Readings

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

SES # TOPICS READINGS
1 Introduction to Arithmetic Geometry

Ellenberg, Jordan S. This resource may not render correctly in a screen reader."Arithmetic Geometry." (PDF)

Poonen, Bjorn. This resource may not render correctly in a screen reader."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.

Buy at Amazon 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

Buy at Amazon Artin, M. Algebra. 2nd ed. Pearson Education, 2010. ISBN: 9780132413770.

Buy at Amazon Knapp, A. Basic Algebra. Birkhäuser, 2006. ISBN: 9780817632489. [Previwe with Google Books]

Buy at Amazon ———. Advanced Algebra. Birkhäuser, 2007. ISBN: 9780817645229.

Milne, J. S. This resource may not render correctly in a screen reader."Fields and Galois Theory." (PDF) 2012.

13 Affine and Projective Varieties Buy at Amazon Knapp, A. Advanced Algebra. Birkhäuser, 2007. ISBN: 9780817645229.
14 Zariski Topology, Morphisms of Affine Varieties and Affine Algebras Buy at Amazon 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

Buy at Amazon Atiyah, M. F., and I. G. Mac Donald. Introduction to Commutative Algebara. Westview Press, 1994. ISBN: 9780201407518. [Preview with Google Books]

Buy at Amazon Bump, D. Algebraic Geometry. World Scienti c Publishing Company, 1998. ISBN: 9789810235611.

17 Tangent Spaces, Singular Points, Hypersurfaces Buy at Amazon Shafarevich, I. R., and Miles Reid. Basic Algebraic Geometry I. 3rd ed. Springer-Verlag, 2013. ISBN: 9783642379550.

Buy at Amazon van der Waerdan, B. L. Algebra Volume I. 7th ed. Springer, 1990. ISBN: 9780387974248.

18 Smooth Projective Curves Buy at Amazon Serre, J. P. Local Fi elds. Springer, 1979. ISBN: 9781475756753.
19 Divisors, The Picard Group Buy at Amazon Stichtenoth, H. Algebraic Function Fi elds and Codes. Springer, 2008. ISBN: 9781475756753. [Preview with Google Books]
20 Degree Theorem for Morphisms of Curves

Buy at Amazon Shafarevich, I. R., and Miles Reid. Basic Algebraic Geometry I. 3rd ed. Springer-Verlag, 2013. ISBN: 9783642379550.

Buy at Amazon Stichtenoth, H. Algebraic Function Fi elds and Codes. Springer, 2008. ISBN: 9781475756753. [Preview with Google Books]

21 Riemann-Roch Spaces Buy at Amazon Stichtenoth, H. Algebraic Function Fi elds and Codes. Springer, 2008. ISBN: 9781475756753. [Preview with Google Books]
22 Proof of the Riemann-Roch Theorem for Curves Buy at Amazon Shafarevich, I. R., and Miles Reid. Basic Algebraic Geometry I. 3rd ed. Springer-Verlag, 2013. ISBN: 9783642379550.

Buy at Amazon Stichtenoth, H. Algebraic Function Fi elds and Codes. Springer, 2008. ISBN: 9781475756753. [Preview with Google Books]

23 Elliptic Curves and Abelian Varieties

Buy at Amazon Shafarevich, I. R., and Miles Reid. Basic Algebraic Geometry I. 3rd ed. Springer-Verlag, 2013. ISBN: 9783642379550.

Buy at Amazon 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 Buy at Amazon Silverman, Joseph H. The Arithmetic of Elliptic Curves. 2nd ed. Springer-Verlag, 2009. ISBN: 9780387094939. [Preview with Google Books]
25 The Mordell-Weil Theorem

Buy at Amazon Serre, J. P. Lectures on the Mordell-Weil Theorem. 3rd ed. Teubner Verlag, 1997. ISBN: 9783663106340.

Buy at Amazon 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 Buy at Amazon Silverman, Joseph H. The Arithmetic of Elliptic Curves. 2nd ed. Springer-Verlag, 2009. ISBN: 9780387094939. [Preview with Google Books]