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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