Below are suggested readings and references for most most of the class sessions.
| SES # | TOPICS | READINGS |
|---|---|---|
| 1 | Introduction to Arithmetic Geometry |
Ellenberg, Jordan S. Poonen, Bjorn. |
| 2 | Rational Points on Conics |
Cremona, J. E., and D. Rusin. "Efficient solution of rational conics." Mathematics of Computation 72 (2003): 1417–41. Cochrane, T., and P. Mitchell. "Small Solutions of the Legendre Equation." Journal of Number Theory 70, no. 1 (1998): 62–6. |
| 3 | Finite Fields |
Berlekamp, Elwyn R. "Factoring polynomials over large finite fields." Mathematics of Computation 24 (1970): 713–35.
O. Rabin, Michael."Probabilistic Algorithms in Finite Fields." SIAM Journal of Computing 9, no. 2 (1980): 273–80. Rousseau, G. "On the Quadratic Reciprocity Law." Journal of the Australian Mathematical Society (Series A) 51, no. 3 (1991): 423–5. |
| 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 |
Milne, J. S. |
| 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 |
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