theory Chelsea Publishing Compagny
  • W. Fulton: Intersection Theory Ergb. Springer Verlag
  • Birger Iversen: Notes on ALGEBRA I, Matematisk Institut, 1993
  • Cremona: Algorithms for modular elliptic curves, Cambridge University Press1992
  • Lang, Lim, Tan: Independent generators for congruence subgroups of Hecke groups, Math. Z. 220, 569-594 (1994)
  • Kulkarni: An arithmetic-geometric method in the study of the subgroups of the modular group, Amer. J. Math. 113, 1053-1133 (1991)

    •  Projects: There is a mandatory student project including notes and student lecture. The project can be done alone or in a group.

    Participants:

  • Morten Skarsholm Risager - risager@imf.au.dk
  • Henrik Gadegaard Spalk - spalk@imf.au.dk
  • Jesper Petersen - jesperp@imf.au.dk
  • Anders Nedergaard Jensen -u950710@daimi.aau.dk
  • Marc Skov Madsen - marc@imf.au.dk
  • Rasmus Faber Larsen - rflarsen@imf.au.dk
  • Lene Søndergaard -

    • Timetabel

    ons. fre.
    9-11  13-15
    koll. G koll. A4
     

    Lectures

    2. september 13-15 in koll. A3. Adjustments of timetabel. What is a modular elliptic curve? Outline of the course. Overheads 25 stk. (postscript) Overheads 25 stk. (Adobe pdf)

    15.-17. september Modular Forms for SL(2,Z). Chap. VIII.1-VIII.4

    Exercise: Make a program in MAPLE to calculate the first N terms in the q-expansion of j. (see p. 227). See also Serre: A Course in Arithmetic , GTM Springer Verlag, p. 90 and the references given there.

    22.-24. september Hecke operators. Chap. VIII.7. Inner products and Hecke operators Chap. VIII.6. and Chap. VIII.8.

    Exercise: Formulate and show that the concept of "holomorf af infinity" used in the book is the same as the concept defined by requiring that f(1/z) is holomorf.

    29. september - 1. october L function of a Cusp form Chap. VIII.5. Dirichlet series and Euler products Chap. VII.2. and with application to the last part in Theorem 8.24.

    The inversion problem, see the attached note, ps-fil pdf-fil

    6.-8. oktober Cubic Curves in Weierstrass form. Singular points, Chap. III. 5. L-function of an elliptic curve, Chap. X.1-3. The actual proof of Hasse's Theorem (Theorem 10.5) was done in the spring course: "Elliptic curves over Q and C."

    Overview of Eichler-Shimura Theory Chap. XI. 1. as an introduction to modular forms for Hecke subgroups, Chap. IX.

    13.-15. oktober Modular forms for Hecke subgroups, Chap. IX.1-2

    Exercise: Draw pictures of fundamental domains for the Hecke groups of level 2, 4 and 8.

    Mortens tegninger:

    Fundamentalomraade for Heckegruppen N=2 (postscript)

    Fundamentalomraade for Heckegruppen N=4 (postscript)

    Fundamentalomraade for Heckegruppen N=8 (postscript)

    20.-22. oktober No lectures.

    27.-29. oktober

  • Henrik Spalk: Student lectures on global minimal Weierstrass equations, Chap. X.1.
  • Marc Skov Madsen: Global minimal Weierstrass forms for the Frey curves, Chap. XII.4
  • Jesper Petersen: Global minimal Weierstrass forms in MAPLE using Apecs.
  • Find in the literature a description of Tate's algorithm for determining a minimal Weierstrass form for p=2.
  • Draw pictures of fundamental domains for the Hecke groups of level 2, 4 and 8. Determine the cusps and their width.
    • 3.-5. november New timetable Timetable

  • Last week's exercises.
  • Modular forms for Hecke subgroups, Chap. IX.2,4. The main result is Hecke's theorem (Theorem 9.8).

    •  10.-12. november
  • Chap. IX.6, Hecke operators, see the attached note, ps-fil
  • Modular forms for Hecke subgroups, Chap. IX.2,4. The main result is Hecke's theorem (Theorem 9.8).

    17.-19. november

    No lectures on 17. nov. Oldforms and newforms.

    24.-26. november

    Generators for Hecke groups, talk by Morten Skarsholm Riasager ps-fil

    Dedekinds eta-function, transformations laws, Corrollary 8.9 p. 235-238

    1.-3. december

    Calculations, Chap. XI.1.

  • Proposition 11.1 p. 303
  • Example 5 (Hecke) p. 267
  • C-program by Anders Nedergaard Jensen for calculating q-expansions C-source code
  • My MAPLE program for calculating the first N terms in the Euler-product (11.5) on p. 304. MAPLE-fil
  • The first 1000 terms in the Euler-product (11.5) on p.304 text-fil

    •

    No lectures on 3. december due to conference on Number Theory and Spectral Theory, Friday December 3 - Saturday December 4, 1999 , Auditorium G1, Department of Mathematical Sciences, University of Aarhus , see Number Theory and Spectral Theory

    8.-11. december

    Calculations. Determination of the L-functions of the curves over Q, Chap. XI.1 p. 306-308. Jesper Petersen En sammenligning mellem Fourier-koefficienterne til en spidsform af niveau 11 og vaegt 2 og twists af visse elliptiske kurver (ps-fil). Riemann surfaces and differentials. Chap. XI.2-4.

    15. december

    Homology, modular symbols Chap. XI.5. and Cremona: Algorithms for modular elliptic curves, Cambridge University Press1992

    Notes

  • The description of sublattices of index n by linear maps of determinat n is treated in:
  • Birger Iversen: Notes on ALGEBRA I 5.4 on page 52.
  • W. Fulton: Intersection Theory in Appendix A. Algebra, Lemma A.2.2
  • For generalities on infinite products and the Gamma function, see Lars V. Ahlfors: Complex Analysis,Chap. 5.2.2-4.
  • The existens of a basis of simultanous eigenvectors of selfadjoint operators is treated in general in Gantmacher: Matrix Theory, Chap. IX, section 10 and 15. We applied this to the Hecke operators.
  • The correspondance between lattices and elliptic curves. The inversion problem ps-fil pdf-fil
  • Hecke operators, note on p. 274, ps-fil

    • Links and software

    Om elliptiske kurver I

    Om elliptiske kurver II

    MATHEMATICA fil til EllipticCurveCalc

    MATHEMATICA INFO fil til EllipticCurveCalc

    Projects

  • q-expansion of G_2k (Prop. 8.1) and rapidly convergent series for g_2 and g_3 (6.50)
  • A modular form of weight 2 and level 11, see Knapp, IX 3. Example 5 p. 267.

    Jesper Petersen En sammenligning mellem Fourier-koefficienterne til en spidsform af niveau 11 og vaegt 2 og twists af visse elliptiske kurver (ps-fil)

  • Generators for the Hecke groups, talk by Morten Skarsholm Riasager Lidt om frembringere (ps-fil), see also
  • Lang, Lim, Tan: Independent generators for congruence subgroups of Hecke groups, Math. Z. 220, 569-594 (1994)
  • Kulkarni: An arithmetic-geometric method in the study of the subgroups of the modular group, Amer. J. Math. 113, 1053-1133 (1991)