Below are suggested readings and references for most most of the class sessions.

SES # | TOPICS | READINGS |
---|---|---|

1 | Introduction to Arithmetic Geometry | Ellenberg, Jordan S. "Arithmetic Geometry." (PDF) Poonen, Bjorn. "Computing Rational Points on Curves" (PDF) |

2 | Rational Points on Conics | Cremona, J. E., and D. Rusin. "Efficient solution of rational conics." Cochrane, T., and P. Mitchell. "Small Solutions of the Legendre Equation." |

3 | Finite Fields | Berlekamp, Elwyn R. "Factoring polynomials over large ﬁnite ﬁelds." Gathen, Joachim von zur, and Jürgen Garhard. O. Rabin, Michael."Probabilistic Algorithms in Finite Fields." Rousseau, G. "On the Quadratic Reciprocity Law." |

4 | The Ring of p-adic Integers | |

5 | The Field of p-adic Numbers, Absolute Values, Ostrowski's Theorem for Q | |

6 | Ostrowski's Theorem for Number Fields | |

7 | Product Formula for Number Fields, Completions | Milne, J. S. "Algebraic Number Theory." 2013. |

8 | Hensel's Lemma | |

9 | Quadratic Forms | |

10 | Hilbert Symbols | |

11 | Weak and Strong Approximation, Hasse-Minkowski Theorem for Q | |

12 | Field Extensions, Algebraic Sets | Artin, M. Knapp, A. ———. Milne, J. S. "Fields and Galois Theory." (PDF) 2012. |

13 | Affine and Projective Varieties | Knapp, A. Advanced Algebra. Birkhäuser, 2007. ISBN: 9780817645229. |

14 | Zariski Topology, Morphisms of Affine Varieties and Affine Algebras | Silverman, Joseph H. The Arithmetic of Elliptic Curves. Springer-Verlag, 2009. ISBN: 9780387094939. [Preview with Google Books] |

15 | Rational Maps and Function Fields | |

16 | Products of Varieties and Chevalley's criterion for Completeness | Atiyah, M. F., and I. G. Mac Donald. Bump, D. |

17 | Tangent Spaces, Singular Points, Hypersurfaces | Shafarevich, I. R., and Miles Reid. Basic Algebraic Geometry I. 3rd ed. Springer-Verlag, 2013. ISBN: 9783642379550. van der Waerdan, B. L. |

18 | Smooth Projective Curves | Serre, J. P. Local Fields. Springer, 1979. ISBN: 9781475756753. |

19 | Divisors, The Picard Group | Stichtenoth, H. Algebraic Function Fields and Codes. Springer, 2008. ISBN: 9781475756753. [Preview with Google Books] |

20 | Degree Theorem for Morphisms of Curves | Shafarevich, I. R., and Miles Reid. Stichtenoth, H. |

21 | Riemann-Roch Spaces | Stichtenoth, H. Algebraic Function Fields and Codes. Springer, 2008. ISBN: 9781475756753. [Preview with Google Books] |

22 | Proof of the Riemann-Roch Theorem for Curves | Shafarevich, I. R., and Miles Reid. Basic Algebraic Geometry I. 3rd ed. Springer-Verlag, 2013. ISBN: 9783642379550. Stichtenoth, H. |

23 | Elliptic Curves and Abelian Varieties | Shafarevich, I. R., and Miles Reid. Silverman, Joseph H. |

24 | Isogenies and Torsion Points, The Nagell-Lutz Theorem | Silverman, Joseph H. The Arithmetic of Elliptic Curves. 2nd ed. Springer-Verlag, 2009. ISBN: 9780387094939. [Preview with Google Books] |

25 | The Mordell-Weil Theorem | Serre, J. P. Silverman, Joseph H. |

26 | Jacobians of Genus One Curves, The Weil-Chatelet and Tate-Shafarevich Groups | Silverman, Joseph H. The Arithmetic of Elliptic Curves. 2nd ed. Springer-Verlag, 2009. ISBN: 9780387094939. [Preview with Google Books] |