# AIC - Local Arithmetic (2021)

## Speakers and Syllabus

Projective limits ; construction of the profinite ring Z p ; Hensel’s lemma ; the structure of the multiplicative group Z×p with its filtration ; squares in Z×p ; the structure of the group(Z /pn Z) × ; the field Qp ; definition and explicit computation of the Hilbert symbol in Qp ;quadratic extensions of Qp ; the ring of integers therein ; the norm map and the Hilbert symbol ;local quadratic reciprocity.

The local fields k((t)) over a finite field k ; Hilbert symbol therein ; quadratic extensions thereof,etc., in analogy with the fields Qp ; raising-to-the-exponent-p map on the units.

Valuations ; equivalence of valuations ; completions ; valuations on finite extensions ; rings of integers ; units ; residual degree and ramification index ; unramified extensions.

Parametrisation of tamely ramified extensions ; characterisation of galoisian or abelian extensions among them ; computation of their total number.

The different and the discriminant ; Serre’s mass formula in the tame case ; finiteness of the number of extensions of bounded degree (and also bounded discriminant in characteristic p).

The ramification filtration ; lower numbering ; upper numbering ; Herbrand’s theorem ; Arf’s theorem.

Cyclotomic extensions ; the raising-to-the-power-p map on the units.

The norm map ; the maximal abelian extension of Qp ; local Kronecker-Weber theorem.

Abelian extensions of exponent p ; degree-p extensions of a local field ; Serre’s mass formula in this case.

## Time Table

 Day Date Time Topic Instructor Monday 18/1 15:00–16:30 Lecture 1 Dalawat Wednesday 20/1 15:00–16:30 Lecture 2 Dalawat Friday 22/1 15:00–16:00 Tutorial 1 Pisolkar & Dalawat Monday 25/1 15:00–16:30 Lecture 3 Dalawat Wednesday 27/1 15:00–16:30 Lecture 4 Dalawat Friday 29/1 15:00–16:00 Tutorial 2 Pisolkar & Dalawat Monday 1/2 15:00–16:30 Lecture 5 Dalawat Wednesday 3/2 15:00–16:30 Lecture 6 Dalawat Friday 5/2 15:00–16:00 Tutorial 3 Pisolkar & Dalawat Monday 8/2 15:00–16:30 Lecture 7 Dalawat Wednesday 10/2 15:00–16:30 Lecture 8 Dalawat Friday 12/2 15:00–16:00 Tutorial 4 Pisolkar & Dalawat Monday 15/2 15:00–16:30 Lecture 9 Dalawat Wednesday 17/2 15:00–16:30 Lecture 10 Dalawat Friday 19/2 15:00–16:00 Tutorial 5 Pisolkar & Dalawat Monday 22/2 15:00–16:30 Lecture 11 Dalawat Wednesday 24/2 15:00–16:30 Lecture 12 Dalawat Friday 26/2 15:00–16:00 Tutorial 6 Pisolkar & Dalawat Monday 1/3 15:00–16:30 Lecture 13 Dalawat Wednesday 3/3 15:00–16:30 Lecture 14 Dalawat Friday 5/3 15:00–16:00 Tutorial 7 Pisolkar & Dalawat Monday 8/3 15:00–16:30 Lecture 15 Dalawat Wednesday 10/3 15:00–16:30 Lecture 16 Dalawat Friday 12/3 15:00–16:00 Tutorial 8 Pisolkar & Dalawat Monday 15/3 15:00–16:30 Lecture 17 Dalawat Wednesday 17/3 15:00–16:30 Lecture 18 Dalawat Friday 19/3 15:00–16:00 Tutorial 9 Pisolkar & Dalawat Day Date Time Topic Instructor Monday 22/3 15:00–16:30 Lecture 19 Dalawat Wednesday 24/3 15:00–16:30 Lecture 20 Dalawat Friday 26/3 15:00–16:00 Tutorial 10 Karmakar & Dalawat Monday 29/3 15:00–16:30 Lecture 21 Dalawat Wednesday 31/3 15:00–16:30 Lecture 22 Dalawat Friday 2/4 15:00–16:00 Tutorial 11 Karmakar & Dalawat Monday 5/4 15:00–16:30 Lecture 23 Dalawat Wednesday 7/4 15:00–16:30 Lecture 24 Dalawat Friday 9/4 15:00–16:00 Tutorial 12 Karmakar & Dalawat Monday 12/4 15:00–16:30 Lecture 25 Dalawat Wednesday 14/4 15:00–16:30 Lecture 26 Dalawat Friday 16/4 15:00–16:00 Tutorial 13 Karmakar & Dalawat Monday 19/4 15:00–16:30 Lecture 27 Dalawat Wednesday 21/4 15:00–16:30 Lecture 28 Dalawat Friday 23/4 15:00–16:00 Tutorial 14 Karmakar & Dalawat Monday 26/4 15:00–16:30 Lecture 29 Dalawat Wednesday 28/4 15:00–16:30 Lecture 30 Dalawat Friday 30/4 15:00–16:00 Tutorial 15 Karmakar & Dalawat Monday 3/5 15:00–16:30 Lecture 31 Dalawat Wednesday 5/5 15:00–16:30 Lecture 32 Dalawat Friday 7/5 15:00–16:00 Tutorial 16 Karmakar & Dalawat Monday 10/5 15:00–16:30 Lecture 33 Dalawat Wednesday 12/5 15:00–16:30 Lecture 34 Dalawat Friday 14/5 15:00–16:00 Tutorial 17 Karmakar & Dalawat Monday 17/5 15:00–16:30 Lecture 35 Dalawat Wednesday 19/5 15:00–16:30 Lecture 36 Dalawat Friday 21/5 15:00–16:00 Tutorial 18 Karmakar & Dalawat

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