Evenness of solutions for ρ = 8nπ

Theorem 5.2. There is a unique even solution within each normalized type II family of solutions of the singular Liouville equationu+e^u= 8nπ δ0 on a torusE,n, wherenis a positive integer. In other words for any normalized type II developing map f of a solution u of the above equation, there exists a unique λ∈R such that the solution

u_λ(z) = log e^2λ|f(z)|² (1 +e^2λ|f(z)|²)² of the same equation satisfies uλ(−z) =uλ(z) ∀z∈C.

Proof. Let f be a normalized type II developing map of a solution u of (0.1.3). It is enough to show that there exists a unique λ ∈ R such that f_λ(z) :=e^λ·f(z) satisfiesf_λ(−z) =c/f_λ(z) for a constant cwith|c|= 1.

Letg:=f/f, the logarithmic derivative off; it is a meromorphic func-tion on E=C/Λ becausef is normalized of type II. It suffice to show that g is even, for then

f(−z) =f(0) exp

−z 0

g(w)dw= f(0)² f(z), and the unique solution of λis given by λ=−log|f(0)|.

We know from Lemma 1.3.7that f is a local unit at points of Λ and g is a meromorphic function on E which has a zero of order 2nat 0 ∈E, no other zeros and 2nsimple poles on E. Moreover the residue of gis equal to 1 atnof the simple poles ofg, and equal to−1 at the othernsimple poles.

Denote byP₁, . . . , P_nthensimples poles ofgwith residue 1 onE=C/Λ, corresponding to zeros of the developing map f, and letQ1, . . . , Qn be the n-simple poles of g with residue −1, corresponding to simple poles of f. Let p₁, . . . , p_n ∈ C be representatives of P₁, . . . , P_n ∈ C/Λ; similarly let q1, . . . , qn ∈ C be representatives of Q1, . . . , Qn ∈ C/Λ. The condition on the poles of g allows us to expressg in terms of the Weierstrassζ-function:

(5.2.1) g(z) = n

i=1

ζ(z−p_i)− n

i=1

ζ(z−q_i) +

i=1

ζ(p_i)−

i=1

ζ(q_i) for a unique constant c, because g(z)−_n

i=1ζ(z −p_i) +_n

i=1ζ(z −q_i) is a meromorphic holomorphic function on E. Of course the constant c is

completely determined by the elementsp1, . . . , pn;q1, . . . , qn∈C:

c=

i=1

ζ(p_i)−

i=1

ζ(q_i).

It remains to analyze the condition thatg(z) has a zero of order 2nat 0∈E, i.e.

(5.2.2) 0 =−g^(r)(0) = −d^rg dz^r

z=0

= n

i=1

℘^(r−1)(p_i)− n

i=1

℘^(r−1)(q_i)

forr= 1, . . . ,2n−1 because

−g^(r)(z) = n

i=1

℘^(r−1)(z−p_i)− n

i=1

℘^(r−1)(z−q_i).

Only the conditions that g^(2s+1)(0) = 0 for s= 0,1, . . . , n−1 will be used for the proof of Theorem5.2. The vanishing of the even-order derivatives of g will be explored in the proof of Theorem 5.6.

By Lemma 5.4 below, relations (5.2.2) for r = 1,3,5, . . . ,2n−1 imply that the sets {℘(p₁), . . . , ℘(p_n)} and {℘(q₁), . . . , ℘(q_n)} are equal as sets with multiplicities. Because P₁, . . . , P_n;Q₁, . . . , Q_n are 2n distinct points on E, it follows that ℘(pi) = ℘(pj) whenever i = j and {Q1, . . . , Qn} = {−P₁, . . . ,−P_n}as subsets ofE{0}withnelements. From the expression (5.2.1) of g and the fact that ζ(z) is an even function on C one sees that g(z) is even. Theorem5.2is proved modulo the elementary Lemma5.4.

We record the following statements from the proof of Theorem5.2.

Corollary 5.3. Let f be a normalize type II developing map of a solution u of the equationu+e^u = 8nπ δ₀ for a positive integern. Then the zeros p_i’s of f modulo Λ correspond ton elements P₁, . . . , P_n∈E{0} and the poles qi’s of f modulo Λ correspond to n elements Q1, . . . , Qn ∈ E{0}. Moreover the following statements hold.

(a) {Q1, . . . , Qn}={−P1, . . . ,−Pn} as subsets ofE{0}withnelements.

(b) ℘(p_i) = 0, or equivalently P_i is not a 2-torsion point of E, for i = 1, . . . , n.

(c) ℘(pi)=℘(pj) for anyi, j= 1, . . . , n such that i=j.

Lemma 5.4. (a) For each positive integer j, there exists a polynomial hj(X)∈C[X] of degree j+ 1such that

℘^(2j)(z) :=

d dz

2j

℘(z) =hj(℘(z)) as meromorphic functions on C.

(b) For every symmetric polynomial P(X1, . . . , Xn) ∈C[X1, . . . , Xn], there exists a polynomial Q(W1, . . . , Wn)∈C[W1, . . . , Wn]such that

P(℘(z1), . . . , ℘(zn)) =Q ⁿ

i=1

℘(zi), n

i=1

℘⁽²⁾(zi), n

i=1

℘⁽⁴⁾(zi), . . . , n

i=1

℘⁽²ⁿ⁻²⁾(zi)

as meromorphic functions on Cⁿ.

Proof. Taking the derivative of the Weierstrass equation

℘(z)²= 4℘(z)³−g2℘(z)−g3, and divide both sides by 2℘(z), we get

℘⁽²⁾(z) = 6℘(z)²−1 2g2.

An easy induction shows that℘^(2j)(z) is equal to hj(℘(z)) for a polynomial h_j(X) ∈ C[X] of degree j+ 1 and the coefficient a_j of X^j+1 is positive.

In fact one sees that a_j = (2j+ 1)! when one compares the coefficient of z^−2j−2 in the Laurent series expansion of ℘^(2j)(z) and hj(℘(z)). We have proved the statement (a).

The statement (b) follows from the fact that every symmetric polynomial inC[X1, . . . , Xn] is a polynomial of the Newton polynomialsp1(X1, . . . , Xn), . . .,p_n(X₁, . . . , X_n), where p_j(X₁, . . . , X_n) :=_n

i=1X_i^j forj= 1, . . . , n.

5.5. Green/algebraic system for ρ = 8nπ. Let u be a type II even solution of the singular Liouville equation u +e^u = 8nπ δ0 on a torus E = C/Λ, where n is a positive integer. As before let f be a normalized developing map of u and let g = (logf) = f/f. Let p₁, . . . , p_n ∈ C be the simple zeros of f modulo Λ and let −p1, . . . ,−pn be the simple poles of f modulo Λ as in Corollary5.3. Let P₁ =p₁mod Λ, . . . , P_n=p_nmod Λ ; they are exactly the blow-up points of the scaling family u_λ of solutions of u+e^u = 8nπ δ0.

5.5.1. Two approaches to the configuration of blow-up points. We will investigate the constraints on the configuration of the pointsP₁, . . . , P_n with two approaches outlined below. The plans are executed in§5.6and§5.7 respectively.

A.In the first approach we have the relations (5.2.2) forr = 2,4, . . . ,2n−2 and also the condition that the period integrals of the meromorphic differ-ential g(z)dz on E are purely imaginary; the latter comes directly from the assumption that the image of the monodromy of the type II developing map lies in the diagonal maximal torus of PSU(2). The first n−1 con-ditions translate into a system of polynomial equations in the coordinates

℘(p1), . . . , ℘(pn), ℘(p1), . . . , ℘(pn) of the npointsP1, . . . , Pn:

(5.5.1) ℘(p₁)℘^r(p₁) +· · ·+℘(p_n)℘^r(p_n) = 0, ∀r= 0, . . . , n−2, while the monodromy constraint becomes

(5.5.2)

n i=1

∂G

∂z(pi) = 0 whereGis the Green’s function on E as in (0.1.1).

The method used in this approach also shows that thenequations (5.5.1) and (5.5.2) are also sufficient: if P1 = p1mod Λ, . . . , Pn = pnmod Λ are n elements inE with distinctx-coordinates℘₁(p₁), . . . , ℘(p_n) satisfying equa-tions (5.5.1) and (5.5.2) and none of P₁, . . . , P_n is a 2-torsion point of E, then there exists a normalized type II developing map f for a solution of u+e^u = 8nπ δ₀ such thatp₁, . . . , p_nis a set of representatives of the zeros off modulo Λ.

B. In the second approach, results on blow-up solutions of a mean field equation on a Riemann surface provides the following constraints

(5.5.3) n∂G

∂z(p_i) =

1≤j≤n, j=i

∂G

∂z(p_i−p_j), for i= 1,2, . . . , n, on the blow-up pointsP1, . . . , Pn.

5.5.2. We will see in5.7that the system of equations (5.5.3) is equivalent to the combination of (5.5.2) and the following system of equations

(5.5.4)

j=i

℘(pi) +℘(pj)

℘(p_i)−℘(p_j) = 0, fori= 1, . . . , n.

Moreover for elements (P1, . . . , Pn)∈(EE[2])ⁿ withdistinct x-coordinates

℘(p1), . . . , ℘(pn), the two systems of equations (5.5.1) and (5.5.4) are equiv-alent.²³ This means that among elements of the subset Bl_n⊂Eⁿconsisting of all n-tuples (P1, . . . , Pn)∈(E{0})ⁿsatisfying the constraints (5.5.2) for blow-up points, those satisfying the non-degeneracy condition

(5.5.5) ℘(p_i)= 0 and ℘(p_i)=℘(p_j) wheneveri=j ∀1≤i, j≤n are indeed blow-up points of the scaling family uλ(z) of a type II solution of the singular Liouville equation u+e^u= 8nπ δ₀ on E.

We recall some properties about period integrals and Green’s functions in lemmas 5.5.3–5.5.4below, before returning to the first approach outlined in 5.5.1.

Lemma 5.5.3. For any y ∈C and anyω ∈Λ = H₁(E;Z), the ω-period of the meromorphic differential _℘(z)^℘(y)_−℘(y)^dz onE =C/Λ is given by

(5.5.6)

Lω

℘(y)

℘(z)−℘(y)dz≡2ω·ζ(y)−2η(ω)·y (mod 2π√

−1Z).

Here Lω : [0,1]→ C is any piecewise smooth path on C such that Lω(1)− L_ω(0) =ω and ℘(z)=℘(y) for allz∈L_ω([0,1]).

Proof. This is a reformulation of [44, Lemma 2.4]. Note that meromorphic differential _℘(z)^℘(y)_−℘(y)^dz on E has poles at 0 and±y (mod Λ), with residues 0 and ±1 respectively, therefore the period integral Iω(y) :=

Lω

℘(y)dz

℘(z)−℘(y) is well-defined modulo 2π√

−1Z. The addition formula for℘(z) gives

℘(y)dz

℘(z)−℘(y) = ℘(z)dz

℘(z)−℘(y) −2ζ(z+y; Λ)dz+ 2ζ(z; Λ)dz+ 2ζ(y; Λ)dz The lemma follows after an easy calculation, using the functional equation for ζ(z; Λ) and the fact that _dz^d logσ(z) =ζ(z) and; see [44, Lemma 2.4] for details.

Alternatively, one computes d

dyI_ω(y) =

Lω

(2℘(z+y)−2℘(y))dz =−2℘(y) + 2η(ω)

23For everym∈N,E[m] denote the subgroup ofm-torsion points onE.

and determine the constant of integration up to 2π√

−1Z by evaluation at 2-torsion points ofE.

Lemma 5.5.4. Let G be the Green’s function on E = C/Λ as in (0.1.1).

The following formulas hold.

(5.5.7) G(z) =− 1

2π logΔ(Λ)¹²¹ ·e^{−z η(z;Λ)/2}·σ(z; Λ) on E,

(5.5.8) −4π ∂G

∂z(z) =ζ(z; Λ)−η(z; Λ) ∀z∈C.

In the first formula (5.5.7),η(z; Λ)is the quasi-period andΔ(Λ)is the non-zero cusp form of weight 12 for SL₂(Z) given by the formula

Δ(Λ) =g2(Λ)³−27g3(Λ)² = (2π√

−1)¹² (ω2)¹² ·qτ ·

∞ m=1

(1−q_τⁿ)²⁴, where q_τ =e^2π√−1τ, τ =ω₂/ω₁ with Im(τ)>0.

Remark 5.5.5. (a) An equivalent form of5.5.4(a) is G(z; Λτ) =− 1

2π log e⁻

πIm(z)2

Im(τ) ·Δ(C/Λτ)⁻¹²¹ ·θ[^1/2_1/2](z;τ)

for the Green’s functionG(z; Λτ) on the elliptic curve C/Λτ, where τ is an element of the upper-half plane and Λτ =Z+Z·τ)). Here we have used the general notation for theta functions with characteristics

θ[^a_b](z;τ) :=

m∈Z

e^{π√−1τ(m+a)2}·e^2π√−1 (m+a)(z+b).

The equivalence of the two formulas follows from the formulas σ(z,Λτ) =−1

πe^η(1,Λτ) · θ[^1/2_1/2](z;τ)

θ[⁰₀](z;τ)θ[_1/2⁰ ](z;τ)θ[^1/2₀ ](z;τ) and

θ[⁰₀](z;τ)θ[_1/2⁰ ](z;τ)θ[^1/2₀ ](z;τ) = 2η_Dedekind(τ)³ = 2

q

1

τ24

∞ m=1

(1−q_τ^m) ₃

.

(b) The function Z(z; Λ) := ζ(z; Λ)−η(z; Λ) = −4π^∂G_∂z appeared in [28, p. 452]; we will call it theHecke form. For any integers a, band N ≥1 such

that gcd(a, b, N) = 1 and (a, b) ≡(0,0) (modN), Z(_N^a+_N^bτ;Z+Zτ) is a modular form of weight one and level N, equal to the Eisenstein series

−N·E₁^N(τ, s;a, b)

s=0

=−N·Im(τ)^s·

(m,n)≡(a,b) modN

(mτ+n)⁻¹· |mτ+n|^−2s s=0

.

See [28, p. 475].

Proof of Lemma 5.5.4. The formula (5.5.7) is proved in [40, II,§5]. The equivalent formula in 5.5.5(a) is proved in [23, p. 417–418]. See also [43,

§7] and [44,§2]. The formula (5.5.8) follows from (a) by an easy computa-tion.

Theorem 5.6. Let n be a positive integer. Let P_i=p_imod Λ, i= 1, . . . , n be ndistinct points onE =C/Λsuch that {P1, . . . , Pn}∩{−P1, . . . ,−Pn}=

∅. In other words℘(p₁), . . . , ℘(p_n)are mutually distinctand none of theP_i’s is a2-torsion point ofE. There exists a normalized type II developing mapf for a solutionuof u+e^u= 8πnδ0 onEsuch thatf(p1) =· · ·=f(pn) = 0 if and only if

(5.6.1)

n i=1

∂G

∂z(p_i) = 0 and

(5.6.2) ℘(p₁)℘^r(p₁) +· · ·+℘(p_n)℘^r(p_n) = 0 for r = 0, . . . , n−2.

Notice that (5.6.1) is the same as (5.5.2) and (5.6.2) is the same as (5.5.1).

Proof. We use the notation in the proof of Theorem5.2 and continue with the argument there. The logarithmic derivative g =f/f of a normalized type II developing map has simple poles at the 2npoints±P₁, . . . ,±P_nand is holomorphic elsewhere on E. Moreover the residues of g at Pi (resp. at

−P_i) is 1 (resp.−1) for each i. Therefore (5.6.3) g(z) = ℘(p₁)

℘(z)−℘(p1) +· · ·+ ℘(p_n)

℘(z)−℘(pn).

becauseg(0) = 0. We know two more properties ofg: (a)g(z) has a zero of

To see what property (a) means, we expand g(z) atz= 0 as a power series in℘(z):

Because g has exactly 2nsimple poles and is holomorphic elsewhere on E, we see that orderz=0g(z) = 2n if and only if all n−1 equations in (5.6.2) hold.

We know that η(z; Λ)y−zη(y; Λ)≡0 (mod √

−1R) for all y, z ∈ C because the left-hand side is R-bilinear, and we know from the Legendre relation that the statement holds wheny, z are both in Λ. By Lemma5.5.3,

Lω

for allω∈Λ. Therefore property (b) holds forg given by (5.6.3) if and only (5.6.1) holds. We have proved the “only if” part of Theorem5.6.

Conversely suppose that equations (5.6.2) and (5.6.1) hold. We have seen that the meromorphic functiong(z) given by (5.6.3) has a zero of order 2n atz= 0 and the period integrals of g dz are all purely imaginary. Therefore f(z) = exp_z

0 g(w)dw is a type II developing map for a solution of the singular Liouville equation u+e^u= 8πn δ₀.

Remark 5.6.1. The property (a) that the order of the meromorphic func-tion

℘(p₁)

℘(z)−℘(p1) +· · ·+ ℘(p_n)

℘(z)−℘(pn)

on E atz= 0 is equal to 2n is also equivalent to: ∃C ∈C^× such that (5.6.4)

n j=1

℘(pj)

i=j

(℘(z)−℘(pi)) =C.

Dans le document Mean field equations, hyperelliptic curves and modular forms: I (Page 76-84)