MAT 536: Algebra III

Instructor: Leon  Takhtajan, Math Tower 5-111, Phone: 632-8287, email: leontak@math.sunysb.edu.

• Elliptic and modular functions
  
• Elementary theory of modular forms for the modular group, Kronecker's limit formulas, Hecke operators
• Representation of integers by quadratic forms
• Congruence subgroups, modular forms of level N, old and new forms
• Elliptic curves over Q, Hasse-Weil L-functions
• Complex multiplication
• Introduction to Eichler-Shimura theory
• Mirror symmetry for elliptic curves

1.  Hurwitz, Adolf Vorlesungen uber allgemeine Funktionentheorie und elliptische Funktionen. (German) Herausgegeben und erganzt durch einen Abschnitt uber geometrische Funktionentheorie von R. Courant. Mit einem Anhang von H. Rohrl. Vierte vermehrte und verbesserte Auflage. Grundlehren der Mathematischen Wissenschaften, Band 3. Springer-Verlag, Berlin-New York, 1964. (Russian translation:   {\cyr Gurvits, A.}; Gurvic, A.; {\cyr Kurant, R.}; Kurant, R. {\cyr Teoriya funktsi\u\i }. (Russian) [Theory of functions]Translated and adapted from the fourth German edition by M. A. Evgrafov. Izdat. ``Nauka'', Moscow, 1968.)
2.  Iwaniec, Henryk. Topics in classical automorphic forms. Graduate Studies in Mathematics, 17. American Mathematical Society, Providence, RI,1997.
3.  Koblitz, Neal. Introduction to elliptic curves and modular forms. Second edition. Graduate Texts in Mathematics, 97. Springer-Verlag, New York, 1993.
   
4.  Knapp, Anthony W. Elliptic curves. Mathematical Notes, 40. Princeton University Press, Princeton, NJ, 1992.
5.  Lang, Serge Introduction to modular forms. With appendixes by D. Zagier and Walter Feit. Corrected reprint of the 1976 original. Grundlehren der Mathematischen Wissenschaften, 222.Springer-Verlag, Berlin, 1995.
6.  Lang, Serge Elliptic functions. With an appendix by J. Tate. Second edition. Graduate Texts in Mathematics, 112. Springer-Verlag, New York, 1987.
   
7.  Serre, J.-P. A course in arithmetic. Translated from the French. Graduate Texts in Mathematics, 7. Springer-Verlag, New York-Heidelberg, 1973.
   
8.  Shimura, Goro Introduction to the arithmetic theory of automorphic functions. Reprint of the 1971 original. Publications of the Mathematical Society of Japan, 11. Kanô Memorial Lectures, 1. Princeton University Press, Princeton, NJ, 1994.
   
9. Siegel, Carl Ludwig Lectures on advanced analytic number theory. Notes by S. Raghavan. Tata Institute of Fundamental Research Lectures on Mathematics, No. 23 Tata Institute of Fundamental Research, Bombay 1965.

DSS advisory. If you have a physical, psychiatric, medical, or learning disability that may affect your ability to carry out the assigned course work, please contact the office of Disabled Student Services (DSS), Humanities Building, room 133, telephone 632-6748/TDGD. DSS will review your concerns and determine what accommodations may be necessary and appropriate. All information and documentation of disability is confidential.