18.782 - Arithmetic Geometry


Lecture Schedule

Roberts
# Date Topic Reading Extra resources
19/2Introduction to arithmetic geometryslidesEllenberg, Poonen, Drew's slides,
Amit (fruit math)
29/4Rational points on conicsnotesCremona-Ruskin
9/7No class (Labor day)
39/9Finite fieldsnotesRabin
49/11Finite fields, the ring of p-adic integersnotes
59/14The field of p-adic numbers, absolute values, Ostrowski's theoremnotesConrad
69/16Product formula for number fieldsnotes (7.1-7.3)
79/18Completionsnotes (7.4)
notes (8.1)
89/21Hensel's lemma and Newton polygonsnotes (8.2-8.3)Conrad, Stoll (§3)
99/23Quadratic formsnotesConway
109/25Hilbert symbolsnotes
119/28Weak and strong approximation, Hasse-Minkowski theoremnotes
129/30Genera of quadratic forms, class groups of binary quadratic formsMartin (4.1-2,4.4)Conway-Sloane (§7)
1310/215 and 290 theorems (Add date)Bhargava (pp 27-38)Henke
1410/5Field extensions, algebraic setsnotes
1510/7Affine varietiesnotes (13.1)
1610/9Projective varietiesnotes (13.2-5)
10/12No class (Indigenous peoples' day); class on Tuesday instead
1710/13Zariski topology, morphisms of affine varietiesnotes
1810/14Affine algebras, rational mapsnotes
1910/16Function fieldsnotes
2010/19Tangent spaces, singular points, smoothness, hypersurfacesnotes
2110/21Smooth projective curvesnotes (18.1-2)
2210/23Function fields and curvesnotes (18.3)
notes (19.1)
2310/26Divisorsnotes (19.2)
2410/28The Picard groupnone
2510/30Degree theorem for morphisms of curvesnotes
2611/2Riemann-Roch spacesnotes
2711/4Genus of curvesnotes (22.1-2)
2811/6Canonical divisors and Riemann-Roch theoremnotes (22.3-22.4)
2911/9Elliptic curves and abelian varietiesnotes
11/11No class (Veteran's day)
3011/13Isogenies and torsion points, the Nagell-Lutz theoremnotes
3111/16The Mordell-Weil theorem 1notes (25.1-4)
3211/18The Mordell-Weil theorem 2 (Drop date)notes (25.5-7)
3311/20Falting's theoremnotes
11/23No class (Thanksgiving)
11/25No class (Thanksgiving)
11/27No class (Thanksgiving)
3411/30Weil conjectures: zeta functionsnotes
3512/2Weil conjectures: rationality and functional equationnotes
3612/4Weil conjectures: etale cohomologynotes
3712/7Weil conjectures: the Riemann hypothesisnotes
3812/9The Honda-Tate theoremnotes