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 Q_p ; definition and explicit computation of the Hilbert symbol in Q_p ;quadratic extensions of Q_p ; 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 Q_p ; 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 Q_p ; 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