Get Algebraic Number Theory PDF

By V. Dokchitser, Sebastian Pancratz

Show description

Read or Download Algebraic Number Theory PDF

Similar algebraic geometry books

New PDF release: Iterated Integrals and Cycles on Algebra

This topic has been of significant curiosity either to topologists and to quantity theorists. the 1st a part of this publication describes many of the paintings of Kuo-Tsai Chen on iterated integrals and the elemental workforce of a manifold. the writer makes an attempt to make his exposition obtainable to starting graduate scholars.

New PDF release: Ramanujan's Lost Notebook: Part IV

​​​​In the spring of 1976, George Andrews of Pennsylvania kingdom collage visited the library at Trinity collage, Cambridge, to ascertain the papers of the past due G. N. Watson. between those papers, Andrews found a sheaf of 138 pages within the handwriting of Srinivasa Ramanujan. This manuscript used to be quickly precise, "Ramanujan's misplaced computer.

Extra resources for Algebraic Number Theory

Example text

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 }| = = . |S5 | 120 6 ❈♦r♦❧❧❛r② ✸✳✷✵✳ ■❢ f (X) ∈ Z[X] ✐s ♠♦♥✐❝ ❛♥❞ ✐rr❡❞✉❝✐❜❧❡ ✇✐t❤ deg f (X) ≥ 2 t❤❡♥ f (X) (mod p) ❤❛s ♥♦ r♦♦t ✐♥ Fp ❢♦r ✐♥✜♥✐t❡❧② ♠❛♥② ♣r✐♠❡s p✳ Pr♦♦❢✳ ■t s✉✣❝❡s t♦ ♣r♦✈❡ t❤❛t t❤❡r❡ ❡①✐sts ❛ g ∈ Gal(f ) t❤❛t ✜①❡s ♥♦ r♦♦t ♦❢ f (X)✳ ❇✉t α:f (α)=0 StabGal(f ) (α) ✐s s♠❛❧❧❡r t❤❛♥ Gal(f ) ❛s ❡❛❝❤ ❤❛s s✐③❡ |Gal(f )|/|{α : f (α) = 0}| ❛♥❞ ❝♦♥t❛✐♥s t❤❡ ✐❞❡♥t✐t② ❡❧❡♠❡♥t✳ Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✸✳✶✽✳ ✭✐✮ ❇② ❊①❛♠♣❧❡ ❙❤❡❡t ✶ ◗✉❡st✐♦♥ ✾✱ ♦♥❧② ✜♥✐t❡❧② ♠❛♥② ♣r✐♠❡s r❛♠✐❢② ✐♥ F/Q✳ ❇② ❈♦r♦❧❧❛r② ✸✳✶✼✱ ✐❢ ρ = I ✐s ❛♥ ✐rr❡❞✉❝✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ G t❤❡♥ Pp (ρ, p−s )−1 L∗ (ρ, s) = p ✉♥r❛♠✐✜❡❞ ✐s ❜♦✉♥❞❡❞ ❛♥❞ ❜♦✉♥❞❡❞ ❛✇❛② ❢r♦♠ ③❡r♦ ♥❡❛r s = 1✳ ✭✐✐✮ ❲r✐t❡ χρ ❢♦r t❤❡ ❝❤❛r❛❝t❡r ♦❢ ρ✱ ✇❤✐❝❤ ✐s ✐rr❡❞✉❝✐❜❧❡✱ ❛♥❞ s❡t χρ (Frobp )p−s .

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 .

R❛❧ ❝❛s❡✳ G IQ/P PP (Ind τ, T ) = det 1 − T FrobQ/P (ResG DQ/P IndH τ ) det 1 − T FrobQ/P D Qi /Si xi i det 1 − T FrobQ/P i i τ xi Resx−1 Hx ∩D Q/P Q/P Indx−1 D Si = i i xi = x−1 Hx D Q/P Indx−1 Hx ∈D = det 1 − T FrobQ/P DQ Resx−1 DQ /S i /P i Si i i IndDQi /S ResH DQ /S τ i i IQ/P i i xi xi τ IQ/P IQi /P i ❤❡♥❝❡✱ ❜② ❙t❡♣ ✶✱ det 1 − T fSi /P FrobQi /Pi = ResH DQ /S τ i IQi /Si i Si PSi (τ, T fSi /P ) = Si Pr♦♣♦s✐t✐♦♥ ✸✳✶✺ ✭❆rt✐♥✬s ❚❤❡♦r❡♠✮✳ ▲❡t G ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✱ ρ ❛ G✲r❡♣r❡s❡♥t❛t✐♦♥✳ ❚❤❡♥ t❤❡r❡ ❛r❡ ❝②❝❧✐❝ s✉❜❣r♦✉♣s Hi , Hj ≤ G ❛♥❞ 1✲❞✐♠❡♥s✐♦♥❛❧ r❡♣r❡s❡♥t❛t✐♦♥s τi , τj ♦❢ Hi , Hj s✉❝❤ t❤❛t ρ⊕n ⊕ IndG Hi τi = i IndG H τj .

Download PDF sample

Algebraic Number Theory by V. Dokchitser, Sebastian Pancratz


by William
4.4

Rated 4.10 of 5 – based on 40 votes