A VIEW ON ELLIPTIC INTEGRALSFROM PRIMITIVE FORMS(PERIOD INTEGRALS OF TYPE A2 , B2 AND G2 )KYOJI SAITO

Elliptic integrals, since Euler’s finding of addition theorem 1751, hasbeen studied extensively from various view points. Present paper gives
a short summery of a view point from primitive integrals of types A2 , B2
and G2 . We solve Jacobi inversion problem for the period maps in thesense explained in the introduction (c.f. [27] Chap.1,13) by introducing certain generalized Eisenstein series of types A2 , B2 and G2 , which
generate the ring of invariant functions on the period domain for the
congruence subgroups Γ1 (N ) (N = 1, 2 and 3). In particular, Eisensteinseries of type B2 includes the case of weight two, and Eisenstein seriesof type G2 includes the cases of weight one and two, which seem to beof new feature. The goal of the paper is a partial answer to the discriminant conjecture, which claims an existence of certain cusp formof weight 1 with character of topological origin, giving a power root of
the discriminant form (Aspects Math., E36,p. 265-320. 2004). See §12Concluding Remarks for more about back grounds of present paper.

Contents
1. Introduction
2. Families of elliptic curves of type A2 , B2 and G2
3. Real elliptic curves and Vanishing cycles
4. Fundamental group and monodromy representation
5. Periods for Primitive forms
6. Jacobian variety
7. Laurent series solutions at infinities
8. Partial fractional expansions
9. Eisenstein series of type A2 , B2 and G2(Primitive automorphic forms)

10. Ring of modular forms and Discriminant
11. Discriminant Conjecture
12. Concluding Remarks
References
1

2
4
7
14
21
25
28
32
-

36
42
47
51
53

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KYOJI SAITO

1. Introduction
We study period integrals for three families of elliptic curves: Weierstrass family, Legendre-Jacobi family and Hesse family. We call thefamilies of type A2 , B2 and G2 , respectively, according to the lattice
structure of vanishing cycles studied in §3. In order to treat these three
types simultaneously, we use notation I2 (p) for the dihedral groups [4]and identify A2 = I2 (3), B2 = I2 (4) and G2 = I2 (6). Then the theory
for the type A2 is nothing but the classical well-known theory of elliptic
integrals for the Weierstrass family of elliptic curves. The purpose of
the present paper is to show that there exist some parallel worlds fortypes B2 and G2 , even in a somewhat deeper manner.
We refer the reader to §12 Concluding Remarks for the motivationand the background of the study. Therefore, in the present introduction, we restrict ourselves to brief explanations of the contents.
As mentioned, we study the three families of elliptic curves of types
A2 , B2 and G2 (2.1). In the first sections §§2-4, we study the geometricand topological aspects of the family. Namely, the geometric family ofaﬃne (i.e. punctured by points at infinity) elliptic curves over the basespace SI2 (p) together with the discriminant loci DI2 (p) is introduced in§2. The number N := [p/2] of points at infinity is called the level.
Then in §3, by the help of a real structure of the family, the lattice ofvanishing cycles in the fiber is described in terms of classical root latticeof type A2 , B2 and G2 . The fundamental group of the complement
SI2 (p) \ DI2 (p) is described in §4 in terms of the Artin group of type
A2 , B2 and G2 . Actually, the image of the monodromy representation
becomes the congruence modular group Γ1 (N ) of level N = 1, 2 and
3 (4.6). In particular, the expression of the modular group leads toa construction of its certain character ϑI2 (p) (4.10) whose k(I2 (p))th
power (4.8) is the sign character: Γ1 (N ) → {±1}. We shall comeback
to the character ϑI2 (p) at the final Theorem 11.1 of the present paper.
The primitive form and associated period map, i.e. the map obtainedby the values of integrals of the primitive form over vanishing cycles
from the monodromy covering space SeI2 (p) of the compliment of thee are introduced in §5. We ask
discriminant to the period domain H,
whether the period map is invertible (in the present paper, we shallcall this question Jacobi inversion problem).
In §§6-9, we study the analytic aspects of the theory. Namely, we fix
a point at infinity and consider the indefinite integral of the primitive
form over open paths on the elliptic curve starting from the point at
infinity. Actually, regarding this integral value z as the time variable,

PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

3

the inverse function of the integral (a meromorphic doubly periodic
function in z) becomes a solution of the Hamilton equation (6.3) of
the motion together with the energy constraint (6.4). Conversely, any
formal meromorphic solutions of the Hamilton equations at a point
infinity are convergent and the set of solutions is in one to one correspondence with the set of points at infinity (which were chosen as theinitial condition for the indefinite integral) (§7 Lemma (7.1)).
This equivalence of geometric solutions and formal solutions is a key
step in the next stage to solve the Jacobi inversion problem. Namely,in the next section §9, by a help of the Hamilton equations (6.3) and(6.4), we can expand those global meromorphic solutions on the fullz-plane into partial fractions by the help of Weierstrass p-functions orezeta-functions, which depends only on the period variable (ω1 , ω0 ) ∈ H.
Then, again by expanding these global meromorphic functions into
Laurent series at the origin z = 0, as coeﬃcients of the expansion,e which we
we obtain an infinite sequence of functions in (ω1 , ω0 ) ∈ H,
call the Eisenstein series of type I2 (p) in §10. Actually, the ring ofe by pulling back by the
Eisenstein series, as functions defined on H,
period map, is identified with ring of the Cartesian coordinate ring of
the parameter space SI2 (p) (Theorem 9.2). Then, those Eisenstein series
e → SI (p) , which finally leads
enable us to construct the inversion map H
2
us to the solution of Jacobi inversion problem in the section §9.
Next in the section §10, we identify the ring of Eisenstein series oftypes A2 , B2 and G2 with the ring of modular forms of the congruencegroup Γ1 (N ), where the ring of modular forms were already determined
explicitly by Aoki-Ibukiyama [2]. In order to determine the homomorphism exactly, we have to determine the exact values of the Eisensteinseries at cusps of the modular group. Actually, Eisenstein series oftype A2 are classical, whose Fourier expansions are well known. Eisenstein series of types B2 and G2 are no-longer classical. However, when
their weights are larger or equal than 3, then their expressions are stillobtained by a “shift” of the constant term of the classical Eisensteinseries (see (9.1)). Then, it is still possible to evaluate the values of theshifted classical Eisenstein series at cusps by using either the classicalRiemann zeta function or Dirichlet’s L-function (see §10 Table 2).
However, Eisenstein series of types B2 and G2 of weights less than orequal to 2 have expression by special values of Weierstrass p-functions
or by diﬀerence of special values of Weierstrass zeta-functions, which
seem to be less standard. We determine their values at cusps in a
separate note joint with Aoki [2] (c.f. [9, 16]). These determinations
lead us to the identification of the ring Eisenstein series of types A2 , B2

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KYOJI SAITO

and G2 with the ring of modular forms of Γ1 (N ) (N = 1, 2 and 3)(Theorem 10.1). It is marvelous that this identification induces further
a one to one correspondence of the set of irreducible components of thediscriminant of our family of elliptic curves with the set of modular form
(up to constant factors) which vanishes exactly once at one Γ1 (N )-orbit
of cusps (§10 Lemma 10.3).
This modular form, generating the ideal vanishing at an equivalenceclass of cusps, on one hand as an equation for an irreducible componente On the other hand asof the discriminant, is nowhere vanishing on H.
a modular form described by theta-series, it has integral Fourier coeﬃcients. Such form can be expressed as suitable quotient of products ofshifted Dedekind eta-functions (see Table 5). Doing this for all cusps,
we determine the eta-product expressions of the discriminant (11.1)
and the reduced discriminant (11.2) of our family of elliptic curves.
These expressions lead us to the final Theorem 11.1:(1) There exists a cusp form of weight 1 of the congruence groupΓ1 (N ) with respect to the character ϑI2 (p) in §4 such the reduced discriminant of the family of elliptic curves for the type I2 (p) is identified
with the 2k(I2 (p))th power of the cusp form (p = 3, 4 and 6).
(2) The discriminants of the families of elliptic curves of all types A2 ,B2 and G2 , up to a rational constant
power π 12 , are identified∏∞ times then 24with the modular discriminant q n=1 (1 − q ) of weight 12.
These two statements give positive answers to the discriminant conjecture 6 posed in [23] §6, and we close the present paper.
2. Families of elliptic curves of type A2 , B2 and G2
We start with the three families, defined by the equations (2.1), ofaﬃne elliptic curves in the (x, y)-plane parametrized by two weighted
homogeneous coordinates g = (gs , gl ) (where s and l stands for small orlarge weights of the coordinates so that the equations become weighted
homogeneous polynomials. See Table 1).
A2 : FA2 (x, y, g) := y 2 − (4x3 − gs x − gl )(2.1)

B2 : FB2 (x, y, g) := y 2 − (x4 − gs x2 + gl + gs2 /8)
G2 : FG2 (x, y, g) := x(y 2 − x2 ) + gs (3x2 + y 2 ) − gl − 2gs3 .

Historically, they are called Weierstrass、Legendre-Jacobi and Hesse
family of elliptic curves (Exactly, compared with historical expressions,some coordinate change is done from a view point of primitive forms.
See Footnote 1 and 2).

PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

5As given in the left side of the equations, we call the families by thenames of root systems of rank 2, i.e. by A2 , B2 and G2 . 1 We shall justify
this renaming in §3 from a view point of vanishing cycles. In order totreat these three cases simultaneously, let us use the notation I2 (p) fordihedral groups [4]. Namely, let us recall the following identifications.
(2.2)

A2 = I2 (3),B2 = I2 (4),G2 = I2 (6)These three cases are exactly the cases when the dihedral group oftype I2 (p) is crystallographic, corresponding to classical root systems
defined over Z which shall play crucial roles in the present paper.
Let us give Table of weights of the variables in the equation FI2 (p) .
We normalize them so that the total weight of FI2 (p) is equal to 1.
wt(FI2 (p) ) wt(x) wt(y) s=wt(gs ) l=wt(gl ) wt(∆I2 (p) ) wt(z)A2
1
1/3 1/2
2/3
1
2
−1/6
B2
1
1/4 1/2
1/2
1
3
−1/4
G2
1
1/3 1/3
1/3
1
4
−1/3

Table 1: Weights of functions and coordinates
Here, ∆I2 (p) and z are the discriminant introduced below (2.5) and the
Hamilton time coordinate z of the elliptic curve ẼI2 (p) introduced in §6(6.1), respectively. Note that the weight wt(z) := wt(x)+wt(y)−wt(FI2 (p) )of the variable z is negative caused from a classification of vanishingcycles, where the negativity plays an essential role in the present paper.
The equations (2.1) define geometric families of aﬃne elliptic curves.
Namely, let us consider the morphisms:(2.3)

πI2 (p) : XI2 (p) −→ SI2 (p) ,p = 3, 4 or 6

where SI2 (p) is the two dimensional complex parameter space of thecoordinates g = (gs , gl ), 2 XI2 (p) is the aﬃne subvariety in C2 × SI2 (p)defined by the equation FI2 (p) = 0, and πI2 (p) is the morphism induced
on XI2 (p) from the projection to the second factor SI2 (p) , respectively.
1To be exact, when we call a family of type A , B

2
2 or G2 , we shall mean thefamily given in the equation (2.1) together with an action of an automorphism givenin (3.11) of the family. They are subfamilies of the bigger families of aﬃne elliptic
curves of type A2 , A3 and D4 which admits an automorphism σ of order 1, 2 and 3,
respectively, so that the present family is the subfamilies over the parameters which
are fixed by σ (see Footnote 9). The study of the periods for the types A3 and D4(unpublished) are beyond the scope of present paper and shall appear elsewhere.
See also Footnote 7 and 14.
2The coordinates g = (g , g ) are, up to constant factor, flat coordinates of thes lfamily [20] [23], whose weights are equal to exponents of I2 (p) plus 1/p [4].

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KYOJI SAITO∂F

I2 (p)The relative critical point set CF of the map πI2 (p) defied by ∂x=∂FI2 (p)= 0 lies proper finite over the base parameter space SI2 (p) . The∂yimage set πI2 (p) (CF ) in SI2 (p) is a one codimensional subvariety(2.4)

DI2 (p) ⊂ SI2 (p) ,called the discriminant loci, which is parametrizing singular elliptic
curves. The defining equation ∆I2 (p) of the discriminant together its
multiplicity (up to a constant factor) is given by (c.f. [19])
√
√
3/2
3/2
∆A2 = −27gl2 + gs3 = ( 27gl + gs )(− 27gl + gs ),3∆B2 = (8gl + gs2 )(−8gl + gs2 )2 ,(2.5)
∆G2 = (gl + 2gs3 )(−gl + 2gs3 )3 .
The fiber EI2 (p),g := πI−1(g) over a point g in SI2 (p) is an aﬃne open
2 (p)curve in the (x, y)-plane which is compactified to an elliptic curve byadding a single, two or three points at infinity according as the types
A2 , B2 or G2 . 4 Using the identification (2.2), we denote by [p/2] thenumber of the points at infinity for the family of the type I2 (p).
Let us denote the points at infinity by(2.6)∞1 , · · · , ∞[p/2](where exact labeling is fixed in §3), the compactified curve by[p/2](2.7) E I2 (p),g = EI2 (p),g ∪ ∪ {∞i } andi=1

E I2 (p),g ∩ XI2 (p) = EI2 (p),gand the fiberwise compactified family by
[p/2]

(2.8)

π I2 (p) : X I2 (p) = XI2 (p) ∪ ∪ {∞i × SI2 (p) } −→ SI2 (p) .
i=1Actually, X I2 (p) is smooth at the points ∞i and π I2 (p) is transversal tothe divisors ∞i × SI2 (p) (see Footnote 4). The weighted homogeneity ofthe equation FI2 (p) implies that there is the t ∈ C× -action (x, y, gs , gl ) 7→
(twt(x) x, twt(y) y, twt(gs ) gs , twt(gl ) gl ) leaving the space XI2 (p) invariant sothat the (2.3) is equivariant with the action. We note that the action
on XI2 (p) extends to X I2 (p) continuously so that the divisor ∞i × SI2 (p)is invariant and the morphism (2.8) is still equivariant.
3The decomposition in case of type A has meaning only when we consider the2
real parameter space SIR2 (p) and it was unnecessary to fix the branch.
4A compactification of E
2I2 (p),g is obtained by the curve defined in P by thehomogenization of the equation FI2 (p) . In case of type A2 , it is tangent of order
3 to the infinite line at a single infinite point, in case of B2 , it is twice tangent
to the infinite line at the same infinite point (so that we need to normalize thecompactified curve at the point to separate branches), and in case of type G2 , it isintersecting transversally with the infinite line at three distinct infinite points.

PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

7

The restriction of the family (2.3) over the complement SI2 (p) \ DI2 (p)of the discriminant loci (i.e. the space of regular values of the family(2.3)) gives a locally topologically trivial family of punctured elliptic
curves, which also induces a topologically locally trivial family of compact smooth elliptic curves.
Note. Compact elliptic curve has well-known abelian group structure(after choosing the origin). Then, we have the following elementary,
but important fact, which we shall reformulate in (5.10) §5.
Fact 1. The diﬀerence of two infinite points [∞i ] and [∞j ] (1 ≤ i, j ≤[p/2]) is a torsion element of order [p/2].
3. Real elliptic curves and Vanishing cycles
We study some real geometry of the family (2.3). It provides a
description of the middle (i.e. one dimensional) homology groups ofthe aﬃne elliptic curves EI2 (p),g in terms of vanishing cycles of rootsystems of types A2 , B2 and G2 .
Let us denote by XIR2 (p) and SIR2 (p) the subspaces of XI2 (p) and SI2 (p)consistings of the points where the coordinates (x, y) and g take realvalues, and call them the real total space and the real parameter
space of the family (2.3), respectively. In the real parameter space,
we are interested in a particular connected component of the compliment SIR2 (p) \ DI2 (p) of the real discriminant loci DI2 (p) ∩ SIR2 (p) , so called,
the totally real component ΓI2 (p) 5 defined by(3.1)

ΓI2 (p):−|gs |p/2 < cgl < |gs |p/2 and gs > 0,

and its boundary as the union of two edges
cgl = ±|gs |p/2 and gs > 0√
and the origin {0} (here c = 27, 8 or 1/2 according as p = 3, 4 or 6(2.5)). See Figure 1 and its following explanations.
(3.2)∂± ΓI2 (p):Explanation: The union of curves pathing through the origin 0 is the real
discriminant DI2 (p) ∩ SIR2 (p) . The union of the lightly and darkly shaded
areas, bounded by the real discriminant, is the totally real component ΓI2 (p) .
The boundary ∂ΓI2 (p) is the union of the upper boundary ∂+ ΓI2 (p) , lower
5Actually, the discriminant loci D

I2 (p) of the family (2.3) is identified with thediscriminant loci in the quotient space of a vector space by the irreducible finitereflection group action of type I2 (p). Then, there is the unique connected component Γ of the complement of the real discriminant loci such that the inverse imageof a point in Γ is totally real in the original vector space of the representation [22].
Then, Γ is homeomorphic to a chamber of type I2 (p) and ∂± ΓI2 (p) are homeomorphic to its walls. These facts are used for a calculation of fundamental group of thecompliment of the discriminant loci (cf. §4).

8

KYOJI SAITO

g+
-

0
g-

a
b

g+
-

g0
-

0
g-

-

g+
-

a
b

g0
-

a
0

-

b

g0
-

g-

Figure 1.Real discriminant DIR2 (p) , totally real component ΓI2 (p) and JI2 (p)boundary ∂− ΓI2 (p) and the origin 0. We choose points g 0 and g ± genericallyin ΓI2 (p) and in ∂± ΓI2 (p) , respectively. See §4 for the role of the other curves,the dark shaded area, called J, and its boundary component a and b.

We observe the following Facts 1-5. of vanishing cycles in the family of aﬃne elliptic curves EI2 (p) . Proof is achieved by explicit direct
calculations (see Figure 2), and we omit its details.
Choose a base point g 0 ∈ ΓI2 (p) (recall Figure 1), and consider theaﬃne elliptic curve EI2 (p),g0 = πI−1
(g).
2 (p)
Fact 1. The real aﬃne elliptic curve EIR2 (p),g := EI2 (p),g0 ∩XIR2 (p) (exhib0
ited in the first row of Figure 2.) consists of a single compact component(=oval), which we call
(3.3)

γ0

and [p/2]-number of non-compact connected components (arcs) δ1 , · · · , δ[p/2] .
The non-compact components are bounded by the points at infinity inthe compactification E I2 (p),g0 . We fix orientations of the arcs δi ’s andthe numbering of the points at infinity such that the cyclic union:(3.4){∞1 } ∪ δ1 ∪ {∞2 } ∪ δ2 ∪ · · · ∪ δ[p/2] ∪ {∞1 }form an oriented closed cycle (i.e. ∂δ1 = ∞2 − ∞1 , · · · , ∂δ[p/2] = ∞1 −∞[p/2] ). Then, this cycle is homologous to γ0 after choosing an orientation of γ0 accordingly. However, this condition determine only a cyclic
ordering of arcs. So, there still remains an ambiguity of reversing theorientation. We resolve this ambiguity in the following observation.
Fact 2. There exists a unique choice of an orientation and a cyclic
numbering of arcs δi ’s which satisfies the initial direction condition(3.5)(δ1 , ∞1 ) ⊂ (R>0 , +∞) × (R<0 , −∞)PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

∞1

∞2

∞2

∞1

9

∞3

∞2

δ3

γ0

δ2

δ1

γ0

δ1

γ0

δ2

∞1

∞1

∞1

∞2

∞3

∞1

∞2

∞2

∞1

∞3

δ1
∞1
∞2

p3
p2

p1

p1

p1
p2

∞1

∞1

∞1

∞2

∞3

∞1

∞2

∞2

∞1

∞3

p0

p0

∞1
∞2

p0

∞1

∞1

∞1

∞1

∞2

∞2

∞1

∞1

∞1

∞2

∞3

∞1

∞1

∞3

∞2

∞2

∞3

∞1

Figure 2. Real elliptic curves of type A2 , B2 and G2
The curves in the second and the third column of the table, called of type B2
and G2 , are also regarded as the real aﬃne elliptic curve of type A3 and D4(see Footnote 1.) together with the action of an automorphism σI2 (p) (3.11).

Here, (δ1 , ∞1 ) in LHS means the germ of the arc δ1 at ∞1 , and the(R>0 , +∞) × (R<0 , −∞) in RHS means a neighborhood of (+∞, −∞)in R>0 × R<0 ⊂ C × C.6
6A priori, there exists neither a guarantee that there exists such pair (δ , ∞ )iisatisfying the condition (3.5), nor a guarantee that the condition (3.5) choose the10

KYOJI SAITO

Fact 3. Let us move the point g 0 inside ΓI2 (p) to a point g + on the upperboundary edge ∂+ ΓI2 (p) (e.g. move along the path a in Figure 1). Then,
accordingly, the cycle γ0 and each arc δi in the fiber EI2 (p),g0 are getting
close to each other, and, finally at g + ∈ ∂+ ΓI2 (p) on the boundary, theyintersect to a node pi of the curve EI2 (p),g+ (i = 1, · · · , [p/2]) (see thesecond row of Figure 2).
Fact 4. Let us move the point g 0 inside ΓI2 (p) to a point g − on thelower boundary edge ∂− ΓI2 (p) (e.g. move along the path b in Figure 1).
Then, accordingly, the cycle γ0 in the fiber EI2 (p),g0 pinches to a Morsesingularity p0 in the fiber EI2 (p),g− (see the third row of Figure 2). Thisimplies that the cycle γ0 is the vanishing cycle generated by the Morsesingularity p0 .
Fact 5. Each nodal point pi generates a vanishing cycle
(3.6)

γi

i = 1, · · · , [p/2], in the nearby elliptic curve EI2 (p),g0 , which intersects
with γ0 and δi (recall Fact 1.) transversally. Since γi lies in the complexification EI2 (p),g , we do not exhibit it in Figure 2, but some conceptual
expression of it shall be given in the first row of Figure 3. We choose
the orientation of γi by the following sign condition on the intersection:(3.7)⟨γ0 , γi ⟩ = ⟨δi , γi ⟩ = 1⟨γi , γj ⟩ = ⟨γj , γi ⟩ = 0(i = 1, · · · , [p/2])(i, j = 1, · · · , [p/2]).

Here we denote by ⟨γ, γ ′ ⟩ the intersection number of paths γ and γ ′whose sign is fixed as follows: The orientation of EI2 (p),g as a real surface0

is fixed by its complex structure. If a path γ ′ crosses another path γ counterclockwisely, 7 then the local intersection number is ⟨γ, γ ′ ⟩ = +1.

Remark 3.1. The most degenerated real curve EIR2 (p),0 is exhibited in
the 4th row of Figure 2.
As a consequence of Facts 1-5., we obtain the following descriptionof the homology group of the aﬃne elliptic curve EI2 (p),g0 .
Fact 6. The classes of γ0 and γi (i = 1, .., [p/2]) in the first homology
group of EI2 (p),g0 form free basis (see the first row of Figure 3). We
pair (δi , ∞i ) uniquely. Therefore, Fact 2. claims that the conditiion actually choose
the unique one (which is easily confirmed from Figure 2). Further more, we remarkthat the choice (3.5) in the real blow up space of P2 (R), in case of type B2 , separates
two infinity points ∞1 and ∞2 , which the P2 (C)-compactification did not separate(recall Footnote 3). Indeed, we have (δ2 , ∞2 ) ⊂ (R<0 , −∞) × (R>0 , +∞).
7A path in the complex-plane crosses other path counter-clockwisely, if and only
if their tangent vectors a and b at the crossing point satisfies Im(a/b) > 0.

PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

11denote the classes by the same notation γi since we shall use notation[γi ] for another meaning below.
[p/2]
⊕
(3.8)
L := H1 (EI2 (p),g0 , Z) = Zγ0 ⊕
Zγi .
i=1

The open embedding EI2 (p),g0 ⊂ E I2 (p),g0 induces a surjective homomorphism(3.9)

L −→ H1 (E I2 (p),g0 , Z)

whose kernel is the radicalrad(L) = {γ ∈ L | ⟨γ, δ⟩ = 0, ∀δ ∈ L}of the lattice L which is additively generated by homologous relations
γ1 ∼ · · · ∼ γ[p/2] .
in the compactification E I2 (p),g . Hence, L/rad(L) is a rank 2 free abeliangroup generated by the equivalence classes [γ0 ] and [γ1 ] = · · · = [γ[p/2] ].
(3.10)

L/rad(L) = Z[γ1 ] ⊕ Z[γ0 ].

Here, we denote by [∗] the equivalence class of ∗ in the quotient module.
The map (3.9) preserves the intersection form, since it is the quotientmorphism by the radical.
We next consider a SL2 -linear automorphism of the (x, y)-plane by 8
(3.11)σA2 (x, y) := (x, y),σB2 (x, y) := (−x, −y),, −3x−y).
σG2 (x, y) := ( y−x22

It fixes the equation FI2 (p) and, hence, induces a fiber automorphism oforder [p/2] of the family (2.3), whose action fixes the base space SI2 (p)point-wise. Since σI2 (p) leaves the real structure XIR2 (p) invariant, it acts
on each complex and real curve EI2 (p),g . One checks directly that σI2 (p) action induces the cyclic permutation of the cycles γi , oriented arcs δiand the points ∞i at infinity for i ∈ Z/[p/2]Z, respectively. However,the cycle γ0 is invariant by the σI2 (p) -action.
Let us consider the sub-lattice of H1 (EI2 (p),g0 , Z) consisting of σI2 (p) fixed elements.
(3.12)

LI2 (p) := LσI2 (p) = H1 (EI2 (p),g0 , Z)σI2 (p)8There is an SL -linear automorphism σ of order 1, 2 and 3 on bigger families
2of type A2 , A3 and D4 (see Footnote 1). Then, the action (3.11) is induced fromσ as fiber-wise action on the subfamilies of type A2 , B2 and G2 , respectively. The
σ-action has not only homological implications as discussed in this section, it has
another important implication on a certain cohomology class called the primitive
form (see Footnote 17).

12

KYOJI SAITO

It is immediate to see that LI2 (p) is a rank 2 sub-lattice generated by∑(3.13)
α := [p/2]andβ := γ0 ,i=1 γiwhose intersection number is counted by (3.7) as(3.14)⟨α , β⟩ = −[p/2],⟨α , α⟩ = 0 and ⟨β , β⟩ = 0.

The composition of the embedding LI2 (p) ⊂ L with the radical quotientmap (3.9) is again an isometric embedding of lattices:9(3.15)

LI2 (p) ⊂ L/rad(L), α 7→ [p/2][γ1 ], β 7→ [γ0 ]of finite index [p/2].
We give below a root lattice theoretic interpretation of what we havecalculated above (see also Footnotes 8, 9 and 11), which answers to thequestion on the naming of the family (2.1), posed in §2.
In the following Figure 3, we exhibit:(1) The first row exhibits a conceptual description of the cycles γ0 and
γi (i = 1, · · · , [p/2]) in the surface EI2 (p),g0 , which is a complexificationof the real curves in the first row of Figure 2. Complexfication of real
curves in the second and the third row of Figure 2. can be obtained fromthis surface EI2 (p),g0 by pinching either the cycles γi (i = 1, · · · , [p/2])
or the cycle γ0 , respectively.
(2) The second row exhibits the intersection diagram of the basis γ0
and γi (i = 1, · · · , [p/2]) of the homology group L := H1 (EI2 (p),g0 , Z)
(actually, they are known as diagrams of types A2 , A3 and D4 , respectively (note Footnote 1 and 9)).
(3) The third row exhibits the folding of the intersection diagramin previous (2) by the action of the automorphism σI2 (p) which is theintersection diagram for the invariant basis α, β of the invariant homology group (3.12) (actually, they are known as diagrams of types A2 , B2
and G2 , respectively).
Consequently, we observe that the σI2 (p) -invariant 1-cycles of the families (2.1) are indexed by the lattices of types A2 , B2 and G2 , respectively. This is the reason why we want to call the families according to
the type of the root systems. 10 The singularity at the origin 0 ∈ C2of the curve EI2 (p),0 for g = 0 (exhibited in the 4th row of Figure 2)
together with the action of σI2 (p) on it is called the singularity of type
9Since we shall no-longer use the lattice L in the present paper, so far as there

is a no-confusion, we shall regard LI2 (p) as a sub-lattice of L/rad(L) (e.g. (4.4)).
10To be exact, what we wrote here needs more explanations in the following∑[p/2]
sense. The intersection form ⟨·, ·⟩ on the free abelian group L :=i=0 Zγi is
skew-symmetric so that the pair (L, ⟨·, ·⟩) is not a root lattice. In order to justify

PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

13

I2 (p) (i.e. of type A2 , B2 and G2 according as p = 3, 4 and 6) and thelattice LI2 (p) is called the lattice of vanishing cycles for the singularity.
Then, the family (2.3) may be regarded also as the universal unfolding(by the parameter space SI2 (p) ) of the singularity of type I2 (p).

EA2 ,g0

EB2 ,g0

EG2 ,g0
γ3

γ2

δ1

∞1

δ3

∞1
γ1

γ1

δ1

γ1

γ2

γ0

γ0
δ2

∞2

∞1

γ0

∞3
γ0

δ1

δ2

∞2

δ1

what we wrote above, we consider a pair (L, J) of the abelian group L with
the non[] [1 −1 −1]
symmetric Seifert form J on it defined by the Seifert matrices 10 −1
1 , 00 10 01[]
1 −1 −1 −1
1 0 0
or 0
0 0 1 0 (which is the table of linking numbers between the ordered basis
0 0 0 1γi (3.8)) according to p = 3, 4 or 6, respectively. Then, the diﬀerence JI2 (p) −t JI2 (p)is a skew symmetric form on L which is identified with the intersection form ⟨·, ·⟩ on
H1 (EI2 (p),g , Z). On the other hand, the sum I := J + t J is a symmetric bilinear
0form so that the pair (L, I) is isomorphic to the root lattice of type A2 , A3 andD4 with the simple root basis γ0 , · · · , γ[p/2] . The geometric automorphism (3.11)
induces an automorphism of the lattice (L, J), denoted again by σI2 (p) . Then the∑[p/2]
invariant sub-lattice LI2 (p) := LσI2 (p) is spanned by α := [i=1 γi and ]β := γ0so that the Seifert form JI2 (p) on the basis gives the matrix [p/2] −[p/2] . Then01
(LI2 (p) , I|LI2 (p) ) is isomorphic to the root lattice of type A2 , B2 and G2 with thesimple root basis α and β according as p = 3, 4 or 6.
There are two constructions which realize the above formal justification: (1)Consider the category of matrix factorization of the singularity FI2 (p) (x, y, 0) (see[14]). Since the simple singularities are self-mirror, the category is isomorphic tothe category of vanishing cycles in the Milnor fiber FI2 (p) (x, y, g 0 ) = 0. Actually,
one finds strongly exceptional collections generating the category such that theirimages γi in the K-group of the category (≃ the ∑middle homology groupoftheMilnor
fiber) gives the simple basis where the Euler form k(−1)kExtk (γi ,γj) is identified with
the Seifert form. (2) Consider the equation FI2 (p) (x, y, g 0 ) + z 2 in three variables(x, y, z). Then the suspensions, denoted by Σγi , of the basis γi (0 ≤ i ≤ [p/2]) of thevanishing cycles in the complex aﬃne curve FI2 (p) (x, y, g 0 ) = 0 form vanishing cycle
basis of the second homology group of the complex aﬃne surface FI2 (p) (x, y, g 0 ) +z 2 = 0. Then the intersection form −⟨Σγi , Σγj ⟩ of the homology classes coincides
with the symmetric form I.

14

KYOJI SAITO

A2 :

γ0

γ1

A3 :

γ0

γ1

γ1

D4 :

γ0

γ2

γ2

/(Z/1Z)
β

A2 :

γ3

/(Z/2Z)

α

β

B2 :

/(Z/3Z)

α

β

α

G2 :

Figure 3. Vanishing cycles and associated diagrams.
4. Fundamental group and monodromy representation
We describe the fundamental group of the compliment of discriminant loci SI2 (p) \ DI2 (p) , and its monodromy representation in the firsthomology groups of EI2 (p),g and that of its compactification E I2 (p),g .
First, we sketch a geometric description of the fundamental groupwhich is valid not only for types A2 , B2 and G2 but also for the regularorbit space of any type finite reflection group by the authors [5],[8],[22].
Since the formulation of [22] is closest to the present setting, we briefly
explain it.
Consider a translation action τε : SI2 (p) → SI2 (p) , (gs , gl ) 7→ (gs , gl +ε)
for ε ∈ C.11 If ε ∈ R, then the action preserves the real space SIR2 (p) .
In Figure 1., we consider two shifted discriminant loci τε (DI2 (p) ) andτ−ε (DI2 (p) ) for some ε ∈ R>0 . Then, the darkly shaded area J in
ΓI2 (p) ⊂ SIR2 (p) exhibit the component (which is homeomorphic to arhombus) in ΓI2 (p) cut by the shifted discriminant loci τε (DI2 (p) ) andτ−ε (DI2 (p) ), where its boundary edges a and b are segments on τε (DI2 (p) )and τ−ε (DI2 (p) ), respectively.
Let ã and b̃ be the path in SI2 (p) \ DI2 (p) defined as follows. We
choose the base point g 0 ∈ ΓI2 (p) at the intersecting point of a and b as
in Figure 1.
Let aC and bC be complexfication of the real segments a and b embedded in the complex shifted discriminants τε DI2 (p) and τ−ε DI2 (p) , respectively, such that aC ∩ DI2 (p) = {g + } and bC ∩ DI2 (p) = {g − }. The
following Figure 4. shows how to choose paths ã and b̃ in the complexification aC and in bC , and then embed them in SI2 (p) \ DI2 (p) .
11The action τ is naturally obtained by integrating the vector field ∂gl , called
the primitive vector field, which is, up to constant factor well-defined (see [22]).

PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

15

Figure 4. Homotopy classes ã and b̃
Fact 7. The fundamental group π1 (SI2 (p) \ DI2 (p) , g 0 ) =: GI2 (p) is generated by the homotopy classes of ã and b̃ and determined by the relation(4.1)

ãb̃ · · · = b̃ã · · ·where the both hand sides are words of alternating sequences of ã, b̃ oflength p, which start either with ã or with b̃.
Sketch of proof. 1. A direct proof is given by Zariski-van Kampen
method [28] w.r.t. the pencils {gs =const.}, which intersects with discriminant loci by two points, giving two generators of the fundamentalgroup easily identified with ã, b̃ ([22]§4.3) and gives the relation (4.1)
by turning gs = eiθ (0 ≤ θ ≤ 2π).
Let us sketch another approach in [5],[8],[22], where we regard thediscriminant loci DI2 (p) as the discriminant loci for the orbit space forthe dihedral group action on a two dimensional vector space. Namely,
let VR be the real 2-space on which the dihedral group W (I2 (p)) acts
as the reflection group. We suitably (up to constant) identify the invariant ring S(VR∗ )W (I2 (p)) with the polynomial ring generated by twovariables gs and gl (recall (2.1)). So, we do W (I2 (p)⧹VC with SI2 (p)and (W (I2 (p))⧹VC )reg = W (I2 (p))⧹(VC \ ∪Hα ) (=regular orbit space)with SI2 (p) \ DI2 (where ∪Hα is the union of complexified reflection hyperplanes). By this identification, the real region ΓI2 (p) (3.1) is homeomorphic to any chamber in VR \ ∪Hα . Then, the inverse image in VRof the closure of ∗ is a curved polygon which is dual to the chamber
decomposition, namely, hexagon, octagon and dodecagon according as
I2 (p) is of type A2 , B2 and G2 . The boundary of the polygon is givenby (ȧḃ)p (see [22] Fig.). Here, each ȧ (resp. ḃ) is a double cover of theclosed interval a (resp. b) which is crossing the reflection hyperplane atits central point. We now consider the inverse image in VC \ ∪Hα of theunion ã ∪ b̃. Actually, the inverse image is √free homotpic to the boundary to the curved dual polygon shifted by −1δ for some δ ∈ VR \ ∪Hα(corresponding to breadth of ã and b̃, Figure 5.), where each component of the inverse image of ã (resp. b̃) is homotopic to a shifted edge

16

ȧ +

KYOJI SAITO√−1δ (resp.
√ ḃ +√−1δ). On the other hand, the shifted real vector

space VR + −1δ does not intersect with any reflection hyperplane sothat the shifted dual polygon is contractible in VC \ ∪Hα . This creates
the homotopy relation (4.1). That there is no-more relations follows
from a dimension argument, which we omit here.
2
The group GI2 (p) with the presentation in Fact 7 is called the Artin
group of type I2 (p) ([6]). The element of the expression (4.1) is theleast common multiple of the generators ã and b̃ inside the positive
monoid of the group and is called the fundamental element ([12],[6]),denoted by 12∆ = ∆I2 (p) .
It is well-known (e.g. [6]) that the center of the group GI2 (p) is an infinite
cyclic group generated by ∆ (types B2 and G2 ) or by ∆2 (type A2 ).
The next task is to determine the monodromy actions of the fundamental group GI2 (p) on the lattices L, LI2 (p) and L/rad(L). Recall
Fact 2, 3, 4 and 5 that the “degeneration” of the curve EI2 (p),g0 moving
the parameter g 0 along the path a (resp. b) pinches the cycles γi (i =1, · · · , [p/2]) (resp. cycle γ0 ) in EI2 (p),g0 to points pi (resp. p0 ). Thenthe actions ρ = ρI2 (p) of ã (resp. b̃) on the lattice L = H1 (EI2 (p),g0 , Z) is
determined by Picard-Lefschetz formula (i.e. the transvections) of thevanishing cycles γ1 , · · · , γ[p/2] (resp. γ0 ), taking the intersection number(3.7) in account. That is,{{∑
γ0
if j = 0γifj=0γ0 − [p/2]ii=1ρ(b̃)(γj ) =ρ(ã)(γj ) =γj + γ0 if j = 1, 2, 3
γj
if j = 1, 2, 3
Then, the other actions on LI2 (p) and L/rad(L), denoted by the sameρ, are induced from this action either by restriction to the sub-latticeor by the quotient out the radical of the lattice. In particular, theembedding (3.15) is equivariant with the monodromy action. Explicit
formulae are given as follows.
Fact 8. 1) The action of GI2 (p) on LI2 (p) = H1 (EI2 (p),g0 , Z)σI2 (p) is given
as follows.
[][](4.2) ρ(ã)(α, β) = (α, β) 10 −1
1

and

1
0 .
ρ(b̃)(α, β) = (α, β) [p/2]112Here, we have an unfortunate coincidence of notation of the fundamental element with that of the discriminant (2.5). Since we use them in diﬀerent places,
there shall be no confusions.

PERIOD INTEGRALS OF TYPE A2 , B2 AND G2172) The action of GI2 (p) on L/rad(L) = H1 (E I2 (p),g0 , Z) is given as
follows.
(4.3)
[
]
[ ]
1
0 and ρ(b̃)([γ ], [γ ]) = ([γ ], [γ ]) 1 1 .
ρ(ã)([γ0 ], [γ1 ]) = ([γ0 ], [γ1 ]) −[p/2]
0
1
0
1
1
0 1

Here, the representations (4.2) and (4.3) are conjugate by the basischange:
[
]
0
1
(4.4)
(α, β) = ([γ0 ], [γ1 ])[p/2] 0(3.15), and, hence, they are equivalent.13 We determine the image andthe kernel of the representations as follows.
Fact 9. Let us identify L/rad(L) with Z2 by the use of the basis [γ0 ]and [γ1 ], and regard the representation (4.3) as a homomorphism ρ :GI2 (p) → SL2 (Z). Then, we have
1. The fundamental element ∆ is represented by ρ as follows.
(4.5)
[
]
[
]
[
]
0 1 , ρB (∆) = −1 0
1 0 .
ρA2 (∆) = −1
and
ρ
(∆)
=
G
2
2
0
0 −1
0 1
In particular, the images Im(ρA2 ) and Im(ρB2 ) contain −id, but Im(ρG2 )does not.
2. The image of the representation ρ in SL2 (Z) is equal to the subgroup
]
}
{[
]
[
] [
a b ∈ SL2 (Z) | a b ≡ 1 b mod [p/2]} ,(4.6) Γ1 ([p/2]) :=
0 1
c d
c d
called a congruence subgroup of SL2 (Z) of level [p/2] (see e.g. [15]):
3. The kernel of the representation ρ is an infinite cyclic group
generated by(4.7)(ãb̃)k(I2 (p))
where k(I2 (p)) ∈ Z≥2 is a number attached to any Coxeter system (see[23] §6, 6.1 ii)) such that
(4.8)
k(A2 ) = 6, k(B2 ) = 4 and k(G2 ) = 3.
13In the sequel, we shall treat these two equivalent representations in parallel (e.g.

(4.12)), which looks a bit redundant and cumbersome. This subtlety was caused
since we, later on, want to study automorphic forms for congruence subgroups (see(4.6) and §6). More precisely, from a view point for the period map of a primitive
form, it is natural to consider the lattice LI2 (p) of vanishing cycles. On the other
hand, from a view point of classical elliptic integrals for a compact elliptic curve andto connect with classical elliptic modular function theory, it is natural to consider
the lattice L/rad(L). Therefore, in the sequel, when we talk about the matrix
expression of the representation ρ, we shall mean those matrices with respect tothe basis [γ0 ] and [γ1 ] but not the basis α and β. These cautious treatments arenecessary for the cases of types B2 and G2 but not for A2 , since the embedding(3.15) is already isomorphic.

18

KYOJI SAITO

14

4. Thus, we have the short exact sequence(4.9)

1 → Z → GI2 (p) → Γ1 ([p/2]) → 1.

That is, the Artin group GI2 (p) is a central extension of an elliptic
congruence modular group Γ1 ([p/2]).
Proof. 1. This is a direct calculation for the cases [p/2] = 1, 2 and 3.
The fact that Im(ρA2 ) does not contain −id is a consequence of ii).
2. Let us show that ρ is surjective to Γ1 ([p/2]). Set A := ρ(ã) andB := ρ(b̃) in (4.3). Clearly A, B ∈ Γ1 ([p/2]). We show that Γ1 ([p/2]) is
generated by A and B. Consider an element C ∈ Γ1 ([p/2]) whose (2, 1)(resp. (2,2)) entry is c (resp. d). By definition, we set c = c[p/2] for somec ∈ Z. We also know by definition that d ̸= 0. Then the (2,1) entry(resp. (2,2) entry) of CAk (k ∈ Z) is equal to c+kd[p/2] = (c+kd)[p/2](resp. is unchanged d). Then, by the Euclidean division algorithm, we
can choose k ∈ Z such that |c − kd| ≤ |d|/2. Next, let us considerC ∈ Γ1 ([p/2]) such that its (2,1) entry c = c[p/2] satisfies the condition|c| ≤ |d|/2. If c = 0 then the diagonal of C is ±(1, 1) where (−1, −1)
cannot occur for the case [p/2] = 3 by the definition of Γ1 (3). By
multiplying −id if necessary for the cases [p/2] = 1 or 2 (recall the
result in 1.), we see that C is already of the form B l for some l ∈ Z.
Suppose c ̸= 0. We consider CB l for l ∈ Z. Then, its (2,1)-entry is
unchanged c, but the (2,2) entry is given by d′ := d − lc. Then after
a suitable choice of l ∈ Z, we have |d − lc| ≤ |c|/2. Combining theabove two procedures, the (2,2) entry d′ of the matrix CAk B l satisfies|d′ | ≤ |c|/2 = |c[p/2]|/2 ≤ |d|[p/2]/4. Since [p/2]/4 < 1 in our case,this means |d′ | < |d|. That is, if C is note generated by A and B,
there are integers k and l such that the (2,2) entry of CAk B l has theabsolute value strictly smaller than that of C. This give an induction
proof of Γ1 ([p/2]) = ⟨A, B⟩.
3. As a result of 1., we observe ρA2 (∆4 ), ρG2 (∆2 ) and ρG2 (∆) are
identity matrices. That is, ∆4A2 , ∆2B2 and ∆G2 belong to the kernel,
where we have relations ∆4A2 = (ãb̃)6 , ∆2B2 = (ãb̃)4 and ∆G2 = (ãb̃)3 .
Thus the element (4.7) is contained in the ker ρ. Conversely, let us
show that ker(ρI2 (p) ) ⊂center of GI2 (p) . Then, we explicitly calculate∆4A2 , ∆2B2 and ∆G2 generate the kernel. To show this, we use thefundamental domain of Γ1 ([p/2]) (see [23] §6 Assertion 5, 6.).
4. This is only the rewriting of the results 2. and 3.
□
14The numbers k(W ) for any finite Weyl group W is defined in [23] 6.1 depending

on Coxeter diagram of W .

PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

19Remark 1. The following characterization of the congruence subgruop
is well-known.
1[γ0 ]}
Γ1 ([p/2])= {m ∈ SL2 (Z) | m preserves the subsets L/rad(L)+ [p/2]= {m ∈ SL2 (Z) | m preserves the subsets LI2 (p) and LI2 (p) +[γ1 ]}= {m ∈ SL(LI2 (p) ) | m extends to SL2 (Z) and preserves LI2 (p) +[γ1 ]}.
A sketch of proof. Reflections by the roots α[and]β, satisfy the conditions. Conversely, the conditions on m = ac db ∈ SL2 (Z) that itpreserves L (resp. LI2 (p) + [γ1 ]) implies c ≡ 0 mod [p/2] (resp. d ≡1 mod [p/2]). 2
As a consequence of the description of the congruence subgroupΓ1 ([p/2]) in Fact 9, we introduce a character (see [23] §6 (6.1.6)) on it,
which shall be used to formulate the discriminant conjecture in §11:
( π √−1 )×(4.10)
ϑI2 (p) : Γ1 ([p/2]) → C , ã, b̃ 7→ exp.
k(I2 (p))
Proof. Obviously, the relation (4.1) is satisfied by the imagesof ϑI2 (p) .
√On the other hand, the ϑI2 (p) -image of (4.7) is equal to exp(2π −1) = 1. 2
Note that the power(4.11)k(I (p))2
θI2 (p) := ϑI2 (p): Γ1 ([p/2]) → {±1}, ã, b̃ 7→ −1
2k(I (p))

is trivial.
defines also a character for the anti-invariants, and ϑI2 (p)2
Actually except for the type G2 , θI2 (p) factor through the sign morphism
WI2 (p) → {±1} of the Weyl group associated with I2 (p).
For a use to describe the period map in the next §5, we prepare
some notations. It is well-known that monodromy data given in Fact
9. is equivalent to data of the local systems of the homology groups
H1 (EI2 (p),g , Z)σI2 (p) and H1 (E I2 (p),g , Z) over g ∈ SI2 (p) \ DI2 (p) . According
to the two local systems, we consider(4.12)L̃I2 (p) := the lifting of the local system H1 (EI2 (p),g , Z)σI2 (p) to SeI2 (p)L̃/rad(L̃) := the lifting of the local system H1 (E I2 (p),g , Z) to SeI2 (p) .
Here, SeI2 (p) is the monodromy ρ covering space of SI2 (p) \ DI2 (p) definedby()∼(4.13)SeI (p) := ker(ρI (p) )⧹ SI (p) \ DI (p) ,
2

2

∼

2

2

where (SI2 (p) \DI2 (p) ) is the universal covering of SI2 (p) \DI2 (p) , on which
the fundamental group GI2 (p) acts from left properly and discontinueI (p) in (SI (p) \ DI (p) )∼ of the base point
ously after choosing a copy Γ222loci ΓI2 (p) ⊂ SI2 (p) \DI2 (p) (3.1) as the base point loci in (SI2 (p) \DI2 (p) )∼ .

20

KYOJI SAITOThen we take the quotient of the universal covering by the kernel ofeI (p) in SeI (p) is again
the monodromy representation. The image of Γ22ee
denoted by ΓI2 (p) and called the base point loci of SI2 (p) .
By definition, L̃I2 (p) and L̃/rad(L̃) are trivial local systems on thespace SeI2 (p) of homotopy type S 1 .
(4.14) L̃I2 (p) ≃ LI2 (p) × S̃I2 (p)and L̃/rad(L̃) ≃ L/rad(L) × S̃I2 (p)with natural inclusion relation L̃I2 (p) ⊂ L̃/rad(L̃). For any element of
LI2 (p) ⊂ L/rad(L), say γ, we shall denote by the same γ the global
section of L̃I2 (p) ⊂ L̃/rad(L̃) associated with it, so far as there may be
no-confusion. So, the free basis α and β (3.12) (resp. [γ1 ] and [γ0 ] (Fact6)) are lifted to global basis α and β (resp. [γ1 ] and [γ0 ]) of L̃I2 (p) (resp.
L̃/rad(L̃)).
Let us show in the next Fact 10 that not only those closed cycles [γi ]are lifted to global sections over SeI2 (p) but also some “special arcs” δion the curve E I2 (p),g , which originally defined only for g ∈ ΓI2 (p) , haveglobal sections over SeI2 (p) . Namely, let us first consider the pullback
familyπ̃I2 (p) : X̃I2 (p) −→ S̃I2 (p)of the family π̃I2 (p) (2.8) to S̃I2 (p) . It carries some additional structures:(1) global sections {∞i × S̃I2 (p) } (1 ≤ i ≤ [p/2]),(2) fiberwise automorphism σ̃I( p) such that global sections are cyclically permutated: σI2 (p) (∞i ) = ∞i+1 (i ∈ Z/[p/2]Z).
Recall further that if the parameter g belongs to the base point
locus ΓI2 (p) , then the fiber curve E I2 (p),g contains oriented arcs δi (i =1, · · · , [p/2]) such thata) ∂(δi ) = ∞i+1 − ∞i for i ∈ Z/[p/2]Z,(4.15)b) σI2 (p) (δi ) = δi+1
for i ∈ Z/[p/2]Z,c) ∑homologous in E I2 (p),g .
i∈Z/[p/2]Z δi ∼ γ0
We extend these δi ’s to global sections over S̃I2 (p) as follows.
Fact 10. 1. There exist global sections for 1 ≤ i ≤ [p/2](4.16)

δi:g ∈ S̃I2 (p)7→δi (g) ∈ C1 (E I2 (p),g , Z)

which coincide with δi in Fact 1. when g ∈ ΓI2 (p) , and satisfy theconditions (4.15). Here, C1 (E I2 (p).g , Z) is the module of singular 1chains on E I2 (p).g .

PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

21

2. The sections are unique up to homologous zero. That is, if there
exists other section δi′ , then δi (g)−δi′ (g) ∼ 0 (homologous in E I2 (p),g )for all g ∈ S̃I2 (p) .
The proof is left to the reader.
5. Periods for Primitive forms
We study integrals over paths and cycles in the curves EI2 (p),g of thefamily (2.3). Since we are interested in period integrals over the cycles
in H1 (EI2 (p),g , Z) (3.8), the integrant form should be holomorphic 1form on XI2 (p) relative to the base space SI2 (p) .15 Still, this condition is
too weak to fix an integrant (e.g. the Betti number of the curves arelarger than 1 so that just one choice of an integrant seems insuﬃcient).
In present paper, we shall focus on the integrals only of the following
form:
]
[
dxdy
(5.1)
ζI2 (p) := Res
FI2 (p) (x, y, g)(see Footnote 16 for explicit descriptions of the residue (5.1) 16), which
has a characterization, up to a constant factor, as the unique primitive
form for the family (2.1). 17
15Forms on X

I2 (p) relative to the base space SI2 (p) means the equivalence classesof forms on XI2 (p) modulo the OXI2 (p) -submodule generated by dFI2 (p) , dgs and dgl .
16For each fixed g ∈ S
I2 (p) , dxdy/FI2 (p) may be considered as a top degree
meromorphic 2-form on (x, y)-plane of simple pole along the curve EI2 (p),g . Then,at the smooth point of EI2 (p),g , the symbol (5.1) defines the residue, which is aholomorphic one form on EI2 (p),g . Actually, using the (x, y)-coordinates, the formdy
is explicitly given by the relative diﬀerential forms ∂FI dx/∂y ∼ − ∂FI2 (p) /∂x (which2 (p)are equivalent modulo dFI2 (p) ). Using these expressions, we confirm that ζI2 (p) , asa 1-form on E I2 (p),g , is well-defined up to the singularity loci {(x, y) ∈ EI2 (p),g |∂FI2 (p)/∂y = ∂FI2 (p)/∂x = 0} =: Sing(EI2 (p),g ) when g ∈ DI2 (p) .
17
As we shall see immediately in Fact 11, the form ζI2 (p) is, so called, the elliptic
integral of the first kind for the compact elliptic curves E I2 (p),g for g ∈ SI2 (p) (see, forinstance, [27]). On the other hand, from a view point of integrals of vanishing cyclesof the universal family (2.3) of type I2 (p), the form ζI2 (p) is, up to an ambiguityof a constant factor, called the primitive form of the family (to be exact, this is a
restriction of the primitive form ζ defined on the bigger family of type A2 , A3 or
D4 . Since the action of σ on the big family (recall Footnote 9) preserves the form
ζ, it induces the form ζI2 (p) (5.1)) (see [19, 23] for primitive forms).
A primitive form in general has distinguished characterizations, which we shall
implicitly (but not explicitly) use in the present paper (actually, we used alreadythe action of the primitive vector field in §4, which is an important ingredient of theprimitive form theory), and which we recall briefly as follows. Let us consider thetwisted relative de-Rham cohomology group RπI2 (p),∗ (ΩXI2 (p) /SI2 (p) ) of the family

22

KYOJI SAITO

In the present section, we introduce the period map (5.3) by integrating ζI2 (p) over closed cycles in EI2 (p),g . The description of its
inversion map (5.6) by the use of Eisenstein series over the lattices
LI2 (p) ⊂ L/rad(L) is the main subject of the present paper. In order tounderstand them, the study of integrals over the closed cycles are not
suﬃcient, but we need to develop a study of integrals of ζI2 (p) over nonclosed paths in E I2 (p),g . Precisely, we study integrals over the arcs δi inthe present section and over indefinite paths in the next section.
Consider the integral of ζI2 (p) over a horizontal family of cycles γ ∈H1 (EI2 (p),g , Z) (where g runs over a simply connected open subset of
SI2 (p) \DI2 (p) )
I
(5.2)
ωγ :=
ζI2 (p) .
γ

As is well-known that ωγ is a holomorphic function in the parameterg (which can be confirmed by a use of Leray’s residue formula) on thedomain where the family γ is defined. On the other hand, ωγ depends
only on its equivalence class [γ] ∈ L/rad(L) due to the following fact.
Fact 11. The form ζI2 (p) can be extended holomorphically on the compactified elliptic curves E I2 (p),g for g ∈ SI2 (p) . The extended form is
nowhere vanishing on the smooth part E I2 (p),g \ Sing(E I2 (p),g ) of thecurve.
Proof. If (x, y) ∈ EI2 (p),g is a non-singlar point, then either of ∂FI2 (p) (x, y, g)/∂x
or ∂FI2 (p) (x, y, g)/∂y is non-zero. Then, one of the two explicit expressions of ζI2 (p) in Footnote 16 gives a non-vanishing and holomorphic
expression of ζI2 (p) at the point (x, y). At the infinity points, we havealready observed in Footnote 3 that E I2 (p),g is smooth. Then usingthat expressions, we check again that ζI2 (p) is holomorphic and nonvanishing. The details are left to the reader.
□This means that ζI2 (p) may be regarded as a relative de Rham cohomology class of the family of compactified elliptic curves E I2 (p),g . The
cycles γ1 , · · · , γ[p/2] are homologous to each other in E I2 (p),g . This imH
Hplies γ1 ζI2 (p) = · · · = γ[p/2] ζI2 (p) , and, ωγ depends only on the classee Therefore, we regard ωγ as a holomorphic function[γ] ∈ L/rad(L).
(2.3).It is a filtered Os -module equipped with the Gauss-Manin connection andhigher residue pairings. Then, ζI2 (p) is an element in the 0th filter satisfying 1)
primitivity, 2) homogeneity, 3) orthogonality, 4) holonomicity (see, for instance,[19]). In particular, the primitivity means that the covariant diﬀerentiations of ζI2 (p)generate all cohomology classes so that just a single choice of ζI2 (p) is suﬃcient.

PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

23

defined on SeI2 (p) . In particular, in view of (3.13), we have expressionsIIωα := ζI2 (p) = [p/2] ωγ1 and ωβ := ζI2 (p) = ωγ0 .
αβThus, in the present paper, we shall integrate either over a cycle ineI (p) and a cycle γ in L̃/rad(L̃).18 (in such situation, we shall say
L2eI (p) ⊂ L/rad(ee“integrate a cycle γ ∈ L
L)”),or over “special arcs” δi2(4.16).
For each point g̃ ∈ SeI2 (p) (4.13), we consider the linear map
IeI (p) ⊂ L/rad(
e
e
(5.3)
γ ∈ L
L)
7→ ωγ := ζI2 (p) ∈ OS̃I (p)2γ2where we set OS̃I (p) = holomorphic functions on S̃I2 (p) .
2eI (p) ⊂ L/e ∼, we obtain a holoUsing the trivialization (4.14) of L2
morpbic map(5.4)
PI2 (p) : SeI2 (p) → HomZ (LI2 (p) , C) = HomZ (L/rad(L), C) (≃ C2 )g7→

γ 7→ ωγ :=H

ζγ(g) I2 (p)where the identification in RHS is canonically given by the change(5.5)ωα = [p/2] ωγ1and ωβ = ωγ0of the basis in both vector spaces. We shall call PI2 (p) the period map
associated with the primitive form ζI2 (p) (5.1).
Fact 12. The period map PI2 (p) is locally bi-holomorphic.
Proof. We show the Jacobian determinant of the map is no-where vanishing. We use an essential property: the primitivity of ζI2 (p) [19]. Since
the proof uses relative de-Rham cohomology theory for the family (2.3)
which is beyond the scope of present paper, a complete proof is left tothe literature but we give here a brief sketch of its idea.
There exists, so called, the Gauss-Manin covariant diﬀerentiation
operator ∇ on the module (over OSI2 (p) ) of relative de Rham cohomology classes of (2.3). Then, one basic property, called the primitivity,of a primitive form is that its covariant diﬀerentiations ∇∂gs ζI2 (p) and∇∂gl ζI2 (p) generate the relative de-Rham cohomology module (here ∂gs
18It is rather restrictive view point to study integrals only over σI2 (p) -invariant
cycles or over equivalent class of cycles. This is caused by the fact that our family(2.1) is already the subfamily of the full families of type A3 or D4 , and is fixed bya cyclic action σ (recall Footnote 1 and 7) where the full (A3 or D4 ) lattice L does
not play role. The studies of period integrals of the primitive form over the fulllattice L in the big families of type A3 and D4 (unpublished) are by themselvesinteresting subject and should appear elsewhere.

24

KYOJI SAITO

and ∂gl stand for the partial derivatives w.r.t. the coordinate systemg). Then, standard duality between the de-Rham cohomology groupand the (σI2 (p) -invariant) singular homology group of the curve EI2 (p),gH][H
H∇∂gs ζI2 (p) Hγ0 ∇∂gl ζI2 (p)γ0H
̸= 0. Since the integral comimplies det∇∂gs ζI2 (p) γ1 ∇∂gl ζI2 (p)γ1mutes with the derivation action ∂g and the [covariant
diﬀerentiationH
H]∂gs Hγ0 ζI2 (p) ∂gl Hγ0 ζI2 (p)∇∂g , we see that the Jacobian determinant det∂gs γ1 ζI2 (p) ∂gs γ1 ζI2 (p)̸= 0 does not vanish.
□
Let us now formulate the first main theorem of the present paper.
Theorem 5.1. The period map (5.4) induces a biholomorphic map(5.6)

PI2 (p) : SeI2 (p)

≃

e ,
H

where the RHS of (5.6), so called the period domain, is given as(5.7)

e := {ω ∈ HomZ (LI (p) , C) | Im(ωα /ωβ ) > 0}H2
= {ω ∈ HomZ (L/rad(L), C) | Im(ωγ1 /ωγ0 ) > 0},(here we used again (5.5) for the identification of the first and thesecond lines). The map (5.4) is equivariant with the action of the groupΓ1 ([p/2]) ⊂ SLZ (LI2 (p) ) ∩ SL2 (Z).
Proof. This result for the case of type A2 = I2 (3), is the well-knownclassic (e.g. see [27]). We want to show the parallel world for the othertypes B2 and G2 exists. The complete proof of this theorem can begiven after solving the Jacobi-inversion problem for those period maps
in Theorem 9.2 in §9. In the present section, we show only the followingclassical well-known fact:
Fact 13. The image of the period map (5.4) is contained in RHS ofee g(5.6). In particular, by the period integral (5.2), the lattice L/rad(L)
for g ∈ SeI2 (p) is embedded into a discrete lattice, called the period lattice:(5.8)ΩL/rad(:= Zωγ0 + Zωγ1
e
e
L),g

in the complex plane C.
Actually, this ∫is well-known as a consequence of (1) the Riemann’s
inequality: 2√1−1 E I (p),g ζI2 (p) ∧ζI2 (p) > 0, due to the positivity of the real
2∫
volume form 2√1−1 ζI2 (p) ∧ ζI2 (p) and (2) the Stokes relation: E I (p),gζI2 (p)∧
2
H
H
H
H
ζI2 (p) = γ1ζI2 (p) γ0ζI2 (p)− γ0ζI2 (p) γ1ζI2 (p) , due to the fact that the cycles γ0and γ1 give a canonical dissection of the real surface E I2 (p),g where ζI2 (p)has no poles (i.e. it is only a topological but not analytical property.
c.f. the first row of Figure 3.).
□

PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

25

Before ending this section, let us consider also the integrals over the
special arcs δi (4.16) constructed in Fact 1 in §2 and Fact 10 in §4.
Namely, for every g ∈ S̃I2 (p) and 1 ≤ i ≤ [p/2], set∫(5.9)ωδi :=
ζI2 (p) .
δi

Recall that the δi ’s are cyclically permuted by σI2 (p) and their sum
is homologous to γ0 (see (4.16) and Fact 10.1). On the other hand,
one sees immediately from the expression (5.1) that the form ζI2 (p) isinvariant by the action of σI2 (p) . These together implies ωδ1 = · · · =ωδ[p/2] and ωδ1 + · · · + ωδ[p/2] = ωγ0 . Thus those arc integrals (5.9) are
expressed in terms of a classical period of a vanishing cycle as follows.
(
)
1
1
ωβ =
ωγ0
(5.10)
ωδ1 = · · · = ωδ[p/2] =
[p/2]
[p/2]
Finally in this section, let us notice some elementary but useful facts.
Fact 14. The periods ωγi and ωδi (i = 1, · · · , [p/2]) are weighted homogeneous functions on S̃I2 (p) of weights given in Table 1. That is, theC× action on SI2 (p) is naturally lifted to that on S̃I2 (p) , which, for an
abuse of notation, we shall denote g 7→ twt(g) g, so that we have thefollowing equivariance w.r.t. the action of t ∈ C× .
ωγi (twt(g) g) = twt(ωγi ) ωγi (g) andωδi (twt(g) g) = twt(ωδi ) ωδi (g)where wt(ωγi ) = wt(ωδi ) = wt(z). Putting negatively graded structure
on RHS’s of (5.4) and (5.6) by multiplication twt(z) for t ∈ C× , themaps (5.4) and (5.6) are equivariant with respect to the C× -action.
Proof. Recall that the C× action on the space XI2 (p) extends to
its partial compactification X I2 (p) in such manner that the restrictionof the action on the divisors ∞i × SI2 (p) are equivariant with that on
SI2 (p) . Then, the action induces an action on the set of paths in E I2 (p),gstarting from ∞1 × g, where the end point of the path is acted by
the C× . We replace the integral (6.1) over a path by the integral
over the “acted” path. Due to the expression (5.1), we have wt(z) =wt(x) + wt(y) − wt(FI2 (p) ). 2.
6. Jacobian variety
We study integrals of the primitive form ζI2 (p) over paths in thesmooth part of the curve E I2 (p),g for each fixed g ∈ SI2 (p) \DI2 (p) which
may not neccesarily be closed. That is, we study the Jacobian variety

26

KYOJI SAITO

of the curve E I2 (p),g .19
point at infinity ∞1 :Precisely, we focus on the integrals from the

(6.1)

z :=

∫ (x,y)∼∞1

ζI2 (p)∈ C

where (x, y)∼ is a point in the universal covering (with respect to thebase point ∞1 ) of the curve E I2 (p)g which lies over a point (x, y) ∈E I2 (p)g , or, equivalently, a homotopy class of rectifiable paths in the
curve E I2 (p)g from ∞1 to (x, y) ∈ E I2 (p)g (the notation (x, y)∼ is ambiguous and we use it only here).
It is a classic that the integral (6.1) induces a biholomorphic map
from the universal covering of E I2 (p),g to the complex plane C (Proof.
The map is locally bi-regular (Fact 11) and is equivariant with the cov∼
ering transformation of π1 (E I2 (p)g , ∞1 ) on E I2 (p),g with the translation
action by the full period lattice ΩL/rad(ee = Zωγ0⊕Zωγ1 on C (Fact 13)).
L)g
For a later use in the study of inversion problem, let us confirm thedirection of the Hamiltonian FI2 (p) at the base point g 0 ∈ ΓI2 (p) as
follows.
Fact 15. By the map (6.1) for g 0 ∈ ΓI2 (p) , the cycle (3.4) is mapped to
the real interval [0, ωβ ] in the z-complex plane.
i
In particular, one has the correspondence: ∞i ↔ [p/2]ωβ mod Zωβ .

Figure 5. Period lattice ΩG2 ,g and its fundamental domain
The big black spots indicate points in ΩG2 ,g , and other small spots indicate otherplaces of the poles of the functions xI2 (p) (z, g) and yI2 (p) (z, g). Shaded area is afundamental domain (=period parallelogram) for translation action of ΩG2 ,g .

Proof. Let’s first observe that the integral (6.1) over the arc δ1 takes
real increasing values. For the purpose, we only need to notify that
19We can study in parallel the case when g belongs to the discriminant D

I2 (p)

by replacing E I2 (p)g by E I2 (p)g \Sing(E I2 (p)g ) and LI2 (p) by the lattices of rank 1
and 0. Details are left to the reader.

PERIOD INTEGRALS OF TYPE A2 , B2 AND G227

δ1 is a real path as in Figure 2. and the integrant ζI2 (p) = ∂FI dx
/∂y
2 (p)
takes positive real values on the arc δ1 , or equivalently, the function∂FI2 (p) /∂y takes negative real values when x is decreasing and takes
positive real values when x is increasing on the path δ1 (near at ∞1 ).
This can be confirmed directly depending on cases using the condition (3.5). The same argument works for the integral over the arcs
(δ2 , ∞2 ), · · · , (δ[p/2] , ∞[p/2] ). Then, recall that the cycle (3.4) is homologous to γ0 = β.
□∫ (x,y)∼We consider the inverse map: z = ∞1 ζI2 (p) ∈ C 7→ (x, y) ∈E I2 (p),g . More precisely, we associate to z ∈ C the coordinate valuesof the corresponding point (x, y) ∈ E I2 (p),g by the relation (6.1) anddenote it by(6.2)

xI2 (p) (z, g) and yI2 (p) (z, g),respectively.20 To be careful, the values of the “functions” xI2 (p) andyI2 (p) may not be defined when (x, y) represents a point at infinity (2.7),
however, the function is obviously holomorphic at other points and we
see easily those undefined points are removable to a holomorphic or
meromorphic function. 21
Recalling the two local expressions in Footnote 16 of the de Rhamclass of ζI2 (p) , we obtain the following Hamilton’s equation of motion:(6.3)∂xI2 (p) (z, g)∂FI2 (p) (x, y, g)∂yI2 (p) (z, g)∂FI2 (p) (x, y, g)
=
and
=−
.
∂z
∂y
∂z
∂x
However, this equation of the motion (which depends only on gs but
not on gl ) alone does not determine the solution (6.2) uniquely. In
order to recover the functions (6.2) as a function in z, we need to put
the following constraint on the energy level (depending on gl ) of the
20Inversion expression of the coordinate (x, y) with respect to the integral value∫ (x,y)√w = ∞ ω over the elliptic integral of the first kind ω = dx/ 4x3 − gs x + gl is well-known to be given by Weierstrass p-function and its derivative as x = p(w), y =p′ (w). Since there exists a factor relation ζA2 = ω/2, we have the relation w = 2z
and the period lattice gets half size. Then, xA2 (z) = 41 p(z), yA2 (z) = 81 p′ (z).
21Here, one should be slightly cautious that x
I2 (p) and yI2 (p) are (at present stage)
as functions in g ∈ SI2 (p) \ DI2 (p) only pointwise. Their holomorphic dependence
on g can be shown again by a use of Leray’s residue formula (details are omitted).
However, their extendability to DI2 (p) is not a priori obvious). Actually combining
with the discussions in Footnote 15, it is possible to show that they actually areextendable to the functions on the whole SI2 (p) . However, without using that logic,
we show directly in Lemma 7.1 in §7.

28

KYOJI SAITO

motion:(6.4)

FI2 (p) (xI2 (p) , yI2 (p) , g) = 0

We remark also that the functions xI2 (p) and yI2 (p) , which are no-longer
polynomials but meromorphic in z, are still weighted homogeneousfunctions if we give the weights to x, y, z and g as given in the Table 1.
That is, we have the following equivariance w.r.t. the action of t ∈ C× .
(6.5)

xI2 (p) (twt(z) z, twt(g) g) = twt(x) xI2 (p) (z, g),yI2 (p) (twt(z) z, twt(g) g) = twt(y) yI2 (p) (z, g)(Proof of (6.5).) Recall that the C× action on the space XI2 (p) extends
to its partial compactification X I2 (p) in such manner that the point ∞1
at infinity of a curve stays at infinity ∞1 of the curve whose parameterg is acted by the C× action. Then, the action induces an action on theset of paths starting from ∞1 to a point in the curve, where the end
point of the path is acted by the C× . We replace the integral (6.1) overa path by the integral over the “acted” path. Due to the expression(5.1), we have wt(z) = wt(x) + wt(y) − wt(FI2 (p) ). 2).
One crucial fact here is that the only variable z has the negative
weight (recall Table 1) and the functions can be (and, actually, is)transcendental in z.
7. Laurent series solutions at infinities
We study formal Laurent series solutions of the equation of the motion (6.3) together with the constraint (6.4) of the energy level andthe weight condition (6.5). The solutions are exactly in one to oneeI (p),g .
correspondence with the set of points at infinity of the curve E
2
More exactly, we do the following shift of center of Laurent expansion. Namely, if xI2 (p) (z) or yI2 (p) (z) has a non trivial pole atz = ω(g) (where ω(g) is a function of g ∈ SeI2 (p) of weight = wt(z),then we consider the Laurent expansion of the pair (x(z), y(z)) :=(xI2 (p) (z + ω(g)), yI2 (p) (z + ω(g))) with respect to the local formal coordinate z at 0. The pair (x(z), y(z)) satisfies the pair of equations(5.3)*∂x(z,g)∂FI2 (p) (x,y,g)
=
∂z
∂y

and

∂y(z,g)∂FI2 (p) (x,y,g)
=
−
∂z
∂x

since the equations (6.3) are invariant by the shift of the center of the
expansion. Next Lemma classify all formal solutions of the equations
with non-trivial poles.
Lemma 7.1. Consider the system of the equations (6.3) together withthe constraint (6.4) and the weight condition (6.5). Then it has exactly[p/2]-pairs of formal Laurent series solutions having non-trivial pole,

PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

29

which are in one to one correspondence with the set {∞1 , · · · , ∞[p/2] }of the points at infinity (2.7) of the curve E I2 (p) .
(7.1){Solutions with non-trivial pole }(x(z), y(z))≃
7→{Points at infinity}lim (x(z), y(z))
z↓0In particular, the linear transformation σI2 (p) (3.11) acts on the setof solutions cyclically equivariant with the bijection (7.1). The coeﬃcients of the Laurent series are Q-coeﬃcients weighted homogeneouspolynomials in g so that the solution is a pair of weighted homogeneousfunctions of weight (wt(x), wt(y)).
Proof. The proof is divided into 4 steps.
Step 1. For each type I2 (p), consider the pair of Laurent series:
(7.2)
∑
∑∞
2n
A
z
and
y
(z,
g)
=
Bn z2n−1xA2 (z, g) = ∑ ∞
n
A
2
n=−a
∑n=−b
∞
∞
2n+1
xB2 (z, g) = ∑and yB2 (z, g) = ∑n=−b Bn z2nn=−a An z∞∞nn
and yG2 (z, g) =xG2 (z, g) =n=−a An zn=−b Bn z
of indeterminate coeﬃcients An and Bn with non vanishing leading
terms A−a B−b ̸= 0 22 (according to the even property of xA2 and yB2
or odd property of yA2 and xB2 caused by the Z/2-symmetry (x, y) →(x, −y), the sum consist either of even or odd powers in z). We assume
that the pair has non-trivial pole, i.e. at least one of a or b is positive.
Here the coeﬃcients are weighted homogeneous functions in g ∈ SeI2 (p) ,a priori not necessarily polynomials, of weightwt(An ) = wt(Bn ) = (1 + n)/dwhere d = 3, 2 or 3 according as I2 (p) = A2 , B2 or G2 (use Table 1 forweights for x, y and z). Therefore, for each type I2 (p), there exists a
positive integer n0 such thatwt(An0 ) = wt(Bn0 ) = 1 = wt(gl ).
(actually, n0 = d − 1 = 2, 1 or 2 according as I2 (p) = A2 , B2 or G2 ).
Step 2. Using only the equations (5.3)∗ double inductively on An (n ≥−a) and Bn (n ≥ −b), one can determine the coeﬃcients until the
degree n < n0 , where we observe two basic facts.
(1) The initial term (A−a , B−b ) are constants independent of the parameter, and we have exactly [p/2]-number of solutions. More exactly,
according to each initial direction condition listed in the following table(7.3), there exists a unique solution satisfying it, where, in the last case,xI2 (p) does not have non-trivial pole. The list of explicit solutions is
22Here, we, unfortunately, use the notation A and B for the coeﬃcients ofnnthe Laurent series, which have nothing to do with the classification names An andBn for root systems. Since they are used only inside present proof, one should
cautiously read them.

30

KYOJI SAITO

given in (7.4), where one confirm that the coeﬃcients are Q-coeﬃcientpolynomials in gs . The calculation is case by case and we omit details.
A2 ∞1 : A−a > 0 and B−b < 0
B2 ∞1 : A−a > 0 and B−b < 0B
2 ∞2 : A−a < 0 and B−b > 0(7.3)
G2 ∞1 : A−a > 0 and B−b < 0
G2 ∞2 : A−a < 0 and B−b < 0
G2 ∞3 : a ≤ 0
and B−b < 0
Already in this initial solutions level (under the assumption that theyshall later extend to full solution), we can confirm the bijection (7.1)
by the use of Figure 2, where the “roots” of paths δi ’s are the infinity
points ∞i . Therefore, we indicated the point at infinity in the table(7.3) of initial conditions and in the table (7.4) of (partial) solutions.
(2) The second nontrivial term ((A1 , B1 ) for type A2 and (A0 , B0 )for types B2 and G2 ) contains the variable gs non-trivial linearly.
Step 3. We next use the energy condition (6.4) to determine the coefficients An0 , Bn0 (actually, the equation (6.3) alone cannot determine
the energy level). We confirm that gl appears non-trivially in An0 , Bn0 .
According to the 6 initial conditions listed in (7.3), the results are
given below. In particular, we confirm that the variable gl appears as
a non-trivial linear term in An0 and/or in Bn0 .
(7.4)
∑
2n
xA2 ,∞1 (z) = 41 z−2 + 15 gs z 2 + 47 gl z4 +∑ ∞n=3 An z
2n−1yA2 ,∞1 (z) = − 14 z−3 + 15 gs z + 87 gl z3 + ∞n=3 Bn z( 1 2 4 ) 3 ∑∞xB2 ,∞1 (z) = 12 z−1 + 13 gs z + 18
A z2n+1g − g z +( 1 2s 65 )l 2 ∑∞n=2 n 2n1 −21yB2 ,∞1 (z) = − 4 z + 6 gs + 12 gs − 5 gl z + n=2 Bn z
∑
1 2
2n+1
xB2 ,∞2 (z) = − 21 z−1 − 31 gs z − ( 18
gs − 54 gl )z3 + ∑ ∞n=2 An z
∞
6
1 −2
1
1 2
2n
2
yB2 ,∞2 (z) = 4 z − 6 gs − ( 12 gs − 5 gl )z + n=2 Bn z∑nxG2 ,∞1 (z) = 21 z−1 + 12 gs + 32 gs2 z + (gs3 − 12 gl )z2 + ∞n=3 An z
∑
∞
3
1 −1
3
3 2
3
2
yG2 ,∞1 (z) = − 2 z + 2 gs − 2 gs z + (3gs − 2 gl )z + n=3 Bn zn∑∞nxG2 ,∞2 (z) = − 21 z−1 + 21 gs − 32 gs2 z + (gs3 − 21 gl )z2 + ∑n=3 An z
∞
3
1 −1
3
3 2 2
3
2
yG2 ,∞2 (z) = − 2 z − 2 gs − 2 gs z − (3gs − 2 gl )z + n=3 Bn zn∑
A2n z2nxG2 ,∞3 (z) = −gs + (−2gs3 + gl )z2 + ∑∞
n=2
∞
2
2n−1
−1
3gs z
yG2 ,∞3 (z) = z ++n=2 B2n−1 z
Step 4. To determine the coeﬃcients An and Bn for n > n0 , we use
again the equation (6.3). By inserting (7.4) in (5.3)∗, compare coefficients of the Laurent expansions in BHS. Let Cn (resp. Dn ) be thecoeﬃcient polynomial of the power of z in the RHS of (6.3) whose degree coincides with the term An (resp. Bn ) in LHS. The Cn and Dn
are rational coeﬃcients weighted homogenous polynomials in gs , gl andPERIOD INTEGRALS OF TYPE A2 , B2 AND G2

31

Am , Bm (m ∈ Z>n0 ). Since the total weight of Cn (resp. Dn ) is equal towt(An ) = wt(Bn ), the coeﬃcient Cn (resp. Dn ) cannot contain Am , Bm
for m > n and An and Bn appear only linearly with constant coeﬃcients. The linear coeﬃcients are independent of n, since such terms
appear in the expansion of RHS of (5.3)∗ only when the term An or Bn
multiplied with the constant coeﬃcient terms of xI2 (p) and yI2 (p) , thatis lowest degree terms A−a z −a and B−b z −b (see (7.4)). But, such pattern does not depends on n ∈ Z>n0 . Calculating explicitly the linear
coeﬃcients, we obtain the equations:A2 type : 2nAn = 2Bn + Cn′ ,(2n − 1)Bn = 6An + Dn′(7.5) B2 type : (2n + 1)An = 2Bn + Cn′ , 2nBn = 3An + Dn′
G2 type : nAn = −An + Bn + Cn′ , nBn = 3An + Bn + Dn′
where Cn′ and Dn′ are the remaining part of Cn and Dn after subtracting the linear terms in An and Bn . We observe immediately that thedeterminant of coeﬃcients of An and Bn in the two equation for thethree types [are given ]by
[
]
[
]
A2 :

2n −2
−2
−1
det −6B2 :det 2n+1G2 :det n+12n−1−3 2n−3 n−1= 2(2n+3)(n−2),= 2(2n−3)(n+2),= (n+2)(n−2)

which takes positive values for n > n0 . Thus An and Bn are uniquely
expressed as rational coeﬃcients polynomials in Am , Bm (n0 ≤ m < n)and gs . This gives the inductive construction of the coeﬃcients An andBn (n ∈ Zn0 ).
This completes the proof of Lemma 7.1 .
□As a result of Lemma 7.1, we can determine the principal parts of
the Laurent expansions of the meromorphic functions xI2 (p) (z, g) andyI2 (p) (z, g) (6.2).
Fact 16. 1. The formal Laurent series solutions in Lemma 6.1 areconvergent.
2. The following substitutions of z in the formal series solution (7.4):
(7.6)

i−1
xI2 (p),∞i (z − ( [p/2]ωγ0 + ωγ )),i−1yI2 (p),∞i (z − ( [p/2]ωγ0 + ωγ ))give the Laurent expansions of the meromorphic functions xI2 (p) (z, g)i−1
ωγ0 + ωγ for any 1 ≤ i ≤ [p/2] andand yI2 (p) (z, g) at the place [p/2]γ ∈ L̃/rad(L̃).
Proof. 1. can be shown as a consequence of the next 2.
2. We know already from geometry (recall a discussion after thedefinition (6.2) and the description of Fact 15) that the functions
i−1
xI2 (p) (z, g) and yI2 (p) (z, g) may have poles only at the places [p/2]ω0 + ωγ
for i = 1, · · · , [p/2] and γ ∈ L̃/rad(L̃). In view of the asymptotic behavior of paths δi ’s at their starting points in the first row of Figure 1,

32

KYOJI SAITO

we observe that all of them become poles except that the function xG2
at the places 23 ω0 + ωγ for any γ ∈ L̃/rad(L̃)) does not have a pole
Obviously, the Laurent expansions at those places should satisfy theequations (6.3) together with the constraint (6.4), which further satisfy the initial constraint (7.3) according to its location. Then, theuniqueness of the solution of the equations under the initial constraint
implies that the formal solution should coincide with the expansion ofxI2 (p) (z, g) or yI2 (p) (z, g).
□
We note that the proof of Lemma 6.1 actually covers also the Laurent series expansions of the cases for g ∈ DI2 (p) . The (7.3) express
the Laurent series expressions of the coordinate (x, y) for the cases ofg ∈ DI2 (p) (recall Footnote 15 and 17.) as trigonometric or rational
functions in z.
Let us give explicitly the first few terms of the Laurent expansion ofxI2 (p) (z, g) and yI2 (p) (z, g) at the origin z = 0 as follows.
A2 case.
4 2 6
48
xA2 (z) = 14 z −2 + 15 gs z 2 + 74 gl z 4 + 75
gs z + 385
gs gl z 8 + · · ·
4 2 5
yA2 (z) = − 41 z −3 + 15 gs z + 87 gl z 3 + 25
gs z + 192
g g z7 + · · ·385 s lB2 case.
(1 2 4 ) 3 (1 3)8xB2 (z) = 21 z −1 + 13 gs z + (18gs − 5 gl )z + (27 gs − 35
gs gl) z 5 + · · ·1 2
5 3
gs − 65 gl z 2 + 54
gs − 74 gs gl z 4 + · · ·yB2 (z) = − 41 z −2 + 61 gs + 12
G2 case.
xG2 (z) = 12 z −1 + 12 gs + 32 gs2 z + (gs3 − 12 gl )z 2 + ( 23 gs4 − 65 gl gs )z 3 + · · ·yG2 (z) = − 21 z −1 + 32 gs − 23 gs2 z + (3gs3 − 32 gl )z 2 − ( 32 gs4 − 65 gl gs )z 3 + · · ·Table 2: Laurent expansions of xI2 (p) and yI2 (p) at z = 0.
8. Partial fractional expansions
We come back to the global study of the meromorphic functions
xI2 (p) (z, g) and yI2 (p) (z, g) in z (6.2). The goal of this section is to givethe partial fractional expansion of them. The study belongs to classicalelliptic function theory. In particular, A2 -type case is well-known as
Weierstrass p-function theory. We generalize it for the other two types
B2 and G2 , since those descriptions in the present section, lead to“generalized Eisenstein series” expression of the modular forms for the
congruence subgroups Γ1 (2) and Γ1 (3) in the next section, which seems
to have be unknown (see also Remark 2 in §9).

PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

33

We recall the classical Weierstrass’s p-function and ζ-function assoe as meromorphic functions on theciated with any point (ω0 , ω1 ) ∈ H
z-plane with double or simple poles (see, e.g. [13]).
)
∑ ( 1
1
p(z) = z12 +
−
2
2
(z−ω)
ω
ω̸=0∈Ω
(
)
∑
1
1
1
z
ζ(z) = z ++ ω + ω2 ,z−ω
ω̸=0∈Ω

where we set Ω := Zω0 + Zω1 . Since they are compact uniforme \ ∪m,n∈Z {z − (mω1 + nω0 ) = 0} , one may
convergent on C × H
derivate them termwisely. In particular, one has the well-known relation: ζ ′ (z) = −p(z).23
Theorem 8.1. The meromorphic functions xI2 (p) (z, g) and yI2 (p) (z, g)have the following partial fractional expansions.
(8.1)
xA2 (z, g) = 14 p(z)yA2 (z, g) = 81 p′ (z)xB2 (z, g) = − 12 ζ( 12 ω0 ) + 21 ζ(z) − 12 ζ(z − 12 ωγ0 )
yB2 (z, g) = − 14 p(z) + 41 p(z − 21 ωγ0 )
xG2 (z, g) = − 16 ζ( 31 ωγ0 ) − 16 ζ( 23 ωγ0 ) + 12 ζ(z) − 12 ζ(z − 31 ωγ0 )
yG2 (z, g) = 12 ζ( 13 ωγ0 ) + 12 ζ( 23 ωγ0 ) − 12 ζ(z) − 21 ζ(z − 13 ωγ0 ) + ζ(z − 32 ωγ0 )where the p-function and the ζ-function in RHS are those associatede and, hence, with the period lattice Ω =with the period (ωγ0 , ωγ1 ) ∈ H,
(5.8).
ΩL/rad(eeL),gProof. Owing to (7.4) and Fact 16, we know already the principal partsof poles of the functions xI2 (p) (z, g) and yI2 (p) (z, g) in z. Since theprincipal parts of poles of p(z), p′ (z) and ζ(z) are 1/(z−ω)2 , −2/(z−ω)3
and 1/(z − ω) for ω ∈ ΩI2 (p),g , respectively, we see that the sum in thebracket of RHS of the following (8.2) give meromorphic functions onz, whose principal parts of poles coincide with those of xI2 (p) (z, g) and[]yI2 (p) (z, g) in z, respectively (here, we denote by f (z) the set of all
23The notation “ ′to z.

” or “ ′′ ” shall mean single or twice derivative with respect

34

KYOJI SAITO

principal parts of a meromorphic function f (z) defined on z-plane).
][] [1[xA2 (z, g)] = [ 14 p(z)]yA2 (z, g) = 8 p′ (z)
(8.2)

]
[
] [1
− 12 ζ(z − 21 ωγ0 ) ][xB2 (z, g)] = [ 2 ζ(z)yB2 (z, g) = − 14 p(z) + 14 p(z − 21 ωγ0 )
[
] [1
]
1
1
x
(z,
g)
=
ζ(z)
−
ζ(z
−
ω
)
G
γ
0
2
3
[ 2
] [2
]
yG2 (z, g) = − 12 ζ(z) − 12 ζ(z − 13 ωγ0 ) + ζ(z − 32 ωγ0 )On the other hand, we remark that the functions in the bracket of
the RHS of (8.2) are periodic functions w.r.t. the period lattice ΩI2 (p),g ,since (1) the functions p(z) and p′ (z) are already periodic, and (2) thesum of coeﬃcients of the linear combinations of the functions of theform ζ(z + ∗) in each formula is equal to zero and, then, it is wellknown that the linear combination is a periodic function (see, e.g. [13]Ch.1§12, this follows from an elementary property of zeta function thatζ(z + mω0 + nω1 ) − ζ(z) = m2ζ(ω0 /2) + n2ζ(ω1 /2) for ω ∈ Ω). Thus,
due to Liouville’s Theorem, the diﬀerence of meromorphic functions in
the brackets of both hand sides of (8.2) are constants.
Actually, the data of the principal parts of poles are not suﬃcient tocontrol the ambiguity of adding constant terms except for the case oftype A2 . Namely, the Laurent expansion for the type A2 at z = 0 (seeTable 2 at the end of §7) does not have constant terms but those forthe other types B2 and G2 contain non-trivial constant terms, which
are linear in gs and which are still to be determined from the data of
the lattice ΩI2 (p) . In order to overcome this issue, we use the Hamilton
equation (6.3) of the motion. This is essentially new feature to be
cautious compared with the classical case of type A2 .
Let us determine the constants depending on the type separately.
A2 type case: Since the constant terms of the Laurent expansions at
the origin of BHS are zero, the diﬀerence is zero, and we obtain already(8.1) for type A2 .
B2 type case: Set, for suitable constants (w.r.t. z) A and B,
xB2 (z, g) = A + 12 ζ(z) − 12 ζ(z − 12 ωγ0 )
yB2 (z, g) = B − 14 p(z) + 41 p(z − 12 ωγ0 )In the first equality, since the constant terms of the Laurent expansionsof xB2 (z, g) and ζ(z) are zero (recall (7.4) and the fact that ζ(z) is an
odd function) the sum of the remaining terms A − ζ(0 − 21 ωγ0 ) is equal
to zero. This determine A = −ζ( 12 ωγ0 ).

PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

35

For the second row of the equality, recall the Hamilton’s equation of∂x
the motion (6.3) ∂zB2 = 2yB2 . The LHS is equal to − 21 p(z) + 12 p(z −1ω ), and substituting yB2 in the RHS, we see that 2B = 0. These2 γ0already gives (8.1) for type B2 . Then, by comparing the constant termsof the Laurent expansion of the BHS of (8.1) in view of (7.4), we obtain1
g = 14 p(0 − 12 ωγ0 ). Hence,
6 s
(8.3)

A = −ζ( 12 ωγ0 )B = 0
gs = 32 p( 12 ωγ0 ).

G2 type case: Set, for suitable constants w.r.t. z, A and B,
xG2 (z, g) = A + 12 ζ(z) − 12 ζ(z − 13 ωγ0 )
yG2 (z, g) = B − 21 ζ(z) − 12 ζ(z − 31 ωγ0 ) + ζ(z − 23 ωγ0 ).
Recall the Hamilton’s equation of the motion (6.3)∂xG2= 2xG2 yG2 + 2gs yG2 .
∂z
The LHS is equal to − 21 p(z)+ 12 p(z − 13 ωγ0 ) so the residue at any pole of
LHS is equal to zero. Thus, we obtain two relations that the residues
at at z = 32 ωγ0 and at at z = 13 ωγ0 of the meromorphic function() ()
2 A( + 12 ζ(z) − 21 ζ(z − 13 ωγ0 ) × B − 12 ζ(z) −) 21 ζ(z − 13 ωγ0 ) + ζ(z − 23 ωγ0 )+ 2 gs B − 12 ζ(z) − 12 ζ(z − 13 ωγ0 ) + ζ(z − 32 ωγ0 )are zero.
(1) Residue at z = 23 ωγ0 : 2A + ζ( 23 ωγ0 ) − ζ( 23 ωγ0 − 13 ωγ0 ) + 2gs = 0(2) Residue at z = 31 ωγ0 : −A − 21 ζ( 13 ωγ0 ) − B + 21 ζ( 13 ωγ0 ) − ζ( 13 ωγ0 −2ω ) − gs = 0
3 γ0
In addition to them, let us consider two more relations:(3) 12 gs = A − 21 ζ(0 − 13 ωγ0 )(4) 32 gs = B − 21 ζ(0 − 13 ωγ0 ) + ζ(0 − 23 ωγ0 )obtained by comparing the constant terms of Laurent expansion of theequalities: xG2 = A + 12 ζ(z) − 12 ζ(z − 13 ωγ0 ) and yG2 = B − 12 ζ(z) −1
ζ(z − 13 ωγ0 ) + ζ(z − 32 ωγ0 ).
2Recalling the fact that ζ is an odd function, we see that (1), (2), (3)and (4) are overdetermined system for A, B and gs , and we obtain thesolution:A = − 16 ζ( 13 ωγ0 ) − 16 ζ( 32 ωγ0 )1B =ζ( 13 ωγ0 ) + 21 ζ( 23 ωγ0 )
(8.4)
2
2
ζ( 13 ωγ0 ) − 13 ζ( 23 ωγ0 )gs =3This completes the proof of Theorem 7.1.
□

36

KYOJI SAITO

Remark 8.2. In order to get the equality (8.1), we have substituted thelattice Ω in the RHS by the period lattice ΩI2 (p),g . However, the expression in the RHS of (8.1) is defined in a self-contained manner for any
e Therefore, we shall hereafter regard RHS of (8.1) as
point (ω0 , ω1 ) in H.
meromorphic functions in z which are holomorphically parametrized bye where the holomorphicity follows from the compact uniform converH,
e regardlessgences of the series p and ζ also in the variable (ω0 , ω1 ) ∈ H,
whether it is in the image of the period map or not.
9. Eisenstein series of type A2 , B2 and G2- (Primitive automorphic forms)

We come back to the solve the inversion problem posed at Theorem
5.1. For the purpose, we use some generalizations of Eisenstein series toobtain inversion maps (see Theorem 9.2). For type A2 , this is classicallywell established theory. Our interest is to show that a generalizationof the theory works for types B2 and G2 (which is the first main goalof the present paper).
Definition 9.1. For each type I2 (p), the coeﬃcients of the Laurent
series expansion at z = 0 of the meromorphic functions in RHS of (8.1),e
as a weighted homogeneous holomorphic functions on (ω0 , ω1 ) ∈ H,
shall be called Eisenstein series of type I2 (p).
In the following, we determine explicitly all Eisenstein series of typeI2 (p). However, such explicit description is un-necessary to solve the inversion problem. So, some readers may skip the present paragraph till
Theorem 9.2. However, the explicit description are unavoidably important in §10, when we study the Fourier expansions of the polynomialsin C[gs , gl ] as modular forms.
e Depending on m ∈ Z≥3 andSet Ω := Zω0 + Zω1 for (ω0 , ω1 ) ∈ H.
a ∈ Rω0 + Rω1 = R ⊗Z Ω, let us consider series:
(9.1)
 ∑
dm−2 (p−z −2 )1(0)
ω −m = (m−1)!
dz m−2ω∈Ω\{0}
dm−1 (ζ−z −1 )1
= − (m−1)!
(0) (if a ∈ Ω)dz m−1Gm (a) := ∑
m−2 p
d
1
−m

(ω + a)=(m−1)! dz m−2 (−a)

ω∈Ω

m−1


=
− 1 d m−1ζ (−a)(if a ̸∈ Ω) .
(m−1)! dz

The first series for a ∈ Ω are the classical well-known classical Eisenstein series of weight m (see, e.g. [H-C,E-Z]). However, the second
series for a ∈/ Ω seem to have not appeared in literature. As we shallPERIOD INTEGRALS OF TYPE A2 , B2 AND G2

37see, since both behaves in parallel to the classical series, we shall callthe latter case shifted classical Eisenstein series of weight m.
It is absolute and locally uniformly convergent so that defines ae of weight −m · wt(z)24 parametrized byholomorphic function on Ha ∈ (Rω0 + Rω1 )/Ω, such that Gm (a) = (−1)m Gm (−a). In particular,
we have the relations:
Gm (a) = 0

for a ∈ 12 Ω and m = odd.

Using (9.1), one get the following Laurent and Tayler expansions.
The first two lines are standard (e.g. [13]), and the latter two for a ∈R ⊗ Ω \ Ω can be shown similarly.25
∑
2n
p(z)
= z −2 + ∞
n=1 (2n + 1)z G2n+2 (0)ζ(z)= z −1 −∑∞(9.2)
p(z − a) = p(a) +

n=1 z

2n+1

G2n+2 (0)∑∞m=1 (m + 1)zζ(z − a) = −ζ(a) − p(a)z −∑∞m

Gm+2 (a)

m=1 z

m+1

Gm+2 (a)Now, let us describe Eisenstein series for each type I2 (p) separately.
The calculation is straight forward from the formula (8.1), and we omit
details of them.
A2 type: Set
(9.3)

∑
2n
xA2 (z) = 41 z −2 + ∑∞n=1 An z∞
1 −3
yA2 (z) = − 4 z + n=1 Bn z 2n−1

Then, we have
(9.4)
B2 type: Set
(9.5)

An =
Bn =

2n+1
G2n+2 (0) (n ≥ 1)
4
(2n+1)nG2n+2 (0) (n ≥ 1)
4

∑∞
2n+1
xB2 (z) = 12 z −1 + ∑n=0 An z
2nyB2 (z) = − 14 z −2 + ∞n=0 Bn z

24We should be cautious about the use of the terminology “weight”. The weight

e comes from the C× -action (recall Fact−m · wt(z) of Gm (a) as a function on H
14). It is proportional to the weight m as the Eisenstein series, but depends on thefactor wt(z) which depends on type I2 (p) (recall Table 1) (c.f. (10.3) and Table 2).
25Actually, the last equality has meaning for the periodic variable a, even though
the zeta function is not periodic, since ζ is still “semi-periodic” (see [13]§11).

38

KYOJI SAITO

Then, we haveA0 =1
p( 12 ω0 )2

An = − 12 G2n+2 (0) + 12 G2n+2 ( 21 ω0 ) (n ≥ 1)
(9.6)
B0 =

1
p( 12 ω0 )4

Bn = − 2n+1
G2n+2 (0) + 2n+1
G2n+2 ( 12 ω0 ) (n ≥ 1)
4
4
G2 type: Set
(9.7)

∑
1 −1
n
xG2 (z) =z + ∑∞
n=0 An z
2∞yG2 (z) = − 21 z −1 + n=0 Bn z nThen, we haveA0 =1
ζ( 13 ωγ0 ) − 16 ζ( 23 ωγ0 )
3

A1 =

1
p( 13 ω0 )2=

1
ζ( 31 ωγ0 ) − 13 ζ( 12 ωγ0 )2An = − 21 Gn+1 (0) + 12 Gn+1 ( 13 ω0 )(9.8)

B0 = ζ( 13 ωγ0 ) − 12 ζ( 23 ωγ0 ) =(n ≥ 2)3
ζ( 13 ωγ0 ) − ζ( 21 ωγ0 )
2

B1 =

1
p( 13 ω0 ) − p( 23 ω0 )2

= − 21 p( 13 ω0 )

Bn =
=

1
G (0) + 12 Gn+1 ( 13 ω0 ) − Gn+1 ( 23 ω0 )2 n+1
1
G (0) + ( 12 + (−1)n )Gn+1 ( 13 ω0 )2 n+1(n ≥ 2)

Remark 2. 1. The infinite sequence of Eisenstein series for each typeI2 (p) are not algebraically independent. More precisely, they are obeying recurrence relation (7.5) in Step 4. of the proof of Lemma 7.1 (whichleads to the isomorphism (9.9), describing relations directly). The relations may be considered as the B2 -type and G2 -type generalizations of
the classically well-known A2 -type relations. However, in the present
paper, we do not go into details of the relations.
2. We note that the first Eisenstein series A0 , B0 in case of type B2
and the first and the second Eisenstein series A0 , B0 and A1 , B1 in caseof type G2 do not have the description using the classical series (9.1).
These exceptional behavior was caused by the fact that the classicalEisenstein series (9.1) do not converge absolutely in those low weights
so that one need to make conditional convergent series by a help of
p-function or ζ-function. This was made possible by the determination

PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

39of the constant terms of fractional expansions in Theorem 8.1 using theenergy condition (6.4) of the Hamilton’s equations of the motion.
In a forthcoming paper [2], we shall study systematically those “exceptional” Eisenstein series from a view point of modular forms.
Now we are to formulate the second main theorem of the present
paper. The proof is essentially done already in previous sections sothat we have only to coordinate them.
Theorem 9.2. Consider the pull-back homomorphism PI∗2 (p) : OHe →e
Oefrom the ring of holomorphic functions on the period domain H
SI2 (p)to that on the monodromy covering space SeI2 (p) of the base space SI2 (p)of the family (2.3) (recall (4.13) for the definition of SeI2 (p) and (5.4) for
the definition of PI2 (p) ).
Then, it induces the ring isomorphism:(9.9)

Q[Eisenstein series of type I2 (p)]≃Q[gs , gl ],where LHS is the ring over Q generated by all Eisenstein series of typeI2 (p) (recall Definition 9.1) and RHS is the coordinate ring of the space
SI2 (p) (recall (2.3)) generated by the flat coordinates gs and gl over Q.
In particular, the generators gs and gl are expressed by Eisensteinseries of type I2 (p) as follows
A2 type:(9.10)

gs =
gl =

15
G4 (0)
435
G6 (0)
16

gs =
gl =

3
p( 12 ω0 )2
5 2 1
p ( 2 ω0 ) + 85 G4 (0) − 58 G4 ( 12 ω0 )32

B2 type:(9.11)
G2 type:(9.12)

gs = ζ( 31 ω0 ) − 23 ζ( 12 ω0 )gs2 = 13 p( 13 ω0 )gl = 2gs3 − G3 ( 31 ω0 )Proof. The explicit description of the function on SeI2 (p) corresponding
to an Eisenstein series by the pull-back morphism PI∗2 (p) is obtainedby the corresponding coeﬃcient of the Laurent expansion at z = 0of the meromorphic functions xI2 (p) (z, g) or yI2 (p) (z, g) (8.1). Then,it was already shown in Lemma 7.1 that they are rational coeﬃcientpolynomials in gs and gl (c.f. Table 2 at the end of §7). This defines
the homomorphism (9.9) from left to right.

40

KYOJI SAITO

The morphism is injective, since the period map PI2 (p) (5.4) is an
open map between connected manifolds (Fact 12).
The morphism is surjective, since (1) the generator gs is, up to a
constant factor, given by the lowest weight Eisenstein series for eachtype (recall Step 2. of the proof of Lemma 7.1 and (7.4)), and (2) thegenerator gl appear non-trivially and linearly in the coeﬃcients An0
and Bn0 of the Laurent expansions of xI2 (p) (z, g) and yI2 (p) (z, g) (recall
Step 3. of the proof of Lemma 7.1 and (7.4)).
□
After the isomorphism (9.9), we shall sometimes identify the ring ofEisenstein series and the polynomial ring in gs and gl .
Proof of Theorem 5.1
We show by 4 steps that the period map PI2 (p) (5.6) is bi-holomorphic.
e belongs to theStep 1. Regardless, whether an element (ω0 , ω1 ) ∈ H
image of the period map or not, let us use the Eisenstein series of typeI2 (p) expressions (9.10),(9.11) and (9.12) to define a holomorphic map(9.13)e −→ SI (p) .
E = (Es , El ) : H
2

The equality (8.1) in Theorem 8.1 implies that the following diagram
is commutative(9.14)

SeI2 (p)PI2 (p)

e
−−−−−
−−→ H
↘
↙E
SI2 (p)

We remark that pull back of the polynomial ring on SI2 (p) by the morphism E (9.13) induces the same isomorphism (9.9), since (i) the period
map PI2 (p) is a non-trivial open map, and (ii) any algebraic dependence
e automaticallyrelation among Eisenstein series on an open domain in H
e by analytic continuation, since H
e is connected.
extends on the whole H
Step 2. Let us show that the image of E is contained in the complimente ⊂ SI (p) \DI (p) . For (ω1 , ω2 ) ∈ H,e using RHS
of the discriminant: E(H)22
of (8.1), we define global meromorphic functions xI2 (p) and yI2 (p) , which
are periodic w.r.t. the lattice Ω = Zω0 + Zω1 . Let us see that the pairsatisfies the relation (6.3) together with (6.4), where the parameter g
is given by (9.13). Actually, the both hand sides give doubly periodic
function of the period Ω = Zω0 + Zω1 , where we can check they havethe same principal parts of poles, and the constant term of Laurent
expansions at 0 coincides.
This means that the time coordinate z is given by the integral (6.1)(up to a shift of a constant). That is, the image of the map (xI2 (p) , yI2 (p)PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

41satisfies the equation (6.4). However, if g belonged to the discriminant,then the associated curve defined by the equation (2.1) is a singularrational curve. The integral (6.1) (avoiding the singularity of the curve
but admitting to go through points at infinity) cannot be doubly periodic (either one periodic for g ∈ DI2 (p) \{0}, or no-periodic for g = 0),e generates rank 2 lattices and xI (p)where as the starting (ω0 , ω1 ) ∈ H
2and yI2 (p) are doubly periodic. A contradiction!
e is conStep 3. Let us show that the period map is surjective. Since Heenected, for any point ω ∈ H, consider any path, say p, in H connecting
e of the base point loci of SeI (p) (reeI (p) ) in H
ω with the image PI2 (p) (Γ22call the definition (4.13)). Then, the projection image E(p) is a path in
SI2 (p) \ DI2 (p) connecting the base point loci ΓI2 (p) with E(ω) (recall the] ofcommutative diagram (9.14)). Then, the monodromy lifting E(p)the path E(p) in the covering space SeI2 (p) is a path connecting the base] which lies over E(ω). Then, the imageeI (p) to a point E(ω)
point loci Γ2] is a monodromy covering in H
e of the path E(p) connecting
PI2 (p) (E(p))] Since p is also is a monodromyeI (p) ) to a point PI (p) (E(ω)).
PI2 (p) (Γ22e of the same path E(p) connecting the base point locicovering in H
] and peI (p) ) to the point p. So the two end points PI (p) (E(ω))
PI2 (p) (Γ22of the paths should coincide each other. In particular, p is in the imageof the period map. This shows also that the modular group Γ1 ([p/2]),
which is the monodromy representation ρ-image of the fundamentalgroup of SI2 (p) \ DI2 (p) (Fact 9, 2.), acts on any fiber of the map E trane is isomorphic
sitively. That is, the modular group action quotient of H
to the discriminant compliment:e ≃ SI (p) \ DI 2)
Γ1 ([p/2])⧹H
2(

Step 4. Finally, let us show that the period map is injective. Since theperiod map is equivariant with the modular group action, it is suﬃcient
e is (generically)that the modular group action on the period domain H
fixed point free. But this is trivially true, since the modular group is
e is ana subgroup of GL2 (Z) so that its fixed points set is thin and H
2open subset of C . Since E is a covering map, if the action modulargroup is fixed point free in one fiber, it is fixed point free for all fibers.
This completes a proof of Theorem 5.1.
Remark 9.3. 1.
(9.15)∂(gs , gl )= c∆redI2 (p)∂(ω0 , ω1 )2

42

KYOJI SAITO

2. In [21, 23] , we posed a general question to describe the inversion
morphism to the period map defined by a primitive form. If a function
on the parameter space of the family is described in terms of the coordinates of the period domain, we call the function (and its description
on the period domain) a primitive automorphic form. In that sense, thegeneralized Eisenstein series of type B2 and G2 in this sections are are
the first examples of primitive automorphic forms beyond the classicalcase of type A2 .
10. Ring of modular forms and Discriminant
We identify the ring of Eisenstein series of type I2 (p) with the ring ofmodular forms of the congruence group Γ1 ([p/2]) (see [1] for M∗ (Γ1 ([p/2]))).
Then we confirm that the set of irreducible components of the discriminant of the family (2.3) is in one to one correspondence with the setof cusps of the congruence group Γ1 ([p/2]).
Theorem 10.1. The ring of Eisenstein series of type I2 (p) is identified
with the ring of holomorphic modular forms of the congruence groupΓ1 ([p/2]), where the identification is given in following (10.3). 26(10.1)

C[Eisenstein series of type I2 (p)]≃M∗ (Γ1 ([p/2])).

The correspondences of generators are given in (10.5), (10.6) and (10.7).
Proof. Proof of the theorem is divided into Steps 1-5.
Step 1. We explain the meaning of “identification”, and fix notation.
e := {(ω0 , ω1 ) ∈ C2 | Im(ω1 /ω0 ) > 0} with
Recall the period domain H
its homogenous coordinates (ω0 , ω1 ). We introduce the inhomogeneous
coordinate(10.2)

τ:= ω1 /ω0e → H := {τ ∈ C | Im(τ ) > 0},
Then the natural projection H(ω0 , ω1 ) 7→ τ := ω1 /ω0 gives a principal C× -bundle, say (L× )−1 , overe ≃ C× H, (ω0 , ω1 ) 7→ (ω0 , τ ).
H, which we trivialize by the morphism He and hence on L.
The modular group Γ1 ([p/2]) acts from the left on Hk
For k ∈ Z>0 , a holomorphic section of the L , say[ s =]s(τ ), such thata bk+bγ ∗ (s) := s · aτ
∈ Γ1 ([p/2]), iscτ +d is equal to (cτ + d) s(τ ) for γ =c dcalled a modular form of weight k of Γ1 ([p/2]) in a wide sense. Then,the correspondence s(τ ) 7→ ω0−k · s(τ ) defines the “identification”:
26This is naturally an expected result. However, this should have been proven,since the Eisenstein series appeared in the context of the geometry of the period
mapping, whereas the modular forms are defined independently by themselves. So,
their coincidence is a non-trivial marvelous fact, which we need to work cautiously.

PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

43(10.3)↔

{modular forms of weight k of the group Γ1 ([p/2]) in a wide sense}
e of weight −k · wt(z) invariant by Γ1 ([p/2])}{holomorphic functions on H

Actually, we study more restricted class of modular forms which are
holomorphic and taking finite values √at cusps, as we explain now.
Recall [15] that a point x ∈ R ∪ { −1∞} (= ∂H) is called a cuspof Γ1 (N ) if it is fixed by a hyperbolic element of Γ1 (N ). The isotropic
subgroup of Γ1 (N ) fixing a cusp is an infinite unipotent group, and
the
[ set
] of all cusps are invariant under the action of Γ1 (N ). For γ =a b∈ SL2 (Z), set (s|k γ)(τ ) := (γ ∗ s)(τ )(cτ + d)−k (so that thec dmodularity propertyof s is equivalent to s|k γ = s for all γ ∈ Γ1 (N )).
√Let γ(x) = −1∞ for a cusp x and γ ∈ SL2 (Z). For a modular
form s (in the wide sense), s|k γ, as a periodic function in τ , develops
into a Fourier series in τ . Then, s is called a holomorphic modular
form if √the Fourier series consists only of non-negative powers of q =exp (2π −1τ ) at all cusps of Γ1 (N ). The constant term of the√Fourier
series is called the value of s at the cusp and denoted by (s|k γ)( −1∞).
Step 2. We recall the results by Aoki and Ibukiyama on the ring ofmodular forms. In [1], Aoki and Ibukiyama gave a simple unified description of the graded ring of holomorphic modular forms of Γ0 (N )for N = 1, 2, 3, 4. From that description, we recover easily the ringM∗ (Γ1 (N )) of holomorphic modular forms of Γ1 (N ). Namely, according as N = 1, 2 and 3, the ring is generated by two (algebraicallyindependent) modular forms e4 , e6 of weight 4 and 6, α2 , β4 of weight
2 and 4, and α1 , β3 of weight 1 and 3, respectively. That is,(10.4)M∗ (Γ1 (1)) = C[e4 , e6 ], M∗ (Γ1 (2)) = C[α2 , β4 ], M∗ (Γ1 (3)) = C[α1 , β3 ]where explicit descriptions of the generators as theta-function, in particular, the first few Fourier coeﬃcients at the cusps are given in [1].
Step 3. We define a morphism from the left hand side to the right handside of (10.1). This is achieved by showing that the Eisenstein series
are holomorphic at all cusps.
More explicitly, recall that, according as N = 1, 2 and 3, the numberof Γ1 (N )-equivalence classes of cusps are 1, 2 and 2, whose representatives are given as follows.
A2 type:√−1∞, B2 type:√−1∞ and 0, G2 type:√−1∞ and 0.

Let us consider the ring of Eisenstein series of type I2 (p). Recall thatthe identification (9.9) was given by the composition with the period

44

KYOJI SAITO

map PI2 (p) , where the period map is Γ1 ([p/2])-equivariant. This meanse So, bythat any Eisenstein series is a Γ1 ([p/2])-invariant function on H.
the identification (10.3), it gives arise a modular form. If we, further,show that the Eisenstein series are holomorphic at all cusps, we obtaina graded ring homomorphism from the left hand side to the right handside of (10.1).
Lemma 10.2. The modular forms associated with Eisenstein series oftype I2 (p) (p = 1, 2, 3) are holomorphic at their cusp(s).
Proof. Let E =√−1∞ and 0 to

[
1
0

√

0
1

]

[

and S =

0
−1

1
0

]

∈ SL2 (Z), which transforms−1∞. We show that the series associated with s|k E

and/or s|k S for an Eisenstein series s with√the weight k converges
absolute uniformly in a “neighborhood” of −1∞. This is classicalfor the case of Eisenstein series s = Gm (0) for m ≥ 3 (see [11]), andsimilar proof works for Eisenstein series of the form s = Gm (a) for asuitable a ∈ ΩQ . In case of Eisenstein series of the form s = p(a) fora suitable a ∈ ΩQ in type B2 and G2 , we may either show directly
the convergence in [2], or alternatively, we use the expression of the
p-function as a proportion of Jacobi forms ([10] Theorem 3.6) to showthat the Fourier expansions at infinity consists only of positive powers.
In case of s = ζ( 13 ω0 ) − 32 ζ( 12 ω0 ), it will be shown in [2].
□Step 4. In the following table, we give the values at cusps of theadditive summands in (9.10), (9.11) and (9.12). Calculations dependon cases: In case of the form Gm (0), it is classical (e.g. [7, 11]). In caseof the form p(a) for some a ∈ ΩQ (see√ [2]). The cases of shifted series
G4 ( 12 ω0 ) and G3 ( 13 ω0 ) at the cusp −1∞ are reduced to Riemann’s
zeta function ζR (4) or Dirichlet’s L-function L(3, χ) of the characterχ given by quadratic residues, respectively (see [3] for explicit values).
Their values at the cusp 0 are directly shown to be zero. In case ofζ( 13 ω0 ) − 23 ζ( 21 ω0 ), it will be shown in [2].
A2 type:( 4)√4
ω0 · G4 (0)|4 E ( −1∞) = π45

B2 type:)√( 66
ω0 · G6 (0)|6 E ( −1∞) = 2π
945

( 2
)√
2
ω0 · p( 12 ω0 )|2 E ( −1∞) = 2π3

)√
( 2
2
ω0 · p( 21 ω0 )|2 S ( −1∞) = − π3)√
( 4 2 14
ω0 · p ( 2 ω0 )|4 E ( −1∞) = 4π9( 4 2 1)√4
ω0 · p ( 2 ω0 )|4 S ( −1∞) = π9

( 4
)√
4
ω0 · G4 (0)|4 E ( −1∞) = π45

( 4
)√
4
ω0 · G4 (0)|4 S ( −1∞) = π45

( 4
)√
4
ω0 · G4 ( 12 ω0 )|4 E ( −1∞) = π3( 4)√
ω0 · G4 ( 21 ω0 )|4 S ( −1∞) = 0

PERIOD INTEGRALS OF TYPE A2 , B2 AND G245

G(2 type:

)√
(
)√
1
2
1
ω0 · (ζ( 13 ω0 ) − 23 ζ( 12 ω0 ))|1 E ( −1∞) ω0 · (ζ(3 ω0 ) − 3 ζ( 2 ω0 ))|1 S ( −1∞)√= √π3
= − −1 π3
( 2
)√
( 2
)√
2
ω0 · p( 13 ω0 )|2 E ( −1∞) = π 2
ω0 · p( 13 ω0 )|2 S ( −1∞) = − π3
( 3
)√
2 3
ω0 · G3 ( 31 ω0 )|3 E ( −1∞) = 23√π3( 3)√
ω0 · G3 ( 31 ω0 )|3 S ( −1∞) = 0

Table 2: Values of Eisenstein series of type I2 (p) at cusps, I
This completes a proof of Theorem 10.1□

Remark 3. In case of type A2 , Fourier coeﬃcients of Eisenstein series are
well-known to be given by the divisor sum function σp (n) with suitable
constant factors given by special values of Riemann’s zeta function (e.g.
[11]VII.1.3). It is natural to ask for similar expressions of the Fouriercoeﬃcients of Eisenstein series for the types B2 and G2 .
Combining above calculations with the expressions (9.10), (9.11) and(9.12), we obtain the following table,
A2 type:()√
144(ω06 · gs |4 E) (√ −1∞) = 2213 π 6,
ω0 · gl |6 E ( −1∞) = 23 33 π ,B2 type:

( 2
)√
( 2
)√
1 2
2
2π ,
(ω04 · gs |2 E) (√ −1∞) = π 1, 4 (ω04 · gs |2 S) (√ −1∞) = −1 4
ω0 · gl |4 E ( −1∞) = − 23 π , ω0 · gl |4 S ( −1∞) = 25 π ,G2 type:
(

√
)√
(
)√
ω0 · gs |1 E ( −1∞) = √π3 ,ω0 · gs |1 S ( −1∞) = − 3−1 π,
√
( 3
)√
( 3
)√
2 −1 32
3
ω0 · gl |3 E ( −1∞) = 3√
−1∞)
=
π
,
ω
·
g
|
S
(
l
3
0
33 π .
3

Table 3: Values of Eisenstein series of type I2 (p) at cusps, II
Step 5. This is the final step to obtain the isomorphism (10.1). We
determine the linear relations between the generators of both handsides by comparing their values at cusps.
First, we compare the weights (of the free generators of) the ringsin both hand sides of (10.1) by the use of “weight factor” wt(z) (recall(10.3)). Comparing Table 1 in §2 and the description of (10.4), we
obtain the following “coincidences” of weights!
A2
B2
G2

−wt(gs )/wt(z), −wt(gl )/wt(z) weights of M∗ (Γ1 ([p/2]))-(2/3)/(-1/6)= 4 , -1/(-1/6)=6
4 , 6-(1/2)/(-1/4)=2 ,-1/(-1/4)=4
2, 4-(1/3)/(-1/3)=1 ,-1/(-1/3)=3
1, 3

46

KYOJI SAITO

Table 4: Weights of the generators of the ring of modular forms
This means that each generators gs and gl are mapped into thegraded vector subspace of M∗ (Γ1 ([p/2])) of the same degree as thecorresponding generators αi , βj etc., respectively. For gs , the dimension of the graded subspace containing it is equal to 1 so that we only
need to fix the constant factor. In case of gl , the dimension is either 1
for type A2 or 2, spanned by gl and a power of gs , for types B2 and G2 .
Let us recall the values of the generators of the modular forms at
cusps. The following values are taken from [1] Internat J. Math. 163(2005) 249-279, B2 : p.270 B3 pp.271-272.
√(e4 |2 E)(√−1∞) = 1,
(e6 |4 E)( −1∞) = 1,√
(α2 |2 E)(√ −1∞) = 1,
(β4 |4 E)( −1∞) = 0,√
(α2 |2 S)(√ −1∞) = −1/2,(β4 |4 S)( −1∞) = 1/256,√√√√(α1 |1 E)(√ −1∞) = 1, (α1 |1 S)(√ −1∞) = −√ −1/ √3,(β3 |3 E)( −1∞) = 1, (β3 |3 S)( −1∞) = − −1/3 3.

Table 5: Values of Modular forms of Γ1 ([p/2]) at cusps
Comparing Table 3 with Table 5, we obtain the following expressionsof the isomorphism (10.1).
A2 type:(10.5)

gs =
gl =

1
π 4 e4 ω0−4
22 3
1
π 6 e6 ω0−6
23 33

B2 type:(10.6)

gs = π 2 α2 ω0−2
gl = 24 π 4 β4 ω0−4 − 213 π 4 (α2 ω0−2 )2G2 type:(10.7)√1 π α1 ω −1
gs =
0
3
2
3
√
gl = − 3 3 π β3 ω0−3

Remark 4. As a consequence of the identification (10.1), we observethat the holomorphicity condition at cusps on modular forms are equivalent to the holomorphic extendability condition on functions on SI2 (p) \DI2 (p) to holomorphic functions on SI2 (p) . Then the cusp form condition“should be” equivalent to the condition to vanish on the discriminant
DI2 (p) . This is the subject discussed in the following lemma.

PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

47

Lemma 10.3. The set of Γ1 ([p/2])-equivalence classes of cusp points corresponds naturally in one to one with the set of irreducible componentsof the discriminant loci DI2 (p) (2.4).
We show that a generator of the ideal of the ring M∗ (Γ1 ([p/2])) ofmodular forms vanishing at each equivalence class of cusp points, which
can be easily found from Table 4, is, by the pull back by the period map(5.6), up to a constant factor, identified with an irreducible componentof the equation ∆I2 (p) ∈ C[gs , gl ] (2.5) of the discriminant.
Here, we may recall Footnote 27 again.
A2 type: Recall that√ there is a unique equivalence class of cusps whichis represented by −1∞. The ideal vanishing at the class is generated
by e34 − e26 . We also recall the equation of the discriminant (2.5) of type
A2 . Then, the identification (10.5) induces the following identity:(10.8)−27gl2 + gs3=

π 12 3(e4 − e26 ) ω0−12
1728

B2 type: Recall that√ there are two equivalence classes of cusps, whicharerepresentedby−1∞ and by 0. The ideal vanishing at the class√−1∞ is generated by β4 , and the ideal vanishing at the class 0 is
generated by α22 −64β 4 . We recall the irreducible factors of the equationof the discriminant (2.5) of type B2 are −8gl + gs2 and 8gl + gs2 . Then,the identification (10.6) induces the following identities.
(10.9)

8gl + gs2
−8gl + gs2 (= −G2 ( 21 ω0 ))==

128π 4 β4 ω0−4
2π 4 (α22 − 64β4 ) ω0−4

G2 type: Recall that√ there are two equivalence classes of cusps, which
are represented by −1∞ and by 0. The ideal vanishing at the class√
1−1∞ is generated by 54(α13 − β3 ), and the ideal vanishing at the class
0 is generated by 12 (α13 + β3 ). We recall the irreducible factors of theequation of the discriminant (2.5) of type G2 are gl + 2gs3 and gl − 2gs3 .
Then, the identification (10.7) induces the following identities.
(10.10)gl + 2gs3
gl − 2gs3 (= −G3 ( 13 ω0 ))= 3√2 3 π 3 (α13 − β3 ) ω0−3
= − 3√2 3 π 3 (α13 + β3 ) ω0−3

11. Discriminant ConjectureThis is the last section of the study of the period map of types A2 ,B2 and G2 . We recover the classical modular discriminant formula for
all types, and answer in Theorem 11.1 to the discriminant conjecture
posed in [23] §6. Let us recall the conjecture in the original form.

48

KYOJI SAITO

Conjecture 6. Let W be a crystallographic finite reflection group. Is
the k(W )th power root of δW , say λW (up to a constant factor), an
automorphic form for the group Γ(W ) with the character ϑW ? Can one
find an infinite product expression for λW compatible with Conjecture 4?

We first explain notation in the conjecture. In [23], W is a Weylgroup for an irreducible finite root system of any type, but in thepresent paper, we restrict ourselves only to the types A2 , B2 or G2 .
The number k(W ) and the character ϑW are defined in [23] §6, but,in the present paper, in (4.8) and (4.10), respectively. The δW is a
generator of anti-invariants whose square is the discriminant ∆I2 (p) . We
shall explain about δW and λW again in Theorem 11.1. The modulargroup Γ(W ) is, in [23] §6, 6.4 Example, unfortunately, wrongly statedto be equal to the congruence group Γ0 ([p/2]). However, according to a
result (4.6) in the present paper, the group Γ(W ) should be corrected
to be Γ1 ([p/2]).27 The Conjecture 4 mentioned in the Conjecture 6
is something about the “liftablity” of the ring of Eisenstein series tothe ring of the “Galois” covering by the Weyl group of the type I2 (p).
However, we will not discuss on this in the present paper.
The answer∏to Conjecture is given by√a use of Dedekind eta functionnη(τ ) := q 1/24 ∞n=1 (1 − q ) (q = exp (2π −1τ )). Recall that a functionin τ ∈ H is∏called an eta-quotient if it has a finite product/quotient
expression si=1 η ri (mi τ ) where s, mi ∈ Z>0 , ri ∈ Z. A holomorphic
modular form that is non-vanishing on H and has integer Fourier coefficients at infinity, is an integer multiple of an eta-quotient (see [18]).
Let us come back to the equalities (10.8), (10.9) and (10.10), andshow that they admit eta-quotient expressions. LHSs, as defining equations of irreducible components of the discriminants, do not vanish on
SI2 (p) \ DI2 (p) . So, after the identification (10.3), they do not vanish one On the other hand, those generators e4 , e6 , α2 , β4 , α1 , β2 in RHS,H.
are described by theta-functions, and, therefore, have integral Fouriercoeﬃcients at infinity (see [1]). Then, we have the following expressions.
e3 −e26
4
Γ1 (1) : 1728
= η(τ )24 ,
η(τ )16
)16
2
,
α
−
64β
=
,
Γ1 (2) : β4 = η(2τ
4
8
2
η(τ )
η(2τ )8

Γ1 (3) :)91(α13 − β3 ) = η(3τ,54
η(τ )3

η(τ )91(α13 + β3 ) = η(3τ2)327In fact, Γ (N ) = Γ (N ) for N = 1, 2, but Γ (3) is a double extension of Γ (3)
0

1

0

1

by ±id. Since in [23], one use the same generators as (4.3) in the present paper as forthe generators of Γ(I2 (p)), practically the calculations in [23] are still meaningful,and we shall use them in the present paper.

PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

49

Table 5: eta-quotients of irreducible components of the disriminant(Proof. Case of type A2 is classical (e.g. [7, 11]).
Case of type B2 : Because of level 2 condition, candidates of eta
quotients are of the form cη(τ )p η(2τ )q for unknowns c ∈ C and√ p, q ∈ Z,
satisfying p+q = 2·weight√= 8. Recall Table 5, so that β4 |4E(√ −1∞) = 0(simple zero √in q), β4 |4S( −1∞) = 1/256 and α22 − 64β4 |4E( −1∞) = 1,2
α2 −64β4 |4S( −1∞) = 0 (simple zero in q). Posing these constraints on
the eta-quotients, we determine c, p and q, and obtain the expression.
Similar proof works for the type G2 . 2)
Applying above expressions for (10.8), (10.9) and (10.10), we are
now able to express the discriminant form ∆I2 (p) (2.5) and its reduced
form ∆red
I2 (p) by some products of eta quotients. As is expected (since
discriminant vanishes on both cusps), the results are no-longer etaquotients, but are eta-products, i.e. they don’t have denominators.
=

π 12 η(τ )24 ω0−12∆B2 = (8gl + gs2 )(−8gl + gs2 )2 =512 π 12 η(τ )24 ω0−12∆A2 =(11.1)−27gl2 + gs3

π 12 η(τ )24 ω0−12∆G2 = (gl + 2gs3 )(−gl + 2gs3 )3 = − 27
4
Table 6. Eta-product expressions of discriminants23∆red
A2 = −27gl + gs =(11.2)π 12 η(τ )12 η(τ )12 ω0−12

−8
8
8
8
4
2
∆red
B2 = −64gl + gs = 256π η(τ ) η(2τ ) ω026∆red
= −4π 6 η(τ )6 η(3τ )6 ω0−6
G2 = −gl + 4gs

Table 7. Eta-product expressions of reduced discriminants
We formulate final Theorem of the present paper, where both formulae (11.2) and (11.1) give answers to the first half and the latter halfof Conjecture, respectively.
Theorem 11.1. 1. For all three types A2 , B2 and G2 , set(11.3)λI2 (p) (τ ) := η(τ )η([p/2]τ ).

Then, (i) λI2 (p) (τ ) is a modular form of weight 1 of the group Γ1 ([p/2])
with respect to the character ϑI2 (p) (4.10),(ii) The power δI2 (p) := λI2 (p) (τ )k(I2 (p)) is a generator of the module
anti-invariant modular forms (recall (4.8) for k(I2 (p))),

50

KYOJI SAITO(iii) The power λI2 (p) (τ )2k(I2 (p)) , up to a non-zero constant factor,
corresponds by (10.3) to the reduced discriminant form ∆red
I2 (p) .
2. For all three types A2 , B2 and G2 , the discriminant form ∆I2 (p)(2.5), up to a non-zero constant factor, corresponds by (10.3) to
∏
n 24
q ∞
n=1 (1 − q )called the modular discriminant (or discriminant function).
28

Proof. 1. (i) The calculations given in [23] to show the modularity ofλI2 (p) with the character ϑ are still valid. For a sake of completeness ofthe present paper, we recall it by adjusting notations.
For simplicity, we introduce a number N ∈ Z>0 called √level, where−1/12) so
N = [p/2] in case of
type
A
,
B
and
G
.
Set
ζ
:=
exp
(π
2
2
2
√
N +1
that ζ
= exp (π −1/k(I2 (p))) (N = 1, 2, 3 and p = 3, 4, 6). In viewof (4.3), (4.6) and (4.10), it is suﬃcient to show the following.
Lemma 11.2. The λN := η(τ )η(N τ ) for N ∈ Z>0 is a modular form ofΓ1 (N ) with a character ϑN , where the character satisfies
[
]
[
]
1 0
1 1
ϑN : ãN :=, b̃N :=
7→ ζ N +1 ∈ C×
−N 1
0 1
Proof of Lemma 10.2. By definition, λN is automatically a modular form with a character of the group Γ1 (N ) (see, e.g. [18]). Recalling (10.2) and (10.3), we have only to show (ã∗N )−1 (λN ω0−1 ) =ζ −N −1 (λN ω0−1 ) and b̃∗N (λN ω0−1 ) = ζ N +1 (λN ω0−1 ). In the following, weshall use a sign convention on the monodromy of the eta-function from[15] p121.
Recall (10.2), b̃∗N (τ ) = τ + 1 and b̃∗N (ω0 ) = ω0 . Then, using thetransformation formula η(τ + 1) = ζη(τ ), we obtain:b̃∗N (λN ω0−1 ) := η(τ + 1)η(N (τ + 1))ω0−1= ζη(τ ) ζ N η(N τ )ω0−1 =ζ N +1 (λN ω0−1 )τand ã∗N (ω0 ) = ω0 − N ω1 = ω0 (1 −Recall (10.2) so that ã∗N (τ ) = 1−Nτ√√τ η(τ ),N τ ). Then, using the transformation formula η(−/τ ) =−1
28 All results are well-known for the type A . However, for the other types B
22

and G2 , at present, there is no geometric explanation why such power roots of the
reduced discriminant ∆red
I2 (p) form modular forms with character, and why all threegeometric discriminants ∆I2 (p) take the same modular discriminant form. What are
λI2 (p) ′ s? Does mirror symmetric interpretation give a hint to understand them?

PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

51

we obtain:
ω −1

τ
Nτ
0
(ã∗a )−1 (λN ω0−1 ) := η(√1+N τ )η( N τ +1 )) 1+N τ
ω0−1−(N τ +1)/τ
1
√
=
η(−N
−
1/τ
)
ζη(−
)
N
τ
+1
1+N
τ
−1
√
ω0−1
−(N τ +1)/τ τ N τ +1 −N
√
√
√
=
ζ
η(τ
)
ζη(N
τ
+
1)
1+N τ
−1
−1
−1
√
−1
1
−N +2
√
= ζ
η(τ )η(N τ )ω0−1= ζ −N −1 ζ(τ )η(N τ )ω0−1=

ζ −N −1 (λN ω0−1 )End of Proof of Lemma 10.2. 2(ii) Recalling the character θ (4.11), δI2 (p) is obviously an anti-invariantof the group Γ1 (N ).
(iii) This a paraphrase of (11.2).
2. This is only a paraphrase of (11.1).
□
Remark 5. In [23] §6, we formulated 6 conjectures. The present paper
gives positive answers to all conjectures for the types A2 , B2 and G2 .
The conjectures seem to be still valid for all types of crystallographic
reflection groups. That is, we ask for a construction of Eisenstein series
for all types to answer to Jacobi inversion problem, where we may needspecial consideration for low weight cases as in the present paper.
12. Concluding Remarks
The findings of the duplication formula for the Lemniscate arc length
integral due to Fagnano (1718) and its generalization to the addition
formula due to Euler (1751) were naturally understandable by inverting the variables, i.e. by parametrizing the Cartesian coordinates of
the curve by the arc length. This led to the finding of new periodic
functions, i.e. the elliptic functions, beyond the trigonometric or exponential functions. It is impressive to see the historical developments
caused by the finding of the elliptic integrals, from the classical AbelJacobi theory through the modern mixed Hodge theory.
I, however, was attracted by other aspects of elliptic integrals. Namely,
covariant diﬀerentiations of the elliptic integral of the first kind giveother kinds of elliptic integrals. That is, the elliptic integral of thefirst kind is a potential for all other periods. I called this property theprimitivity of the elliptic integral of the first kind.
The primitivity is combined with another remarkable property of thetheory. Namely, the 2 of the rank of the lattice of cycles used for theelliptic integrals is equal to the 2 of the dimension of the unfolding
parameters (gs , gl ) of the elliptic curve. That is, the map from the52

KYOJI SAITO

space of parameters to the space of periods defined by the period integrals of first kind becomes a morphism between the spaces of the same
dimension. The Schottky type problem, an unsolved problem in classical Abelian integral theory to determine the image set of periods (c.f.
[27]), is resolved automatically in this setting! I called this property
the equi-dimensionality of the period map of elliptic integrals.
Then, it was again natural to ask for a description of the inverse
morphism from the space of periods to the original parameter space.
Actually, it was done by theta-series by Jacobi and later by Eisensteinseries. We call this procedure “solutions to Jacobi’s inversion problem”.
Following those works, I was inspired to look for (higher dimensional)
analogs of the elliptic integral of the first kind, which carries the primitivity and the equi-dimensionality. That is the theory of integrals ofprimitive forms over Lefschetz vanishing cycles [19]. If we look backsome historical works from the view point of primitive forms, the works
by E. Picard [17] (1883) and by G. Shimura [25, 26] (1963,1964) canbe regarded already as some part of period integrals of primitive forms(1,1)
of type E6 and E8 , E8 and some others, respectively.
Nowadays, primitive forms become a driving force for constructing
new integrable hierarchies, and play a role in mirror symmetry fromcomplex geometric side. However, this is one half of the primitive form
theory, i.e. algebraic analytic aspects. The transcendental aspects, i.e.
the period integral theory over Lefschetz vanishing cycles is missing
still. Primitive forms can play their full original power only after they
are integrated to period maps, and the solutions to the Jacobi inversion
problem leads us to the study of new transcendental functions [24].
However, the integral theory over closed cycles is a quite hard subject, since they form a closed world which is rigid and inflexible. We
first need to embed them in a big ocean of integrals over open cycles,
where we have wide freedom of making new pictures and theories, as
was done in the original works of arc-length integrals by Fagnano andEuler. Also, the classical abelian integral theory by Riemann was successful through integrals over open intervals, called the Jacobian variety
theory, where the inversion maps are described by theta functions [27].
Thus, it was my pleasure to reinterpret the classical elliptic integralsof the families for Weierstrass, Legendre-Jacobi and Hesse in termsof integrals of primitive forms over vanishing cycles of types A2 , B2
and G2 as in the present paper (types D4 and A3 are ongoing). We
tried to make clear the importance of integrals over open paths byshowing that their inversion functions are the solutions of Hamilton’s
equation of motion in §6. That is, open integrals are inverse to certain

PERIOD INTEGRALS OF TYPE A2 , B2 AND G2

53

Dynamical systems. Actually, this fact was the key reason, why we
could determine the inversion map in §7-9 to solve Jacobi’s inversion
problem by introducing generalized Eisenstein series for each type.
We do not know yet what are the higher dimensional analog of them:
how to invert the period map to answer Jacobi’s inversion problem, how
to generate the inversion functions ([21]). When the vanishing cycles
form a finite root system, there are some conjectural descriptions ([23]).
The positive answer to the conjectures for types A2 , B2 and G2 in the present

paper by a use of the Dedekind eta-function is a toy model and did notgive any new transcendental function. However, we may get enoughhope to expect that the conjectures still hold for all other types of rootsystems.
One may expect the mirror symmetry and the study of d-branes onopen string theory in high energy physics may give some suggestions
for the understanding of the period maps for primitive forms, since thestudy of the power roots of the discriminant, as done in the present
paper, is explained not from the primitive form side by itself but fromthe structure of the root system of the vanishing cycles [23] which
belongs to the mirror side, i.e. the symplectic geometry side.
Acknowledgements. The author expresses his gratitudes to Yoshihisa Obayashi, who draw Figures 1- 5 of the present paper. Particular thanks go to Hiroki Aoki, without whose helps the author may
have not accomplish the identification of the ring of Eisenstein seriesof type I2 (p) with the ring of modular forms of the congruence groupΓ1 ([p/2]). He expresses also gratitudes to Yoshihisa Saito, Kenji Iohara,
Takashi Takebe, Tomoyoshi Ibukiyama, Hiroyuki Yoshida, Masanobu
Kaneko and Akio Fujii for helpful and inspiring discussions, and to Yota
Shamoto for careful reading of manuscripts. This work was partially
supported by JSPS KAKENHI Grant Number 18H01116.
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Institute for Physics and Mathematics of the Universe, the University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan
E-mail address: kyoji.saito@ipmu.jp
Research Institute for Mathematical Sciences, Kyoto University,
Sakyoku Kitashirakawa, Kyoto, 606-8502, Japan
Laboratory of AGHA, Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141700, Russian Federation