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❑r✐❝❤❡✈❡r ❢♦r♠❛❧ ❣r♦✉♣s ❛♥❞ t❤❡ ❞❡❢♦r♠❡❞ ❇❛❦❡r✲❆❦❤✐❡③❡r ❢✉♥❝t✐♦♥✳

❊❧❡♥❛ ❨✉✳ ❇✉♥❦♦✈❛

❜✉♥❦♦✈❛❅♠✐✳r❛s✳r✉

❙t❡❦❧♦✈ ▼❛t❤❡♠❛t✐❝❛❧ ■♥st✐t✉t❡✱ ❘✉ss✐❛♥ ❆❝❛❞❡♠② ♦❢ s❝✐❡♥❝❡s✱ ▲❛❜♦r❛t♦r② ♦❢ ❣❡♦♠❡tr✐❝ ♠❡t❤♦❞s ✐♥ ♠❛t❤❡♠❛t✐❝❛❧ ♣❤②s✐❝s✱ ▼❙❯ ▼♦s❝♦✇

✐♥ ❝♦❧❧❛❜♦r❛t✐♦♥ ✇✐t❤ ❱✳▼✳❇✉❝❤st❛❜❡r

■♥t❡r♥❛t✐♦♥❛❧ ❝♦♥❢❡r❡♥❝❡ ”●❡♦♠❡tr②✱ ■♥t❡❣r❛❜✐❧✐t② ❛♥❞ ◗✉❛♥t✐③❛t✐♦♥“✱ ✽ ❥✉♥❡ ✷✵✶✶

◆❡✇ r❡s✉❧ts ❛r❡ ♣✉❜❧✐s❤❡❞ ✐♥ ❬✶❪ ❱✳ ▼✳ ❇✉❝❤st❛❜❡r ❛♥❞ ❊✳ ❨✉✳ ❇✉♥❦♦✈❛✱ ❑r✐❝❤❡✈❡r ❢♦r♠❛❧ ❣r♦✉♣ ❧❛✇✱ ❋✉♥❝t✐♦♥❛❧✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✹✺✿✷ ✭✷✵✶✶✮✳ ❬✷❪ ❊✳ ❨✉✳ ❇✉♥❦♦✈❛✱ ❚❤❡ ❛❞❞✐t✐♦♥ t❤❡♦r❡♠ ❢♦r t❤❡ ❞❡❢♦r♠❡❞ ❇❛❦❡r✲❆❦❤✐❡③❡r ❢✉♥❝t✐♦♥✱ ❘✉ss✐❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙✉r✈❡②s✱ ✷✵✶✵✱ ✻✺✿✻✳ ❬✸❪ ❱✳ ▼✳ ❇✉❝❤st❛❜❡r✱ ❚❤❡ ❣❡♥❡r❛❧ ❑r✐❝❤❡✈❡r ❣❡♥✉s✱ ❘✉ss✐❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙✉r✈❡②s✱ ✷✵✶✵✱ ✻✺✿✺✱ ✾✼✾✕✾✽✶✳ ❬✹❪ ❱✐❝t♦r ▼✳ ❇✉❝❤st❛❜❡r✱ ❊❧❡♥❛ ❨✉✳ ❇✉♥❦♦✈❛✱ ”❊❧❧✐♣t✐❝ ❢♦r♠❛❧ ❣r♦✉♣ ❧❛✇s✱ ✐♥t❡❣r❛❧ ❍✐r③❡❜r✉❝❤ ❣❡♥❡r❛ ❛♥❞ ❑r✐❝❤❡✈❡r ❣❡♥❡r❛“✱ ❛r❳✐✈✿✶✵✶✵✳✵✾✹✹ ❬✺❪ ❱✳ ▼✳ ❇✉❝❤st❛❜❡r ❛♥❞ ❊✳ ❨✉✳ ❇✉♥❦♦✈❛✱ ”❆❞❞✐t✐♦♥ ❚❤❡♦r❡♠s✱ ❋♦r♠❛❧ ●r♦✉♣ ▲❛✇s ❛♥❞ ■♥t❡❣r❛❜❧❡ ❙②st❡♠s“✱ ❆■P ❈♦♥❢❡r❡♥❝❡ Pr♦❝❡❡❞✐♥❣s ❱♦❧✉♠❡ ✶✸✵✼✱ ❳❳■❳ ❲♦r❦s❤♦♣ ♦♥ ●❡♦♠❡tr✐❝ ▼❡t❤♦❞s ✐♥ P❤②s✐❝s✳

✶

❚❤❡ ❢♦r♠❛❧ ❣r♦✉♣✳ ❆ ❝♦♠♠✉t❛t✐✈❡ ♦♥❡✲❞✐♠ ❢♦r♠❛❧ ❣r♦✉♣ ❧❛✇ ✭♦r s❤♦rt❧② ❢♦r♠❛❧ ❣r♦✉♣✮ ♦✈❡r ❛ ❝♦♠♠✉t❛t✐✈❡ ❛ss♦❝✐❛t✐✈❡ r✐♥❣ A ✐s t❤❡ ❢♦r♠❛❧ s❡r✐❡s F(u, v) = u + v +

• ai,juivj,

ai,j ∈ A, i > 0, j > 0, s❛t✐s❢②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥s F(u, v) = F(v, u), F

• u, F(v, w)
• = F
• F(u, v), w
• .

▲❡t Fa(u, v) = u + v✳ ❋♦r ❛♥② ❢♦r♠❛❧ ❣r♦✉♣ F(u, v) ∈ A[[u, v]] t❤❡r❡ ❡①✐sts ❛♥ ✐s♦♠♦r♣❤✐s♠ f : Fa → F ♦✈❡r A ⊗ Q✳ ❚❤❡ s❡r✐❡s f(t) ∈ A ⊗ Q[[t]] ✉♥✐q✉❡❧② ❞❡✜♥❡❞ ❜② t❤❡ ❝♦♥❞✐t✐♦♥s f(t1 + t2) = F

• f(t1), f(t2)
• ,

f(0) = 0, f′(0) = 1 ✐s t❤❡ ❡①♣♦♥❡♥t✐❛❧ ♦❢ t❤❡ ❢♦r♠❛❧ ❣r♦✉♣ F(u, v)✳

✷

❊①❛♠♣❧❡s ♦❢ ❢♦r♠❛❧ ❣r♦✉♣s ❛♥❞ t❤❡✐r ❡①♣♦♥❡♥t✐❛❧s✳ F(u, v) = u + v − µ1uv, f(t) = 1

µ1

• 1 − exp(−µ1t)
• .

F(u, v) =

u+v 1+µ2uv,

f(t) =

1 õ2 th

õ2 t

• ✳

F(u, v) = u √

1−2δv2+εv4 + v

√

1−2δu2+εu4 1−εu2v2

, f(t) = sn(t)✱ ✇❤❡r❡ sn(t) ✐s t❤❡ ❏❛❝♦❜✐ ❡❧❧✐♣t✐❝ s✐♥❡✿ (f′)2 = 1 − 2δf2 + εf4.

✸

❚❤❡ ❣❡♥❡r❛❧ ❲❡✐❡rstr❛ss ♠♦❞❡❧ ♦❢ t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡ Y 2Z + µ1XY Z + µ3Y Z2 = X3 + µ2X2Z + µ4XZ2 + µ6Z3 ❞❡♣❡♥❞s ♦♥ 5 ♣❛r❛♠❡t❡rs µ = (µ1, µ2, µ3, µ4, µ6)✳ ❚❤❡ ❣❡♦♠❡tr✐❝ ❣r♦✉♣ str✉❝t✉r❡ ”✰“ ♦♥ t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡✿ ❚❤❡ ♣♦✐♥ts P✱ Q✱ R ♦❢ t❤❡ ❝✉r✈❡ ❛r❡ ♦♥ ❛ str❛✐❣❤t ❧✐♥❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ P + Q + R = 0✳ ▲❡t O = (0 : 1 : 0) ❜❡ t❤❡ ③❡r♦ ♦❢ t❤❡ ❣❡♦♠❡tr✐❝ ❣r♦✉♣ str✉❝t✉r❡✳ ❋♦r P + Q + R = 0 ❛♥❞ R + R + O = 0 ✇❡ ❣❡t P + Q = R✳

✹

■♥ t❤❡ ❝❤❛rt Z = 0 ✐♥ ❲❡✐❡rstr❛ss ❝♦♦r❞✐♥❛t❡s x = X/Z ❛♥❞ y = Y/Z t❤❡ ❝✉r✈❡ t❛❦❡s t❤❡ ❢♦r♠ y2 + µ1xy + µ3y = x3 + µ2x2 + µ4x + µ6. ■♥ t❤❡ ❝❤❛rt Y = 0 ✐♥ ❚❛t❡ ❝♦♦r❞✐♥❛t❡s u = −X/Y ❛♥❞ s = −Z/Y t❤❡ ❝✉r✈❡ t❛❦❡s t❤❡ ❢♦r♠ s = u3 + µ1us + µ2u2s + µ3s2 + µ4us2 + µ6s3. ■♥ t❤❡ ❝❤❛rt X = 0 ✇✐t❤ ❝♦♦r❞✐♥❛t❡s v = Y/X ❛♥❞ w = Z/X t❤❡ ❝✉r✈❡ t❛❦❡s t❤❡ ❢♦r♠ vw(v + µ1 + µ3w) = 1 + µ2w + µ4w2 + µ6w3. ❚❤❡ ❣r❛❞✐♥❣s ❛r❡ deg X = −4✱ deg Y = −6✱ deg Z = 0✱ deg µi = −2i✳

✺

❚❤❡ ❚❛t❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡✳ ■♥ t❤❡ ❝❤❛rt {Y = 0} ✇✐t❤ u = −X/Y ❛♥❞ s = −Z/Y t❤❡ ❝✉r✈❡ t❛❦❡s t❤❡ ❢♦r♠ s = u3 + µ1us + µ2u2s + µ3s2 + µ4us2 + µ6s3. ❲❡ ❣❡t t❤❡ ❢♦r♠❛❧ s❡r✐❡s s(u) ∈ Z[µ][[u]]✿ s = u3 + µ1u4 + (µ2

1 + µ2)u5 + (µ3 1 + 2µ1µ2 + µ3)u6+

+ (µ4

1 + 3µ2 1µ2 + µ2 2 + 3µ1µ3 + µ4)u7 + ...

❚❤❡ ❝♦♦r❞✐♥❛t❡s (u, s(u)) ❛r❡ t❤❡ ❛r✐t❤♠❡t✐❝ ❚❛t❡ ❝♦♦r❞✐♥❛t❡s✳ ❚❤❡② ❞❡✜♥❡ t❤❡ ❚❛t❡ ✉♥✐❢♦♠✐③❛t✐♦♥ ♦❢ t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡✳

✻

✼

❚❤❡ ❡❧❧✐♣t✐❝ ❢♦r♠❛❧ ❣r♦✉♣✳ ▲❡t P =

• u, s(u)
• ✱ Q =
• v, s(v)
• ✱ R =
• w, s(w)
• ❛♥❞ R =
• ¯

w, s( ¯ w)

• ✳

❚❤❡ ❣❡♦♠❡tr✐❝ ❣r♦✉♣ str✉❝t✉r❡ ♦♥ t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❞❡✜♥❡s t❤❡ s❡r✐❡s FEl(u, v) ♦✈❡r Z[µ]✿ FEl(u, v) = ¯ w. ❚❤❡♦r❡♠✳ ✭●❡♥❡r❛❧ ❡❧❧✐♣t✐❝ ❢♦r♠❛❧ ❣r♦✉♣✮ FEl(u, v) =

• u + v − uv(µ1 + µ3m) + (µ4 + 2µ6m)k

(1 − µ3k − µ6k2)

• ×

× (1 + µ2m + µ4m2 + µ6m3) (1 + µ2n + µ4n2 + µ6n3)(1 − µ3k − µ6k2), ✇❤❡r❡

• u, s(u)
• ∈ Vµ

❛♥❞ m = s(u) − s(v) u − v , k = us(v) − vs(u) u − v , n = m + uv(1 + µ2m + µ4m2 + µ6m3) (1 − µ3k − µ6k2) .

✽

❋♦r♠❛❧ ❣r♦✉♣s ❢♦r ❝✉r✈❡s ✇✐t❤ ❛✉t♦♠♦r♣❤✐s♠s✳ ❚❤❡ ♠❛♣ u → αu ❞❡✜♥❡s ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ❛♥ ❡❧❧✐♣t✐❝ ❢♦r♠❛❧ ❣r♦✉♣✳ α = 2✱ µ = (0, 0, µ2, µ4, µ6)✳ ❚❤❡♥ F(u, v) = u + v + b (µ2 + 2µ4m + 3µ6m2) (1 + µ2m + µ4m2 + µ6m3). α = 3✱ µ = (0, 0, µ3, 0, µ6)✳ ❚❤❡♥ F(u, v) = (u + v)(1 + µ6m3) + µ3m2 + 3µ6m2b (1 + µ6m3)(1 − µ3b) − bm

uv µ3(1 − µ3b − µ6b2)

. α = 4✱ µ = (0, 0, 0, µ4, 0) = 0✳ α = 6✱ µ = (0, 0, 0, 0, µ6) = 0 f2(f′2 + 4µ4f4 − 1) = 0, f′3 + 3f′2 + 27µ6f6 − 4 = 0, F(u, v) = u + v + 2µ4mb 1 + µ4m2. F(u, v) = u + v + 3µ6m2b 1 + µ6m3.

✾

❆♥ ❡❧❧✐♣t✐❝ ❢✉♥❝t✐♦♥ ✐s ❛ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥ ♦♥ t❤❡ t♦r✉s T = C/Γ✱ ✇❤❡r❡ Γ ∈ C ✐s ❛ ❣r✐❞ ♦❢ r❛♥❦ 2✳ f(t + 2ω1) = f(t), f(t + 2ω2) = f(t), ✇❤❡r❡

Im(ω1

ω2 ) = 0. ❊❧❧✐♣t✐❝ ❢✉♥❝t✐♦♥s ❢♦r♠ ❛ ❞✐✛❡r❡♥t✐❛❧ ✜❡❧❞✱ ❛❧❣❡❜r❛✐❝❧② ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❢✉♥❝t✐♦♥s ℘(t) ❛♥❞ ℘′(t)✳ ❚❤❡ ❲❡✐❡rstr❛ss ❢✉♥❝t✐♦♥ ℘(t) ✐s t❤❡ ✉♥✐q✉❡ ❡✈❡♥ ❡❧❧✐♣t✐❝ ❢✉♥❝t✐♦♥ ♦♥ C ✇✐t❤ ♣❡r✐♦❞s 2ω1✱ 2ω2 ❛♥❞ ❞♦✉❜❧❡ ♣♦❧❡s ❛t ❣r✐❞ ♣♦✐♥ts s✉❝❤ t❤❛t lim

t→0

• ℘(t) − 1

t2

• = 0.

❲❡ ✇✐❧❧ ♥❡❡❞ t❤❡ ❲❡✐❡rstr❛ss σ✲❢✉♥❝t✐♦♥✱ ✇❤✐❝❤ ✐s ❛♥ ❡♥t✐r❡ ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ✉♥✐✈❡rs❛❧ ❝♦✈❡r✐♥❣ ♦❢ T ❛♥❞ ✐s ❞❡✜♥❡❞ ❜② t❤❡ ❝♦♥❞✐t✐♦♥ (ln σ)′′ = −℘ ❛♥❞ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s σ(0) = 0✱ σ′(0) = 1✳

✶✵

❚❤❡♦r❡♠✳ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ♦❢ t❤❡ ❣❡♥❡r❛❧ ❡❧❧✐♣t✐❝ ❢♦r♠❛❧ ❣r♦✉♣ ✐s fEl(t) = −2 ℘(t) − 1

12(µ2 1 + 4µ2)

℘′(t) − µ1(℘(t) − 1

12(µ2 1 + 4µ2)) − µ3

✇❤❡r❡ ℘(t) = ℘(t; g2(µ), g3(µ))✳ fEl(t) ✐s ❛♥ ❡❧❧✐♣t✐❝ ❢✉♥❝t✐♦♥ ♦❢ ♦r❞❡r ✸ ❢♦r µ6 = 0 ❛♥❞ ♦❢ ♦r❞❡r ✷ ❢♦r µ6 = 0 ✭✐♥ t❤❡ ❝❛s❡ ♦❢ ❛ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ❝✉r✈❡✮✳ ❊①❛♠♣❧❡ ♦❢ ❛ ❞❡❣❡♥❡r❛t❡ ❝✉r✈❡ µ = (µ1, µ2, 0, 0, 0)✳ ℘(t) = (a − b)2 4

 

• eat + ebt

eat − ebt

2

− 2 3

  = 1

t2 + (a − b)4 240 t2 + ... ✇❤❡r❡ µ1 = a + b, µ2 = −ab✳ ❚❤❡ ❢♦r♠❛❧ ❣r♦✉♣ ✐s r❛t✐♦♥❛❧ F(u, v) = u + v − µ1uv 1 + µ2uv , ✇❤❡r❡ f(t) = eat − ebt aeat − bebt.

✶✶

❲❡ ❝❛♥ r❡✇r✐t❡ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❛s 1 fEl(t) = µ1 2 − 1 2 ℘′(t) + ℘′(w) ℘(t) − ℘(v) , ✇❤❡r❡ ℘(t) = ℘(t; g2(µ), g3(µ))✱ ❛♥❞ v ❛♥❞ w ❛r❡ ❞❡t❡r♠✐♥❡❞ ❜② ℘′(w) = −µ3✱ ℘(v) = 1

12(4µ2 + µ2 1)✳

■♥ t❤❡ ❝❛s❡ v = ±w t❤❡ ❝♦♠♣❛t✐❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ❧❛st s②st❡♠ ✐s µ6 = 0✳ ❋♦r µ6 = 0 t❤❡ s♦❧✉t✐♦♥ ♦❢ (ln Φ(t))′ = µ1 2 − 1 fEl(t) ✭✶✮ ✐s t❤❡ ✇❡❧❧✲❦♥♦✇♥ ❇❛❦❡r✲❆❦❤✐❡③❡r ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡✳ ❚❤❡ s♦❧✉t✐♦♥ ♦❢ ❡q✳ ✭✶✮ ✐♥ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ✐s ❝❛❧❧❡❞ t❤❡ ❞❡❢♦r♠❡❞ ❇❛❦❡r✲❆❦❤✐❡③❡r ❢✉♥❝t✐♦♥✳

✶✷

❚❤❡ ❇❛❦❡r✲❆❦❤✐❡③❡r ❢✉♥❝t✐♦♥ Φ(t) = Φ(t, τ) = σ(τ + t) σ(t)σ(τ)e−ζ(τ)t ❣✐✈❡s ❛ s♦❧✉t✐♦♥ t♦ t❤❡ ▲❛♠❡ ❡q✉❛t✐♦♥

• d2

dt2 − 2℘(t)

• Φ(t) = ℘(τ)Φ(t).

■♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t = 0 ✇❡ ❤❛✈❡ Φ(t; τ) = 1 t − 1 2℘(τ)t − 1 6℘′(τ)t2 + (t3). ▲❡t 2ωk✱ k = 1, 2❜❡ t❤❡ ♣❡r✐♦❞s ♦❢ t❤❡ ℘✲❢✉♥❝t✐♦♥✳ ❚❤❡♥ Φ(t + 2ωk; τ) = Φ(t; τ) exp(−2(ζ(τ)ωk − ηkτ)), Φ(t + 2ωk; ωk) = Φ(t; ωk), Φ(t; τ + 2ωk) = Φ(t; τ).

✶✸

▲❡♠♠❛✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ♦♣❡r❛t♦rs ❛♥♥✉❧❛t❡ t❤❡ ❇❛❦❡r✲❆❦❤✐❡③❡r ❢✉♥❝t✐♦♥✿ H0 = 4g2 ∂ ∂g2 + 6g3 ∂ ∂g3 − t ∂ ∂t − τ ∂ ∂τ − 1, H2 = 6g3 ∂ ∂g3 + 1 3g2

2

∂ ∂g3 − ζ(t) ∂ ∂t − ζ(τ) ∂ ∂τ − (℘(t) + ℘(τ) + 1 2t℘′(τ)), L1 = ∂ ∂t + P, ✇❤❡r❡ P = −1 2 ℘′(t) − ℘′(τ) ℘(t) − ℘(τ) , L2 = ∂2 ∂t2 − 2℘(t) − ℘(τ), L3 = 2 ∂3 ∂t3 − 6℘(t) ∂ ∂t − 3℘′(t) − ℘′(τ).

✶✹

❚❤❡♦r❡♠✳ ❚❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❛❧❣❡❜r❛✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠ ♦♥ C4 ✇✐t❤ ❝♦♦r❞✐♥❛t❡s (z1, z2, x, y)✿ ∂tz1 = z2, ∂τz1 = 1 2(z2

1 + z2 − 3x),

∂tz2 = 2z3

1 − 6z1x − 2y,

∂τz2 = z3

1 − 3z1x − y + z1z2,

∂tx = 0, ∂τx = y, ∂ty = 0, ∂τy = 6x2 − 1 2(z2

2 − z4 1 + 6xz2 1 + 4yz1 + 3x2).

♣❛ss✐♥❣ ❛t t = 0, τ = 0 t❤r♦✉❣❤ (z0

1, z0 2, x0, y0)

✇❤❡r❡ 3x0 = (z0

1)2 − z0 2 ✐s

(−(ln Φ(t+t0; τ+τ0))′, −(ln Φ(t+t0; τ+τ0))′′, ℘(τ+τ0), ℘′(τ+τ0)).

✶✺

❋♦r t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ❲❡✐❡rstr❛ss σ✲❢✉♥❝t✐♦♥ σ(u) = u

• i,j≥0

ai,j (4i + 6j + 1)!(g2u4 2 )i(2g3u6)j, t❤❡r❡ ✐s t❤❡ ❲❡✐❡rstr❛ss r❡❝✉rs✐♦♥ ai,j = 3(i+1)ai+1,j−1+16 3 (j+1)ai−2,j+1−1 3(4i+6j−1)(2i+3j−1)ai−1,j ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s a0,0 = 1, ❛♥❞ ai,j = 0 ❢♦r i < 0 ♦r j < 0. ❈♦♥❥❡❝t✉r❡✳ ❋♦r a(i, j) = 2k3ls(i, j)✱ ❛♥❞ (4i + 6j + 1)! 23i+4j 3i+j i! j! = 2k13l1s1(i, j), ✇❤❡r❡ s(i, j) ❛♥❞ s1(i, j) ∈ Z ❛r❡ ❝♦♣r✐♠❡ ✇✐t❤ 2 ❛♥❞ 3 ✇❡ ❤❛✈❡ k = k1✱ l = l1✳

✶✻

❚❤❡ ❑r✐❝❤❡✈❡r ❣❡♥✉s✳ ■✳ ▼✳ ❑r✐❝❤❡✈❡r ✐♥tr♦❞✉❝❡❞ ✐♥ ✶✾✾✵ t❤❡ ❍✐r③❡❜r✉❝❤ ❣❡♥✉s Lf✱ ✇❤❡r❡ f(t) = fKr(t) = exp(µ1

2 t)

Φ(t, τ) , ❛♥❞ ♣r♦✈❡❞ t❤✐s ❣❡♥✉s t♦ ❤❛✈❡ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ r✐❣✐❞✐t② ♣r♦♣❡rt② ♦♥ ♥♦r♠❛❧❧② ❝♦♠♣❧❡① S1✲❡q✉✐✈❛r✐❛♥t SU✲♠❛♥✐❢♦❧❞s✳ ❚❤❡♦r❡♠✳ ❚❤❡ s❡r✐❡s fKr(t) ✐s ❛ ❍✉r✇✐ts s❡r✐❡s ♦✈❡r Z[µ1

2 , ℘(τ), ℘′(τ), g2 2 ]✱

t❤❛t ✐s fKr(t) =

• k≥0

fk tk+1 (k + 1)!, fk ∈ Z[µ1 2 , ℘(τ), ℘′(τ), g2 2 ].

✶✼

❚❤❡ ❛❞❞✐t✐♦♥ t❤❡♦r❡♠ ❢♦r t❤❡ ❇❛❦❡r✲❆❦❤✐❡③❡r ❢✉♥❝t✐♦♥✳ Φ(t + q) = Φ(t)Φ′(q) − Φ′(t)Φ(q) ℘(t) − ℘(q) . ❋♦r ❛♣♣❧✐❝❛t✐♦♥s t♦ t❤❡ ❑♣ ❤✐❡r❛r❝❤② s❡❡ ✐♥ ■✳▼✳ ❑r✐❝❤❡✈❡r✱ ❋❆❆ ✭✶✾✽✵✮✳ ❈♦rr♦❧❛r②✳ Φ(t + q) Φ(t)Φ(q) = −1 2

• 1

1 1 ℘(t) ℘(q) ℘(τ) ℘′(t) ℘′(q) ℘′(τ)

• 1

1 1 ℘(t) ℘(q) ℘(τ) ℘(t)2 ℘(q)2 ℘(τ)2

• .

✶✽

❚❤❡ ❑r✐❝❤❡✈❡r ❢♦r♠❛❧ ❣r♦✉♣✳ ❙❡t B = Z[χk : k = 1, 2, ...]✳ ❈♦♥s✐❞❡r t❤❡ s❡r✐❡s ♦❢ t❤❡ s♣❡❝✐❛❧ ❢♦r♠

• F(u, v) = ub(v) + vb(u) − b′(0)uv + b(u)β(u) − b(v)β(v)

ub(v) − vb(u) u2v2, ✇❤❡r❡ b(u) = 1 + biui✱ ❛♥❞ β(u) = b′(u)−b′(0)

2u

=

k0 βk+2uk✳

❍❡r❡ b1 = χ1✱ b2i = χ2i b2i+1 = 2χ2i+1 ❛♥❞ β2k = kχ2k✱ β2k+1 = (2k + 1)χ2k+1✳ ❚❤✐s s❡r✐❡s ❞❡✜♥❡s ❛ ❢♦r♠❛❧ ❣r♦✉♣ FKr(u, v) ∈ A[[u, v]] ✇❤❡r❡

• A = B/

J ❛♥❞ J ✐s t❤❡ ❛ss♦❝✐❛t✐✈✐t② ✐❞❡❛❧✳ ❚❤❡♦r❡♠✳ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ♦❢ t❤❡ ❑r✐❝❤❡✈❡r ❢♦r♠❛❧ ❣r♦✉♣ ✐s fKr(t)✳

✶✾

❚❤❡♦r❡♠✳ ❆♥ ❡❧❧✐♣t✐❝ ❢♦r♠❛❧ ❣r♦✉♣ ♦✈❡r t❤❡ r✐♥❣ A ✇✐t❤ ♥♦ ③❡r♦ ❞✐✈✐s♦rs ✐s ❛ ❑r✐❝❤❡✈❡r ❢♦r♠❛❧ ❣r♦✉♣ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐♥ A ✇❡ ❤❛✈❡✿ µ2µ3 − µ1µ4 = 0, µ2

3 + 3µ6 = 0,

µ3(µ1µ3 + µ4) = 0. ❈♦r♦❧❧❛r②✳ ❚❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠ ❛r❡ ❡q✉✐✈❛❧❡♥t t♦ µ6 = 0 ✐♥ t❤❡ ❝❛s❡ µ1 = 0, µ3 = 0; µ2 = 0, µ2

3 = −3µ6,

µ4 = 0 ✐♥ t❤❡ ❝❛s❡ µ1 = 0, µ3 = 0; µ4 = 0, µ6 = 0 ✐♥ t❤❡ ❝❛s❡ µ1 = 0, µ3 = 0; µ2 = −µ2

1, µ4 = −µ1µ3, −3µ6 = µ2 3

✐♥ t❤❡ ❝❛s❡ µ1 = 0, µ3 = 0.

✷✵

❊①❛♠♣❧❡s ♦❢ ❡❧❧✐♣t✐❝ ❑r✐❝❤❡✈❡r ❢♦r♠❛❧ ❣r♦✉♣s ✭❛♥❞ ✐♥t❡❣r❛❧ ❑r✐❝❤❡✈❡r ❣❡♥❡r❛✮✳ ▲❡t µ1 = µ3 = µ6 = 0 ❛♥❞ δ = µ2✱ ε = µ2

2 − 4µ4✱ t❤❡♥

FEl(u, v) = FKr(u, v) = uρ(v) + vρ(u) 1 − εu2v2 ❢♦r ρ2(u) = 1 − 2δu2 + εu4✳ ■♥ t❤✐s ❝❛s❡ fKr(t) = sn(t)✳ ▲❡t µ1 = µ2 = µ4 = 0 ❛♥❞ µ2

3 = −3µ6✱ t❤❡♥

FEl(u, v) = FKr(u, v) = u2r(v) − v2r(u) ur2(v) − vr2(u) ❢♦r r3(u) = 1 − 3µ3u3✳ ▲❡t µ3 = µ4 = µ6 = 0✱ t❤❡♥ FEl(u, v) = FKr(u, v) = u + v − µ1uv 1 + µ2uv .

✷✶

❚❤❡ ❞❡❢♦r♠❡❞ ❇❛❦❡r✲❆❦❤✐❡③❡r ❢✉♥❝t✐♦♥✳ ▲❡t fEl(t) ❜❡ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ♦❢ FEl(u, v)✳ ❚❤❡♥ 1 fEl(t) − µ1 2 = φ(t; v, w) = −1 2 ℘′(t) + ℘′(w) ℘(t) − ℘(v) , ✭✷✮ ✇❤❡r❡ ℘(t) = ℘(t; g2(µ), g3(µ))✱ ℘′(w) = −µ3✱ ℘(v) =

1 12(4µ2 +

µ2

1)✳

❉❡✜♥❡ t❤❡ ❞❡❢♦r♠❡❞ ❇❛❦❡r✲❆❦❤✐❡③❡r ❢✉♥❝t✐♦♥ ❛s t❤❡ s♦❧✉t✐♦♥ ♦❢ Ψ′(t) + φ(t)Ψ(t) = 0 s✉❝❤ t❤❛t Ψ(t) = Ψ(t; v, w) = 1/u + (r❡❣✉❧❛r ❢✉♥❝t✐♦♥)✳ ▲❡♠♠❛✳ ❋♦r α = ℘′(w)

℘′(v)

Ψ(t) = σ(t + v)

1 2(1−α)σ(v − t) 1 2(1+α)

σ(t)σ(v) exp

• (−µ1

2 + αζ(v))t

• .

✷✷

❚❤❡ ❛❞❞✐t✐♦♥ ❢♦r♠✉❧❛✳ ❚❤❡♦r❡♠✳ Ψ(t + q) = =

• Ψ(t)

Ψ(q) Ψ′(t) Ψ′(q)

• ℘(t) − ℘(q) ×
• 1

1 1 ℘(t) ℘(q) ℘(v) ℘′(t) ℘′(q) ℘′(v)

• 1−α

2

• 1

1 1 ℘(t) ℘(q) ℘(−v) ℘′(t) ℘′(q) ℘′(−v)

• 1+α

2

1−α 2

• 1

1 1 ℘(t) ℘(q) ℘(v) ℘′(t) ℘′(q) ℘′(v)

• + 1+α

2

• 1

1 1 ℘(t) ℘(q) ℘(−v) ℘′(t) ℘′(q) ℘′(−v)

• .

✷✸

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❞❡❢♦r♠❡❞ ❇❛❦❡r✲❆❦❤✐❡③❡r ❢✉♥❝t✐♦♥ ❢♦r µ1 = 0✳ ✶✳ ■♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t = 0 ✇❡ ❤❛✈❡ Ψ(t; v, w) = 1 t − 1 2℘(v)t + 1 6℘′(w)t2 + (t3). ✷✳ ❚❤❡ ❢✉♥❝t✐♦♥ Ψ(t; ±v, w) ❣✐✈❡s ❛ s♦❧✉t✐♦♥ t♦ t❤❡ ❞❡❢♦r♠❡❞ ▲❛♠❡ ❡q✉❛t✐♦♥ Ψ′′(t) − UΨ(t) = ℘(v)Ψ(t), ✇❤❡r❡ U = 2℘(t) − ℘′(v)2 − ℘′(w)2 4(℘(t) − ℘(v))2.

✷✹

✸✳ ❚❤❡ ♣❡r✐♦❞✐❝ ♣r♦♣❡rt✐❡s ❛r❡ Ψ(t + 2ωk; v, w) = Ψ(t; v, w) exp(2α(ζ(v)ωk − ηkv)); Ψ(t; v + 2ωk, w) = Ψ(t; v, w). ❚❤❡ ❢✉♥❝t✐♦♥ Ψ(t; ωk, w) = Ψ(t; ωk) ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ w ❛♥❞ Ψ(t + 2ωk; ωk) = Ψ(t; ωk). ✹✳ Ψ(t; v, ωk) =

• ℘(t) − ℘(v)✳

✺✳ ❲❡ ❤❛✈❡ t❤❡ r❡❧❛t✐♦♥s Ψ(t; v, w) = Ψ(t; −v, −w) = −Ψ(−t; v, −w).

✷✺

▲❡t L+

1 = L1 = ∂ + φ(t) ❛♥❞ L− 1 = ∂ − φ(t) ✇❤❡r❡

φ(t) = −1 2 ℘′(t) + ℘′(w) ℘(t) − ℘(v) . ❲❡ ❤❛✈❡ L2Ψ(t) = 0✱ ✇❤❡r❡ L2 = ∂2 − U − ℘(v) = L−

1 L+ 1 ❛♥❞

U = 2℘(t) − ℘′(v)2 − ℘′(w)2 4(℘(t) − ℘(v))2. ❙❡t V = (1−α2)

16

℘′(v)2T ✱ ✇❤❡r❡ T = (3℘′(t)+℘′(w))

(℘(t)−℘(v))3

è (1 − α2)℘′(v)2 = ℘′(v)2 − ℘′(w)2✳ ❚❤❡ ❛❞❞✐t✐♦♥ ❢♦r♠✉❧❛ ❣✐✈❡s t❤❡ ♦♣❡r❛t♦r L3 = 2∂3 − 3U∂ − U0✱ ãäå U0 = 3

2U′ + 2V − ℘′(w)✱ s✉❝❤ t❤❛t L3Ψ(t) = 0✳ ❲❡ ❤❛✈❡

[L2, L3] = −1 4(1 − α2)℘′(v)2

∂

∂tT

• L1.

✷✻

❚❤❡ ❍✐r③❡❜r✉❝❤ ❣❡♥❡r❛✳ ▲❡t A = A−2k ❜❡ ❛ ❝♦♠♠✉t❛t✐✈❡ ❛ss♦❝✐❛t✐✈❡ ❣r❛❞❡❞ t♦rs✐♦♥✲ ❢r❡❡ r✐♥❣ ❛♥❞ ❧❡t f(u) = u +

k≥1 fkuk+1✱ ✇❤❡r❡ fk ∈ A−2k ⊗ Q✳

❙❡t Lf(σ1, ..., σk, ...) =

N

• i=1

ui f(ui), ✇❤❡r❡ σk ✐s t❤❡ k✲t❤ ❡❧❡♠❡♥t❛r② s②♠♠❡tr✐❝ ♣♦❧②♥♦♠✐❛❧ ♦❢ u1, ..., uN✳ ❲❡ ❤❛✈❡ Lf(σ1, ..., σn) = 1 − f1σ1 + (f2

1 − f2)σ2 1 + (2f2 − f2 1)σ2 + . . .

❚❤❡ ❍✐r③❡❜r✉❝❤ ❣❡♥✉s Lf ♦❢ ❛ st❛❜❧② ❝♦♠♣❧❡① ♠❛♥✐❢♦❧❞ M2n ✇✐t❤ t❛♥❣❡♥t ❈❤❡r♥ ❝❧❛ss❡s ci = ci(τ(M2n)) ❛♥❞ ❢✉♥❞❛♠❡♥t❛❧ ❝②❝❧❡ M2n ✐s ❞❡✜♥❡❞ ❜② t❤❡ ❢♦r♠✉❧❛ Lf(M2n) = (Lf(c1, ..., cn), M2n) ∈ A−2n ⊗ Q.

✷✼

❚❤❡♦r❡♠✳ ❚❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❛❧❣❡❜r❛✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠ ♦♥ C4 ✇✐t❤ ❝♦♦r❞✐♥❛t❡s (z1, z2, x, y) ✭s❡❡ ❜❡❧♦✇✮ ✐s (−(ln Φ(t+t0; τ +τ0))′, −(ln Φ(t+t0; τ +τ0))′′, ℘(t+t0), ℘′(t+t0)). ∂tz1 = z2, ∂τz1 = 3x + 2z2 − z2

1,

∂tz2 = 2z1z2 − 2y, ∂τz2 = −y + 2z1z2, ∂tx = y, ∂τx = 0, ∂ty = −2z2

2 + 2z2 1z2 − 2z1y − 6xz2,

∂τy = 0. ∂0z1 = z1, ∂2z1 = z3

1 − z2z1 − 4xz1 − 3

2y ∂0z2 = 2z2, ∂2z2 = 4xz2 + z1z2 − 2z2

1z2 + 2z2 2

∂0x = 2x, ∂2x = −2x2 − 4 3z2

2 + 4

3z2

1z2 − 4

3z1y − 4xz2 ∂0y = 3y, ∂2y = 3xy

✷✽