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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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