A Book of Curves - download pdf or read online

By E. H. Lockwood

ISBN-10: 0521044448

ISBN-13: 9780521044448

ISBN-10: 0521055857

ISBN-13: 9780521055857

This publication opens up a tremendous box of arithmetic at an simple point, one within which the portion of aesthetic excitement, either within the shapes of the curves and of their mathematical relationships, is dominant. This publication describes equipment of drawing aircraft curves, starting with conic sections (parabola, ellipse and hyperbola), and occurring to cycloidal curves, spirals, glissettes, pedal curves, strophoids etc. regularly, 'envelope tools' are used. There are twenty-five full-page plates and over 90 smaller diagrams within the textual content. The e-book can be utilized in faculties, yet can be a reference for draughtsmen and mechanical engineers. As a textual content on complex aircraft geometry it may attract natural mathematicians with an curiosity in geometry, and to scholars for whom Euclidean geometry isn't really a primary examine.

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Example text

S ❛❧s♦ ❛♣♣❧✐❡s t♦ t❤❡ ✜♥✐t❡ ❡①t❡♥s✐♦♥s ♦❢ ❈♦r♦❧❧❛r② ✹✳✸✳ Tr(F m ) = |X(km )| ❢♦r ❡✈❡r② k✳ ❍❡♥❝❡ ✿ m 1✳ ❘❡♠❛r❦s✳ ✶✮ ❙✐♥❝❡ F :X→X ✐s ❛ r❛❞✐❝✐❛❧ ♠♦r♣❤✐s♠✱ ✐t ✐s ❛♥ ❤♦♠❡♦♠♦r♣❤✐s♠ ❢♦r t❤❡ ét❛❧❡ t♦♣♦❧♦❣②✳ ❍❡♥❝❡ ❡✈❡r② ❡✐❣❡♥✈❛❧✉❡ ♦❢ F ♦♥ Hci (X, Q ) ✐s ♥♦♥✲③❡r♦ ❀ ❢♦r ❛ ♠♦r❡ ♣r❡❝✐s❡ st❛t❡♠❡♥t✱ s❡❡ ❚❤❡♦r❡♠ ✹✳✺ ❜❡❧♦✇✳ ✷✮ ❚❤❡ t❤❡♦r❡♠ ♣r♦✈❡❞ ❜② ●r♦t❤❡♥❞✐❡❝❦✱ ❧♦❝✳❝✐t✳✱ ✐s ♠♦r❡ ❣❡♥❡r❛❧ t❤❛♥ ❚❤❡♦r❡♠ ✹✳✷ ✿ ✐t ❛♣♣❧✐❡s t♦ ❡✈❡r② ❝♦♥str✉❝t✐❜❧❡ ❛s ❛ s✉♠ ♦❢ ❧♦❝❛❧ tr❛❝❡s ❛t t❤❡ ♣♦✐♥ts ♦❢ ✸✮ ❆ss✉♠❡ k = Fp ✱ Q ✲s❤❡❛❢✱ ❛♥❞ ❣✐✈❡s Tr(F ) X(k)✳ t♦ s✐♠♣❧✐❢② ♥♦t❛t✐♦♥s✳ ❚❤❡♥ ❈♦r♦❧❧❛r② ✹✳✸ ✐s ❡q✉✐✈❛✲ ❧❡♥t t♦ s❛②✐♥❣ t❤❛t t❤❡ ❉✐r✐❝❤❧❡t s❡r✐❡s ❞❡♥♦t❡❞ ❜② ζX,p (s) ✐♥ ➓✶✳✺ ✐s ❡q✉❛❧ ✸✹ ✹✳ ❘❡✈✐❡✇ ♦❢ −s F |Hci (X, Q i det(1 − p ▼♦r❡♦✈❡r✱ ♦♥❡ ❤❛s t♦ i+1 ))(−1) NX (pe ) = ✲❛❞✐❝ ❝♦❤♦♠♦❧♦❣② ✱ ✇❤✐❝❤ ✐s ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ♦❢ p−s ✳ (−1)i Tri (F e ) i e ∈ Z ✭❛♥❞ ♥♦t ♠❡r❡❧② ❢♦r e 1✮✳ ■♥ ♣❛rt✐❝✉❧❛r✱ NX (p0 ) ✐s ❡q✉❛❧ t♦ i i (−1) dim Hc (X, Q )✱ ✇❤✐❝❤ ✐s t❤❡ ❊✉❧❡r✲P♦✐♥❝❛ré ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ X ✳ ❢♦r ❡✈❡r② i ✹✳✹✳ ❚❤❡ ❝❛s❡ ♦❢ ❛ ✜♥✐t❡ ✜❡❧❞ ✿ t❤❡ ❣❡♦♠❡tr✐❝ ❛♥❞ t❤❡ ❛r✐t❤♠❡t✐❝ ❋r♦❜❡♥✐✉s ❑❡❡♣ t❤❡ ♥♦t❛t✐♦♥ ♦❢ ➓✹✳✸✳ ❚❤❡ ●❛❧♦✐s ❣r♦✉♣ i ❝♦❤♦♠♦❧♦❣② ❣r♦✉♣ Hc (X, Q ♦❢ Γk )✳ Γk = Gal(k/k) ❛❝ts ♦♥ ❡❛❝❤ σ = σq σ ✮✱ t❤❛t ✐s ❝❛❧❧❡❞ t❤❡ ❛r✐t❤✲ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ♥❛t✉r❛❧ ❣❡♥❡r❛t♦r ❛❝ts ❜② ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ✭st✐❧❧ ❞❡♥♦t❡❞ ❜② ♠❡t✐❝ ❋r♦❜❡♥✐✉s ❛✉t♦♠♦r♣❤✐s♠ ✐♥ ♦r❞❡r t♦ ❞✐st✐♥❣✉✐s❤ ✐t ❢r♦♠ t❤❡ ❣❡♦♠❡tr✐❝ ❋r♦❜❡♥✐✉s F ❞❡✜♥❡❞ ❛❜♦✈❡✳ ❚❤❡s❡ t✇♦ ❦✐♥❞ ♦❢ ✏❋r♦❜❡♥✐✉s ❛✉t♦♠♦r♣❤✐s♠s✑ ❛r❡ r❡❧❛t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ s✐♠♣❧❡ r❡s✉❧t ✭s❡❡ ❬❙●❆ ✺✱ ♣✳✹✺✼❪✱ ♦r ❬❑❛ ✾✹✱ ✷✹✲✷✺❪✮ ✿ ❚❤❡♦r❡♠ ✹✳✹✳ ❚❤❡ ❛r✐t❤♠❡t✐❝ ❋r♦❜❡♥✐✉s ❛♥❞ t❤❡ ❣❡♦♠❡tr✐❝ ❋r♦❜❡♥✐✉s ❛r❡ ✐♥✈❡rs❡s ♦❢ ❡❛❝❤ ♦t❤❡r✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ σ(F ξ) = F (σξ) = ξ ❢♦r ❡✈❡r② ❆ s✐♠✐❧❛r r❡s✉❧t ❤♦❧❞s ❢♦r t❤❡ ❝♦❤♦♠♦❧♦❣② ❣r♦✉♣s ξ ∈ Hci (X, Q )✳ H i (X, Q )✱ ❜✐tr❛r② s✉♣♣♦rt ✭❛♥❞ ❛❧s♦ ❢♦r t❤❡ ❝♦❤♦♠♦❧♦❣② ✇✐t❤ ❝♦❡✣❝✐❡♥ts ✐♥ ❊①❛♠♣❧❡✳ ❙✉♣♣♦s❡ t❤❛t ❚❛t❡ Q X ✐s ❛♥ ❛❜❡❧✐❛♥ ✈❛r✐❡t② ♦✈❡r ✇✐t❤ ❛r✲ Z/ n Z✮✳ k ✱ ❛♥❞ ❧❡t V (X) ❜❡ ✐ts ✲♠♦❞✉❧❡✳ ❬❘❡❝❛❧❧ t❤❛t V (X) = Q ⊗ lim X[ n ]✱ ✇❤❡r❡ X[ n ] ✐s t❤❡ ❣r♦✉♣ ♦❢ t❤❡ n ✲❞✐✈✐s✐♦♥ ←− ♣♦✐♥ts ♦❢ X(k)✱ ✐✳❡✳ t❤❡ ❦❡r♥❡❧ ♦❢ n : X(k) → X(k✮ ❀ ✐t ✐s ❛ Q ✲✈❡❝t♦r s♣❛❝❡ ♦❢ ❞✐♠❡♥s✐♦♥ 2dim X ✳❪ ❚❤❡ ❋r♦❜❡♥✐✉s ❡♥❞♦♠♦r♣❤✐s♠ ♠❡t✐❝ ❋r♦❜❡♥✐✉s F s F : X → X ❛❝ts ♦♥ V (X) ❀ t❤❡ ❛r✐t❤✲ ❛❧s♦ ❛❝ts✱ ❛♥❞ ✐ts ❛❝t✐♦♥ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ❛❝t✐♦♥ ♦❢ F ❛♥❞ s ❛❝t ✐♥ t❤❡ s❛♠❡ ✇❛② ♦♥ X(k)✮✳ ❚❤❡ ✜rst ❝♦❤♦♠♦❧♦❣② H 1 (X, Q ) ✐s t❤❡ ❞✉❛❧ ♦❢ V (X) ❀ t❤❡ ❛❝t✐♦♥ ♦❢ F ♦♥ ✐t ✐s ❞❡✜♥❡❞ ❜② ❢✉♥❝t♦r✐❛❧✐t②✱ ✐✳❡✳ ❜② tr❛♥s♣♦s✐t✐♦♥ ❀ t❤❡ ❛❝t✐♦♥ ♦❢ s ✐s ❞❡✜♥❡❞ ❜② tr❛♥s♣♦rt ♦❢ ✭❜❡❝❛✉s❡ ❣r♦✉♣ str✉❝t✉r❡✱ ✐✳❡✳ ❜② ✐♥✈❡rs❡ tr❛♥s♣♦s✐t✐♦♥✳ ❚❤✐s ❡①♣❧❛✐♥s ✇❤② t❤❡ t✇♦ ❛❝t✐♦♥s ❛r❡ ✐♥✈❡rs❡ ♦❢ ❡❛❝❤ ♦t❤❡r✳ ✸ ❲❤❛t ✸ t❤✐s ❡①❛♠♣❧❡ s✉❣❣❡sts ✐s t❤❛t✱ ✐❢ ét❛❧❡ t♦♣♦❧♦❣② ✇❡r❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ❤♦♠♦❧♦❣② ✐♥st❡❛❞ ♦❢ ❝♦❤♦♠♦❧♦❣②✱ t❤❡ t✇♦ t②♣❡s ♦❢ ❋r♦❜❡♥✐✉s ✇♦✉❧❞ ❜❡ t❤❡ s❛♠❡✳ ✹✳✺✳ ✳ ❚❤❡ ❝❛s❡ ♦❢ ❛ ✜♥✐t❡ ✜❡❧❞ ✿ ❉❡❧✐❣♥❡✬s t❤❡♦r❡♠s ✸✺ ✹✳✺✳ ❚❤❡ ❝❛s❡ ♦❢ ❛ ✜♥✐t❡ ✜❡❧❞ ✿ ❉❡❧✐❣♥❡✬s t❤❡♦r❡♠s ❲❡ ❦❡❡♣ t❤❡ ♥♦t❛t✐♦♥ ❛♥❞ ❤②♣♦t❤❡s❡s ♦❢ ➓✹✳✹ ❛❜♦✈❡✳ q ✲❲❡✐❧ ✐♥t❡❣❡r ♦❢ ✇❡✐❣❤t w ∈ N ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ✐♥t❡❣❡r α |ι(α)| = q w/2 ❢♦r ❡✈❡r② ❡♠❜❡❞❞✐♥❣ ι : Q(α) → C✳ ❋♦r ✐♥st❛♥❝❡ ❛ ❘❡❝❛❧❧ t❤❛t ❛ s✉❝❤ t❤❛t q ✲❲❡✐❧ ✐♥t❡❣❡r ♦❢ ✇❡✐❣❤t ✵ ✐s ❛ r♦♦t ♦❢ ✉♥✐t② ✭❑r♦♥❡❝❦❡r✮✳ q ✲❲❡✐❧ ✐♥t❡❣❡r ♦❢ ✇❡✐❣❤t ✇❡✐❣❤t w r❡❧❛t✐✈❡❧② t♦ q ✑✳ ❘❡♠❛r❦✳ ■♥ ❉❡❧✐❣♥❡ ❬❉❡ ✽✵✱ ➓✶✳✷✳✶❪✱ ✇❤❛t ✇❡ ❝❛❧❧ ❛ w ✐s ❝❛❧❧❡❞ ✏ ❛♥ ❛❧❣❡❜r❛✐❝ ✐♥t❡❣❡r t❤❛t ✐s ♣✉r❡ ♦❢ ❚❤❡♦r❡♠ ✹✳✺✳ ✭❉❡❧✐❣♥❡✮ ▲❡t d = dim X ✳ α ♦❢ t❤❡ ❣❡♦♠❡tr✐❝ ❋r♦❜❡♥✐✉s F ❛❝t✐♥❣ ♦♥ Hci (X, Q ) ✐s ❛ q ✲❲❡✐❧ ✐♥t❡❣❡r ♦❢ ✇❡✐❣❤t i ; ✐❢ i d✱ t❤❡♥ α ✐s ❞✐✈✐s✐❜❧❡ ❜② q i−d .

Ss✉♠❡ t❤❛t X ✐s ♣r♦♣❡r ❛♥❞ s♠♦♦t❤✳ ❚❤❡♥ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②✲ i ♥♦♠✐❛❧ ♦❢ F ❛❝t✐♥❣ ♦♥ Hc (X, Q ) ❤❛s ❝♦❡✣❝✐❡♥ts ✐♥ Z ❛♥❞ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ ; ✐ts r♦♦ts ❛r❡ q ✲❲❡✐❧ ✐♥t❡❣❡rs ♦❢ ✇❡✐❣❤t i✳ ❛✮ ❊✈❡r② ❡✐❣❡♥✈❛❧✉❡ ❆ss❡rt✐♦♥ ❜✮ ✐s t❤❡ ❝❡❧❡❜r❛t❡❞ ❲❡✐❧ ❝♦♥❥❡❝t✉r❡✳ ■t ✐s ♣r♦✈❡❞ ✐♥ ❬❉❡ ✼✹❪ X ✉♥❞❡r t❤❡ s❧✐❣❤t❧② r❡str✐❝t✐✈❡ ❛ss✉♠♣t✐♦♥ t❤❛t ✐s ♣r♦❥❡❝t✐✈❡✱ ✐♥st❡❛❞ ♦❢ ♠❡r❡❧② ♣r♦♣❡r ❀ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬❉❡ ✽✵❪✳ ❚❤❡ ♣r♦♦❢ ♦❢ ❛✮ ✐s ❣✐✈❡♥ ✐♥ ❬❉❡ ✽✵✱ ➓✸✳✸❪ ❀ s❡❡ ❛❧s♦ ❬❑❛ ✾✹✱ ♣♣✳✷✻✲✷✼❪✳ ❚❤❡ ❞✐✈✐s✐❜✐❧✐t② ♦❢ α ❜② q i−d ✐s ♥♦t ❡①♣❧✐❝✐t❧② st❛t❡❞ ✐♥ ❬❉❡ ✽✵❪✱ ❜✉t ✐t ❢♦❧❧♦✇s ❢r♦♠ ❈♦r✳ ✸✳✸✳✽ ✇❤✐❝❤ s❛②s t❤❛t✱ ✐❢ ❡✈❡r② p✲❛❞✐❝ ✈❛❧✉❛t✐♦♥ v ♦❢ i d✱ ♦♥❡ ❤❛s v(α) (i − d)v(q) ❢♦r Q(α)✳ = p✱ ❛♥❞ ❞❡✜♥❡ t❤❡ i✲t❤ ❇❡tt✐ ♥✉♠❜❡r Bi ♦❢ X ❛s Hci (X, Q ) ✭r❡❝❛❧❧ t❤❛t ✐t ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ ✐♥ ❝❛s❡ NX (q) ❜❡✱ ❛s ✉s✉❛❧✱ t❤❡ ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts ♦❢ X(k)✳ ❇② ▲❡t ✉s ✜① ❛ ♣r✐♠❡ t❤❡ Q ✲❞✐♠❡♥s✐♦♥ ♦❢ ❜✮ ❛❜♦✈❡✮✳ ▲❡t ❝♦♠❜✐♥✐♥❣ ❚❤❡♦r❡♠ ✹✳✷ ❛♥❞ ❚❤❡♦r❡♠ ✹✳✺ ✇❡ ❣❡t ✿ ❚❤❡♦r❡♠ ✹✳✻✳ NX (q) = i=2d i i=0 (−1) νi ✱ ✇❤❡r❡ i/2 ❛♥ ❛❧❣❡❜r❛✐❝ ✐♥t❡❣❡r s✉❝❤ t❤❛t |νi | q Bi ✳ ✭▼♦r❡ ♣r❡❝✐s❡❧②✱ ❛❧❧ t❤❡ ●❛❧♦✐s ❝♦♥❥✉❣❛t❡s ♦❢ ❞❡❞ ❜② νi = Tr(F |Hci (X, Q )) ✐s νi ❤❛✈❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ❜♦✉♥✲ q i/2 Bi ✳✮ ◆♦t❡ t❤❛t t❤❡ ✏♠❛✐♥ t❡r♠✑ ✐♥ t❤✐s ❢♦r♠✉❧❛ ✐s ν2d ✱ ✇❤❡r❡ d = dim X ✳ ■ts IX t❤❡ s❡t ♦❢ ✐rr❡❞✉❝✐❜❧❡ ✈❛❧✉❡ ✐s ❡❛s② t♦ ❝♦♠♣✉t❡✳ ■♥❞❡❡❞✱ ❧❡t ✉s ❞❡♥♦t❡ ❜② X ♦❢ ❞✐♠❡♥s✐♦♥ d✳ ■t ✐s ♣r♦✈❡❞ ✐♥ ❬❙●❆ ✹✱ ❳❱■■■✳✷✳✾❪ t❤❛t Hc2d (X, Q ) ✐s ❝❛♥♦♥✐❝❛❧❧② ✐s♦♠♦r♣❤✐❝✹ t♦ Q (−d)IX ✳ ❙✐♥❝❡ F ❛❝ts ❜② q d d ♦♥ Q (−d)✱ t❤✐s s❤♦✇s t❤❛t ✇❡ ❤❛✈❡ ν2d = eq , ✇❤❡r❡ e ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts ♦❢ IX t❤❛t ❛r❡ k ✲r❛t✐♦♥❛❧✱ ✐✳❡✳ ✐♥✈❛r✐❛♥t ✉♥❞❡r σq ✳ ❇② ❛♣♣❧②✐♥❣ ❝♦♠♣♦♥❡♥ts ♦❢ ❚❤❡♦r❡♠ ✹✳✺ ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡ ✿ ✹ ❘❡❝❛❧❧ Q ✲❞✉❛❧ ♦❢ Q (−d) ✐s t❤❡ d✲t❤ t❡♥s♦r ♣♦✇❡r ♦❢ Q (−1)✱ ❛♥❞ t❤❛t Q (−1) ✐s t❤❡ Q (1) = Q ⊗ lim µ n ✱ ✇❤❡r❡ µ n ✐s t❤❡ ❣r♦✉♣ ♦❢ n ✲t❤ r♦♦ts ♦❢ ✉♥✐t② ✐♥ k✳ ←− t❤❛t ✸✻ ✹✳ ❘❡✈✐❡✇ ♦❢ ❈♦r♦❧❧❛r② ✹✳✼✳ |NX (q) − eq d | 1 (B − B2d )q d− 2 , ✇❤❡r❡ ✲❛❞✐❝ ❝♦❤♦♠♦❧♦❣② B= ❤❛s ♦♥❧② ♦♥❡ ❡❧❡♠❡♥t✱ ❤❡♥❝❡ Bi .

Tr(F ) = i ❚❤✐s ✐s t❤❡ ▲❡❢s❝❤❡t③ ♥✉♠❜❡r ♦❢ F✱ r❡❧❛t✐✈❡ t♦ t❤❡ ✲❛❞✐❝ ❝♦❤♦♠♦❧♦❣② ✇✐t❤ ♣r♦♣❡r s✉♣♣♦rt✳ ❆ ♣r✐♦r✐✱ ✐t ❞❡♣❡♥❞s ♦♥ t❤❡ ❝❤♦✐❝❡ ♦❢ ✳ ■♥ ❢❛❝t✱ ✐t ❞♦❡s ♥♦t✱ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ♦❢ ●r♦t❤❡♥❞✐❡❝❦ ✭❬●r ✻✹❪✱ s❡❡ ❛❧s♦ ❬❙●❆ 4 21 ✱ ♣✳✽✻✱ t❤✳✸✳✷❪✮ ✿ ❚❤❡♦r❡♠ ✹✳✷✳ Tr(F ) = |X(k)|. ❚❤✐s ❛❧s♦ ❛♣♣❧✐❡s t♦ t❤❡ ✜♥✐t❡ ❡①t❡♥s✐♦♥s ♦❢ ❈♦r♦❧❧❛r② ✹✳✸✳ Tr(F m ) = |X(km )| ❢♦r ❡✈❡r② k✳ ❍❡♥❝❡ ✿ m 1✳ ❘❡♠❛r❦s✳ ✶✮ ❙✐♥❝❡ F :X→X ✐s ❛ r❛❞✐❝✐❛❧ ♠♦r♣❤✐s♠✱ ✐t ✐s ❛♥ ❤♦♠❡♦♠♦r♣❤✐s♠ ❢♦r t❤❡ ét❛❧❡ t♦♣♦❧♦❣②✳ ❍❡♥❝❡ ❡✈❡r② ❡✐❣❡♥✈❛❧✉❡ ♦❢ F ♦♥ Hci (X, Q ) ✐s ♥♦♥✲③❡r♦ ❀ ❢♦r ❛ ♠♦r❡ ♣r❡❝✐s❡ st❛t❡♠❡♥t✱ s❡❡ ❚❤❡♦r❡♠ ✹✳✺ ❜❡❧♦✇✳ ✷✮ ❚❤❡ t❤❡♦r❡♠ ♣r♦✈❡❞ ❜② ●r♦t❤❡♥❞✐❡❝❦✱ ❧♦❝✳❝✐t✳✱ ✐s ♠♦r❡ ❣❡♥❡r❛❧ t❤❛♥ ❚❤❡♦r❡♠ ✹✳✷ ✿ ✐t ❛♣♣❧✐❡s t♦ ❡✈❡r② ❝♦♥str✉❝t✐❜❧❡ ❛s ❛ s✉♠ ♦❢ ❧♦❝❛❧ tr❛❝❡s ❛t t❤❡ ♣♦✐♥ts ♦❢ ✸✮ ❆ss✉♠❡ k = Fp ✱ Q ✲s❤❡❛❢✱ ❛♥❞ ❣✐✈❡s Tr(F ) X(k)✳ t♦ s✐♠♣❧✐❢② ♥♦t❛t✐♦♥s✳ ❚❤❡♥ ❈♦r♦❧❧❛r② ✹✳✸ ✐s ❡q✉✐✈❛✲ ❧❡♥t t♦ s❛②✐♥❣ t❤❛t t❤❡ ❉✐r✐❝❤❧❡t s❡r✐❡s ❞❡♥♦t❡❞ ❜② ζX,p (s) ✐♥ ➓✶✳✺ ✐s ❡q✉❛❧ ✸✹ ✹✳ ❘❡✈✐❡✇ ♦❢ −s F |Hci (X, Q i det(1 − p ▼♦r❡♦✈❡r✱ ♦♥❡ ❤❛s t♦ i+1 ))(−1) NX (pe ) = ✲❛❞✐❝ ❝♦❤♦♠♦❧♦❣② ✱ ✇❤✐❝❤ ✐s ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ♦❢ p−s ✳ (−1)i Tri (F e ) i e ∈ Z ✭❛♥❞ ♥♦t ♠❡r❡❧② ❢♦r e 1✮✳ ■♥ ♣❛rt✐❝✉❧❛r✱ NX (p0 ) ✐s ❡q✉❛❧ t♦ i i (−1) dim Hc (X, Q )✱ ✇❤✐❝❤ ✐s t❤❡ ❊✉❧❡r✲P♦✐♥❝❛ré ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ X ✳ ❢♦r ❡✈❡r② i ✹✳✹✳ ❚❤❡ ❝❛s❡ ♦❢ ❛ ✜♥✐t❡ ✜❡❧❞ ✿ t❤❡ ❣❡♦♠❡tr✐❝ ❛♥❞ t❤❡ ❛r✐t❤♠❡t✐❝ ❋r♦❜❡♥✐✉s ❑❡❡♣ t❤❡ ♥♦t❛t✐♦♥ ♦❢ ➓✹✳✸✳ ❚❤❡ ●❛❧♦✐s ❣r♦✉♣ i ❝♦❤♦♠♦❧♦❣② ❣r♦✉♣ Hc (X, Q ♦❢ Γk )✳ Γk = Gal(k/k) ❛❝ts ♦♥ ❡❛❝❤ σ = σq σ ✮✱ t❤❛t ✐s ❝❛❧❧❡❞ t❤❡ ❛r✐t❤✲ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ♥❛t✉r❛❧ ❣❡♥❡r❛t♦r ❛❝ts ❜② ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ✭st✐❧❧ ❞❡♥♦t❡❞ ❜② ♠❡t✐❝ ❋r♦❜❡♥✐✉s ❛✉t♦♠♦r♣❤✐s♠ ✐♥ ♦r❞❡r t♦ ❞✐st✐♥❣✉✐s❤ ✐t ❢r♦♠ t❤❡ ❣❡♦♠❡tr✐❝ ❋r♦❜❡♥✐✉s F ❞❡✜♥❡❞ ❛❜♦✈❡✳ ❚❤❡s❡ t✇♦ ❦✐♥❞ ♦❢ ✏❋r♦❜❡♥✐✉s ❛✉t♦♠♦r♣❤✐s♠s✑ ❛r❡ r❡❧❛t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ s✐♠♣❧❡ r❡s✉❧t ✭s❡❡ ❬❙●❆ ✺✱ ♣✳✹✺✼❪✱ ♦r ❬❑❛ ✾✹✱ ✷✹✲✷✺❪✮ ✿ ❚❤❡♦r❡♠ ✹✳✹✳ ❚❤❡ ❛r✐t❤♠❡t✐❝ ❋r♦❜❡♥✐✉s ❛♥❞ t❤❡ ❣❡♦♠❡tr✐❝ ❋r♦❜❡♥✐✉s ❛r❡ ✐♥✈❡rs❡s ♦❢ ❡❛❝❤ ♦t❤❡r✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ σ(F ξ) = F (σξ) = ξ ❢♦r ❡✈❡r② ❆ s✐♠✐❧❛r r❡s✉❧t ❤♦❧❞s ❢♦r t❤❡ ❝♦❤♦♠♦❧♦❣② ❣r♦✉♣s ξ ∈ Hci (X, Q )✳ H i (X, Q )✱ ❜✐tr❛r② s✉♣♣♦rt ✭❛♥❞ ❛❧s♦ ❢♦r t❤❡ ❝♦❤♦♠♦❧♦❣② ✇✐t❤ ❝♦❡✣❝✐❡♥ts ✐♥ ❊①❛♠♣❧❡✳ ❙✉♣♣♦s❡ t❤❛t ❚❛t❡ Q X ✐s ❛♥ ❛❜❡❧✐❛♥ ✈❛r✐❡t② ♦✈❡r ✇✐t❤ ❛r✲ Z/ n Z✮✳ k ✱ ❛♥❞ ❧❡t V (X) ❜❡ ✐ts ✲♠♦❞✉❧❡✳ ❬❘❡❝❛❧❧ t❤❛t V (X) = Q ⊗ lim X[ n ]✱ ✇❤❡r❡ X[ n ] ✐s t❤❡ ❣r♦✉♣ ♦❢ t❤❡ n ✲❞✐✈✐s✐♦♥ ←− ♣♦✐♥ts ♦❢ X(k)✱ ✐✳❡✳ t❤❡ ❦❡r♥❡❧ ♦❢ n : X(k) → X(k✮ ❀ ✐t ✐s ❛ Q ✲✈❡❝t♦r s♣❛❝❡ ♦❢ ❞✐♠❡♥s✐♦♥ 2dim X ✳❪ ❚❤❡ ❋r♦❜❡♥✐✉s ❡♥❞♦♠♦r♣❤✐s♠ ♠❡t✐❝ ❋r♦❜❡♥✐✉s F s F : X → X ❛❝ts ♦♥ V (X) ❀ t❤❡ ❛r✐t❤✲ ❛❧s♦ ❛❝ts✱ ❛♥❞ ✐ts ❛❝t✐♦♥ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ❛❝t✐♦♥ ♦❢ F ❛♥❞ s ❛❝t ✐♥ t❤❡ s❛♠❡ ✇❛② ♦♥ X(k)✮✳ ❚❤❡ ✜rst ❝♦❤♦♠♦❧♦❣② H 1 (X, Q ) ✐s t❤❡ ❞✉❛❧ ♦❢ V (X) ❀ t❤❡ ❛❝t✐♦♥ ♦❢ F ♦♥ ✐t ✐s ❞❡✜♥❡❞ ❜② ❢✉♥❝t♦r✐❛❧✐t②✱ ✐✳❡✳ ❜② tr❛♥s♣♦s✐t✐♦♥ ❀ t❤❡ ❛❝t✐♦♥ ♦❢ s ✐s ❞❡✜♥❡❞ ❜② tr❛♥s♣♦rt ♦❢ ✭❜❡❝❛✉s❡ ❣r♦✉♣ str✉❝t✉r❡✱ ✐✳❡✳ ❜② ✐♥✈❡rs❡ tr❛♥s♣♦s✐t✐♦♥✳ ❚❤✐s ❡①♣❧❛✐♥s ✇❤② t❤❡ t✇♦ ❛❝t✐♦♥s ❛r❡ ✐♥✈❡rs❡ ♦❢ ❡❛❝❤ ♦t❤❡r✳ ✸ ❲❤❛t ✸ t❤✐s ❡①❛♠♣❧❡ s✉❣❣❡sts ✐s t❤❛t✱ ✐❢ ét❛❧❡ t♦♣♦❧♦❣② ✇❡r❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ❤♦♠♦❧♦❣② ✐♥st❡❛❞ ♦❢ ❝♦❤♦♠♦❧♦❣②✱ t❤❡ t✇♦ t②♣❡s ♦❢ ❋r♦❜❡♥✐✉s ✇♦✉❧❞ ❜❡ t❤❡ s❛♠❡✳ ✹✳✺✳ ✳ ❚❤❡ ❝❛s❡ ♦❢ ❛ ✜♥✐t❡ ✜❡❧❞ ✿ ❉❡❧✐❣♥❡✬s t❤❡♦r❡♠s ✸✺ ✹✳✺✳ ❚❤❡ ❝❛s❡ ♦❢ ❛ ✜♥✐t❡ ✜❡❧❞ ✿ ❉❡❧✐❣♥❡✬s t❤❡♦r❡♠s ❲❡ ❦❡❡♣ t❤❡ ♥♦t❛t✐♦♥ ❛♥❞ ❤②♣♦t❤❡s❡s ♦❢ ➓✹✳✹ ❛❜♦✈❡✳ q ✲❲❡✐❧ ✐♥t❡❣❡r ♦❢ ✇❡✐❣❤t w ∈ N ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ✐♥t❡❣❡r α |ι(α)| = q w/2 ❢♦r ❡✈❡r② ❡♠❜❡❞❞✐♥❣ ι : Q(α) → C✳ ❋♦r ✐♥st❛♥❝❡ ❛ ❘❡❝❛❧❧ t❤❛t ❛ s✉❝❤ t❤❛t q ✲❲❡✐❧ ✐♥t❡❣❡r ♦❢ ✇❡✐❣❤t ✵ ✐s ❛ r♦♦t ♦❢ ✉♥✐t② ✭❑r♦♥❡❝❦❡r✮✳ q ✲❲❡✐❧ ✐♥t❡❣❡r ♦❢ ✇❡✐❣❤t ✇❡✐❣❤t w r❡❧❛t✐✈❡❧② t♦ q ✑✳ ❘❡♠❛r❦✳ ■♥ ❉❡❧✐❣♥❡ ❬❉❡ ✽✵✱ ➓✶✳✷✳✶❪✱ ✇❤❛t ✇❡ ❝❛❧❧ ❛ w ✐s ❝❛❧❧❡❞ ✏ ❛♥ ❛❧❣❡❜r❛✐❝ ✐♥t❡❣❡r t❤❛t ✐s ♣✉r❡ ♦❢ ❚❤❡♦r❡♠ ✹✳✺✳ ✭❉❡❧✐❣♥❡✮ ▲❡t d = dim X ✳ α ♦❢ t❤❡ ❣❡♦♠❡tr✐❝ ❋r♦❜❡♥✐✉s F ❛❝t✐♥❣ ♦♥ Hci (X, Q ) ✐s ❛ q ✲❲❡✐❧ ✐♥t❡❣❡r ♦❢ ✇❡✐❣❤t i ; ✐❢ i d✱ t❤❡♥ α ✐s ❞✐✈✐s✐❜❧❡ ❜② q i−d .

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A Book of Curves by E. H. Lockwood


by Kenneth
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