Orbifold points on Teichmiiller curves and
Jacobians with complex multiplication
by
TECHN,,'OLOGY
byOF
Ronen E. Mukamel
SEP 022011
Submitted to the Department of Mathematics
LIBRARIE-S
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2011
@ Ronen E. Mukamel, MMXI. All rights reserved.
The author hereby grants to MIT permission to reproduce and
distribute publicly paper and electronic copies of this thesis document
in whole or in part.
Author.... ....
....
. .........................
r.....
Department of Mathematics
April 29, 2011
C ertified by ..........
..........................
Curtis T. McMullen
Professor
Thesis Supervisor
Accepted by .
Bjorn Poonen
Chairman, Department Committee on Graduate Theses
4:7
2
Orbifold points on Teichm6ller curves and Jacobians with
complex multiplication
by
Ronen E. Mukamel
Submitted to the Department of Mathematics
on April 29, 2011, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
For each integer D > 5 with D = 0 or 1 mod 4, the Weierstrass curve WD is an
algebraic curve and a finite volume hyperbolic orbifold which admits an algebraic
and isometric immersion into the moduli space of genus two Riemann surfaces. The
Weierstrass curves are the main examples of Teichm5ller curves in genus two.
The primary goal of this thesis is to determine the number and type of orbifold
points on each component of WD. Our enumeration of the orbifold points, together
with [Ba] and [Mc3], completes the determination of the homeomorphism type of WD
and gives a formula for the genus of its components. We use our formula to give
bounds on the genus of WD and determine the Weierstrass curves of genus zero. We
will also give several explicit descriptions of each surface labeled by an orbifold point
on WD.
Thesis Supervisor: Curtis T. McMullen
Title: Professor
Acknowledgments
First and foremost, I would like to thank my advisor, Curt McMullen, for his help
and guidance during the writing of this thesis. He has been mathematically very
generous to me, and I deeply appreciate all that he has taught me. I would also like
to thank my other readers-Dick Gross, Abhinav Kumar, and Bjorn Poonen-as well
as Anatoly Preygel and Vaibhav Gadre for their useful comments and suggestions. I
have had many other inspiring mathematics teachers over the years-including Fred
Cohen, Joe Harris, Peter Kronheimer and Arnold Ross-and I would like to thank
them as well. I would also like to thank my family-Amelia, Eran, Dana and Shaulfor their continued love and support. Finally, I would like to thank my dear friends
who have made the years I have spent graduate school very special. In no particular
alphabetical order they include: Ailsa, Alex, Amelia, Bea, Christa, Dustin, Ethan,
Griselda, Helen, Liz, Pat, Pat, Jacob, Jacob, Jim, Nick, Olga, Toly and Won.
6
Contents
1 Introduction
2 Background
2.1
Abelian varieties ......
2.2
Riemann surfaces, Jacobians and Automorphisms . . . . . . . . . .
2.3
The W eierstrass curve
............................
28
. . . . . . . . . . . . . . . . . . . . . . . . .
3 Orbifold points on Hilbert modular surfaces
39
4
The D 8-family
45
5
Endomorphisms
57
6
Spin
63
7 Genus
67
A The D 12 -family
75
A.1
The D 12 -family . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
A.2
Homeomorphism type of WD . . . . . .
79
- - - - - - - - - - - -
-.
8
Chapter 1
Introduction
Let Mg be the moduli space of genus g Riemann surfaces. The space Mg can be
viewed as both a complex orbifold and an algebraic variety and carries a complete
Teichmfiller metric. A Teichmiller curve is an algebraic and isometric immersion of
a finite volume hyperbolic Riemann surface:
f : C = H/F -+ Mg.
The modular curve M 1 is the first example of a Teichmiiller curve. Other examples
emerge from the study of polygonal billiards [Ve, MT] and square-tiled surfaces. While
the Teichmiiller curves in M
2
have been classified [Mc5], much less is known about
Teichmiller curves in Mg for g > 2 [BaM, BM2].
For each integer D > 5 with D = 0 or 1 mod 4, the Weierstrass curve WD is
the moduli space of Riemann surfaces whose Jacobians have real multiplication by
the quadratic order
OD
=
Z
D+2]
stabilizing a holomorphic one form with double
zero up to scale. The curve WD is a finite volume hyperbolic orbifold and the natural
immersion:
WD -+
M 2,
is algebraic and isometric and has degree one onto its image [Ca, Mc1]. The curve WD
is a Teichmaller curve unless D > 9 with D = 1 mod 8 in which case WD = W Li WD
is a disjoint union of two Teichmaller curves distinguished by a spin invariant in
D mod 16
e2(WD)
1,5,9, or 13
NE(-4D)
0
-!(h(-D) + 2h(-D/4))
4
0
8
ih(-D)
12
-!((-D)+ 3h(-D/4))
Table 1.1: For D > 8, the number of orbifold points of order two on WD is given by a weighted
sum of class numbers. The function h(-D) is defined below.
Z/2Z [Mc3]. A major challenge is to describe WD as an algebraic curve and as a
hyperbolic orbifold. To date, this has been accomplished only for certain small D
[BM1, Mc1, Lo].
The purpose of this thesis is to study the orbifold points on WD. Such points
label surfaces with automorphisms commuting with OD. The first two Weierstrass
curves W5 and W8 were studied by Veech [Ve] and are isomorphic to the (2,5, oo)- and
(4, oo, oo)-orbifolds. The surfaces with automorphisms labeled by the three orbifold
points are drawn in Figure 1-1.
Our primary goal is to give a formula for the number and type of orbifold points
on WD. Together with [Mc3] and [Ba], our formula completes the determination of
the homeomorphism type of WD and gives a formula for the genus of WD. We will
use our formula to give bounds for the genera of WD and WL and list the components
of UD WD of genus zero. We will also give several explicit descriptions of the surfaces
labeled by orbifold points on WD, giving the first examples of algebraic curves labeled
by points of WD for most D.
Main results.
Our main theorem determines the number and type of orbifold
points on WD:
Theorem 1.1. For D > 8, the orbifold points on WD all have order two, and the
number of such points e2 (WD) is the weighted sum of class numbers of imaginary
quadratic orders shown in Table 1.1.
We also give a formula for the number of orbifold points on each spin component:
Theorem 1.2. Fix D > 9 with D =1 mod 8. If D =f
2
is a perfect square, then all
of the orbifold points on WD lie on the component with spin (f + 1)/2 mod 2:
e 2 (W f+1)/2) =h(-4D) and e 2 (Wif1)/2)
Otherwise, e 2 (WD)
e2 (Wj1) = h(-4D).
When D is not a square and WD is reducible, the spin components of WD have
algebraic models defined over Q(v/D) and are Galois conjugate [BM1]. Theorem 1.2
confirms that the spin components have the same number and type of orbifold points.
The class number h(-D) is the order of the ideal class group H(-D) for O-D
and counts the number of elliptic curves with complex multiplication by
0
-D up to
isomorphism. The weighted class number
h(-D)
-
2h(-D)/| 0DI
appearing in Table 1.1 is the number of elliptic curves with complex multiplication
weighted by their orbifold order in M 1 . Note that h(-D)
=
h(-D) unless D
=
3 or
4. When D is odd, the orbifold points on WD are labeled by elements of the group
H(-4D)/[P] where [P] is the ideal class in O-4D representing the prime ideal with
norm two.
The orbifold Euler characteristics of WD and WL were computed in [Ba] and the
cusps on WD were enumerated and sorted by component in [Mc3]. Theorems 1.1 and
1.2 complete the determination of the homeomorphism type of WD and give a formula
for the genera of WD and its components.
Corollary 1.3. For any e > 0, there are constants C, and N such that:
CD3/2+, > g(V) > D 3 / 2 /650,
whenever V is a component of WD and D > N,.
Figure 1-1: The first two Weierstrass curves W5 and W 8 are isomorphic to the (2, 5, oo) and
(4, oo, oo)-orbifolds. The point of order two is related to billiards on the L-shaped table (left)
corresponding to the golden mean - = 1+". The points of order five (center) and four (right)
are related to billiards on the regular pentagon and octagon.
Modular curves of genus zero play an important role in number theory [Ti]. We
also determine the components of Weierstrass curves of genus zero.
Corollary 1.4. The genus zero components of UD WD are the 23 components of
UD<41
WD and the curves W 0
W'
49
_49Y
and W\.
81'
We include a table listing the homeomorphism type of WD for D < 250 in §A.2.
Orbifold points on Hilbert modular surfaces.
Theorem 1.1 is closely related
to the classification of orbifold points on Hilbert modular surfaces we prove in §3.
The Hilbert modular surface XD is the moduli space of principally polarized abelian
varieties with real multiplication by OD. The period map sending a Riemann surface
to its Jacobian embeds WD in XD.
Central to the story of the orbifold points on XD and WD are the moduli spaces
M 2 (D8 ) and M 2 (D12 ) of genus two surfaces with actions of the dihedral groups of
orders 8 and 12:
D8 - (r, J: r 2
Jr)2 _ j
4
1) and D12 = (r, Z : r 2 = (Zr)2
Z6 -
).
The surfaces in M 2 (D8 ) (respectively M 2 (D12 )) whose Jacobians have complex mul-
tiplication have real multiplication commuting with J (respectively Z). The complex
multiplication points on M 2 (D8 ) and M 2 (D1 2 ) give most of the orbifold points on
UD XD:
Theorem 1.5. The orbifold points on UD XD which are not products of elliptic
curves are the two points of order five on X 5 and the complex multiplication points
on M 2 (D8 ) and M 2 (D1 2 )Since the Z-eigenforms on D12-surfaces have simple zeros and the J-eigenforms
on D8 -surfaces have double zeros (cf. Proposition 3.2), we have:
Corollary 1.6. The orbifold points on UD WD are the point of orderfive on W5 and
the complex multiplication points on M 2 (D8 ).
Corollary 1.6 explains the appearance of class numbers in the formula for e2 (WD)The involutions r and Jr on a Ds-surface X have genus one quotients X/r and X/Jr
whose Jacobians are related by a degree two isogeny and the family M 2 (D 8 ) is birational to the modular curve Yo(2). The Jacobian Jac(X) has complex multiplication
Q(vD, i) if and only if Jac(X/r) has complex multiplication by an
Q(Vf 7). The formula for e (WD) follows by sorting the 3h(-D) surfaces
by an order in
order in
2
with D 8-action covering elliptic curves with complex multiplication by OD by their
orders for real multiplication.
The product locus PD. A recurring theme in the study of the Weierstrass curves
is the close relationship between WD and the product locus PD C XD. The product
locus PD consists of products of elliptic curves with real multiplication by OD.
The cusps on WD were first enumerated and sorted by spin in [Mc3] and, for
non-square D, are in bijection with the cusps on PD (cf. §7). The Hilbert modular
surface XD has a meromorphic modular form with a simple pole along PD and a
simple zero along WD. This modular form can be used to give a formula for the Euler
characteristic of WD and, for non-square D, the Euler characteristics of WD, XD and
PD satisfy ([Ba], Cor. 10.4):
X(WD)
x(PD) - 2X(XD).
Our classification of the orbifold points on XD and WD in Theorem 1.5 and Corollary
1.6 show that all of the orbifold points of order two on XD lie on WD or PD.
Theorem 1.7. For non-square D, the homeomorphism type of WD is deteTrined by
the homeomorphism types of XD and PD and D mod 8.
The D8 -family.
A secondary goal of our analysis is to give several explicit descrip-
tions of D8 -surfaces and to characterize those with complex multiplication. We will
outline a similar discussion for M 2 (D1 2 ) in §A.1. For a genus two surface X E M ,
2
the following are equivalent:
1. Automorphisms. The automorphism group Aut(X) admits an injective homomorphism p: D 8 -+ Aut(X).
2. Algebraic curves. The field of meromorphic functions C(X) is isomorphic to:
Ka = C(z, x) with z2
(X 2 _
1)(x 4 - ax 2 + 1),
for some a E C \ {±2}.
3. Jacobians. There is a number T E H such that the Jacobian Jac(X) is isomorphic to the principally polarized abelian variety:
A= C2 /AI
where A, = Z
plectic form
(2
1 ),(
((b), ())
1
1 ),(r$ ),
I"(
(1_+)) and A, is polarized by the sym-
)
4. Pinwheels. The surface X is isomorphic to the surface X, obtained from the
polygonal pinwheel P, (Figure 1-2) for some r in the domain:
U--,r
1
~
2
2 >-and |Re r < 2
-2
-2J
1+i)/2
Figure 1-2: For r in the shaded domain U, the pinwheel Pr has vertices at z = 1g:,
2,
tr, and tir. Gluing together opposite sides on P, by translation gives a genus two surface
admitting an action of D8 . The one form induced by dz is a J-eigenform and has a double zero.
5. Parallelograms. The surface X is isomorphic to the surface Y" obtained from
the parallelogram
Q.
(Figure 1-3) for some w in the domain:
V = {w E H: 0 < Rew < 2 and lw - 1 > 1}.
It is straightforward to identify the action of D8 in each of the descriptions above.
The field Ka has automorphisms r(z, x)
linear transformations r
=
=
(Q 0 ) and J =
(z, -x) and J(z, x) = (iz/x 3 , 1/x). The
(0
1)
preserve the polarized lattice A,.
The action of D8 on the surface Y. is identified in Figure 1-3. The surface X, satisfies
C(X,) ~ Ka for some a because the order four automorphism obtained by rotating
P, acts as a product of two disjoint transpositions on the Weierstrass points of X,
(cf. Propositions 3.2 and 3.3).
The function relating the number -r determining the polygon P, and abelian variety A, to the number a determining the field Ka is the modular function:
1
a(r) - -2 ± A(r)A(-r
+ 1)
The function A(-r) is modular for the group F(2) = ker(SL 2 (Z) -+ SL 2 (Z/2Z)) and
descends to the isomorphism A : H/F(2) ~'+C \ {O, 1} sending the cusps r(2) - 0,
F(2) - 1 and F - oo to 0, 1 and oo respectively. The function relating r and w is
the Riemann mapping w : U -+ V whose extension to &U has w ((-1 + i)/2) = 0,
w ((1 + i)/2) = 2 and w (oo) = oo. In Sections 4 and 5 we will prove:
Theorem 1.8. Fix r G U. The surface X, obtained from the polygon P, admits a
faithful D8 -action, is isomorphic to the surface obtained from the parallelogramQ,(,)
and satisfies:
Jac(X,) e A, and C(X,)
Ka(,-
The JacobianJac(X,) has complex multiplication if and only if r is imaginaryquadratic.
Teichmiiller and Shimura curves.
There are very few examples of Teichmiiller
curves parametrizing surfaces whose Jacobians lie on a Shimura curve [Mo].
The
families M 2 (D8 ) and M 2 (D1 2 ) are examples of Teichmiiller-Shimura curves and are
related to the known examples by a branched covering construction.
The families M 2 (D8 ) and M 2 (D1 2 ) are Teichmialler curves arising from squaretiled surfaces (see Figures 1-3 and A-2). The Jacobians of D8 - and D 12 -surfaces lie on
different components of the transverse self-intersection of the immersion of X 4 into
the moduli space of principally polarized abelian varieties. The Jacobians of D8 - and
D 12 -surfaces admit proper actions of the involutive rings
RD8 =
Z[D 8 ]/(J 2 + 1) and RD12 = Z[D 1 2 ]/(Z 3 + 1),
such that the actions of r and Jr in D8 and r and Zr in D 12 are self-adjoint. The
Weierstrass curve WD, by contrast, is not a Shimura curve ([Mc1], Cor. 10.2) be-
W
0
1
Figure 1-3: For w in the shaded domain V, the parallelogram Q. has vertices {O, 1, 4w, 4w + 1}.
Gluing the marked segments together as indicated gives a genus two surface Y, with a degree
four cyclic covering map to the double of a parallelogram. The surface Yw has an action of
D 8 generated by the Deck transformation and the involution covered by z '-+ -z + 2w + 1.
The quadratic differential q on Y. induced by dz 2 has two double zeros and is a product of
J-eigenforms.
cause of the condition on the OD-eigenform. The abelian varieties in XD which are
Jacobians of surfaces in WD cannot be distinguished by their endomorphism rings.
Algebraic models for D 8 -surfaces with complex multiplication.
The func-
tion a(ir) is a transcendental function of r and modular for the group Fo(2)
(F(2), (11)). Setting q =e2
=
r, we have:
a(r) = -2 - 256q - 6144q 2 - 76800q3 - 671744q 4 +
...
Usually, at most one of r and a(-r) is algebraic. When r is imaginary quadratic,
the number a(r) generates a finite abelian extension of Q(ir). The D8 -surfaces with
complex multiplication give the first explicit examples of algebraic curves labeled by
points on WD for most D.
For instance, when r
1+2T
X' is labeled by an orbifold point on W76 and
a(r) generates a degree three abelian extension of
Q(w/E9).
Computer experiment
indicates that the number a(-r) = -1.999710899 ... is the unique real root of:
x3 + 3x2 + 3459x + 6913 = 0.
Similarly, the point of order two on W5 is obtained from the polygon Pi(1+jg=
a ((I
+ v/15))
and
34 - 16v/5. The point of order four on Ws is obtained from the
polygon Pvfz22/ 2 and a(v/~ 2/2) = -6.
Outline.
We conclude this Introduction with an outline of the proofs of our main
results.
1. An automorphism
4
on a surface X E M 2 induces a permutation og on the
Weierstrass points. A brief analysis of the possibilities for og shows that the
most of the orbifold points on UD XD which are not products of elliptic curves
lie on M 2 (D8 ) and M 2 (D1 2) and that most of the orbifold points on UD WD
lie on M 2 (D8 ) (§3).
2. In §4 we study the D8 -family. For a genus two surface X with faithful D8 -action
p, the correspondence X -+ X/r x X/Jr induces a degree two isogeny between
elliptic curves:
cp: Jac(X/r) -
Jac(X/Jr).
We compute C(X) and Jac(X) in terms of the isogeny c, and show that the
map (X, p) i-+ c, embeds M 2 (D8 ) as the complement the degree two endomorphism of the square torus in the modular curve Y(2) parametrizing degree two
isogenies between elliptic curves. We then prove Theorem 1.8 by computing
the isogenies associated to the D8 -actions on the surfaces X, and Y,(,).
The
outer automorphism o- of D8 acts on M 2 (D8 ) and induces the Atkin-Lehner
involution on Yo(2) sending an isogeny to its dual. Note that the domain U for
pinwheels is a fundamental domain for the group lo(2), o-
( 0
1
. The
family M 2 (D1 2 ) admits a similar analysis, which we outline in §A.1.
3. Let (X, p) be a D8 -surface and let E = Jac(X/r). In §5 we use the isogeny
Jac(X)
-±
E x E induced by the map X -+ X/r x X/r
as an order in M 2 (End(E) 0
Q).
to embed End(Jac(X))
It follows that Jac(X) has complex mul-
tiplication if and only if E has complex multiplication and that the complex
multiplication points on M 2 (D8 ) give orbifold points on Weierstrass curves. We
then sort the 3h(-D) surfaces with D8 -action covering elliptic curves with complex multiplication by
0
-D by their orders for real multiplication commuting
with J giving the formula for e 2 (WD) in Theorem 1.1.
4. In §6 we sort the orbifold points on WD by spin component when D = 1 mod 8.
For such discriminants, the orbifold points on WD correspond to ideal classes
for
0
-4D-
These form a group H(-4D) and there is a spin homomorphism:
Eo : H(-4D) -+ Z/2Z.
Fix a proper ideal I for
0
-4D and let (X, p) be the D8 -surface with X/r ~ C/I
and with real multiplication by OD- The spin-invariant for the corresponding
orbifold point on WD is given by:
E
f
2
+ Co(I) mod 2,
f
is the conductor of OD, i.e. the index of OD in the maximal order of
Q(V'/5).
The spin homomorphism is the zero map iff D is a square giving the
where
formula in Theorem 1.2.
5. In §7 we use the formula for the number and type of orbifold points on WD to
give bounds on the genera of the components of WD.
Open problems.
While the homeomorphism type of WD is now understood, de-
scribing the components of WD as Riemann surfaces remains a challenge.
Problem 1. Describe WD as a hyperbolic orbifold and as an algebraic curve.
Our analysis of the orbifold points on WD have given explicit descriptions of some
complex multiplication points on WD. By the Andr6-Oort conjecture [KY], there are
only finitely many complex multiplication points on WD and it would be interesting
to find them.
Problem 2. Describe the complex multiplication points on WD.
The complex multiplication points on M 2 (D8 ) generate Teichmnller curves and
the complex multiplication points on M 2 (D12 ) generate complex geodesics with infinitely generated fundamental group. It would be interesting to find other examples
of Shimura varieties whose complex multiplication points lie on interesting complex
geodesics.
Problem 3. Find other Shimura varieties whose complex multiplication points generate Teichmiller curves.
The divisors supported at cusps on modular curves generate a finite subgroup of
the associated Jacobian [Ma]. It would be interesting to know if the same is true for
Teichmiiller curves. The first Weierstrass curve with genus one is W44 .
Problem 4. Compute the subgroup of Jac(W4) generated by divisors supported at
the cusps and points of order two.
Algebraic geometers and number theorists have been interested in exhibiting explicit examples of algebraic curves whose Jacobians have endomorphisms. A parallel
goal is to exhibit Riemann surfaces whose Jacobians have endomorphisms as polygons
in the plane glued together by translations as we did for the complex multiplication
points on M 2 (D8 ) and M 2 (D1 2).
Problem 5. Exhibit surfaces whose Jacobians have complex multiplication as polygons in C glued together by translation.
Notes and references.
For a survey of results related to the Teichmiller geodesic
flow, Teichmiiller curves and relations to billiards see [KMS, MT, KZ, Zo]. Background about abelian varieties, Hilbert modular surfaces and Shimura varieties can
be found in [vdG], [Sh2], [BL], and [Sh1]. The orbifold points on XD are studied in
[Pr] and the family M 2 (D8 ) was studied in [Si].
Chapter 2
Background
2.1
Abelian varieties
In this section we will collect background and definitions about abelian varieties, endomorphisms of abelian varieties, Shimura varieties, Hilbert modular surfaces and modular curves. For more background on abelian varieties, endomorphisms and Shimura
varieties see [BL], for more background on Hilbert modular surfaces see [vdG] and
for more background on modular curves see [DS].
Abelian varieties.
A g-dimensional complex torus B
=
V/A is the quotient of a
g-dimensional C-vector space V by a cocompact lattice A. Any connected, compact
complex Lie group is a complex torus. The vector space V is naturally isomorphic to
the dual to the space of holomorphic one forms on B and the lattice A is naturally
isomorphic to the first integral homology:
B n Q(B)*/H 1 (B, Z).
A polarization on B is the first Chern class of an ample line bundle on A, or equivalently, an integral symplectic form EB(-, -) on H1 (B, Z) with the property that
EB(i-,
-) + iEB(-, -) is a positive definite Hermitian form on O(B)* - H1 (B, Z) 0 R.
A complex torus is an abelian variety if it admits a polarization. A polarization on
B is principalif the symplectic form is unimodular on H1(B, Z). We will denote by
Ag the moduli space of principally polarized abelian varieties of dimension g.
Homomorphisms and endomorphisms of abelian varieties.
A homomor-
phism of abelian varieties f : B 1 -+ B 2 is a holomorphic map which is also a homomorphism. Any holomorphic map between abelian varieties is the composition of
a homomorphism with a translation. A homomorphism of abelian varieties
mined by its lift to the universal cover, which is a C-linear map
with f(H 1 (A, Z)) C H1 (B, Z). The map
f
f
f is deter-
: Q(B 1 )* a
Q(B2)*
is an isogeny if it has finite kernel and
f* : B 2 -+
cokernel. Principal polarizations on B1 and B 2 determine a dual map
B1
characterized by:
EB
2
(fX, Y)
= EBi
(,fN),
whenever x E H1 (B 1 , Z) and y E H1 (B2 , Z).
Now fix a principally polarized abelian variety B. The endomorphisms of B are
the holomorphic self-homomorphisms and form a Z-algebra End(B) called the endomorphism ring of B. An endomorphism
polarization, i.e.
End(B) 0
Q.
f* = f-1.
f of B is an
automorphism if it preserves the
The rational endomorphism ring of B is the Q-algebra
The ring End(B) is a subring with unit of End(B) 0
a lattice, i.e. an order in End(B) 0
Q.
Q
which is also
The ring End(B) has an analytic represen-
tation <ba on Q(B)*, a rational representation b, on H1 (B, Q) and a representation
<bo on the Q-vector space End(B) 0
Q.
The anti-involution
f
*
f* of
End(B)
is called the Rosati involution. The Rosati involution is positive in the sense that
(f, g) -+ Tr(<Qb(fg*)) is a positive definite bilinear form.
Every abelian variety has a self-dual action of Z and a typical abelian variety has
End(B) = Z. The subgroup B[n] of B is the kernel of the multiplication by n map
and B[n]* is set of points of order n. The endomorphism ring of the product of elliptic
curves E x F is given by:
a EEnd(E), b e Hom(F, E)
ba
J(
Eld(E xF
c d
c E Hom(E, F) and d E End(F)
)*
The Rosati involution induced by the product polarization is ()
Families of abelian varieties with endomorphisms.
=( *
Let 0 be a Z-algebra with
positive anti-involution *. An 0-action on an abelian variety B E Ag is an injective
homomorphism:
t:
0 -+ End(B) with t(x)* = t(x*).
Q.
The family:
parametrizes principally polarized abelian varieties with an 0-action.
Two pairs
The 0-action i is proper if it does not extend to a larger ring in 0 0
Ag(O) = {(B,
t)
: t is a proper O-action on B}
(B1 , ti) and (B2 , L2) are equivalent, i.e. (B 1 , ti)
-
/
~,
(B 2 , t2), if there is an isomorphism
of polarized abelian varieties B 1 -+ B 2 intertwining ti.
The components of Ag(O) can be distinguished by the rational representation of
0. A unimodular 0-module is an 0-module M with a unimodular symplectic form
Em(-, -) which is positive, i.e. the trace of (xx*) E End(M) is positive for reach
x E 0, and which is compatible with *, i.e. EM(xmi,m2)
=
EM(mi,x*m2) for every
x G 0 and mi C M. A unimodular 0-module M is an ideal if it is a submodule of
00
Q and M is properif the action of 0
does not extend to a larger ring in 0 0
Q. A
proper, unimodular 0-module M with Z-rank 2g determines a component of Ag(O):
Ag(M)
1 (B, t,):
(B, t) c Ag(0) and # : M + H1(B, Z)
is a symplectic 0-module isomorphism
The triple (B 1 , t i,
#1) is
equivalent to (B 2 , t2,
02)
if there is an isomorphism B 1
-+
B2
intertwining tj and
45.
Shimura varieties.
A Shimura variety is the quotient of a Hermitian symmetric
domain by an arithmetic group. The family Ag(M) is a prototypical example of
a Shimura variety. Complex structures on M @z R compatible with EM form a
Hermitian symmetric domain 71(M). A point r c -(M) turns the real torus M @z
R/M into an abelian variety B, with 0-action
t,.
Two complex structures in -(M)
give the same point in Ag(M) if they differ by an element of the arithmetic group
SL(M) of symplectic 0-module automorphisms M.
Let G be the subgroup of SL(M) which fixes every point in 7-(M). The group G
acts by automorphisms commuting with 0 on each abelian variety in Ag(M) and the
group PSL(M) = SL(M)/G acts faithfully on 7-(M). The space Ag(M) is presented
as a complex orbifold and Shimura variety by:
Ag(M)
Real and complex multiplication.
? (M)/PSL(M).
Let L be a commutative Q-algebra with pos-
itive anti-involution * and let LO denote the set of self-adjoint elements of L. Since *
is positive, L0 is a direct sum of totally real fields and either L
L' or L is a degree
two totally imaginary extension of L0 (see [Sh1], Prop. 1).
When L = LO and 0 is an order in L, we will say that a g
[L : Q] dimensional
abelian variety B E Ag has real multiplication by 0 if B admits a proper O-action.
For a proper unimodular module M over 0, the space 7-(M) is isomorphic to (H)
and the group SL(M) is commensurable to SL2 (0).
When [L : L0 ] = 2, [L : Q] = 2g and 0 is an order in L, we will say B E Ag has
complex multiplication by the order 0 if B admits a proper O-action. For a proper
unimodular module M over 0, the space 7-(M) is a point.
Real quadratic orders.
Each integer D = 0 or 1 mod 4 determines a quadratic
ring:
Z[t]
(t - Dt + D(D - 1)/4)
2
Every quadratic ring is isomorphic to OD for some D. The integer D is called the
discriminantof
OD
and OD is totally real whenever D > 0. Let KD
0
- OD
Q, let
OKD be the maximal order in KD and let vD be the square root of D equal to 2t - D.
The conductorf of OD is the index of OD in OKD- The inverse different
=
7
is the fractional ideal dual to OD under the trace pairing. When D > 0, let o-+ and
o_ denote the two homomorphisms KD -+ R satisfying o-+(v D) > 0 > o- (V
will also write o-+ and o-_ for the induced homomorphisms M 2 (KD)
-+
). We
M 2 (R) and
SL 2 (KD) -± SL 2 (R).
Our analysis works equally well when D is a square, even though the ring OD is no
longer a domain and the homomorphisms oa are no longer injective. When D = f2,
Q X Q,
we have KD
OKD
= Z x Z and OD is the subring of Z x Z consisting of
pairs (a, b) with a = b mod f.
Hilbert modular surfaces.
The OD-module
OD
E0
has a unimodular symplec-
tic form:
((x
2,y 2 )) = TrK (X1y 2 - X2 y 1 )1,y1 ),(x
Up to isomorphism of symplectic OD-modules, OD G O is the unique unimodular
and proper OD-module isomorphic to Z' as a Z-module. The Hermitian symmetric
domain 7l(OD E OD) is isomorphic to H x 181 with a point r = (T 1 ,r 2 ) corresponding
to the abelian variety:
B,
(+
where
a(X)
0
4, (x,
0
y)
O
C2 /0T(OD E OD)
+ ri-+(y), o--(x) +
= (o+(x)
OD on
o(x))f
=
T20-_ (y)).
The diagonal action x i-+
C2o
2
covers a proper OD-action t, on B, giving a point (Br, tr) E
A2(OD E O). Two points in H x H determine the same point in Ag(OD P OD) if
and only if they differ by an element of the group:
SL(OD 9
V
)
a b
c
d
ad - bc = 1, a, d E OD,
b E v/DOD and c E 0vD
The matrix A
c
xby
XH
SL(ODED O ) acts on
where the matrices
0-a (A)
A-
(Ti, T2)
= (o+(A)r1, o-_ (A)r2)
c SL2 (R) act on H by linear fractional transformation.
The group G fixing every point in H x H is generated by the elements of order two
and G has order two unless D
=
1 or 4, in which case G has order four. The Hilbert
modular surface:
XD
=1(H x H)/ PSL(OD (D
'j
is in natural bijection with the points in the Shimura variety A2(OD @ O)
presents A2(OD
Quaternion
and
0O ) as a complex orbifold.
algebras.
Now fix two positive discriminants D and E. Let 0 be an
order in the quaternion algebra:
(DE) = Q (a, b)/(a
2
= D, b2 = E, ab =-ba),
which is invariant under the anti-involution generated by a* = a, b* = b and (ab)* = ba
and has o n Q(a)
= OD
and 0 0 Q(b) =OE. Let M be a proper, unimodular
0-module. The family A 2 (M) parametrizes a family of abelian varieties with real
multiplication by OD and OE and the projection A 2 (M) -+ A 2 covers a component of
the intersection of the image of XD with the image of XE. The Hermitian symmetric
domain 7-(M) is isomorphic to H and the arithmetic group SL(M) is isomorphic to
the group of units in 0.
Square discriminants.
When D is a square, the Hilbert modular surface XD
parametrizes abelian varieties which are isogenous to a product of elliptic curves.
The Hilbert modular surface X1 is isomorphic to M
1
x M 1 and parametrizes abelian
varieties which are isomorphic to products of elliptic curves.
Elliptic transformations and orbifold points.
is elliptic if it fixes a point in ?I(M). For
-
An element A / 1 in PSL(M)
E 7(M), let Stab(T) denote the stablizer
of T in SL(M). The group Stab(T) acts by automorphisms commuting with t,(O) on
the associated abelian variety Br. When Stab(T) is strictly larger than G, the point
(B,, t,) E Ag(M) is called an orbifold point and the cardinality of Stab(r)/G is called
the orbifold order of (B,, t,).
An elliptic transformation in A E PSL(OD
09)
O
fixes either a unique point in
H x H or every point on a complex geodesic of the form -r x H or H x r. It is easy
to check that, when D > 4 the elliptic elements of PSL(OD E O)
have unique fixed
points. For such discriminants, the following are equivalent:
* A has finite order,
I| Tr(o (A))| < 2 and ITr(o-_ (A)) < 2,
* A fixes a point in H x H, and
" A fixes a unique point in H x H.
There is a natural bijection between the elements of order n > 2 in SL(ODeOD) up to
conjugacy and abelian varieties with complex multiplication by orders 0 containing
OD[n] with o n (OD 0
Modular curves.
Q)
OD and (, a primitive nth root of unity.
The modular curve M1 ~ A1 is the moduli space of elliptic
curves and is isomorphic to the hyperbolic Riemann surface H/ SL 2 (Z). A number
,r c H corresponds to the elliptic curve E, = C/Z G -rZ. Let F(n) be the kernel of
the natural homomorphism SL2 (Z)
Fo(n) ={(a
b)
-+
SL 2 (Z/nZ), let F1(n)
(17(n), (Q1)) and let
: c = 0 mod n}. The quotients:
Y(n) = H/F(n), Y(n) = H/1(n) and Yo(n) =H/Fo(n),
parametrize elliptic curves distinguished points of order n. Note that Fo(2) = 1(2).
We will be most concerned with the curve Yo(n) which parametrizes cyclic degreen isogenies i : E -+ F between elliptic curves up to isomorphism (see Figure 4-1).
The point
T
E H corresponds to the isogeny i, on E, whose kernel is generated by
the image T, of 1/n in C/Z E rZ. The modular curve Yo(n) has an Atkin-Lehner
involution sending the isogeny i, to its dual i*= i-1/n
Class numbers of imaginary quadratic fields.
Fix a negative discriminant D <
0. The following sets are in natural bijection:
" Elliptic curves with complex multiplication by OD up to isomorphism,
* Proper OD-ideal classes, and
" Triples of integers (a, b, c) with D = b - 4ac, gcd(a, b, c) = 1, |bi < a < c, and
if |bi
=
a or a = c then b > 0.
The triple (a, b, c) corresponds to the ideal class of I
=
aZ E -b+vZ, which in turn
corresponds to the elliptic curve C/I. The conditions on (a, b, c) ensure that I is a
proper OD-ideal and the number r = (-b+N
VI)/(2a) is in the standard fundamental
domain for the action of the group SL 2 (Z) on H.
The class number h(-D) is the cardinality the set of OD-ideal classes. Using
ID|
=
4ac - b2 > 4a 2 - a 2 , we have that |bj, a < -v//3and (cf. [Coh], pg. 232):
h(-D) < 2D/3.
2.2
Riemann surfaces, Jacobians and Automorphisms
In this section we will collect background and definitions about the moduli space of
curves, Jacobians, the period map and the Teichmniller geodesic flow. References for
this section include [BL] (especially §11) for Jacobians and the period map and [MT]
and [Zo] for the Teichmiiller geodesic flow.
Jacobians of Riemann surfaces.
Let Mg denote the moduli space of genus g
Riemann surfaces. The space Mg is both an algebraic variety and a complex orbifold.
The space Mg has a Deligne-Mumford compactificationMg by stable algebraic curves.
For X E Mg, let Q(X) denote space of holomorphic one forms on X. The Jacobian
of X is the g-dimensional principally polarized abelian variety:
Jac(X) = Q(X)*/H
1 (X,
Z).
f
A holomorphic map
:X
-±
Y between Riemann surfaces induces a holomorphic
homomorphism Jac(X) -+ Jac(Y). The automorphisms of X give automorphisms
of the polarized abelian variety Jac(X) and when g = 2, the automorphism groups
coincide Aut(X) = Aut(Jac(X)).
A point P E X determines a map Op : X -+ Jac(X) and any holomorphic
map from X to an abelian variety B factors through Op.
The Jacobian Jac(X)
is isomorphic to the identity component Pic"(X) of the Picardvariety. The abelian
variety Pico(X) parametrizes degree zero line bundles on X.
The period map X
The period map.
Jac(X) embeds Mg in Ag and extends to
C
Mg of stable algebraic curves with compact Jacobians. The
Mg
consists of polarized products of Jacobians of smooth curves
the moduli space Mg
boundary of Mg in
i-+
of lower genus.
When g < 2, the period map is dominant and extends to an isomorphism of
complex orbifolds on Mg. The space M
2
is the partial compactification of M
2
by
the moduli space of pairs of elliptic curves E V F joined at a single simple node:
M
2=
2
U Sym 2 (M1).
The Jacobian of E V F is the product E x F with the product polarization. The
complement of M
2
in A 2 is the image of the Hilbert modular surface X 1 :
M2
~> A2 \X
I
I
M2
Surfaces with automorphisms.
~k
1
A2
For a finite group G, the family of genus g sur-
faces with a faithful action of G is parametrized by the space:
(G) = (X, p):,
C
and p: G
isinjective.
Aut(X)
Two pairs (Xi, pi) and (X 2 , P2 ) are equivalent, i.e. (X1, p1)
-
(X 2 , P2 ), if there is an
isomorphism X1 -+ X 2 intertwining the pi. The set Mg(G) has a natural topology
and a unique holomorphic structure making the map Mg(G) -+ Mg holomorphic.
We will denote by Mg(G) the subspace of M(G) parametrizing the family of
smooth surfaces with an action of G. The action of G on Jac(X) extends to a (not
necessarily faithful) action of the group ring Z[G]. The Rosati-involution is generated
by g* = g-1 for g G G.
An injective group homomorphism G1 -+ G2 gives a holomorphic map M 2 (G2 )
-+
M 2 (G1 ). The automorphism group of G acts on M 2 (G) and the inner automorphisms act trivially. This gives an action of the outer automorphism group Out(G) =
Aut(G)/Inn(G) on M 2 (G). Note that Out(Ds) is isomorphic to Z/2Z and the nontrivial outer automorphism o- has o-(r) = Jr and o-(J) = J.
Correspondences.
f : Y -+ X1
and f. : Jac(Y)
Suppose
between Riemann surfaces
and g : Y -+ X 2 are holomorphic maps
-+
Jac(Xi) and g* : Jac(Y) -+ Jac(X 2 )
are the maps induced on Jacobians. The map Y -+ X 1 x X 2 is called a correspondence
between X 1 and X 2 and induces a holomorphic homomorphism g*(f.)*
Jac(X 1 )
-+
Jac(X 2 ).
Under an identification of Jac(Xi) with Pic"(Xi), the holomorphic homomorphism
g*(f*)* is obtained by taking a degree zero divisor on X 1 , pulling it back to Y and
pushing it forward to X 2 :
(g*)(f.)*: :(Pi
-
Qi)
g(f'-(P))
X-
gf-
2
for any points P and
Qj are
points in X 1 .
The hyperelliptic involution and Weierstrass points.
with compact Jacobian X E M
2
Every genus two surface
has a hyperelliptic involution y acting by w
'-4
-w
on Q(X).
When X is smooth, the quotient X/ is a sphere naturally isomorphic to PQ(X)*.
The surface X has six distinguished Weierstrass points fixed by q and we will denote
the set of Weierstrass points by Xw
Fix(7). For each P E X, there is a holomorphic
-
one form in Q(X) with zeros at P and y(P) and the Weierstrass points are the zeros
of holomorphic one-forms with double zero.
When X = E1 V E 2 is singular, 7 restricts to the elliptic involution on each
component of X fixing the node. Taking the node to be the base point for E1 and
E 2 , the set of Weierstrass points XW
=
E1 [2]* L E2 [2]* consists of the points of order
two. The node is not a Weierstrass point even though it is fixed by 77.
For any X E M 2 , the hyperelliptic involution q is in the center of the automorphism group Aut(X) and the group Aut(X) acts naturally on both Xw and X/.
The group Sym(Xw) is isomorphic to the symmetric group on six letters S6 and the
conjugacy classes in Sym(Xw) are naturally labeled by partitions of six:
[nin,...,nk] with
Involutions of genus two surfaces.
ni = 6.
Fix an order two automorphism
two surface X E M 2 . There are three possibilities for
4,
4 on a genus
distinguished by the action
of Z[4] ~ 0 on Jac(X):
" Z[#] is not faithful. The map
# acts by
-1 on Jac(X) and
4 is the
hyperelliptic
involution.
* Z[4] is faithful but not proper. The action of Z[4] on Jac(X) extends to real
multiplication by the maximal order 01 = Z
#
[ ]. The surface X is nodal and
restricts to the identity on one component of X and the elliptic involution on
the other. The eigenspaces for
#
in Q(X) are one dimensional and
4
fixes the
components of X.
" Z[#] is proper. The quotient X/# has genus one. This is true for nodal X since
#
interchanges the components of X because it does not commute with 01.
For smooth X,
4
is an involution different from the hyperelliptic involution,
so the quotient must have genus one. The eigenspaces for
4 on
dimensional and Jac(X) has real multiplication by 04 = Z[0].
Q(X) are one-
Fix a surface X with involution
proper. Since
involution
nE
4
d
such that the induced action of Z[#] on Jac(X) is
commutes with the hyperelliptic involution q, 'q induces an elliptic
on the quotient E = X/.
#-orbit and that
It is easy to check that Fix(94) forms a single
97E fixes its image. Setting this point to be the base point, i.e. E[1]
Fix(y4)/4, identifies E with its Jacobian via
OE[1]
=
: E -+ Jac(E). The remaining
points fixed by nE are covered by the Weierstrass points, giving an identification:
E[2]*
Similarly, the quotient F
point F[1]
=
=
Xw/0.
X/n#q has genus one, a natural elliptic involution qF, base
Fix(4)/y# and identification F[2]* = Xw/.
Abelian and quadratic differentials.
For a Riemann surface X E Mg, the space
of holomorphic quadratic differentials Q(X) consists of sections of the square of the
cotangent bundle on X. The space of holomorphic one-forms on X admits a quadratic
map Q(X) -+ Q(X) and when g < 3, the space Sym 2 (Q(X)) is naturally isomorphic
to Q(X).
The bundle of abelian differentials QMg
+
Mg parametrizes Riemann
surfaces with a non-zero holomorphic one-form. The bundle of quadratic differentials
QMg
-4
Mg parametrizes Riemann surfaces with a non-zero quadratic differential
and is isomorphic to the complement of the zero section in the cotangent bundle
T*Mg.
A typical way to give a point in (X, q) E QMg is to glue polygons P in C together
along their sides by affine maps of the form z
i-+
±z + c as in Figures 1-2 and 1-3. The
quadratic differential dz 2 E Q(C) is invariant under such maps and gives a quadratic
differential on the quotient:
(X, q) = U(Pi,dz2 )
A quadratic differential (X, q) is the square of a one-form, i.e. q = w2 , if (X, q) can be
written as a union of polygons glued together by affine maps of the form z
-+
z + c.
Complex geodesics.
The real geodesic flow on Mg gives R-actions on QMg
and QMg which extend to SL2 (R)-actions. For the quadratic differential (X, q) =
Uj(P, dz 2 )/
-
obtained from polygons Pi C C, the matrix A E SL 2 (R) acts on
C by real-linear transformations giving a new surface with quadratic differential
A - (X, q) = Uj(A(P), dz 2 )/
_.
The subgroup S0 2 (R) acts on (X, q) by leaving X fixed and rotating q. The
SL 2 (R)-orbit SL 2 (R) - (X, q) covers a holomorphic and isometric immersion of the
hyperbolic plane called a complex geodesic:
f(x,q) : IH SL 2 (R)/ SO 2 (R) -+ Mg.
The map f(X,q) further factors through the quotient by the Veech group SL(X, q)
Stab (X, q) in SL 2 (R) and the projectivizations PQMg = QMg/C* and PQMg
QM,/C* are foliated by complex geodesics. The SL 2 (R)-action preserves the quadratic
differentials which are squares of holomorphic one-forms. The bundle QMg extends
to the bundle OMg whose fiber over the nodal surface with compact Jacobian X
with components Xi is isomorphic to ®iQ(Xi). The SL 2 (R)-action extends naturally
to QMg.
The bundle QGMg(G) -+ Mg(G) parametrizing G-invariant quadratic
differentials is closed under the foliation by complex geodesics.
Teichmiller curves.
When SL(X, q) is a lattice, i.e. SL(X, q) has finite covolume
in SL 2 (R), the complex geodesic f(x,,q) factors through an algebraic and isometric immersion of the quotient C = H/ SL(X, q). Such an immersion is called a Teichmiiller
curve. If P are polygons whose vertices lie in the lattice Z[i] C C, then the quadratic
differential (X, q) = Uj(Pi, dz 2 )/ ~ is square-tiled, SL(X, q) is a lattice in SL 2 (Z) and
(X, q) generates a Teichmiller curve.
Topological invariants of hyperbolic Riemann surfaces.
The topological in-
variants of a two real dimensional finite volume hyperbolic orbifold V with real dimension two include the number of components h0 (V), the genus g(V)
=
dimc Q(V),
the number of cusps C(V) and the number of orbifold points en(V) of order n for
n > 1. These numbers are combined to give an orbifold Euler characteristic x(V):
x(V)
2h 0 (V)
-
2g(V)
-
C(V)
en(V)
-
i - I).
nn
When V is connected (i.e. h 0 (V) = 1) the numbers g(V), e,(V) and C(V) determine
the homeomorphism type of V.
2.3
The Weierstrass curve
In this section we will collect background related to the Weierstrass curve and its
components. References for this section include [Mc1, Mc4, Ba] for the definition
of WD, [Mc3] for the components of WD and [At] for spin structures on Riemann
surfaces.
Eigenforms and the Weierstrass curve.
Fix a surface X E M
2
and a proper
OD-action t: OD - End(Jac(X)). The space of one-forms up to scale PQ(X) has a
distinguished point corresponding to the line of eigenforms [w] with:
t(z)w = o-+(x)W,
where o
is the homomorphism OD --+ R with o±(v') > 0. The map sending (X, L)
to its og-eigenform up to scale gives an embedding of the Hilbert modular surface XD
in PQM
2
(cf. [Ba], Section 4.3). The images of XD and XE in PQM 2 are disjoint
whenever D
#
E.
The Weierstrass curve WD parametrizes the family of surfaces with real multiplication by OD whose eigenforms have a double zero:
X E M 2 , t is a proper OD-action on
WD
(X, t):
Jac(X), and the og-eigenform up to
scale [w] E PQ(X) has a double zero
/
~-
The period map embeds WD as a suborbifold of
XD-
The Hilbert modular surface XD and the Weierstrass curve WD are foliated by
Teichmiiller geodesics [Ca, Mc1]:
Theorem 2.1 (Calta, McMullen). The images of WD and XD in PQM 2 are closed
under the foliation by complex geodesics.
Orbifolds points on WD.
Since WD is a suborbifold of XD, the orbifold points on
WD correspond to pairs (X, t) where
tiplication by an order containing
t : OD
OD[(n]
-+ End(Jac(X)) extends to complex mul-
for some n > 2. Since the automorphisms
of Jac(X) coincide with automorphisms of X, (X, t) is an orbifold point on WD if and
only if X has an automorphism commuting with t(OD). The orbifold order of (X, t)
is the cardinality of the group Aut(X)'/r/ of automorphisms commuting with
t(OD)
modulo the hyperelliptic involution.
Components of WD.
The components of WD were enumerated in [Mc3]. Usually
WD is irreducible; when WD is reducible the components of WD can be distinguished
by a spin invariant which we describe below.
A spin structure on a topological surface X is a quadratic form
q: H 1 (X,Z/2Z) -+ Z/2Z
satisfying q(x + y)
=
q(x) + q(y) + (x, y) where
(,)
is the intersection pairing. The
parity of q is given by the Arf-invariant:
Arf(q) =
q(Ai)q(Bi) E Z/2Z
for a standard symplectic basis {Aj, Bi} of H1 (X, Z/2Z).
The number Arf(q) is
independent of basis.
If X E M
2
is a Riemann surface, a one-form w E Q(X) with double zero deter-
mines an odd spin structure q, i.e. Arf(q) = 1. Any x c H 1 (X, Z/2Z) is the image of
the fundamental class of the circle S' under a smooth immersion C : S1 -+ X that
avoids the zero Z(w) of w. The Gauss map is the map S' -+ S1 given by:
0 (C' (0))
lo(C'(6))|'
and the function q(x) = deg G2 + 1 mod 2 does not depend on the immersion C and
defines an odd spin structure on X.
Now fix D > 1 with D _ 1 mod 8 and a point (X, t) E WD and let
f
be the
conductor of OD- Let q be the spin structure defined by the o--eigenform up to
scale [w]. The Jacobian endomorphism (f + vD)/2 gives a linear transformation on
H1 (X, Z/2Z) whose image V is a two dimensional subspace. The spin invariant of
(X, t) is defined to be the Arf-invariant of q restricted to V:
E(X, w)= e(X, t) = Arf (gly).
The spin component associated to EE Z/2Z is given by:
WL = f{(X, t): (X,
t)
c WD
and c(X, t) = c}.
McMullen showed that the discriminant for real multiplication and the spin invariant
distinguish the irreducible components of UD WD ([Mc3], Thm. 1.1):
Theorem 2.2 (McMullen).
The curves W and W 4 are empty. For discriminants
D > 9 with D = 1 mod 8, the spin components WD and WD are non-empty and
irreducible. For all other discriminants,WD is irreducible.
Bouw and M611er showed the spin components are Galois conjugate and homeomorphic ([BM1], Thm. 3.3):
Theorem 2.3 (Bouw-M6ller). Fix D =1 mod 8 with D not a square. The curves
W and WD have algebraic models defined over Q(VTD) and are Galois conjugate.
When D
=
f2
is an odd square, the spin invariant has a more elementary inter-
pretation. For such discriminants, (X, t) E WD is a degree
f
branched cover of an
elliptic curve E branched over a single point B, and the number N =
determines the spin invariant c(X, t)
= N.
f
1 (B)
n Xw
38
Chapter 3
Orbifold points on Hilbert modular
surfaces
In this section, we will show that most of the orbifold points on UD XD are Jacobians
of D8 - and D 12-surfaces with complex multiplication. In §4, 5 and A.1 we will show
that every complex multiplication point on M 2 (D8 ) and M 2 (D1 2 ) gives orbifold point
on UD XD establishing Theorem 1.5.
Automorphisms of abelian surfaces.
Let 34 and D 12 be the automorphism
groups of the polarized abelian varieties Av'--2/ 2 andA,/3 defined in Sections 1
and A. 1. The groups
S4
and D 12 are Z/2Z-central extensions of the groups S4 and
D 12 respectively. The following proposition is well known (cf. [BL], pg. 340):
Proposition 3.1. The automorphism group of a two dimensional principally polarized abelian variety which is not a product of elliptic curves is isomorphic to one of
the following: Z/2Z, Z/10Z, D4 , D8 , D 12 , S 4 or D 1 2 .
Note that the group S4 contains D and the group D 12 contains both D8 and D 12 Orbifold points on Hilbert modular surfaces.
We will prove the following
variant of Proposition 3.1:
Proposition 3.2. Fix a discriminantD > 0, an integer n > 2 and an abelian variety
B G A 2 with OD[n] -action t. One of the following holds:
e
B is a product of elliptic curves,
" (B, t) is a point of orbifold order five on X,
" An iterate of t(n) extends to a faithful action of D8 on B by automorphisms,
or
" An iterate of t((n) extends to a faithful action of D 12 on B by automorphisms.
Recall that the orbifold points of order n on XD correspond to abelian varieties
with complex multiplication by orders containing OD [(2] (§2.1).
Proposition 3.2
shows that the orbifold points on UD XD which are not products of elliptic curves are
the points of order five on X5 and Jacobians of D8 - and D 12 -surfaces with complex
multiplication.
Proof. Every two dimensional principally polarized abelian variety is a product of
elliptic curves or the Jacobian of a smooth surface (cf. 2.2).
Jacobian of X E M 2 . Let
#
Suppose B is the
be the automorphism of X inducing
t((n)
and let o-j be the induced permutation of the Weierstrass points Xw
# and each
=
of its iterates can fix at most three points on the sphere X/,
on Jac(X),
Fix(i). Since
the conjugacy
class of o-, is one of the following:
-> C with x o
* [2,2,2]: There is a coordinate x : X/
=
-x and x(Xw) _
{±1, ±a,±b} for some a and b E C*. The field of functions on X satisfies:
and
4(z, x)
C(z) [x]
X
2C(Z
~-
C(X)
(z
2
-
(x
2
1)(X 2 - a2 )(X2-
-
b2)),
= (z, -x) or (-z, -x). This is impossible since
42
* [2,2,1,1]: Let P 1 , P 2 , Q1 and Q2 be Weierstrass points so
XW = {P 1 , P2,Q 1,
4(Q1 ),Q 2, 4(Q 2 )}.
4 1.
#(P)
= P and
There is a unique coordinate x : X/q -+
C with x(Pi) = 1, x(P2)
=
-1 and x(Q1) = -x(Q 2 ). This coordinate satisfies
x o #= 1/x and x(Xw)
=
{±1, ±x(Q1), ±x(Q1)-1}. The field of functions on
X satisfies:
C(z) [x]
C(X)
(Z2
_ (X2 -
1) (X4-
aX2
+
1)
where a
=
x(Q)
2+
x(Qi)- 2 . The surface X has a faithful D8 -action generated
by r(z, x) = (z, -x) and J
=
(iz/x 3 , 1/x). Since x o # = 1/x,
#
=
J or J 3 and
t((n) extends to a faithful action of D8 on B.
[/3,3]: Let P and
x : X/Ir
-+
have x(Xw)
Q be points
=
{a,(3a,
There is a coordinate
x(Q)- 1 . Setting a
(3x and x(P)
C with x o q
#-orbits.
in XW in different
x(P), we
a, a- 1 , ( 3 a- 1 , (3a- 2 } and the field of functions on X
satisfies:
C(z)x]-
C(X)
(X3 - as) (X3-
(z2 -
a-3))
The surface X has a faithful D 12-action generated by r(z, x)
Z(z, x) = (-z, ( 3 x). Since x o # = (3x,
#
=
(z/x 3 , 1/x) and
is in the subgroup of D 12 generated
by Z and t((,) extends to an action of D 12 on B.
" [4,1,1] or [4,2]: The automorphism
#2
acts on XW by a permutation in the
conjugacy class [2,2, 1, 1] and B has a faithful D8 -action by automorphisms
extending t(Q). The surface X is the unique genus two surface whose automorphism group is S4 :
C(X) ~
(z2 _ (x2
-
" /5,1]: There is a coordinate x : X/r/
C(z)[x]
4
1)(x
-~
+6x+2
1))
C with x o # = ( 5x and x(Xw)
oo}. The surface X is the unique genus two surface with order five
automorphism and satisfies:
{1, (=
,(2,
C(z)[x]
C(X)
(Z2
_ (XI _ 1))'
The order for real multiplication OD is 05
* /6]: The automorphism
#2 induces
=
Z[( 5] n R.
a permutation of XW in the conjugacy class
[3, 3] and B has a faithful D12 -action extending tQ). The surface X is the
unique genus two surface whose automorphism group is D 12 and X satisfies:
C(X)
(z2 (C(z)[x]
(Z2 -
(X6 + 1
We record the following consequence of the proof of Proposition 3.2 which will be
useful in our analysis of the D8 -family:
Proposition 3.3. Suppose X G M
2
has an orderfour automorphism$. The induced
permutation o E Sym(Xw) is in the conjugacy class [2,2,1,1] and # extends to an
action of D8 .
Proof. Since the kernel of Aut(X) -+ Sym(XW) is generated by the hyperelliptic
involution, the permutation o-4 has order two or four. Since the automorphisms inducing permutations in the conjugacy class [2,2,2] have order two and those inducing
permutations in [4, 1, 1] and [4,2] have order eight, the permutation o-O must be in
[2,2, 1, 1].
5
Orbifold points on WD.
Our characterization of the orbifold points on XD which
lie on WD follows readily from Proposition 3.2.
Proposition 3.4. Fix X
E
M
2
with OD[Cn]-action t on Jac(X) and suppose an
eigenform for t has a double zero. Either:
e X is the genus two surface with orderfive automorphism, or
" X admits a faithful action of D8 extending an iterate of ((cn).
Proof. If
#
is the automorphism inducing t((n) on Jac(X) and o-0 is the induced
permutation on Xw, the conjugacy class of o-O in Sym(XW) is [2,2, 1,1], [4, 1, 1] or
[5,1] since
#
fixes at least one point in XW, namely the zero of the eigenform.
O
Remark. The orbifold points on XD which are products of elliptic curves are also easy
to characterize. For D
=
1, the complex orbifold X1 is isomorphic to the product
M 1 x M 1 and the product E x F with 01 generated by (1 0) is an orbifold point if
and only if E or F is an orbifold point on M 1 . For D > 4, if the product E x F is
labeled by an orbifold point on XD, then E and F are isomorphic and have complex
multiplication by Z[(3], Z[i] or an order in
Q(V
).
44
Chapter 4
The D 8-family
The family M 2 (D8 ) parametrizes pairs (X,p) where X is in M
2
and p :D
--
Aut(X) is an injective homomorphism (cf. §2.2). In this section, we will identify the
components of M 2 (D8 ) and study the component whose general member is smooth.
For (X, p) C M 2 (D8 ) we will show that the correspondence X -+ X/r x X/Jr induces
a degree two isogeny between elliptic curves:
cp: Jac(X/r)
-+
Jac(X/Jr).
We will compute Jac(X) and C(X) in terms of c, to show:
Proposition 4.1. The map (X, p)
-+
c, gives an embedding:
M 2 (D 8 ) -+ Yo(2),
whose image is the complement of the degree two endomorphism of the square torus.
The action of the non-trivial outer automorphism o- of D8 on M 2 (D8 ) is induced by
the Atkin-Lehner involution on Y(2).
In §1, we defined the domain U, the field K, the abelian variety A, the surfaces
X, and Yw and the functions a(T) and w(r). We will also show that the surfaces X,
and Y(,) admit faithful Ds-actions inducing the isogeny i, on E, = C/Z
'rZ
whose
kernel is generated by the image T, of 1/2 in C/Z E rZ (see Figure 4-1), establishing:
T
ET
I
(-I+i)/2
E-1/2T
T -12
IT
~/
0
r
TT
Figure 4-1: The domain V obtained from U and the image of U under r '-4 -1/2r is a
fundamental domain for the group Io(2). A point r E V determines an elliptic curve E, =
C/Z E -rZ and a degree two isogeny i, whose kernel is generated by the point of order two T,.
The isogeny i-1/2r is dual to i, since i-1/2 o i, is the multiplication by two map on E,.
Proposition 4.2. Fix r E U. The surface X, admits a faithful D8 -action p with
Cp= i, is isomorphic to the surface Y,(,) and satisfies:
C(X,)
Components of M 2 (D8 ).
K,,(, and Jac(X,) r A,.
Fix a surface X E M
2
and let p : D8
-+
Aut(X) be
a faithful representation of D8 . We will call p proper if the induced actions of Z[r]
and Z[Jr] on Jac(X) are proper. We will call p improper if it is not proper. We will
denote the corresponding components of M 2 (D8 ) by M 2 (D8 )r and M 2 (D8 )im.
A faithful representation on a smooth surface is automatically proper and we will
show that M 2 (Ds)
is the closure of M 2 (D8 ). For a nodal surface X = E V F, an
action of D8 is proper if and only if both r and Jr interchange the components of
X. For any genus one surface E E M 1 , the singular surface E V E has a faithful
D8 -action which is improper.
Nodal surfaces.
We now characterize the nodal surfaces in M 2 (D8 ).
Proposition 4.3. The nodal surface X = E V F admits an improperD8 -action iff E
is isomorphic to F. The nodal surface X = E V F admits a proper D 8 -action if and
only if both E and F are isomorphic to the square torus.
Proof. It is easy to show that E V E has an improper action of D8 for any E E MI.
It is also easy to check that when E = C/Z[i], the automorphism group of E V E
contains a proper action of Ds, which is unique up to conjugation in Aut(E V E).
For the converse, suppose p : D 8 -+ Aut(X) is injective. Since the subgroup of
Aut(X) fixing the components of X is the commutative group Aut(E) x Aut(F),
either r or Jr interchanges the components of X and E is isomorphic to F. Now
suppose p is also proper. Both r and Jr interchange the components of X and the
composition J = (Jr)(r)fixes the components of X. Since J4 = 1, both E and F are
isomorphic to the square torus.
Remark. Proposition 4.3 can be used to show that M 2 (D8 )i' has two components
both of which are isomorphic to M 1 (the components are distinguished by whether
r or Jr generates a proper 0 4-action). Also, the proper D 8 -action on E V E when
E = C/Z[i] is unique up to ~ since it is unique up to conjugation in Aut(E V E).
Weierstrass points.
Let
4
: D8
S6
-+
be the representation into the symmetric
group on six letters given by:
O(r) = (12)(34)(56) and
4(J)
= (13)(24).
For a proper D8 -surface, the action D8 induces on the Weierstrass points is isomorphic
to #.
Proposition 4.4. Fix a surface X G M
a bijection XW
-+
S 6 /S
5
2
with a proper D8 -action p.
inducing a homomorphism
following diagram commutes:
Ds
D8
P
> Sym(Xw)
S6
f
There is
: Sym(Xw) -+ S 6 so the
Proof. We will show that J2 is the hyperelliptic involution and that the involutions r
and Jr fix no point in XW. As a consequence, the action D8 induces on Xw factors
through the commutative group D4 =~D8 /J 2 and the proposition follows easily.
Since Z[r] and Z[Jr] act faithfully on Jac(X), the eigenspaces for r and Jr in
Q(X) are one-dimensional. Since r and Jr do not commute, their eigenspaces are
distinct. The automorphism J2 commutes with both r and Jr and fixes every line in
Q(X). The automorphism J2 is the hyperelliptic involution.
Since Z[r] and Z[Jr] are proper, the quotients X/r and X/Jr have genus one. For
smooth X, the fixed points of r and Jr are the simple zeros of the one-forms pulled
back from X/r and X/Jr, so they are disjoint from Xw. For nodal X, r and Jr
interchange the components of X and again the fixed points of r and Jr are disjoint
5
from Xw.
We will denote by Fixw(J) the set of Weierstrass points fixed by J. We will need
the following fact in our discussion of surfaces obtained from polygons:
Proposition 4.5. Two proper D8 -surfaces (X 1 , pi) and (X 2 , P2) have an isomorphism
X 1 -± X 2
X:
intertwining pi(J) if and only if:
(X1, pI)
-
(X 2 , p 2 ) or (X1,p1)-
(X 2 , p2 o 0-).
Proof. Let t be the automorphism of X1 pulled back from p2 (r) under V). By Proposition 4.4, the action of t on X1w coincides with that of r or Jr and, as a consequence,
t is one of r, Jr,rr or rJr.If t is r or Jr,# gives the desired equivalence. Otherwise,
4 o p1(J) gives the desired
equivalence.
Proper D 8-actions, correspondences and isogeny.
5
We will now show that a
proper D8 -surface (X, p) E M 2 (D8 )Pr determines a degree two isogeny c, between the
genus one surfaces Jac(X/r) and Jac(X/Jr). The non-trivial outer automorphism 0of D8 has o-(r)
=
Jr and o-(J)
determines the dual isogeny C*.
=
J. We will also show that the surface (X, p o 0-)
Recall that the quotients E
involutions
TiE
and
F
=
X/r and F = X/Jr have genus one, elliptic
induced by q, natural base points E[1]
= Fix(Tr)/r and
F[1] = Fix(r)/qr and identifications E[2]* = Xw/r and F[2]* = Xw/Jr (see §2.2).
Proposition 4.6. Suppose X G M 2 has a proper D8 -action p. The correspondence
X -+ X/r x X/Jr induces an isogeny:
c,,: Jac(X/r) --+ Jac(X/Jr)
whose kernel is generated by the point of order two [Fixw(J)/r - Fix(ijr)/r]. The
isogeny c, is dual to the isogeny c,,,.
Proof. We will show that ker(c,) contains the point of order two labeled by Fixw(J)/r
and that c, o c,
is the multiplication by two map on Jac(X/r). It follows that both
c, and c,,, have degree two and c* = c,,.
The isogeny c, is given by summing the images of the fibers of the map X
under the map X
-4
-4
X/r
X/Jr. For the point in X/r labeled by the r-orbit {Q, rQ} we
have:
c, ({Q, rQ})
=
{Q, JrQ} +F
The point Fixw(J)/r is in ker(c,) since {rQ, JQ}
=
{rQ,
F
JQ}.
({Q, JrQ}) whenever J(Q)
-
Q.
The non-trivial outer automorphism o- of D8 is generated by o-(J) = J and o(r) =
Jr. Using the formula above for c, and the relations on D8 , we have that c,,
o
c,({Q, rQ}) is equal to:
{Q, rQ} +E {JrQ, J3Q} +E
Since J2 =',
we have {JrQ, J3 Q}
=
{rQ, Q} +E
{JQ1, rJQ}.
r/E ({JQ, rJQ}) and the second and fourth
terms cancel, giving c, o co, ({Q, rQ}) = {Q, rQ} +E {Q, rQ}.
The unique nodal and proper D8 -surface (X, p) has:
Jac(X/r) = Jac(X/Jr) = C/Z[i],
D
and c, = i(+i)/2 is point of orbifold order two on the modular curve Y(2) (see §2.1).
Jacobians.
We now compute the Jacobian Jac(X) for a proper Ds-surface (X, p)
in terms of the isogeny c,.
Proposition 4.7. Fix a Ds-surface (X,p) E -M 2 (D8 )P' and a number T Ei H so
c,= i4. The Jacobian of X satisfies:
Jac(X) - A,.
The polarized abelian variety A, was defined in §1 and the isogeny i, is the map
on E, = C/Z E rZ whose kernel is generated by the image T, of 1/2 in C/Z E rZ.
Proof. Since the lattice A, contains the lattice Z ((2), (27-), (0), (4)), the abelian
variety A, satisfies:
A_
(E. x Er)/FT- ,
where FT C E,[2]* x E,[2]* is the graph of the transposition fixing T,. Twice the
product polarization on E, x E, is pulled back from the principal polarization on A,
under the isogeny E, x E, -+ A,.
Let E = Jac(X/r) and E
Jac(X/r/r). Since c, = i, the elliptic curve E, is
isomorphic to E. The automorphism J induces an isomorphism:
#: X/r ~> X/rr,
since J-'rJ=r/r so E is also isomorphic to E,. The equality Xw/r
=
Xw/r/r gives
a bijection on the points of order two 0 : E[2]* -+ E[2]* and the composition
#-' o V
is the action J induces on XW/r = E[2]*. We computed the action of p on XW in
Proposition 4.4 and
#-' o #
is the transposition fixing T,.
The map X -+ X/r x X/r/r factors through a degree four map on Jacobians
Jac(X) -+ E x
. The image of Xw in E x5
E, x E, is the graph
FT-
of 0-1 o #
and the image of Jac(X)[2] in E, x E, is the subgroup generated by FT. The quotient
map:
Jac(X) -> (E, x E,)/FTE
has degree sixteen, vanishes on the two torsion Jac(X)[2] and factors through multiplication by two to give an isomorphism. The polarization is a product since the
self-adjoint operator r has eigenspaces pulled back from Q(E) and Q(E). Unimodularity implies the polarization is twice the product polarization.
l
Remark. More generally, if Y E M2 has an involution r generating a proper 04action on Jac(Y), the quotients Ei
=
Y/r and E 2 = Y/rr have genus one, natural
elliptic involutions and base points. The points of order two on E1 and E2 have an
identification coming from Yw:
E1[2|* _ Yw/
Yw/r/r
= E2[2]*.
The middle equality comes from the fact that y fixes Yw pointwise. The Jacobian of
Y satisfies:
Jac(Y) ~ (E1 x E2 )/w,
where Iw is the graph of this identification and the right hand side is polarized by
twice the product polarization. Proposition 4.7 is a special instance of this computation.
Since the Jacobian of a D8 -surface can be computed from the isogeny c,, we have
established that the map (X, p)
'-*
c, embeds M 2 (D8 ) in Yo(2):
Proof of Proposition4.1. We will show that the map M
(X, p)
-+
2
(D 8 )pr -
Yo(2) given by
c, is an isomorphism. The remaining claims follow from Proposition 4.6
and our observation that the unique nodal and proper D8 -surface maps to the isogeny
z(1+i)/2.
We will define an inverse for the map (X, p)
-
c,. Fix an isogeny i, in Y(2) and
let X be the (possibly nodal) surface with Jacobian A,. Let p, : D8 - Aut(X) be the
action of D8 inducing the Jacobian automorphisms pr(r) = (' _0) and p,(J) = (1).
The map i,
+
(X,, pr) is the desired inverse by Proposition 4.7.
0
Remark. The family M 2 (Ds8)P
is connected and one dimensional. Since there is a
unique nodal and proper D8 -surface, M 2 (D8 )pr is the closure of M 2 (D8 ) in M 2 (D8 ).
-2 +
The function a(T)
Algebraic curves.
for Fo(2) and covers an isomorphism a : Yo(2) -+ C
defined in §1 is modular
\ {-2, oo}.
Using A (-1+)
A (2-) = 1, the isomorphism a is uniquely determined by:
a (Fo(2) - 0) = oo, a (F0 (2) - oo) = -2,
and a (F0 (2) - (1 + i)/2) = 2.
The number a(T) is characterized by:
C (Er) ~lI2C(Z
(z2
-
(W
1) (W2-
a(r)w + 1))
rZ and T, = -
where as usual E, =C/Z
with T, = [w- 1(1) - w-1(oo)],
W
+Z
D rZ.
Proposition 4.8. Fix a smooth Ds-surface (X, p) and r G H with cp
=
iT. The field
of functions on X satisfies:
C(X) ~-'K,(,)Proof. Let Y be the genus two surface with:
C(Y) ~ Ka,()
= C(z, x) with Z2
(X2 _ 1)(X 4 - a(r)x2 + 1),
and let @be the D8 -action on Y with O(r)(z, x) = (z, -x) and @(J)(z, x) = (iz/x 3 , 1/X).
The quotient Y/r has C(Y/r) ~ C(z, x 2 ), a field isomorphic to C(E,). The action
O(J) induces on the points of order two E,[2]* is the transposition fixing [w-'(1) w-1 (oo)]. The isogeny c p is isomorphic to i, and Proposition 4.1 implies that (Y, @) ~
E
(X, p).
Remark. The field
K(-2a+1 2 )/(a+ 2 )
is isomorphic to Ka. Equipping K, with the D 8 -
actions Pa(z, x) = (z, -x) and J(z, x)
=
(iz/x 3 , 1/x), the D8 -surfaces associated to
(Ka, Pa) and (K(-2a+1 2)/(a+ 2), P(-2a+12)/(a+2))
D8 .
differ by the outer automorphism o- of
Cusps on M 2 (Ds)/o-.
There are two cusps on
M2(Ds)/o-,
one of which maps to the
cusp of the (2, 4, oo)-orbifold M 2 (Ds)Pr/o-. For (X, p) E M 2 (D8 ), let |q,| be the flat
metric with singularities induced by a quadratic differential on X/p(J) with simple
poles at Xw/p(J) and unit area fl(j) q,|= 1.
Proposition 4.9. For a sequence of Ds-surfaces (Xi, pi) G M 2 (D8 ), the following
are equivalent:
1. The D 8 -surfaces (Xi, pi) diverge in M 2 (Ds )pr/o-.
2. The quotients Xi/pi(r) diverge in M 1 .
3. The surfaces Xi tend to a stable limit with geometric genus zero.
4. The diameter of Fixw(pi(J)) in the metric space (Xi/pi(J), jq, |) tends to zero.
Proof. Condition (1) and (2) are equivalent since the two maps M 2 (D8 )Pr
M
1
given by (X, p)
i-+
X/r and (X, p)
'-+
Y(2) --+
X/Jr are proper. Now suppose (2) holds
and Xi/r diverge in M 1 . This happens if and only if there is a sequence of numbers
a tending to -2 so C(Xi)
Ka,. The stable limit X,, has:
C(X,) 2 K 2 = C(z, x) with z 2
(X2 - 1)(x
and the geometric genus of X,, is zero since K 2 = C
2+
( (2+.(X-1)
1)2,
Conversely, a
surface with geometric genus zero does not have compact Jacobian so (2) and (3) are
equivalent.
Finally, the fields C(Xi/J) C Ka, are generated by the function g = x + 1/x and
g(Fixw(J)) = {±2}. If ai tends to -2, the remaining points in XW tend to 0 under
g and the diameter of Fixw(pi(J)) tends to zero in the metric space (Xi/J, Iq, I). If
ai tends to 2, the remaining points in XW tend to ±2 under g and the diameter of
Fixw(pz(J)) tends to oo in (Xi/J, Iq, ). This shows that (2) and (4) are equivalent.
0
PT+ I
Figure 4-2: The surfaces X, and X,+1 are isomorphic since the polygons P, and P,+1 differ
by a cut-and-paste operation.
Polygons.
We now turn to the surfaces obtained from the polygons P, and
Q".
The following proposition, together with Propositions 4.7 and 4.8 completes the proof
of Proposition 4.2:
Proposition 4.10. Fix r
U. The surfaces X, and Y(.,,) have proper D 8-actions
px and py with cp, = cp, = i,.
Proof. Let J, denote the order four automorphism on X, obtained from counterclockwise rotation the polygon P, and let J, denote the order four Deck transformation
for the map on Y depicted in Fig. 1-3. We will show that the maps r -+ (X,, J,)
and w -+ (Y., J.) give maps:
gU : U/ -U-+ M 2 (Ds)/o- and gv : V/
with -r
-
y-V-
M 2 (Ds)/o-
r+1, -r ~u -1/2r, w ~V w+2 and w ~y w/(w+ 1). Since U/ ~u, V/ -yv
and M 2 (Ds)/o- are all isomorphic to the (2, oo, oo)-orbifold, the maps gu and gV are
isomorphisms. We will also show that gu(-r) and gv(w) diverge in M 2 (Ds8)/o as
Im(r) and Im(w) tend to oo, determining the isomorphisms gu and gv and showing
that X, and Y,(,) have D 8 -actions px and py with c,, = c, = i,.
By Proposition 3.3, the automorphisms J, and J, extend to actions of D8 on
X, and Y.
By Proposition 4.5, the maps -r -
(X,, Jr) and w
i-±
(Y., J.) give
holomorphic maps gu : U
-+
M 2 (Ds)/- and gv: V -+ M 2 (Ds)/a.
The polygons P, and P,+1 differ by a cut-and-paste operation (see Figure 4-2) and
the polygons P, and P-1/2, differ by Euclidean similarity. The surfaces Xr, X,+ 1 and
X-1/2,
are isomorphic with isomorphisms intertwining the actions of Jr, J,+1 and
J- 1 /2, and the map gu descends to the quotient U/ ~l-'.
As Im r -+ o, the stable
limit of X, has geometric genus zero and gu(T) diverges in M 2 (Ds8)P/j.
It is easy to check that the Veech group of (Y, qi)
=
(Qj, dz 2 )/ ~ contains
)
(12
and (1 ?). The surfaces Y., Y,+ 2 and Y/(+1) are all isomorphic with isomorphisms
intertwining J,, J,+ 2 and Jw/(w+1) and the map gv descends to the quotient V/ ~.
The flat metric Iq,| on Y,/J. is induced by |dz2j / Im(w) (Fig. 1-3) and the diameter
of Fixw(J.) in (Y/J., qJ) is 1/w/Im(w) which tends to zero as Im(w) - oo.
El
56
Chapter 5
Endomorphisms
Fix a smooth D8 -surface (X, p) and let E = Jac(X/r). In this section we embed
End(Jac(X)) as an order in M 2 (End(E) 9
Q),
characterize the proper D8 -surfaces
with complex multiplication and sort them by their orders for real multiplication to
compute e2(WD) and prove Theorem 1.1.
Theorem 1.8 describing the D8 -surfaces with complex multiplication follows from
Proposition 4.2 and:
Proposition 5.1. The abelian variety A, has complex multiplication if r is imaginary quadratic.
Whenever E has complex multiplication, p(J) extends to complex multiplication
by an order in End(E) 0 Q[i] giving an orbifold point on UD WDProposition 5.2. Fix a Ds-surface (X, p) E M 2 (D8 ) and suppose Jac(X) has complex multiplication. The action of J on Jac(X) extends to complex multiplication t
by an order in
Q(vW, i) and the pair (X,
t)
gives an orbifold point on UD WD-
Propositions 3.4 and 5.2 complete our characterization of the orbifold points on
UD WD asserted in Corollary
1.6. The analogous results for D 12 -surfaces show that the
complex multiplication points of M 2 (D1 2 ) are labeled by orbifold points on UD XD
(see §A.1). Together with Proposition 3.2, this establishes our characterization the
orbifold points on UD XD in Theorem 1.5.
The endomorphism rings of isogenous abelian va-
Isogeny and endomorphism.
rieties are related by the following proposition.
Proposition 5.3. Suppose
f
: A -± B is an isogeny between principally polarized
abelian varieties and n-times the polarization on A is the polarization pulled back
from B.
The ring End(B) is isomorphic as an involutive algebra to the subring
Rf C End(A) 0
Q
given by:
Rf =-
1
{#
n
End(A): 4 (ker(nf)) C ker(f)}.
E
The ring Rf satisfies 1-End(A) D R1 D nEnd(A). Proposition 5.3 shows that
End(A) is an order in End(B) 0
Q
and that the rational endomorphism ring is an
isogeny invariant. Propositions 5.1 and 5.2 follow:
Proof of Proposition 5.1. The abelian variety A, has a degree four isogeny to E, x E,
and twice the polarization on A, is pulled back from the product polarization. When
is imaginary quadratic, the ring End(A,) is an order in M 2 (Q(-r)) with (a
and A, has complex multiplication by an order in
Q (r, i
=
( _01)).
b)*=
T
(QE)
When -r is not
imaginary quadratic, End(A,) is an order in M2 (Q) with (a b)* = (' c) and A, does
not have complex multiplication since every commutative subalgebra of M2(Q) has
rank two over
Q.
l
Proof of Proposition 5.2. Suppose (X, p) is a smooth D 8 -surface and Jac(X) has complex multiplication. The center of End(A,) is an order in Q(T) and the order four
automorphism p(J) extends to complex multiplication t by an order ( in Q(r, i).
The O-eigenforms have double zeros by Proposition 3.3, so the pair (X, t) gives an
orbifold point on UD WD-
El
Remark. The Jacobian Jac(X) for a D 12-surface (X, p) is isogenous to E x E where
E - Jac(X/r) (§A.1).
The same argument shows that the Z-eigenforms on D12-
surfaces with complex multiplication are --eigenforms for orbifold points on UD XD,
establishing our characterization of the orbifold points on Hilbert modular surfaces
of Theorem 1.5.
Proof of Proposition 5.3. We will show that the map:
p: End(B) -4 End(A) 9 Q
4qt--f*o # o f
n
is an injective homomorphism and has image p(End(B)) = Rf. The condition on
the polarization implies that
f* o f
and
f o f* are
the multiplication by n maps on
A and B respectively and it is easy to check that p is a homomorphism. The map
p is injective since rationally it in an isomorphism. The inverse for p is given by
p-
) = if
o
of*.
The image of End(B) is contained in R1 since:
ff
ker(nf) + B[n]
for any
#
-
B[n]
ker(f)
-+
E End(B). To see that the image of p is all of R 1 , fix 4 E End(A) 9
satisfying @(ker(nf)) C ker(f). The endomorphism
#=
Q
1f o4 o f* is integral on B
since:
B[n2 ] 1
Since p()
ker(nf) i
ker(f)
-+
0.
= ,, the image of p is all of Rf.
The isomorphism between End(B) and Rf has p(#)* = p(#*) since (f*of)*
*O*f-E
Invertible modules over finite rings.
isogeny
f
: Jac(X) -+ E x E with
The Jacobian of X has a degree four
f o f* acting by
multiplication by two on E x E.
In light of Proposition 5.3, if E has complex multiplication by Oc, the order for real
multiplication on X commuting with J has discriminant D
=
-C/4, -C or -4C.
The actual order for real multiplication on Jac(X) commuting with J depends only
on End(E), the M 2 (End(E))-module E[4] x E[4] and the Z-submodules ker(f) and
ker(2f).
To sort the complex multiplication points on M 2 (D8 ) by their orders for real
multiplication we need to determine the possible Oc-modules E[4]:
Proposition 5.4. Let 0 be a quadratic order and I be a proper 0-ideal. The module
I/nI is isomorphic to O/nO as 0-modules.
For imaginary quadratic orders, Proposition 5.4 can be deduced from the fact that
C/I and C/J have algebraic models which are Galois conjugate whenever I and J
and proper 0-ideals.
Proof. Since 0 is quadratic, the Galois conjugate I' is an inverse for I and I
I' is
isomorphic to 0. The quotient I/nI is also invertible and the invertible modules over
R
O/nO form a group Pic(R). We will show Pic(R) is trivial.
The ring R is an Artin ring and by the structure theorem there are local Artin
rings Ri with:
R = fl R and Pic(R) = j Pic(Ri).
i
t
The group Pic(Ri) is trivial by Nakayama's lemma: if I is an invertible Ri module,
and mi is the maximal ideal, Ii/mi is an invertible module over the field RI/mi. The
Rj/mi-vector space I2 /m;I2 is one dimensional and if x E I generates this vector
= Ii which implies I -=Rix.
space, Rx + miI
2
We are now ready to sort the complex multiplication points on M 2 (D8 ) by their
orders for real multiplication.
Proof of Theorem 1.1. The only D8 -surface whose automorphism group is larger than
D8
is the surface obtained from the regular octagon labeling the point of order four
on W 8 (cf. Proposition 3.1). For discriminants D > 8, all of the orbifold points on
WD are labeled by complex multiplication points on M 2 (D8 ) and have order two.
To complete our proof, we need to determine the number of D 8 -surfaces with real
multiplication by OD commuting with p(J).
For each C = 0 or 1 mod 4, set:
C
Z[x]
2 -
CX
+ C(C-1)
4
C mod 16
D1
D2
D3
0
-4C
-C
-4C
4
-4C
-4C
-C
8
-4C
-C
-4C
12
1 or 9
-4C
-4C
-4C
-4C
-4C
-4C
-C/4
-4C
-4C
5 or 13
Table 5.1: The elliptic curve E = C/Oc is covered by three Ds-surfaces (Xi,pi). The
discriminant Di of the order for real multiplication on Jac(Xi) commuting with J is computed
using Proposition 5.3.
The ring Oc/40c depends only on C mod 16. Let P1 = 1, P2
=
x and P 3 =x +1 be
the points of order two in Oc/ 2 Oc and let (Xi, pi) be the Ds-surface in M 2 (D8 )f
with Xi/pi(r) - C/Oc and ker(cp,) generated by P.
In Table 5.1 we compute
the discriminant Di of the order for real multiplication on Jac(Xi) commuting with
J using Proposition 5.3. Most of the entries in Table 5.1 are determined by the
observation that Jac(Xi) has complex multiplication by an order containing Oc[i iff
the Oc-module POc/2Cc is a proper submodule of Oc/20c.
For any genus one surface E with complex multiplication by Oc, we have E[4] a
Oc/ 4 by Proposition 5.4. With an appropriate ordering, the three D8 -surfaces covering E have the same orders for real multiplication as the three D8 -surfaces covering
C/Oc by Proposition 5.3. The formula for e2(WD) in Table 1.1 follows easily. Note
that the factor of two in the formula for e2 (WD) comes from the fact that the D8 surfaces (X, p) and (X, p o o) label the same orbifold point on UD WD-
LI
62
Chapter 6
Spin
For a number r E U, let w, E Q(X,) be the holomorphic one-form with double zero
on the surface X, obtained by gluing together opposite sides on P:
(X,
wr) = (Pr,dz)/
~ .
In this section we compute the spin structure q : H1 (X,, Z/2Z)
-+
Z/2Z coming from
the form w and then, for D = 1 mod 8, sort the orbifold points on WD by their spin
component.
Spin structure.
Let a be the number
-1+i
and
xt
be the homology class in
H1 (X,, Z) with period t E Z[i] ® aZ[i] under the one-form w,.
Proposition 6.1. The spin structure q on X, obtained from the form with double
zero w, has:
q (klx1 + k 2 xi + k 3 Xa + k4 Xai)
-
k2 + k2 + k1 k3 + k 2 k 4 + k 3 k 4 .
Proof. Isotoping representatives away from the zero Z(wo) and computing the degree
of the Gauss map shows that the spin structure q associated to w, satisfies:
q(x1 ) = q(xi)
-
1 and
q(Xa)
= q(xai) = 0.
From the relation q(x + y) = q(x) + q(y) + (x, y), any basis x, of H1(X,, Z/2Z) has:
k2 q(xn) +
=(
knx)
q (
n
(
kik. (xi, xn).
I<n
n
Computing the intersection pairing on the basis {x1, x
gives the stated for}Xai
xa,
mula for q.
0
Spin homomorphism.
Now fix an integer D > 9 with D =1 mod 8 and let
be the conductor of D. An ideal I C O-4D has norm Nm(I) =
2 kl
f
with 1 odd. We
define:
c(I)=
2
mod 2.
Proposition 6.2. The number co(I) depends only on the ideal class of I and defines
a spin homomorphism:
Eo : H(-4D) -+ Z/2Z.
The spin homomorphism eo is the zero map iff D is a square.
Proof. Any x E O-4D has norm Nm(x) - X± x2D = 2
ideals I and J are in the same ideal class, they satisfy xI
with 1 = 1 mod 4. If the
=
yJ for some x and y in
O-4D and co(I) = co(J). The map Eo is a homomorphism since the norm of ideals is
a homomorphism.
Now suppose D
the form I
=
=
f2
is a square. Any 0- 4 D-ideal class has a representative of
xZ D (fi - y)Z. Since I is an ideal, x divides
f2 +
y2 and since I is
proper gcd(x, y, (f2 + y 2 )/x) = 1. If an odd prime p divides Nm(I), then p divides
x 2 , f 2 + y2 , and y. Since
f2
_y 2 mod p and p does not divide both
f and y, -1 is
a square mod p, p - 1 mod 4 and Eo(I) = 0.
If D is not a square, D = pkpi2
kpk ...
with pi distinct odd primes and ki odd.
By Dirichlet's theorem, there is a prime p with:
"p
3 mod 4,
"p
1 modpi for 1 > 1, and
) = -1.
Quadratic reciprocity gives
1. If x 2
is an O- 4D-ideal and has norm p and Eo(I)
=
-D mod p, then I = pZe(V/7-Dx)Z
1.
Remark. When D -1 mod 8, the ideal (2) ramifies in
ideal P with P 2
represented by I
0
-4D and there is a prime
(2). Since Nm(P) = 2, we have eo(P) = 0. The ideal classes
Z E rZ and J
=
Z D-1/
2 rZ
satisfy [I] = [PJ]. This is related to
the fact the polygonal stars P, and P-1/2, give the same point on WD and so must
have the same spin invariant.
Spin and orbifold points.
and conductor
As before, fix a discriminant D > 9 with D - 1 mod 8
f.
Proposition 6.3. Suppose (X,, w,) is an eigenform for real multiplication t, by OD
obtainedfrom the polygon P, and let I = Z E rZ. The spin invariantfor (X,, t,) is
given by:
C(X,
t)
=
2
+ Eo(I) mod 2.
The formula for e2 (WL) stated in Theorem 1.2 follows from Propositions 6.2 and
6.3.
Proof. According to Table 5.1, Jac(X,) has real multiplication by OD with eigenform
w, iff I is a proper O- 4 D-ideal and 1 + V/-IU E 21.
Since I is an ideal, vy17= x-r+y for some x and y E Z, x divides D+y
2
and I has
the same class as Io = xZ E (V-7D - y)Z. Since I is proper, gcd(x, y, (D + y 2 )/X)
and the norm of Io is x up to a factor of two. The condition 1 + v
x = 2 mod 4 and co(I) = y
=
7 E 21 ensures
mod 2.
To compute the spin invariant c(X,, w,), we need to determine the subspace V
Im (f--)
1
=
of H 1 (X, Z/2Z) and evaluate Arf(qv). The subspace V is spanned by v
and iv where:
f - iv ~Di7
2f +x
2
4
.x - 2y
4
.
+azmod2.
Since q(v) = q(iv), we have by Proposition 6.1:
e(Xr,r) = q(v)2
2
Remark. For square discriminants D=
shows e2 (WL-1)/2)
=
f2 ,
+ co (I) mod 2.
there is an elementary argument that
0. When the surface Xr is labeled by an orbifold point on
WD, the number r is in Q(i) and rescaling Pr, we can exhibit X, as the quotient of a
polygon with vertices in Z[i] and area
f.
The area determines the number of points
in Xw lying on the lattice Z[i] which in turn determines the spin invariant.
Chapter 7
Genus
Together with [Ba, Mc3], Theorems 1.1 and 1.2 complete the determination of the
homeomorphism type of WD. For non-square D, we will show that the homeomorphism type of WD can be determined from the homeomorphism type of the Hilbert
modular surface XD and the product locus PD:
Proposition 7.1. For non-square D > 8, the homeomorphism type of WD is determined by the orbifold Euler characteristicx(XD), the number of cusps C(PD) on PD,
the number of orbifold points of order two e2(XD) and e 2 (PD) and D mod 8.
We will use our formula to give bounds for the genus of WD and its components.
Proposition 7.2. For any e > 0, there are positive constants C, and N so:
CD3/2+c
>
g(V)
whenever V is a component of WD and D > Nr.
We will also give effective lower bounds.
Proposition 7.3. Suppose D > 0 is a discriminant and V is a component of WD.
If D is not a square, the genus of V satisfies:
g(V) > D 3 / 2 /600 - D/6 - D 3/ 4 /2
-
75.
If D is a square, the genus of V satisfies:
g(V) > D 3 / 2 /240 - D - D 3 / 4 /2
The components of UD WD with genus g
-
75.
4 are all listed in §A.2.
Corollary 7.4. The components of UD WD with genus g < 4 all lie on UD<121 WD.
Proof. The bounds in Proposition 7.3 show that g(V) > 4 whenever D > 17500 for
non-square D and D > 2502 for square D. The remaining discriminants were checked
by computer.
The Hilbert modular surface XD has a
Euler characteristic of XD and WD.
meromorphic modular form with a simple zero along WD and simple pole along PD.
This gives a simple relationship between the orbifold Euler characteristics of WD, PD
and XD and a modular curve SD in the boundary of XD. The curve SD is empty
unless D
=
f2 is a square,
in which case SD E X 1 (f).
Theorem 7.5 ([Ba] Cor. 10.4). The Euler characteristicof WD satisfies:
x(WD)
=
x(PD) - 2x(XD) - X(SD)-
For a discriminant D, define:
F(D) =
fl (1
-
(p
P-2).
PIf
f is the conductor of OD, Do = D/f 2 is the discriminant of the
Q(/f) and the product is over primes dividing f. The number
where
maximal order
in
F(D) satisfies
1 > F(D) > (Q( 2 )-l > 6/10.
For square discriminants, X(SD)
=
-f
2 F(D)/12 and
the Euler characteristic of
WD and its components are given by ([Ba] Thm. 1.4):
1)F(D)/16,
X(Wf2) =f2(f2
x(WO2 ) =-f
x(W
2
(f
1)F(D)/32, and
- 3)F(D)/32.
) =f2(f
For non-square discriminants, X(SD)
-
=
0 and x(PD)
-jx(XD) giving x(WD)
=
-X(XD). The Euler characteristic x(XD) can be computed from ([Ba] Thm. 2.12):
x(XD) = 2f3(Do (-1)F(D).
Here (Do is the Dedekind-zeta function and can be computed from Siegel's formula
([Bru], Cor. 1.39):
(Do (- 1
1
4Do
- e2J
e2 <Do,e=Do mod 2
where o-(n) is the sum of the divisors of n. For reducible WD, the spin components
satisfy X(WD,) = x(WD)
=
jX(WD)
The well-known bound o(n)
-
([Ba] Thm. 1.3).
o(nl+e) gives constants C, and N, so:
CD 3/2+e
> x(XD)
whenever D > N. Using o(n) > n + 1 and F(D) > 6/10 gives:
x(XD) > D 3/ 2 /300.
We can now prove the upper bounds for the genus of WD:
Proof of Proposition 7.2. For square discriminants D =
f 2 , we
have jX(WD) <
For non-square discriminants, the bounds for X(XD) and the formula X(WD)
-9x(XD)/2 gives IX(WD)l = O(D3/2+e).o Since
WD
f 3.
=
has one or two components,
g(WD) = O(\x(WD)) -
The modular curve PD.
The modular curve PD is isomorphic to:
(OnYo(m)
)g,
where the union is over triples of integers (e, 1,m) with:
D = e2 +41 2 m, l,m>0, and gcd(e,l)= 1
and g is the automorphism sending the degree m isogeny i on the component labeled
by (e, l, m) to the isogeny i* on the (-e,l,m)-component (cf. [Mc3] Thm. 2.1). The
isogeny i : E
B
-
F on the (e, 1,m)-component corresponds to the abelian variety
E x F with OD generated by t (e+
-
)o
The components of PD are labeled by triples (e, l, m) as above subject to the
additional condition e > 0. We will need the following bound on the number of such
triples:
Proposition 7.6. The number of components of PD satisfies:
ho(PD)
D3/ 4 + 150.
Proof. Let 1(n) denote the largest integer whose square divides n and let f(n) =
f is multiplicative and the
.e2There is a finite
(e, 1,m) with e fixed is bounded above by f (D
d(l(n)) be the number of divisors of 1(n). The function
number of triples
set S of natural numbers n for which f(n) > n 1 /4 (they are all divisors of 212365472112)
since d(n) is o(nE) for any E > 0 and it is easy to check that EnES f(n)
The asserted bound on h0 (PD) follows from:
ho(PD)
z
e=D mod 2
O<e<v/
fD
D
e2
2)D
e
< 150 +
(D
)
1/4
-
n 1/ 4 < 150.
Cusps on WD and PD.
Let C1(WD) and C2(WD) be the number of one- and two-
cylinder cusps on WD respectively and C(WD) = C1(WD) - C2(WD) be the total
number of cusps. The cusps on WD were first enumerated and sorted by component
in [Mc3]:
Proposition 7.7. For non-square discriminants, the number of cusps on WD is equal
to the number of cusps on PD:
C(WD) = C2(WD)
C(PD),
and C(WD) = C(Wh) when WD is reducible. For square discriminantsD
=
f 2,
the
number of one- and two-cylinder cusps satisfy:
C2(Wf2) < C(P
When
f
is odd, C(W 2 )
-
2
)
and C1(Wf2) < f 2 /3.
C(W 2 ) < 7f 2/12.
Proof. Except for the explicit bounds on C1(Wf2) and C(W 2 )
-
C(W ) , the claims
2
in the proposition follow from the enumeration of cusps on PD and WD in [Ba],
Section 3.1. We will sketch another argument for the relationship between C(WD) and
C(PD). The cusps on WD are labeled by splitting saddle connections for eigenforms
([Mc3], §3). For a point (X, t) E WD near a cusp, there is a unique short splitting
saddle connection and the splitting gives a point on PD near a cusp. This gives a
correspondence between the two-cylinder cusps on WD and the cusps on PD. One
can show that this correspondence is always injective and is bijective whenever D is
not a square by showing there is an inverse correspondence defined on a subset of the
cusps of PDWe now turn to the bounds on Cl(Wf2) and C(W 2 ) - C(Wf2) . When D is not a
square, there are no one-cylinder cusps and when D = f 2 is a square, the one-cylinder
cusps are parametrized by cyclically ordered triples (a, b, c) with (cf. [Mc3] Theorem
f = a+b+c,a,b,c> 0 and gcd(a,b,c) = 1.
Reordering (a, b, c) so a < b and a < c ensures that a < f/3 and b <
f,
giving
C1(WD) < f2 /3= D/3. The difference in the number of two cylinder cusps is given
by (Theorem A.4 in [Mc3]):
3
C 2 (W) - C2 (WD) =
4(gcd(b, c)),
b+c=f,0<c<b
which is smaller than D/4 using c < f /2 and
#(gcd(b, c))
< f /2. The bound asserted
for |C(WD) - C(W)I follows.
Our formula for e2(WD) gives:
Orbifold points on WD.
Theorem 7.8. The number of points of order two on WD satisfies:
e2(WD)
-
e2(XD) - e 2 (PD)
4
and e2(WD) < -D.
3
For non-square D, Theorem 7.8 establishes Proposition 7.1 together with Proposition 7.7 and the enumeration of the irreducible components of WD in [Mc3].
Proof. Theorem 1.5 and Corollary 1.6 say that every orbifold point of order two on
XD is
an orbifold point on PD or WD, giving e2(WD)
= e2(XD) - e 2 (PD)-
The bound
for e2(WD) follows from the bound h(-D) < 2D/3 (§2) for discriminants -D < 0
0
and the formula in Theorem 1.1.
Lower bounds, non-square discriminants.
Now suppose D is a non-square dis-
criminant and D > 8 so all of the orbifold points on WD have order two. Using
the formula for X(WD), the equality C(PD) = C(WD), and x(SD) = 0 and ignoring
several terms which contribute positively to the g(WD) gives:
g(WD) > x(XD)
-
h0 (PD)
-
e2(WD)/4.
Combining the bound above with x(XD) > D 3 / 2 /300, ho(PD) < D 3/4+ 150, e2(WD) <
D and g(V) > Ig(WD) whenever V is a component of WD gives the bound in
Proposition 7.3.
Lower bounds, square discriminants.
Now suppose D =f 2 . Using the formula
for x(WD) in terms of x(XD), x(PD) and x(SD), the bound C2(WD) < C(PD) and
ignoring some terms which contribute positively to g(WD) gives:
g(WD)
X(XD) - ho(PD) - e2(WD)/4
+ x(SD)/2 - C1(WD)/2.
As before we have ho(PD) < D 3 / 4 + 150, e2(WD) < !D and C1(WD) < D/3. By
Theorem 2.12 and Proposition 10.5 of [Ba] and using (Q(2 ) > 6/10, we have X(XD)
+
x(SD)/2 > D 3 / 2 /120 - D/40 so long as D > 36, giving:
g(WD) > D 3 / 2 /120 - 3D/5 - D3/ 4
-
150.
Finally, to bound g(V) when V is a component of WD, we bound the difference:
|g(WD ) - g(WH)| <
We have seen that IC(Wh)
of [Ba] gives |x(Wg)
-
-
x(W)
2x(WD ) +
C(WD'-c(Wg)
+
e 2 (WD)/4
C(WD)I < 7D/12 and e2(WD)/4 < D/3. Theorem 1.4
x(WA)| < D/16 and |g(WD) - g(Wg)| < 7D/10. The bound
asserted for g(V) in Proposition 7.3 follows.
O
74
Appendix A
The D 12-family
A.1
The D 12-family
In this section we will describe the surfaces in M 2 (D1 2 ). For a smooth surface X E
M 2 , the following are equivalent:
" Automorphisms. The automorphism group Aut(X) admits an injective homomorphism p : D12 -+ Aut(X).
" Algebraic curves. The field of functions C(X) is isomorphic to the field:
Ka
C(z, x) with z 2
- ax 3 + 1,
-
for some a E C \ {±2}.
*
Jacobians. The Jacobian Jac(X) is isomorphic to the principally polarized
abelian variety:
where X =rZ
plectic form (()
e
1/
(d)
(-)
,f--
,
= Jm(a +b)
21m-r
V3ij
,
(_') )
and is polarized by the sym-
Hexagonal pinwheels. The surface X is isomorphic to the surface X, obtained
by gluing the hexagonal pinwheel H, (Fig. A-1) to -H,
for some r in the
Figure A-1: For r in the shaded domain U, the hexagonal pinwheel H, has vertices lying on
two equilateral triangles. Gluing together sides on H, and -H, by translation gives a genus
two surface admitting an action of D12 . The one form induced by dz is a Z-eigenform.
domain:
U={
H :-r 4 (
{
or 52,|Re r| 5 -- and Ir I1
.
2
Parallelograms. The surface X is isomorphic the degree six cyclic cover Y of
the double of a parallelogram with vertices {0, 1, w/2, (w + 1)/2} depicted in
Fig. A-2 for some w in the domain:
(1E
Re-rI : ,2I
1 and 1T-11
1
It is straightforward to identify the action of D12 on the surfaces described above.
The field K, has automorphisms Z(z, x) = (-z, ( 3 x) and r(z, x)
polarized lattice A, is preserved by the linear transformations r
=
=
(z/X 3 , 1/x). The
(0 _S) and Z
=
(
. The surface obtained from Z7 satisfies C(Z,) a Ka for some a since Xr
~
has an order six automorphism acting on the Weierstrass points by a permutation
2
the conjugacy class [3, 3]. The surface Yw has an order six Deck transformation for
the map depicted in Fig. A-2 and an involution covered by z
'-4
-z + 5w + 1.
...............
6:J
0
Figure A-2:
For w in the shaded domain V, the parallelogram Q. has vertices
£0, 1, 6w, 6w + 1}. Gluing together the marked segments as indicated gives a genus two surface
Yw = Qw/ ~ with a degree six cyclic covering map to the double a parallelogram. The surface
Yw has an action of D 12 generated by the Deck transformation and the involution obtained by
rotating Qw about z = 3w + 1/2.
The isogeny c, : Jac(X/r) -+ Jac(X/Zr) induced by the correspondence X
-
X/r x X/Zr is a degree three isogeny between elliptic curves and the map:
M 2 (D 12 )
given by (X, p)
'-+
-+
Xo(3)
c, is an isomorphism. The non-trivial outer automorphism of D 12
given by o(Z) = Z and o-(r)
=
Zr induces the Atkin-Lehner involution on M 2 (D 12 )
and the unique nodal D 12 -surface corresponds to the point of order three on Xo(3).
The function relating r and a is modular and the function relating -r and w is a
Riemann mapping between U and V.
The surface obtained from H, has complex multiplication and is labeled by an
orbifold point on a Hilbert modular surface iff -r is imaginary quadratic. When -ris in
Q((3),
the vertices of H, lie in a lattice and the Teichmniller geodesic generated by the
Z-eigenform for X, is a Teichmiiller curve associated to a square tiled surface. When
r is imaginary quadratic but not in Q((3), the complex geodesic in M
2
generated by
the Z-eigenform on X, is an example of the type studied in [Mc2] and has infinitely
generated fundamental group.
79
A.2 Homeomorphism type of WD
DI g(WD) e2 (WD) C(WD)
X(WD)
D
g(WD)
52
e 2 (WD)
C(WD)
X(WD)
1
0
15
-15
5
0
1
1
8
0
0
2
-3
53
2
3
7
-2
9
0
1
2
-1
56
3
2
10
-15
12
0
1
3
-3
13
0
1
3
-
57
60
{1, 1}
3
{1, 1}
4
16
0
1
3
-1
61
2
3
13
-3
17
-18
3
{10, 10} {2-,
-2}
-18
12
17
{0, 0}
{1, 1}
{3, 3}
{-j,-2}
64
1
2
20
0
0
5
-3
65
{1, 1}
{2,2}
21
0
2
4
-3
68
3
0
14
-18
24
0
1
6
-.
69
4
4
10
-18
25
{0, 0}
{0, 1}
{5,3}
{-3,-}
72
4
1
16
28
0
2
29
0
3
7
5
-6
-2
73
76
{1,1}
4
{1, 1}
3
32
0
2
7
-6
77
5
4
8
-18
33
{0, 0}
{1,1}
{6,6}
{-,-2}
80
4
4
16
-24
36
0
0
8
-6
37
0
1
9
-1
81
84
{2, 0}
7
{0, 3}
0
{16,14}
18
40
0
1
12
-2
85
6
2
16
-27
41
{2,2}
3
{7,7}
9
{-6, -6}
-9
1
2
8
-9
7
{3,3}
8
22
45
88
89
92
1
44
{0, 0}
1
{3,3}
6
{14, 14}
13
48
1
2
11
-12
93
8
2
12
-27
49
{0, 0}
{2, 0}
{10, 8}
{-9, -6}
96
8
4
20
-36
-2
{11,11} {-12,-12}
{16,16} {21
2
-7
{-18, -2}
-30
{9
2
9
-30
Table A.1: The Weierstrass curve WD is a finite volume hyperbolic orbifold and for D > 8 its
homeomorphism type is determined by the genus g(WD), the number of orbifold points of order
two e2(WD), the number of cusps C(WD) and the Euler characteristic X(WD). The values of
these topological invariants are listed for each curve WD with D < 250 as well as several larger
discriminants. When D > 9 with D = 1 mod 8, the curve WD is reducible and the invariants are
listed for both spin components with the invariants for WD appearing first.
D
97
100
101
g(WD)
e2(WD)
C(WD)
{4, 4}
4
{1,1}
{19, 19}
0
30
15
6
7
104
9
3
20
105
{6, 6}
{2,2}
{16, 16}
x(WD)
g(WD)
e2(WD)
CWD)
x(WD)
-23
2
152
22
3
18
-36
153
{10, 10}
{2,2}
{26, 26}
57
156
25
8
26
75
2
157
20
3
25
160
22
4
40
-84
81
2
81
2
161
{14, 14}
{4,4}
{20,20}
{-48, -48}
164
20
0
34
-72
{-L,
-L}
-7
{-27, -27}
{-45, -45}
-78
129
2
108
10
3
21
109
8
3
25
112
10
2
29
-48
165
24
4
18
-66
113
{6, 6}
{2,2}
{16, 16}
{-27, -27}
168
29
2
24
-81
116
11
0
25
-45
169
{14, 7}
{0,3}
117
10
4
16
-36
172
29
3
{37, 39}
37
120
16
2
20
-51
173
22
7
13
121
{6, 3}
{3, 0}
{26, 26}
{-, -30}
176
27
6
29
-84
124
15
6
29
-60
{1, 1}
{26, 26}
75
{-i9, -i9}
-90
125
11
5
15
128
13
4
22
129
{8, 8}
{22, 22}
132
15
{3, 3}
0
26
133
15
2
136
17
137
{9,9}
177 {17, 17}
}
189
2
117
2
180
28
0
36
181
26
5
33
171
2
184
37
2
38
-111
-54
185
{17, 17}
{23,23}
22
-51
188
31
{4,4}
10
19
{-57, -57}
-84
2
36
-69
189
27
6
26
-81
{2,2}
{19, 19}
{-36, -36}
192
31
4
34
-7
-48
140
19
6
18
-57
193
{19, 19}
{1,1}
141
18
4
18
-54
196
25
0
{37, 37}
60
144
11
4
38
-60
197
26
5
21
{2,2}
{29, 29}
{-48, -48}
200
31
3
36
-75
201
{20, 20}
204
38
{3,3}
6
{34, 34}
40
145 {10, 10}
{-63,-
148
20
0
37
149
16
7
19
105
2
-96
{
147
2'
-217
-108
147
2
195
147
-117
L
D
g(WD)
e2(WD)
C(WD)
x(WD)
213
36
4
18
-90
216
38
3
46
-2432
217
{25, 25}
{2, 2}
{38, 38}
{-87, -87}
220
46
8
44
-138
221
32
8
30
-96
224
42
8
34
-120
225
{21, 16}
{4, 0}
{42, 42}
{-84, -72}
228
43
0
42
-126
229
42
5
37
243
232
49
1
52
297
233
{27,27}
{26, 26}
236
45
{3,3}
9
237
42
6
20
-105
240
52
4
40
-144
241
{31, 31}
244
53
{3,3}
0
{45, 45}
61
245
40
6
24
-105
248
52
4
22
-126
249
{32, 32}
{40, 40}
41376
164821
{3,3}
112
1552
-331248
{28, 28}
{1442, 1442}
{-228006, -228006}
-359352
41377 {113276, 113276}
-- 2}
{-2
255
35
{-
, -9
}2
-165
{- T,
-i }
41380
178100
0
3154
41381
119380
89
665
478935
2
41384
145957
68
884
-292830
{24, 24}
{1284, 1284}
{-217032, -217032}
41385 {107869, 107869}
41388
155386
54
1188
-311985
41389
146346
81
1475
588411
2
41392
173203
48
2340
-348768
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