Qualifying Examination Syllabus

Name: Frank Calegari Committee: Ken Ribet, Bjorn Poonen, Edward Frenkel, Susan Graham.

Major Topic: Elliptic Curves

Objects attached to an elliptic curve: j-invariant, invariant differential, Tate module, Weil paring, endomorphism ring and automorphism group. Isogenies and the dual isogeny. Elliptic curves over ℂ. Elliptic curves over finite fields, local fields, and global fields. Mordell Weil threorem. Selmer and Tate– Shavarevich groups. 2-Descent.

Main Reference: Silverman, The Arithemetic of Elliptic Curves, Chapters I–III, V–VIII & X; A. Knapp, Elliptic Curves.

Major Topic: Algebraic Geometry

Sheaves. Schemes. First properties of Schemes. Separated and proper morphisms. Valuative criteria. Varieties. Sheaves of modules. Divisors. Projective morphisms. Differentials. Cohomology of Noetherian schemes.

Main Reference: Hartshorne, Algebraic Geometry, II.1–II.8 & III.1–III.5.

Minor Topic: Complex Analysis

Residue theorem, Liouville’s theorem. The fundametal theorem of algebra. The Maximum modulus theorem. Taylor and Laurent series. Analytical continuation. Analytical propeties of the Riemann zeta function and its applications. Modular functions.

Main Reference: Ahlfors, Complex Analysis.