18.782 - Arithmetic Geometry

Lecture Schedule

# Date Topic (references) Materials
19/5Introduction to arithmetic geometry (Ellenberg, Poonen)slides, worksheet
29/10Rational points on conics (Cremona-Rusin)notes
39/12Finite fields (Rabin)notes, worksheet
49/17The ring of p-adic integersnotes
59/19The field of p-adic numbers, absolute values, Ostrowski's theorem for Qnotes
69/24Ostrowski's theorem for number fields (Conrad)
79/26Product formula for number fields, completionsnotes
810/1Hensel's lemmanotes
910/3Quadratic formsnotes
1010/8Hilbert symbolsnotes
1110/10Weak and strong approximation, Hasse-Minkowski theorem for Qnotes
1210/17Field extensions, algebraic setsnotes
1310/22Affine and projective varietiesnotes
1410/24Zariski topology, morphisms of affine varieties and affine algebrasnotes
1510/29Rational maps and function fieldsnotes
1610/31Products of varieties and Chevalley's criterion for completenessnotes
1711/5Tangent spaces, singular points, hypersurfacesnotes
1811/7Smooth projective curvesnotes
1911/12Divisors, the Picard groupnotes
2011/14Degree theorem for morphisms of curvesnotes
2111/19Riemann-Roch spacesnotes
2211/21Proof of the Riemann-Roch theorem for curvesnotes
2311/26Elliptic curves and abelian varietiesnotes
2412/3Isogenies and torsion points, the Nagell-Lutz theoremnotes
2512/5The Mordell-Weil theoremnotes
2612/10Jacobians of genus one curves, the Weil-Chatelet and Tate-Shafarevich groupsnotes