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." 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. Gathen, Joachim von zur, and Jürgen Garhard. Modern Computer Algebra. 3rd ed. Cambridge University Press, 2013. ISBN: 9781139856065. [Preview with Google Books] 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 |
Artin, M. Algebra. 2nd ed. Pearson Education, 2010. ISBN: 9780132413770. Knapp, A. Basic Algebra. Birkhäuser, 2006. ISBN: 9780817632489. [Previwe with Google Books] ———. Advanced Algebra. Birkhäuser, 2007. ISBN: 9780817645229. 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. Introduction to Commutative Algebara. Westview Press, 1994. ISBN: 9780201407518. [Preview with Google Books] Bump, D. Algebraic Geometry. World Scientic Publishing Company, 1998. ISBN: 9789810235611. |
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. Algebra Volume I. 7th ed. Springer, 1990. ISBN: 9780387974248. |
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. Basic Algebraic Geometry I. 3rd ed. Springer-Verlag, 2013. ISBN: 9783642379550. Stichtenoth, H. Algebraic Function Fields and Codes. Springer, 2008. ISBN: 9781475756753. [Preview with Google Books] |
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. Algebraic Function Fields and Codes. Springer, 2008. ISBN: 9781475756753. [Preview with Google Books] |
23 | Elliptic Curves and Abelian Varieties |
Shafarevich, I. R., and Miles Reid. Basic Algebraic Geometry I. 3rd ed. Springer-Verlag, 2013. ISBN: 9783642379550. Silverman, Joseph H. The Arithmetic of Elliptic Curves. 2nd ed. Springer-Verlag, 2009. ISBN: 9780387094939. [Preview with Google Books] |
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. Lectures on the Mordell-Weil Theorem. 3rd ed. Teubner Verlag, 1997. ISBN: 9783663106340. Silverman, Joseph H. The Arithmetic of Elliptic Curves. 2nd ed. Springer-Verlag, 2009. ISBN: 9780387094939. [Preview with Google Books] |
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] |