Get Algebraic Number Theory PDF

By V. Dokchitser, Sebastian Pancratz

Show description

Read Online or Download Algebraic Number Theory PDF

Similar algebraic geometry books

Invariant Factors, Julia Equivalences and the (Abstract) by Karsten Keller PDF

This booklet is principally dedicated to the combinatorics of quadratic holomorphic dynamics. The conceptual kernel is a self-contained summary counterpart of attached quadratic Julia units that is outfitted on Thurston's idea of a quadratic invariant lamination and on symbolic descriptions of the angle-doubling map.

Equidistribution in Number Theory: An Introduction - download pdf or read online

Written for graduate scholars and researchers alike, this set of lectures presents a based creation to the idea that of equidistribution in quantity thought. this idea is of starting to be significance in lots of parts, together with cryptography, zeros of L-functions, Heegner issues, best quantity thought, the speculation of quadratic types, and the mathematics elements of quantum chaos.

Read e-book online Geometry of Subanalytic and Semialgebraic Sets PDF

Actual analytic units in Euclidean house (Le. , units outlined in the community at every one element of Euclidean area by way of the vanishing of an analytic functionality) have been first investigated within the 1950's through H. Cartan [Car], H. Whitney [WI-3], F. Bruhat [W-B] and others. Their method was once to derive information regarding genuine analytic units from homes in their complexifications.

Extra resources for Algebraic Number Theory

Sample text

Dn ❤❛s ❉✐r✐❝❤❧❡t ❞❡♥s✐t② |{g ∈ Gal(f ) ✇✐t❤ ❝②❝❧❡ t②♣❡ (d1 , . . , dn ) ✐♥ t❤❡ ❛❝t✐♦♥ ♦♥ r♦♦ts}| . |Gal(f )| Pr♦♦❢✳ f (X) (mod p) ❤❛s ❛ r❡♣❡❛t❡❞ r♦♦t ✐♥ F¯ p ❢♦r ♦♥❧② ✜♥✐t❡❧② ♠❛♥② p✳ ❋♦r t❤❡ r❡st✱ Frobp ❛❝ts ❛s ❛♥ ❡❧❡♠❡♥t ♦❢ ❝②❝❧❡ t②♣❡ (d1 , . . , dn ) ✇❤❡r❡ t❤❡s❡ ❛r❡ t❤❡ ❞❡❣r❡❡s ♦❢ t❤❡ ✐rr❡❞✉❝✐❜❧❡ ❢❛❝t♦rs ♦❢ f (X) (mod p)✱ ❜② ❈♦r♦❧❧❛r② ✷✳✺ ❛♥❞ ✐ts ♣r♦♦❢✳ ❊①❛♠♣❧❡✳ ❙✉♣♣♦s❡ f (X) ✐s ❛♥ ✐rr❡❞✉❝✐❜❧❡ q✉✐♥t✐❝ ✇✐t❤ Gal(f ) = S5 ✳ ✸✹ L✲❙❡r✐❡s • ❚❤❡ s❡t ♦❢ ♣r✐♠❡s s✉❝❤ t❤❛t f (X) (mod p) ✐s ❛ ♣r♦❞✉❝t ♦❢ ❧✐♥❡❛r ❢❛❝t♦rs ❤❛s ❞❡♥s✐t② 1/120✳ • ❚❤❡ s❡t ♦❢ ♣r✐♠❡s s✉❝❤ t❤❛t f (X) (mod p) ❢❛❝t♦r✐s❡s ✐♥t♦ ❛ ❝✉❜✐❝ ❛♥❞ ❛ q✉❛❞r❛t✐❝ ❤❛s ❞❡♥s✐t② 1 20 1 |{❡❧❡♠❡♥ts ♦❢ t❤❡ ❢♦r♠ (··)(· · ·) ✐♥ S5 }| = = .

P ♣r✐♠❡ ◆♦✇ ✇r✐t❡ Ca ❛s ❛ s✉♠ ♦❢ ❉✐r✐❝❤❧❡t ❝❤❛r❛❝t❡rs✳ Ca , χ = = ❍❡♥❝❡ Ca = χ(a) χ φ(N ) χ✳ 1 φ(N ) Ca (n)χ(n) n∈(Z/N Z)× χ(a) . φ(N ) ❙♦ p∈Pa 1 = ps χ χ(a) φ(N ) χ(p)p−s . p ♣r✐♠❡ ✷✽ L✲❙❡r✐❡s ❊❛❝❤ t❡r♠ ♦♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s ❜♦✉♥❞❡❞ ❛s s → 1 ❡①❝❡♣t ❢♦r t❤❡ ❝♦♥tr✐❜✉t✐♦♥ ❢r♦♠ χ = I✱ s♦ 1 1 1 1 ∼ p−s ∼ log ps φ(N ) φ(N ) s−1 p∈Pa pN ❛s s → 1 ❜② ❈♦r♦❧❧❛r② ✸✳✾✳ ✸✳✹ ❉✐r✐❝❤❧❡t ❈❤❛r❛❝t❡rs ❘❡❝❛❧❧ t❤❛t ∼ (Z/N Z)× − → Gal Q(ζN )/Q a → σa a σa (ζN ) = ζN p → σp p σp (ζN ) = ζN ■❢ Q ✐s ❛ ♣r✐♠❡ ♦❢ Q(ζN ) ❛❜♦✈❡ p N t❤❡♥ σP = FrobQ/P ✳ ◆♦t❛t✐♦♥✳ ■❢ F/K ✐s ❛ ●❛❧♦✐s ❡①t❡♥s✐♦♥ ♦❢ ♥✉♠❜❡r ✜❡❧❞s ✇✐t❤ Gal(F/K) ❛❜❡❧✐❛♥✱ ❛♥❞ P ✐s ❛ ♣r✐♠❡ ♦❢ K ✉♥r❛♠✐✜❡❞ ✐♥ F/K ✱ ✇r✐t❡ FrobP ∈ Gal(F/K) ❢♦r t❤❡ ❋r♦❜❡♥✐✉s ❡❧❡♠❡♥t ♦❢ ❛♥② ♣r✐♠❡ ❛❜♦✈❡ P ✱ ✐♥❞❡♣❡♥❞❡♥t ♦❢ Q ❛❜♦✈❡ P ❛s t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❣r♦✉♣s ❛r❡ ❝♦♥❥✉❣❛t❡✱ ❛♥❞ I = 1 ❛s P ✐s ✉♥r❛♠✐✜❡❞✳ ❚❤❡♦r❡♠ ✸✳✶✶ ✭❍❡❝❦❡✱ ✶✾✷✵✱ ❈❧❛ss ❋✐❡❧❞ ❚❤❡♦r② ✮✳ ▲❡t F/K ❜❡ ❛ ●❛❧♦✐s ❡①t❡♥s✐♦♥ ♦❢ ♥✉♠❜❡r ✜❡❧❞s ✇✐t❤ Gal(F/K) ❛❜❡❧✐❛♥✱ ❛♥❞ ψ : Gal(F/K) → C× ❛ ❤♦♠♦♠♦r♣❤✐s♠✳ ❚❤❡♥ 1 L∗ (ψ, s) = 1 − ψ(FrobP )N (P )−s p ♣r✐♠❡s ♦❢ K ✉♥r❛♠✐✜❡❞ ✐♥ F/K ❤❛s ❛♥ ❛♥❛❧②t✐❝ ❝♦♥t✐♥✉❛t✐♦♥ t♦ C✱ ❡①❝❡♣t ❢♦r ❛ s✐♠♣❧❡ ♣♦❧❡ ❛t s = 1 ✇❤❡♥ ψ = I✳ Pr♦♦❢✳ ❖♠✐tt❡❞✳ ❘❡♠❛r❦✳ ❲❤❡♥ K = Q✱ F = Q(ζN ) t❤✐s r❡❝♦✈❡rs ❚❤❡♦r❡♠ ✸✳✻✱ ❛♥❞ ♠♦r❡✳ ✸✳✺ ◆♦t❛t✐♦♥✳ ❆rt✐♥ L✲❋✉♥❝t✐♦♥s ■❢ I ≤ D ❛r❡ ✜♥✐t❡ ❣r♦✉♣s✱ ρ ❛ D✲r❡♣r❡s❡♥t❛t✐♦♥✱ ✇r✐t❡ ρI = {v ∈ ρ : ∀g ∈ I gv = v} ❢♦r t❤❡ s✉❜s♣❛❝❡ ♦❢ I ✲✐♥✈❛r✐❛♥t ✈❡❝t♦rs✳ ❘❡♠❛r❦✳ ■❢ I D t❤❡♥ ρI ✐s ❛ D✲s✉❜r❡♣r❡s❡♥t❛t✐♦♥✳ ■❢ v ∈ ρI ✱ g ∈ D✱ i ∈ I t❤❡♥ i(gv) = g(i v) = gv ❢♦r s♦♠❡ i ∈ I ✱ s♦ gv ∈ ρI ✳ ✷✾ ❉❡✜♥✐t✐♦♥✳ ▲❡t F/K ❜❡ ❛ ●❛❧♦✐s ❡①t❡♥s✐♦♥ ♦❢ ♥✉♠❜❡r ✜❡❧❞s✱ ❧❡t ρ ❜❡ ❛ Gal(F/K)✲ r❡♣r❡s❡♥t❛t✐♦♥✳ ▲❡t P ❜❡ ❛ ♣r✐♠❡ ♦❢ K ✱ ❛♥❞ ❝❤♦♦s❡ Q ❛ ♣r✐♠❡ ♦❢ F ❛❜♦✈❡ K ✱ ❝❤♦♦s❡ FrobP t♦ ❜❡ ❛♥ ❡❧❡♠❡♥t ♦❢ DQ/P ✇❤✐❝❤ ✐♥ DQ/P /IQ/P ✐s FrobQ/P ✱ ✐✳❡✳✱ FrobP ❛❝ts ♦♥ t❤❡ r❡s✐❞✉❡ ✜❡❧❞ ❛s ❋r♦❜❡♥✐✉s✳ ❚❤❡♥ t❤❡ ❧♦❝❛❧ ♣♦❧②♥♦♠✐❛❧ ♦❢ ρ ❛t P ✐s PP (ρ, T ) = PP (F/K, ρ, T ) ρIP = det 1 − T FrobP Gal F/K ✇❤❡r❡ IP = IQ/P ❛♥❞ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s det(1 − T FrobP ) ❛❝t✐♥❣ ❛t ResDQ/P ▲❡♠♠❛ ✸✳✶✷✳ ρ IP ✳ PP (ρ, T ) ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❝❤♦✐❝❡ ♦❢ Q ❛♥❞ FrobP ✳ Pr♦♦❢✳ ❋♦r ✜①❡❞ Q✱ ✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ ❝❤♦✐❝❡ ♦❢ FrobP ✐s ❝❧❡❛r✿ ❛♥♦t❤❡r ❝❤♦✐❝❡ ❞✐✛❡rs ❜② ❛♥ ❡❧❡♠❡♥t ♦❢ IQ/P ✇❤✐❝❤ ❛❝ts ❛s t❤❡ ✐❞❡♥t✐t② ❛t ρIQ/P ✳ ■❢ Q = gQ ✐s ❛♥♦t❤❡r ♣r✐♠❡✱ g ∈ Gal(F/K)✱ t❤❡♥ ✇❡ ❝❛♥ t❛❦❡ FrobP ❢♦r Q t♦ ❜❡ −1 g FrobP g −1 ❛♥❞ ♦❜s❡r✈❡ t❤❛t ❡✐❣❡♥✈❛❧✉❡s ✭✇✐t❤ ♠✉❧t✐♣❧✐❝✐t✐❡s✮ ♦❢ g FrobP g −1 ♦♥ ρgIP g ❛❣r❡❡ ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡s ♦❢ FrobP ♦♥ ρIP ✱ s♦ ❤❛✈❡ t❤❡ s❛♠❡ ♠✐♥✐♠❛❧ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ❤❡♥❝❡ ❣✐✈❡ t❤❡ s❛♠❡ ❧♦❝❛❧ ❢❛❝t♦rs✳ ❉❡✜♥✐t✐♦♥✳ ▲❡t F/K ❜❡ ❛ ●❛❧♦✐s ❡①t❡♥s✐♦♥ ♦❢ ♥✉♠❜❡r ✜❡❧❞s✱ ❛♥❞ ρ ❜❡ ❛ Gal(F/K)✲ r❡♣r❡s❡♥t❛t✐♦♥✳ ❚❤❡ ❆rt✐♥ L✲❢✉♥❝t✐♦♥ ♦❢ ρ ✐s ❞❡✜♥❡❞ ❜② t❤❡ ❊✉❧❡r ♣r♦❞✉❝t L(F/K, ρ, s) = L(ρ, s) = K ♣r✐♠❡ ♦❢ K 1 .

Dn ❤❛s ❉✐r✐❝❤❧❡t ❞❡♥s✐t② |{g ∈ Gal(f ) ✇✐t❤ ❝②❝❧❡ t②♣❡ (d1 , . . , dn ) ✐♥ t❤❡ ❛❝t✐♦♥ ♦♥ r♦♦ts}| . |Gal(f )| Pr♦♦❢✳ f (X) (mod p) ❤❛s ❛ r❡♣❡❛t❡❞ r♦♦t ✐♥ F¯ p ❢♦r ♦♥❧② ✜♥✐t❡❧② ♠❛♥② p✳ ❋♦r t❤❡ r❡st✱ Frobp ❛❝ts ❛s ❛♥ ❡❧❡♠❡♥t ♦❢ ❝②❝❧❡ t②♣❡ (d1 , . . , dn ) ✇❤❡r❡ t❤❡s❡ ❛r❡ t❤❡ ❞❡❣r❡❡s ♦❢ t❤❡ ✐rr❡❞✉❝✐❜❧❡ ❢❛❝t♦rs ♦❢ f (X) (mod p)✱ ❜② ❈♦r♦❧❧❛r② ✷✳✺ ❛♥❞ ✐ts ♣r♦♦❢✳ ❊①❛♠♣❧❡✳ ❙✉♣♣♦s❡ f (X) ✐s ❛♥ ✐rr❡❞✉❝✐❜❧❡ q✉✐♥t✐❝ ✇✐t❤ Gal(f ) = S5 ✳ ✸✹ L✲❙❡r✐❡s • ❚❤❡ s❡t ♦❢ ♣r✐♠❡s s✉❝❤ t❤❛t f (X) (mod p) ✐s ❛ ♣r♦❞✉❝t ♦❢ ❧✐♥❡❛r ❢❛❝t♦rs ❤❛s ❞❡♥s✐t② 1/120✳ • ❚❤❡ s❡t ♦❢ ♣r✐♠❡s s✉❝❤ t❤❛t f (X) (mod p) ❢❛❝t♦r✐s❡s ✐♥t♦ ❛ ❝✉❜✐❝ ❛♥❞ ❛ q✉❛❞r❛t✐❝ ❤❛s ❞❡♥s✐t② 1 20 1 |{❡❧❡♠❡♥ts ♦❢ t❤❡ ❢♦r♠ (··)(· · ·) ✐♥ S5 }| = = .

Download PDF sample

Algebraic Number Theory by V. Dokchitser, Sebastian Pancratz


by Edward
4.2

Rated 4.78 of 5 – based on 24 votes