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The study of classical Diophantine problems from the modern perspective of Arithmetic Geometry helps you equate the sum of two or more monomials, each of degree 1 in one of the variables, to a constant.

From this [course_title] along with its vivid examples and explanation, you will understand the topics like p-adic numbers, The Riemann-Roch theorem, Affine and projective varieties, Rational points on elliptic curves etc.

Assessment

This course does not involve any written exams. Students need to answer 5 assignment questions to complete the course, the answers will be in the form of written work in pdf or word. Students can write the answers in their own time. Each answer needs to be 200 words (1 Page). Once the answers are submitted, the tutor will check and assess the work.

Certification

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Course Credit: MIT

Course Curriculum

 Module: 01 Introduction to Arithmetic Geometry 01:00:00 Rational Points on Conics 00:45:00 Finite Fields 00:45:00 The Ring of p-adic Integers 00:30:00 The Field of p-adic Numbers, Absolute Values, Ostrowski’s Theorem for Q 00:30:00 Ostrowski’s Theorem for Number Fields 00:30:00 Product Formula for Number Fields, Completions 00:45:00 Hensel’s Lemma 00:30:00 Quadratic Forms 00:30:00 Hilbert Symbols 00:30:00 Module: 02 Weak and Strong Approximation, Hasse-Minkowski Theorem for Q 00:30:00 Field Extensions, Algebraic Sets 00:45:00 Affine and Projective Varieties 00:45:00 Zariski Topology, Morphisms of Affine Varieties and Affine Algebras 00:30:00 Rational Maps and Function Fields 00:45:00 Products of Varieties and Chevalley’s criterion for Completeness 01:00:00 Tangent Spaces, Singular Points, Hypersurfaces 00:30:00 Smooth Projective Curves 00:45:00 Module: 03 Divisors, The Picard Group 00:45:00 Degree Theorem for Morphisms of Curves 00:30:00 Riemann-Roch Spaces 00:30:00 Proof of the Riemann-Roch Theorem for Curves 01:00:00 Elliptic Curves and Abelian Varieties 00:45:00 Isogenies and Torsion Points, The Nagell-Lutz Theorem 01:00:00 The Mordell-Weil Theorem 01:00:00 Jacobians of Genus One Curves, The Weil-Chatelet and Tate-Shafarevich Groups 01:00:00 Assessment Submit Your Assignment 00:00:00 Certification 00:00:00

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