By Kenji Ueno

Algebraic geometry performs a huge position in numerous branches of technological know-how and know-how. this can be the final of 3 volumes by way of Kenji Ueno algebraic geometry. This, in including Algebraic Geometry 1 and Algebraic Geometry 2, makes a great textbook for a path in algebraic geometry.

In this quantity, the writer is going past introductory notions and provides the speculation of schemes and sheaves with the objective of learning the houses precious for the whole improvement of contemporary algebraic geometry. the most subject matters mentioned within the e-book comprise measurement idea, flat and correct morphisms, normal schemes, delicate morphisms, of entirety, and Zariski's major theorem. Ueno additionally provides the idea of algebraic curves and their Jacobians and the relation among algebraic and analytic geometry, together with Kodaira's Vanishing Theorem.

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**Additional resources for Algebraic geometry 3. Further study of schemes**

**Example text**

1 − T )dim ρ ❍❡♥❝❡ |aP j ||N (P )−js | ≤ P ♣r✐♠❡ ❛❜♦✈❡ p j≥0 ✇❤❡r❡ σ = [K:Q] 1 (1 − p−σ )dim ρ (s) ❛♥❞ ✇❡ ♥♦t❡ a(1) = 1✱ ✇❤❡♥❝❡ |aP j ||N (P )−js | ≤ p P j≥0 1 1 − p−σ (dim ρ)[K:Q] = ζ(s)(dim ρ)[K:Q] ❛s (s) > 1✳ ❊①❛♠♣❧❡✳ ✭✐✮ ▲❡t K = Q✱ F ❛r❜✐tr❛r②✱ ρ = I✳ ❚❤❡♥ ❢♦r ❛ ♣r✐♠❡ P = (p) ♦❢ K ✱ ρIP = ρ ❛♥❞ FrobP ❛❝ts ❛s t❤❡ ✐❞❡♥t✐t② ♦♥ ρIP ✳ ❙♦ PP (ρ, T ) = det(1 − T |I) = 1 − T ✳ ❚❤✉s L(I, s) = p 1−p1 −s = ζ(s)✳ ✭✐✐✮ ▲❡t K, F ❜❡ ❛r❜✐tr❛r②✱ ρ = I✳ ❚❤❡♥ L(I, s) = P 1 = ρK (s) 1 − N (P )−s t❤❡ ❉❡❞❡❦✐♥❞ ρ✲❢✉♥❝t✐♦♥ ♦❢ K ✳ ✭✐✐✐✮ ▲❡t K = Q✱ F = Q(ζN )✱ ✇❤❡r❡ N ✐s ♣r✐♠❡✱ ❛♥❞ ρ 1✲❞✐♠❡♥s✐♦♥❛❧ ♥♦♥tr✐✈✐❛❧✳ ❚❤❡♥ L(ρ, s) = LN (ψ, s) ✇❤❡r❡ ψ ✐s t❤❡ ❉✐r✐❝❤❧❡t ❝❤❛r❛❝t❡r ♠♦❞✉❧♦ N ❞❡✜♥❡❞ ❜② ψ(n) = ρ(σn ) ✇❤❡r❡ n✳ σn ∈ Gal Q(ζN )/Q ✇✐❝❤ σn ζN = ζN ◆♦t❛t✐♦♥✳ ■❢ ρ : G → GLn (C) ✐s ❛ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♥ ✇r✐t❡ Tr λi gi ρ = Tr det λi ρ(gi ) = λi gi ρ = det λi χρ (gi ), λi ρ(gi ) .

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 }| = = .

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 }| = = .