Table Of ContentON THE GEOMETRY OF ELLIPTIC MODULAR SURFACES AND _REPRESENTATIONS
OF FINITE GROUPS
D. Burns*
DEPARTMENT OF MATHEMATICS
University of Michigan
Ann Arbor, Michigan 48109
Introduction. Let F CSI2(~) be a torsion-free subgroup of
finite index. Associated to F is an elliptic surface E(F) fibered
over X(F) : = F\~, where ~ = {T e ~IIm(T) > 0} , and F operates on
in the usual manner. (cf 7 for the construction). If F is
normal in 'F , S12(~)~ F'~ F , then the finite group G = F\F' acts
on X(F) and E(F) , and X(F) , E(F) have G-equivariant models in
various ~(V) , for V a suitable representation space of G . Con-
versely, one may try to reconstruct X(F) , E(F) and the presentation
1 ÷ F ÷ 'F ÷ G ÷ 1 from the geometry of the representation V (and
its tensor representations). More specifically, one could try to find
a G-invariant curve X in ~(V) , and construct a G-invariant
surface E over X whose classifying map to the modular stack for
elliptic curves could be used to intertwine G and S12(~) , or a
subgroup of it. This latter was our initial motivation for the present
paper, and remains largely unfulfilled, except for some elementary
examples at the end of the paper. Nevertheless, we feel that some
progress has been made in this direction, since several characteristic
features of the geometry of E(F) as an elliptic surface over X(F)
can be specified intrinsically.
The approach taken here is to consider the ruled surface over
X(F) obtained by quotienting E(F) by the involution ~(z) = -z ,
where the sign refers to the group law in the fibers of E(F) We
can construct E(F) over X(F) , if we can construct the ruled sur-
face over X(F) , the branch curve and divisor class of the double
covering.
Our first object in the paper is to construct a standard minimal
model S(F) for the ruled surface <~>~E(F) This is done for any
F CS12(~ ) The point is that this ruled surface can be specified
over X(F) purely geometrically, i.e., without reference to F
Next, for F normal in S12(~) , (i.e., F' = S12(~)) , we
characterize the image B in S(F) of the branch curve of E(F) over
*A. P. Sloan Fellow. Partially supported by the National Science
Foundation.
<o>\E(F) as the unique G-invariant divisor in its linear equivalence
class. This can be generalized slightly, i.e., to F' ~ S12(~) , if
X(F') has genus 0 , and one knows enough about the cusps of X(F')
Finally, for certain F , we calculate that there is only one
G-invariant square-root of the line bundle L(B) on S(F)
Concretely, given any projective immersion X(F) + ~(V) , S(F) is
the tangential ruled surface of X(F) : given x e X(F) , the fiber
S(F) x is the projective tangent line to X(F) at x . The zero-
section of S(F) is simply the distinguished point x itself in the
tangent line. If V is a G-representation space, and X(F) is a G-
invariant curve, to reconstruct the branch curve B , it suffices to
choose the remaining three points (besides x) of B~S(F) x in a
G-equivariant way. By the previous results, there is only one way to
do this, and it is here that one uses most strongly the representation
theory of G . In the examples we can calculate (F = F(5),F(7) ,
Fo(2)~F(5) ; G = A 5 , PSL(2;7), A5) , one uses the decomposition of
the tensor representations of V to construct G-equivariant families
of hypersurfaces which cut out B~S(F) as x moves on X(F) The
X
examples can be carried out in varying degrees of detail. The construc
tions carried out here should be compared with a construction of
Naruki's 6 for the elliptic modular surface of level 5.
It is a pleasure to take this opportunity to thank I. Dolgachev
for many long and fruitful discussions of the present topic. The
difference between his contributions to this paper and those of a
co-worker are negligible. Thanks are also due to R. Griess and
R. Gunning, for useful remarks and references. The author also wishes
to thank the Institute for Advanced Study and l'Unlverslte . . J de Paris
XI (Orsay) for their hospitality during the preparation of this paper.
§i. Standard Models for <~>\E(F)
As above E(F) denotes the elliptic modular surface associated
to FCSI2(~ ) If F is torsion free, then the singular fibers of
E(F) occur exactly over the cusps of X(F) These singular fibers
are of type I b or I{ , b >_ 1 , in Kodaira's notation, as the
corresponding cusps are of the first or second kind, respectively,
in the terminology of 7, for example. Recall that by construction,
any g e S12(~) which normalizes F acts on E(F) preserving the
fibers and the 0-section. Let ~ denote the involution of E(F)
induced by -I 8 S12(~) Over F\~, o fixes the zero section O E
and a smooth 3-fold section ~E composed of all the non-trivial points
of order two in the fibers. Thus, to construct non-singular minimal
models of <~> \E(F) , it is only necessary to study the behavior of
near the singular fibers, as in 6. We treat the cases I b and
I{ separately. (We call b the width of the cusp.)
i) Ib:
The cases b even or odd are slightly different. As in 6,
we show the array of curves in a fiber of type I b , and indicate the
fixed points of o :
b odd: b even:
/ , /'f"~
:
/ /
i, :,:/.i."
"'/"",, E~'
Each straight line represents a rational curve of self-intersection -2.
In either case, ~ has no isolated fixed points, and so <~> \E(F) is
a smooth surface ~(r) over x(r) , with special fiber over v :
A A
- XZ___
-2
A A
@
Here is the image in ~(F) of 0 E , and 4~ is the image of ~E
The components of the special fibers are again rational curves, with
self-intersections as indicated. Note that the fibers E(F) v and
A
S(F) are reduced.
V
The canonical model 'S )F( of <o> \E(F) is the ~l-bundle
over X(F) obtained by blowing down the "top-most" exceptional curve
of the first kind (ECFK) and continuing until the final situation below
is realized:
l l °
,
The "lower" local component of ~' crosses S' )F( v transversally, and
is the image of the local component of ~E which passed through the
same component of E(F) v as 0 E Note that ~' .~' = 0 Also
note that 'S )F( is canonical in the sense that any g 8 SI2(Z) which
normalizes F will operate on 'S )F( preserving the ruling, ~9" and
~' Finally, note that reversing the above procedure amounts to
the minimal succession of monoidal transforms needed to resolve the
singularities of ~'
ii) I~ :
Separate b even from b odd:
b odd: b even:
j °.2 .F"4
"" "~E
+4
Cv,b+4 ...
c ,"l I
4,vC
The irreducible components of E v are labelled Cv,0,...,Cv,b+ 4 ,
where E v = C v ,U _ + Cv,l + Cv,2 + Cv, 3 + 2C v ,4 +." .. + 2Cv0b+ 4 .
0E'Cv, 0 = 1,~E,Cv, 1 = 1 . If b even, Cv,2-~ E Cr,3,4/- E = 1 ,
and Cv,b+4"~E= 0 , and ~ fixes Cv,4,Cv,6,...,Cv,b+ 4 pointwise.
If b odd, Cv, 2.~E = Cv, 3.~E= 0 , Cv,b+ 4-~E = 1 , and ~ fixes
Cv,4,Cv,6,...,Cv,b+ 3 pointwise. Thus, ~ has no isolated fixed
points, and the quotient <0> ~E(F) is non-singular, and the fiber
over v looks like:
b odd: b even:
J A
-4
-4 :
4-
-t -t
-4 -4
A °° .-A °. /k
A "0
~'-°.
.
The self-intersections are marked. The components with self-intersec-
A
tion -4 have become simple components of the fiber S(F) v All
double components are ECFK. Blow these (disjoint) curves down, to get:
b odd : b even:
A A
Next, blow-down again all the (disjoint) ECFK's, to get
b odd: b even:
• /k
Finally, blow-down ECFK successively, starting at the curve farthest
from ~, and continue until we have a ruled surface S' (F) , with
S' )F( v of the form:
b odd: b even:
• . .. Jt
"" O~..~J ..
- J ,
~ "" "0" p
The local equations for the double points of ~ ''~. are again y2 = x b
AS before, this S' )F( is canonical, and ~' = 0 .
Consider, next, another model S(F) of <o>\E(F) , which can be
defined only when each singular fiber of E(F) is of type I b or I{ ,
with b > 1 . We will call this the geometric model, for reasons that
will be clear in 52. For this model, for fibers of type I b , one
first blows down the ECFK in ~(F) which crosses ~ , and then
proceeds to blow-down ECFK's, starting from the one farthest from
One gets:
b odd: b even:
The local equation for 4~ near a point of intersection with S(F) v
2 b-2
other than where it crosses '9~ simultaneously, is y = x (When
b = 2 , there are two such distinct transverse intersections.)
Similar adjustments can be made in the blowing down of S(F) near
a cusp v with E(F) v of type I~ . The resulting intersection
diagrams for -~ ,~ and S([)v are exactly as for the corresponding
b I (same b) as above.
Note that for an elliptic: modular surface with all cusp widths
> 1 , in the geometric ruled surface S(F) , the divisor ~ intersects
the section ~" transversally, exactly at the cusps. Finally, note
that the only automorphism of S' (F) over X(F) , of finite order and
fixing a cusp of width 1 , is the identity.
§2. Intrinsic characterization of S' , (F) S(F)
Recall that if S is a ruled surface over a curve X with pro-
jection ~ and a section ~9" , then we have a short exact sequence on
S:
0 ÷ s ÷ 0 0S(~) ÷ N~ ÷ 0 (2.1)
where N~ is the normal sheaf to 7~ in S . Taking direct images
via ~ , one has
0 + 0X ÷ E ÷ ~ ÷ 0 (2.2)
and S is isomorphic to the projectivization ~?(E) of the rank 2
vector bundle E on X associated to the sheaf E . E is determined
by its extension class in HI(x,~)
For any variety, @X will denote the tangent sheaf, and for
X = X(F) as above, < will denote the divisor of cusps, each cusp
counted with multiplicity i.
Theorem : i' For F as in ~i, S = S' )F( , then ~91N a 9X(-<) The
extension (2.2) splits. If F is normal in S12(~) , the extension
splits in a unique G = F\SI2(~) equivariant way.
Theorem i: For F as above, with all cusp widths > 1 , and
S = S(F) , then N~ x = 0 . The extension (2.2) does not split.
Note that the statement of Theorem 1 determined E in (2.2),
and hence S(F) , uniquely up to isomorphism, since
HI(x(F),H~)__ = HI(x(F),~Ix(F)) : { , and the extension class is well-
determined modulo isomorphisms only up to multiplication by non-zero
complex numbers: hence, the extension class is either 0 (split) or
non-zero (unique non-split extension). The latter case arises quite
naturally geometrically as the tangential ruled surface for any
projective immersion of X(F) in pn
To calculate ~ as in theorems 1 and i' , we pass to an
auxiliary subgroup 0 F . Let ° F be a normal subgroup of finite
index in S12(~) , with F 0CF . If F is torsion-free, then
O 0 = G F \F acts on S' (F0) and (F) S' is birational to G0\S' 0) (F ,
the map being biregular over X(F) - {cusps} We claim that this
birational map from G0~S' (F0) to S' (F) is biholomorphic. It
suffices to check the behavior at the cusps.
Note, first of all, that the group O G has fixed points on
X(F )0 only at the cusps. If ¥ 6 O G fixes the cusp v e X(F 0) ,
it must act as the identity on S' (F0) v . Indeed, g preserves
~' and -£~' in S' (F )0 , hence it fixes these points in S' (F0) v :
where S' (F0) v meets ~'' , where it meets ~' transversally,
and where it meets ' with multiplicity 2 Since S' (F0) v ,
g fixes S' (F0) v pointwise. In particular, the quotient space
G0\S' F( 0) is non-singular, and a smooth ruled surface over X(F)
Let S" = G0\S' 0) (r , and let -O ''~ , ~" denote the images of the
curves ~' and ~' in S' (F0) in S" Let q : S" ÷ S' (F) ,
be birational and biregular except possibly over the cusps. At a
cusp v of X(F) , trivialize S' )F( by coordinates (z,~) where z
is the coordinate on X(F) , with {z : 0} = v , and ~ an affine
coordinate on ~' , so that -~" is given locally by {% = 0}
Assume, further that ~ is so normalized that the local component of
~' which crosses S'(F) v transversally is given locally by ~ :
I
and that the double intersection of ~' with S'(F)v occurs at z = O,
= 1 . We can similarly find coordinates z , ~" on S" so that
{~" = O} is the image of --~' F( )O in S" , and the transverse local
component of the image of 4~'(F )O is given by ~" = ~ , and the
double intersection occurs at {z = 0 , ~" = q} With these normali-
zations, the map q is given locally by
(z,~") I q÷ (z,a(z)-~") = (z,)
where a(z) is a meromorphic function defined on a small disc about
z = 0 and non-vanishing away from z : 0 . Since q must send ~"
to ~' =~' )F( , we see that a(z) + 1 as z ÷ 0 Hence, q extends
biholomorphically across z : 0 , and S" + ~ S' )F( Let N~. )F( be
the normal bundle to '~d- in S' )F( , on X(F) , and N~. F( )0 the
analogous bundle in X(F )0
Similarly, if p : X(F )0 ÷X(F) denotes the quotient map, we see
that S' F( )0 ~ ÷ p*S' )F(
Next, note that if @X(F0 ) (-<(F0)) and @X(F) (-<(F)) are the
sheaves for S' F( )0 and S' )F( in the statement of Theorem i' , and
TX(F0) (-<(F0)) and TX(F) (-<(F)) the corresponding line bundles,
then the differential dp. establishes isomorphisms
TX(F0 ) (-<(F0)) + P*Tx(F) (-<(F)) , on X(F )0 )i(
~ on X(F) )2(
G0\Tx(F0 ) (-<(F0)) ÷ TX(F) (-<(F)) ,
For (2), note that the isotropy action of G O on TX(FN) (-<(F0)) at
a cusp is trivial, so there is a well-defined holomorphic line bundle
G0\Tx(F0) (-<(F0)) on X(F) Then the group G O acts on N~. F( )0 ,
again with trivial isotropy action over a fixed point v in X(F )0
(since all S' (F0) v will be fixed by the isotropy group of v in
G O ) , and we get isomorphisms:
01
N~,(F )0 ÷ ~ p*N~,(r) , on x(r )0 , and
G0\N~,(r )0 ~ ÷ N~,(r) , on x(r)
As a result of this sequence of observations, we claim that if
we can find one r I normal, torsion-free, of finite index in SI2(Z),
with N~, (r )I ÷ ~ TX(FI) (-<(FI)) in a G 1 : FI\SI2(~ ) - equivariant way,
then we have an isomorphism N~, )F( ÷ TX(F) (-<(r)) , for any r torsion
free, of finite index in SI2(~) , which is natural in the sense that
the isomorphism is equivariant for the action of the normalizer of F
in SI2(~) on N~, )r( and TX(F) (-<(F)) Indeed, one only has to
choose F 0 above sufficiently small that it is contained normally in
both r and r I . The above observations say the desired isomorphism
"pulls back" from F 1 to r 0 , and "descends" from r 0 to r
However, the work of Naruki shows Theorem i' is true for F 1 = r(5) ,
the principal congruence subgroup of level 5. (This follows from 6,
§2, 3 ; it also follows from the construction of E(F(5)) below in
4, and the remarks below on the relationship of S' )F( and S(F).)
Thus, we know N~, )F( ÷ Ox(r) (-<(F)) , for all FCSI2(~) as in
Theorem 'i
Consider now the extension (2.2):
(2.3)
0 ÷ 0x(r) + ÷ 0x(r)(-<(r)) + 0
The extension class lies in Hl(x(r) , ~ix(r) (<(r))) , which is zero,
since deg <(r) is always positive. Thus splits. To complete
the proof of Theorem I', consider more closely the case when rcsl2(~)
is normal. Tensor (2.3) with fiX(F) 1 (<(r)) and take cohomology to
get:
(2.4
0 ÷ H°(X(r) , ~ix(r) (<(r)))
~H°(x(r) , (Kx(r)+<(r)))
~-~H°(X(F) , OX(F) ÷ 0
G = F\SI2(~ ) operates equxvariantly on (2.4). A G-invariant splitting
of (2.3) is a G-invariant element % e H°(X(F), (Kx(F)+<(F)))
