AIC Local Arithmetic (2021)

School Convener(s)

Name of the speaker

Position

Affiliation and address

Chandan Singh Dalawat

Professor H+, Mathematics

Harish-Chandra Research Institute

Chhatnag Road, Jhunsi

Allahabad-211019

INDIA

Name of the tutor

Position

Affiliation and address

Supriya Pisolkar

Associate Professor, Mathematics

Indian Institute of Science Education and Research (IISER)

Dr. Homi Bhabha Road,

Pashan, Pune 411 008

INDIA

Debasish Karmakar

PhD Student

Harish-Chandra Research Institute

Chhatnag Road, Jhunsi

Allahabad-211019

INDIA

Name of the Speaker

:

Detailed Syllabus

Chandan Singh Dalawat

Lecture 1 : A discussion of the syllabus, inverse systems and direct systems, projective limits and inductive limits, initial objects and final objects.

Lecture 2 : The ring Z_p of p-adic integers, p-adic expansions, the analogy with F_p[[t]], multiplication by p in Z_p, the ideals and quotients of the ring Z_p, invertible elements in Z_p, the p-adic valuation, the p-adic distance, dense subsets in Z_p.

Lecture 3 : Hensel's lemma in Z_p, squares and quadratic characters of Z_p^*, multiplicative representatives, structure of the group Z_p^*.

Lecture 4 : The field of Q_p of p-adic numbers, squares and quadratic characters of Q_p^*, structure of the group Q_p^*.

Lecture 5 : Revision of lectures 1 to 4.

Lecture 6 : The Hilbert symbol, Steinberg property, explicit formulae, the conic ax^2+by^2=1.

Lecture 7 : Quadratic extensions of Q_p, their compositum, bijection with the set of quadratic characters of Q_p^*, Kummer theory, Artin-Schreier theory.

Lecture 8 : Power series over a finite field F_p, analogy with Z_p, the field of fractions F_p((t)), squares in F_p((t))^*, quadratic characters of F_p((t))^*, the Hilbert symbol, Steinberg property, explicit formulae, the conic ax^2+by^2=1, quadratic extensions of F_p((t)), bijection with the set of quadratic characters.

Lecture 9 : Algebraic preliminaries, Galois theory, Artin-Schreier theory, reformulation of local quadratic reciprocity.

Lecture 10 : Structure of the filtered space k((t))^* modulo p-th powers, structure of k((t))^+ modulo the Artin-Schreier map (k finite field), the snake lemma.

Lecture 11 : p-fields, rings of integers, residue fields, uniformisers, Hensel's lemma, multiplicative representatives.

Lecture 12 : Structure theorem for p-fields, ramification index, residual degree, raising to the power p in p-fields of characteristic 0, the filtration on the quotient modulo p-th powers.

Lecture 13 : Valuations, equivalence of valuations, rings of integers, residue fields, discrete valuations, uniformisers, ideals in the ring of integers, norms, equivalence of norms, completions.

Lecture 14 : Hensel's lemma, extension of a complete discrete valuation to a finite extension field, integral closure of the ring of integers.

Lecture 15 : Krasner's lemma, ramification index and residual degree, multiplicativity in towers.

Lecture 16 : Relation between the degree, the ramification index, and the residual degree, the norm map, monogenity of the ring of integers in a finite extension, unramified extensions, totally ramified extensions, tamely ramified extensions.

Lecture 17 : Eisenstein polynomials, totally ramified extensions, structure of tamely ramified extensions, parametrisation of tamely ramified extensions of a given degree.

Lecture 18 : Maximal tamely ramified extension, galoisian and abelian tamely ramified extensions, maximal tamely ramified abelian extension, the number of tamely ramified extensions of various kinds.

Lecture 19 : The trace map, the different and the discriminant, transitivity formulae, the different and the ramification, Serre's mass formula in the tame case.

Lecture 20 : Inertia subgroup and ramification subgroup, higher ramification groups, lower numbering, ramification filtration on a subgroup.

Lecture 21 : The ramification filtration on a quotient group, Hasse-Herbrandt functions, transitivity, roots on unity.

Lecture 22 : Ramification filtration on cyclotomic extensions, ramification jumps, cyclic extensions, abelian extensions of prime exponent, Hasse-Arf.

Lecture 23 : cyclic extensions of p-fields of characteristic 0, raising to the exponent p, filtration on the units modulo p-th powers, primitive roots of unity, the unramified F_p-line, ramification breaks of cyclic extensions.

Lecture 24 : ramification breaks of cyclic extensions in the absence of a primitive p-th root of unity, contribution of cyclic extensions to Serre's mass formula.

Lecture 25 : Parametrisation of solvable extensions of prime degree over an arbitrary field.

Lecture 26 : Specialisation to the case of p-fields, the maximal abelian extension of exponent dividing p-1, the galois module structure of units modulo p-th powers, Serre's mass formula in prime degree.

Lecture 27 : The norm map and the ramification filtration.

Lecture 28 : The local Kronecker-Weber theorem.

Lecture 29 & 30 : Primitive solvable extensions of an arbitrary field.

Lectures 31 & 32 : Irreducible F_p-representations of the absolute Galois group of a p-field.

Lectures 33 & 34 : The Galois module structure of units modulo p-th powers in a tamely ramified split extension.

Lecture 35 & 36 : Primitive extensions of p-fields.