LECTURE TWO

Theorem (Periods integrals and type II solutions)

Consider the mean field equation4u+e^u=ρδ₀on E=C/Λ.

I If solutions exist forρ=8nπ, then there is a unique even solution within each type II scaling family. (`=2n, a_n+i=−^ai.)

I The n-th equation is given byR

L_ig∈√

−1R, which is equivalent to

∑

n i=1

∇^G(ai) =0.

I The n−1 algebraic equations:

Theorem (Green/polynomial system)

Forρ=8nπ, n∈N, the n equations for a={^a1, . . . , an}are precisely

℘⁰(a1)℘^r(a1) +· · · + ℘⁰(an)℘^r(an) =0, where r=0, . . . , n−^{2, and}∇^G(a1) +· · · + ∇^G(an) =0.

Theorem (Hyperelliptic geometry/Lam´e curve)

For xi := ℘(ai), yi:= ℘⁰(ai), the first n−1 algebraic equations

∑

^{yixri =0, r=}0, . . . , n−2,

defines an affine hyperelliptic curve under the 2 to 1 map a7→∑℘(ai): Xn:={(^xi, y_i)} ⊂^SymnE−→ (^x1+· · · +^xn)∈P¹.

I The proof relies on its relation to Lam´e equations:

I This is a long calculation via the polynomial system (omitted).

I Idea of proof on the hyperelliptic structure on Xn. I Consider y²=p(x) =4x³−^g2x−^g3, where

(x, y) = (℘(z),℘⁰(z)), and we set(x_i, y_i) = (℘(a_i),℘⁰(a_i)). Consider a basis of solutions to the Lam´e equation

w⁰⁰= (n(n+1)℘(z) +B)w (for some B) given by wa(z)and w_−a(z).

I Let X(z) =wa(z)w_−a(z). By the addition theorem, X(z) = (−1)ⁿ

∏

n i=1

σ(z+a_i)σ(z−^ai)

σ(z)²σ(a_i)² = (−1)ⁿ

∏

n i=1

(℘(z)− ℘(ai)).

That is, X(x) = (−1)ⁿ∏ⁿ_i=1(x−xi)is a polynomial in x.

I Key: X(z)satisfies the second symmetric power of the Lam´e equation:

d³X

dz³ −4(n(n+1)℘ +B)^dX

dz −2n(n+1)℘⁰X=0.

I Hence X(x)is a polynomial solution, in variable x, to

p(x)X⁰⁰⁰+³₂p⁰(x)X⁰⁰−4((n²+n−3)x+B)X⁰−2n(n+1)X=0.

I X is determined by B and certain initial conditions.

I Write X(x) = (−¹)ⁿ(xⁿ−^s1xⁿ⁻¹+· · · + (−¹)ⁿsn), this translates to a linear recursive relation for µ=0,· · ·^{, n}−^1:

0=2(n−^µ)(2µ+1)(n+µ+1)sn−µ

−⁴(µ+1)Bs_n−µ−1

+¹₂g2(µ+1)(µ+2)(2µ+3)sn−µ−2

−^g3(µ+1)(µ+2)(µ+3)s_n−µ−3. I We set s0=1.

I For µ=n−^{1 we get B}= (2n−¹)s₁as expected.

I Thus all s₂,· · ·^{, s}n, X(z), are determined by s₁, i.e. by B, alone.

I In fact, a slightly more work shows that the set a={^ai}^{is also} determined by B up to sign. Hence a7→^Bais 2 to 1. QED

Theorem (Chai-Lin-W 2012)

I There is a natural projective compactification ¯Xn ⊂SymⁿE as a, possibly singular, hyperelliptic curve defined by

C²= `n(B, g₂, g₃) =4Bs²_n+4g₃sn−2sn−^g2s_n−1sn−^g3s²_n−1, in affine coordinates(B, C), where

s_k=s_k(B, g2, g3) =r_kB^k+· · · ∈^Q[B, g2, g3]

is an universal polynomial of homogeneous degree k with deg g2=2, deg g3=3, and B= (2n−1)s1.

I Thus deg`n=2n+1 and ¯Xnhas arithmetic genus g=n.

I The curve ¯Xnis smooth except for a finite number ofτ, namely the discriminant loci of`<sub>n</sub>(B, g<sub>2</sub>, g3), so that`_n(B)has multiple roots. In particular ¯Xnis smooth for rectangular tori.

(Continued.)

I The 2n+2 branch points a∈X^¯n\Xnare characterized by−a=a.

I {−ai} ∩ {ai} 6=^∅⇒ −a=a.

I Also 0∈ {^ai} ⇒^a= (0, 0,· · ·, 0).

I By setting(x_i, y_i) = (t³_i, 2t²_i), the limiting system at a=0ⁿ:

∑

ⁿi=1t^2r+1_i =0, r=1, . . . , n−^1,

has a unique solution with ti6=0 and ti6= −^tjinPⁿ⁻¹up to permutations.

I Meaning of C (I):Applying Cramer’s rule to the n−^{1 linear} equations∑ⁿ_i=1x^k_iyi=0 in yi’s, there is a constant C∈C^×such that

y_i= ^C

∏_j6=i(x_i−^xj)^{, i}=1, . . . , n.

I Meaning of C (II):Let w₁, w₂be two ind. solutions of w⁰⁰=Iw.

I From

I The constant terms give the hyperelliptic equation in(B, C). I In particular, C=0 if and only if wa=w_−a, i.e. a=−^{a. These}

are the branch points of ¯Xn.

I Definition:Denote by Yn=X^¯n\{⁰ⁿ}the affine hyperelliptic curve defined by

C²= `n(B, g2, g3).

I Now we study the last equation on ¯Xn:

I Consider the rational function on Eⁿ:

zn(a₁, . . . , an):=ζ(a₁+· · · +^an)−

∑

n i=1

ζ(a_i). (It is periodic in each variable.)

I Let ai=tiω1+siω2, then

I It is thus crucial to study the branched covering map

σ : ¯Xn →^{E, a}7→^σ(a):=

∑

n i=1

a_i.

Theorem (Lin–W 2013, new pre-modular functions)

(1) The mapσ has degree equals ¹₂n(n+1).

(2) There is a universal (weighted homogeneous) polynomial Wn(x)∈C[g2, g3,℘(σ),℘⁰(σ)][x]of degree ¹₂n(n+1)such that

Wn(zn) =0.

In fact, zn∈K(X^¯n)is a primitive generator for the field extension K(X^¯n)over K(E).

(3) The function Zn(σ; τ):=Wn(Z)is pre-modular of weight

1

2n(n+1). That is, it is modular wrt.Γ(N)ifσ∈^Eτ[N].

I Idea of proof for (1):Apply Theorem of the Cube: For any three morphisms f , g, h : Vn−→^{E and L}∈^{Pic E,}

(f+g+h)^∗L∼= (f+g)^∗L⊗ (^g+h)^∗L⊗ (^h+f)^∗L

⊗f^∗L⁻¹⊗g^∗L⁻¹⊗h^∗L⁻¹.

I Apply to the case Vn⊂Eⁿwhich is the ordered n-tuples so that Vn/Sn=X^¯n, and deg L=1. We prove inductively that the map

fk(a):=a1+· · · +ak

has degree ¹₂k(k+1)n!. It is not hard to check for k=1, 2.

I From k to k+1, we let f =f_k−1, g(a) =a_k, and h(a) =a_k+1. I Then fk+1has degree n! times

1

2k(k+1) +3+¹₂k(k+1)−¹₂(k−¹)k−¹−¹= ¹₂(k+1)(k+2).

I Idea of proof of (2):Major tool: tensor product of two Lam´e equations w⁰⁰=I1w and w⁰=I2w, where I=n(n+1)℘(z), I1=I+Baand I2=I+Bb.

I For ¯Xn(τ)smooth, and a general point σ0∈E, we need to show that the ¹₂n(n+1)points on the fiber of ¯Xn→^{E above σ}0has distinct znvalues. It is enough to show that for σ(a) =σ(b) =σ0, the condition∑ ζ(ai) =∑ ζ(bi)implies Ba=B_b(and then a=b).

I If w⁰⁰₁ =I1w1and w⁰⁰₂ =I2w2, then the product q=w1w2satisfies q⁰⁰⁰⁰−²(I₁+I₂)q⁰⁰−^6I0q0+ ((Ba−^Bb)²−^2I00)q=0.

I If a6=b, by addition law we find that Q=waw_−b+w_−awbis an even elliptic function solution, namely a polynomial in x= ℘(z). This leads to strong constraints on the corresponding 4-th order ODE in variable x, and eventually leads to a contradiction for generic choices of σ₀.

Indeed,

As an even elliptic function, Q takes the form

Q(x) =C The xⁿ⁺²terms agree automatically. The xⁿ⁺¹degree gives

∑

^{℘(ci) =s1= 1}2

Ba+B_b

2n−¹ = ¹₂(

∑

^{℘(ai) +}

∑

^℘(bi)).

I Inductively the xⁿ⁺²⁻ⁱcoefficient in (4) gives recursive relations to solve siinterns of Ba+B_b,(Ba−^Bb)²and g2, g3for i=1, . . . , n.

I Indeed

s_i=s_i(Ba+B_b,(Ba−^Bb)², g₂, g₃) =C_i(Ba+B_b)ⁱ+· · · is homogeneous of degree i if we assign deg Ba=deg B_b=1 and deg g2=2, deg g3=3.

I There are two remaining consistency equations F1=0, F0=0 coming from the x¹and x⁰coefficients in (4).

I In fact(Ba−B_b)²is a factor of both equations and we may write F₁(Ba, B_b) = (Ba−^Bb)^2d1G₁(Ba, B_b)and

F0(Ba, B_b) = (Ba−^Bb)^2d0G0(Ba, B_b). I If Ba6=^Bb(i.e∑℘(a_i)6=∑℘(b_i)), then

G1(Ba, B_b) =0, G0(Ba, B_b) =0,

which has only a finite number of solutions(Ba, B_b)’s, i.e. Eτ’s.

Example (of compatibility equations for n = 2)

For n=2 we have s₁= ¹₆(Ba+B_b)and

s2= ₃₆¹(Ba+B_b)²+₇₂¹(Ba−^Bb)²−¹₄^g2.

The first compatibility equation from x¹is (after substituting s₁)

1

6(Ba−B_b)²(Ba+B_b) =0.

The second compatibility equation from x⁰is

(Ba−^Bb)²(₃₆¹(Ba+B_b)²+₇₂¹(Ba−^Bb)²−¹₆^g2) =0.

If Ba 6=^Bbthen B_b=−^Baand then we can solve Ba, B_b: B²_a=3g2=⇒ ℘(^a1) + ℘(a2) =±^pg2/3.

Such a∈^X¯2indeed lies at the branch loci of the Lam´e curve.

Example (of new pre-modular forms for n = 2)

For z2(a1, a2) =ζ(a1+a2)−^ζ(a1)−^ζ(a2), on X2: z³₂(a)−³℘(a1+a2)z2(a)− ℘⁰(a1+a2) =0.

On E²it has one more term−¹₂(℘⁰(a₁) + ℘⁰(a₂)). Thus, Z2(σ; τ) =W2(Z) =Z³−³℘(σ)Z− ℘⁰(σ).

Example (n = 3)

For z=z3(a) =ζ(a1+a2+a3)−^ζ(a1)−^ζ(a2)−^ζ(a3), on X3: z⁶−¹⁵℘_z⁴−²⁰℘⁰_z³+ (²⁷₄g₂−⁴⁵℘²)z²−¹²℘⁰℘_z−⁵₄℘⁰² =0.

Thus, Z₃(σ; τ) =W₃(Z).

I Key point: Z1≡^Z=−4π∇G is the Hecke modular function.

The critical point equation (⇐⇒type II solutions of MFE) is transformed into zero of pre-modular forms.

I For general n≥1, we have the equivalences:

• Solution u to MFE for ρ=8πn.

• Periods integral Z

L_jg∈√

−1R (=ω_jcoordinates of∑ ai.)

• Green equation

∑

n i=1

∇^G(a_i) =0 on Xn.

• zn(a) =Z(σ(a)).

• Non-trivial zero of Zn(σ; τ):=Wn(Z).

I Need to prove the last one. Notice that the branch point a∈^Yn\^Xn(a6= −a) satisfies the Green equation trivially.

I The second technique used in ρ=8π is to use the method of continuity to connect to the known case ρ=4π by establishing the non-degeneracy of linearized equations.

I For general ρ, such a non-degeneracy statement is out of reach.

However, since solutions uηalways exist for ρ=8πη, η 6∈N, it is natural to study the limiting behavior of uηas η→^{n. If the} limit does not blow up, it converges to a solution u for ρ=8πn.

I For the blow-up case, we have the connection between the blow-up set and the hyperelliptic geometry of Yn→P¹: I

Theorem

Suppose that S={^p1,· · ·^{, p}n}is the blow-up set of a sequence of solutions u_kto withρ_k→^{8πn as k}→^{∞, then S}∈^Yn. Moreover,

(1) Ifρ_k6=8πn then S is a branch point (a=−a) of Yn. (2) Ifρ_k=8πn for all k, then S is not a branch point of Yn.

I To go deeper, need to know the converse statement: for which p∈^Yn\^Xncan we construct a blow-up sequence with blow-up set p? The Morse type of p is fundamental.

Theorem

Suppose that the pair of non half-period critical points{±^p}of G exists, the

±p are the minimal points of G.

I In fact our proof shows that any solution for ρ=8π must be a minimizer of the non-linear functional

J8π(u) =¹ 2 Z

E|∇^u|²−^{8π log} Z

Ee^−8πG+u on u∈H¹(E)∩ {u|R

Eu=0}.

Corollary

Forτ∈^Ω5, all the three half periods are (non-degenerate) saddle points.

I If ukis a blow-up sequence with ρ=ρk→8π (as k→^∞), ρ_k6=8π for large k, then the blow-up point q is a half period.

I Asymptotically

ρ_k−8π= (D(q) +o(1))e^−λk (5) where λ_k=max_Eτu_kand

D(q):= Z

E_τ

h(z)e^{8π( ˜G(z,q)−φ(q))}−h(q)

|^z−^q|⁴ − Z

E^c_τ

h(q)

|^z−^q|^4. Here h(z) =e^−8πG(z), ˜G(z, q)is the regular part of the Green function, and φ(q) =G^˜(q, q).

I The sign of D(q)determines the direction where the bubbling may take place, namely ρ_k <8π or ρ_k>8π.

Theorem (Lin–W)

For any half period q∈^Eτ,τ=a+bi, we have

D(q) =−4π²be^−8πG(q)det D²G(q). (6)

I Hence D(q) >0 if q is a saddle point. In particular if τ∈^Ω5

then D(q) >0 for all half-periods since they are all saddle.

I Since the extra critical point p (reps.−p) is a discrete minimal point, the index of∇G at p (reps.−p) is 1. By the Hopf–Poincar´e index theorem,

−1=χ(Eτ\{0}) =2+

∑

3 i=1

ind1 2ω_i∇^G.

Since¹₂ω_iis non-degenerate,∇G has index±1 at it. Hence the index must be−1 for all i=1, 2, 3. This implies that¹₂ω_iis a saddle point for all i.

I Combining with a recent technique in analyzing uniqueness of blow-up solutions by Lin–Yan, we may classify all solutions to the mean field equation for ρ∈ (^{0, 8π}+e₀)for some e₀>0:

Theorem (Lin–W)

(i) Ifτ∈^Ω3then the MFE has only one solution forρ<8π, no solution forρ=8π, and two solutions for 8π<ρ<8π+e0. (ii) Ifτ∈^Ω5then the MFE has only one solution forρ<8π,

infinitely many solutions forρ=8π, and four solutions for 8π<ρ<8π+e0.

I MFE with ρ=12π has exactly two solutions on Eτfor τ6=^eπi/3. Hence when τ∈^Ω5the bifurcation diagram is complicate for ρ∈ (^{8π, 12π}). It is a natural question whether MFE has exactly two solutions for ρ∈ (^{8π, 16π})when τ∈^Ω3.

I The Theorem also reflects the difficulty in the study the corresponding Lam´e equation for the case η6∈ ¹₂N.

I The hyperelliptic curve Ynis parametrized by(B, C)with

I As in the case n=1, these two invariants are related to a geometric quantity D(p)derived from the blow-up analysis of solutions u_kwith ρ_k→^{8πn. Let p}= (p₁,· · ·^{, p}n)with

{^p1,· · ·^{, p}n}being the blow-up set of u_k. Then ρ_k−^8πn= (D(p) +o(1))e^−λk, λ_k:=max

E u_k. I The analytic expression of D(p)is rather complicate. However,

its geometric meaning is reflected in the following

Theorem

Moreover, cp>0 except for a finite set of tori.

(This has been verified for n=1, 2.)

在文檔中 Mean Field Equations, Hyperelliptic Curves and Modular Forms I, II (頁 32-58)