Lectures in course MATH-494: Topics in arithmetic geometry

Explores p-adic expansion, unique representations, and power series for computing p-adic expansions.

Explores integration theory over real numbers and Berkovich spaces, revealing intriguing asymmetries and unsolved conjectures.

Covers the construction of Qp, norms, valuations, and Ostrovski's theorem.

Explores combinatorial proofs for the Frobenius Theorem, demonstrating mathematical reasoning and calculations.

Covers the p-adic expansion theorem and Hensel's Lemma in Qp.

Explores a 'global' view of Dwork's proof, emphasizing the vanishing of a determinant.

Explores the definition of Q_p and the completion of Q,1(p) to form Qp.

Covers matrix operations in linear algebra, including determinants and row operations.

Discusses the application of Hensel's lemma in determining nth roots and elements that are nth powers.

Covers the uniqueness of norm extension in finite fields and the construction of norms on finite extensions of Qp.

Explores p-adic integration, Igusa zeta function rationality, and monodromy conjectures.

Covers the proof of Hensel's Lemma and a review of field theory, including Newton's approximation and p-adic complex numbers.

Introduces analytic manifolds, Berkovich spaces, and multiplicative semi-norms in K-algebras.

Covers manifolds, charts, compatibility, and submanifolds with smooth analytic equations.

Explores ramified extensions and Eisenstein polynomials, showcasing their applications in mathematical contexts.

Covers the concept of P-adic manifolds and their analytical properties in mathematical computations.

Explores ramification and structure of finite extensions of Qp, including unramified extensions and Galois properties.

Covers differential forms, invariant measures, and integration on manifolds with examples and illustrations.

Covers the extension of fields, defining the norm of an element from one field to another, and introducing the p-adic valuation.

Covers the classification of compact p-adic manifolds using the C.o.V-formula and explores smooth algebraic varieties and Hensel's lemma.