18.782 | Fall 2013 | Undergraduate
Introduction to Arithmetic Geometry

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. “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 Scientific 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]

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