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Introduction to arithmetic geometry

Introduction TO arithmetic geometry (NOTES FROM , FALL 2009)BJORN POONENC ontents1. What is arithmetic geometry ?32. Absolute values on fields33. Thep-adic absolute value onQ44. Ostrowski s classification of absolute values onQ55. Cauchy sequences and completion86. Inverse limits107. DefiningZpas an inverse limit108. Properties ofZp119. The field ofp-adic expansions1311. Solutions to polynomial equations1412. Hensel s lemma1413. Structure ofQ p1514. Squares inQ The case of The casep= analytic functions1816. Algebraic closure1917. Finite fields2018. Inverse limits in general2219. Profinite Topology on a profinite Subgroups2520. Review of field theory2621. Infinite Galois Examples of Galois groups2822. Affine varieties29 Date: December 10, Affine Affine Irreducible Smooth varieties3223. Projective Projective Projective Projective varieties as a union of affine varieties3424.

41. Height functions on elliptic curves 67 42. Descent 70 43. Faltings’ theorem 71 Acknowledgements 71 References 71 1. What is arithmetic geometry? Algebraic geometry studies the set of solutions of a multivariable polynomial equation (or a system of such equations), usually over R or C. For instance, x2 + xy 5y2 = 1 de nes a hyperbola.

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