The Hilbert modular fourfold determined by the totally real quartic number field k is a desingularization of a natural compactification of the quotient space Gamma(k)\H-4, where Gamma(k) = PSL2(O-k) acts on H-4 by fractional linear transformations via the four embeddings of k into R. The arithmetic genus, equal to one plus the dimension of the space of Hilbert modular cusp forms of weight (2, 2, 2, 2), is a birational invariant useful in the classification of these varieties. In this work, we describe an algorithm allowing for the automated computation of the arithmetic genus and give sample results

Grundman, Helen G

Lippincott, L. E

English

Scholarship;  Research;  and Creative Work at Bryn Mawr College

Bryn Mawr College
Scholarship, Research, and Creative Work at Bryn Mawr
College
Mathematics Faculty Research and Scholarship Mathematics
2006
Computing the Arithmetic Genus of Hilbert
Modular Fourfolds
Helen G. Grundman
Bryn Mawr College, hgrundma@brynmawr.edu
L. E. Lippincott
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Custom Citation
Grundman, H.G., and L.E. Lippincott. "Computing the Arithmetic Genus of Hilbert Modular Fourfolds." Math. Comp. 75 (2006):
1553-1560.
MATHEMATICS OF COMPUTATION 
Volume 75, Number 255, July 2006, Pages 1553-1560 
S 0025-5718(06)01842-4 
Article electronically published on March 21, 2006 
COMPUTING THE ARITHMETIC GENUS 
OF HILBERT MODULAR FOURFOLDS 
H. G. GRUNDMAN AND L. E. LIPPINCOTT 
ABSTRACT. The Hilbert modular fourfold determined by the totally real quar- 
tic number field k is a desingularization of a natural compactification of the 
quotient space Fk\H4, where Fk = PSL2(Ok) acts on H4 by fractional linear 
transformations via the four embeddings of k into R. The arithmetic genus, 
equal to one plus the dimension of the space of Hilbert modular cusp forms of 
weight (2, 2, 2, 2), is a birational invariant useful in the classification of these 
varieties. In this work, we describe an algorithm allowing for the automated 
computation of the arithmetic genus and give sample results. 
1. INTRODUCTION 
Let k be a totally real algebraic number field of degree n over Q with ring of 
integers Ok and discriminant dk. For i = 1,..., n and a E k, let a 
- 
a(i) denote 
the ith embedding of k into the real numbers. The group SL2 (0k) acts on tln, the 
product of n copies of the complex upper half plane, by ( 
c 
b ( , ( (1)Zl 
) 
(2)(1 c (2) 2 a(2) a"(n)zn+ 
~2b(n) 
c d) (ZlZ2 Zn)= c(l)zl + d(l) 
' C(2)z2 + 
d(2),'.' c(n)z 
+ d(n) 
The Hilbert modular group of the field k is Fk = PSL2(0k). The Hilbert modular 
variety of k is a desingularization of the natural compactification of Fk\7knR. For 
more detailed discussions of the construction of these varieties, see [7] and [12]. 
The arithmetic genus, in the sense of Hirzebruch [8], is a birational invariant 
useful in the classification of nonsingular varieties. Specifically, it is equal to one 
plus the dimension of the space of Hilbert modular cusp forms of weight (2, 2, 2, 2). 
By [4], the arithmetic genus of the Hilbert modular variety of k is given by 
(-1)"vol(Fk\THn) + C E(rk, ) + L(rk, 4), 
where the first summation is over a complete set of representatives of the I7k- 
equivalence classes of fixed points of Fk and the second is over the set of cusps. 
Precise definitions of E(rk, 7) and L(rk, K) are given below. 
Hirzebruch [7] provided a formula for computing the arithmetic genus in the 
cases where the ring Ok contains a unit of norm -1. This has been widely used 
for computing the arithmetic genus of Hilbert modular varieties of odd dimension, 
since -1 is a unit of norm -1 in any field of odd degree. For varieties of even 
Received by the editor April 23, 2004 and, in revised form, May 10, 2005. 
2000 Mathematics Subject Classification. Primary 11F41, 14E08; Secondary 14J10, 14J35. 
The first author wishes to acknowledge the support of the Faculty Research Fund of Bryn 
Mawr College. 
(2006 American Mathematical Society 
Reverts to public domain 28 years from publication 
1553 
1554 H. G. GRUNDMAN AND L. E. LIPPINCOTT 
dimension, the computation is more complicated. In the 1970's, Hirzebruch, van 
de Ven, and Zagier [9, 10] demonstrated how to carry out these computations for 
surfaces, but until recently little progress was made with higher degrees. It must 
be said that part of this is due to the earlier lack of computational tools. For 
example, a challenge as little as a decade ago, the computation of the volume term, 
(-1)nvol(Fk\ n), is all but trivialized by now-available software packages. 
In [6], the current authors focused on the degree four case and determined how to 
compute the fixed point term, E. E(Fk, ri). This left the cusp term, , L(rk, ), 
as the final roadblock to completing the computations. In this paper, we offer a 
now complete solution to the problem, providing an algorithm for computing the 
arithmetic genus of the Hilbert modular variety of an arbitrary quartic field k, and 
describing our implementation thereof. 
We develop the necessary mathematics and describe our algorithm for computing 
the cusp term in Section 2. In Section 3, we review the needed results from [6] and 
describe our algorithms for computing the volume and fixed point terms. Finally, in 
Section 4, we describe our specific implementation and give a table of the arithmetic 
genera of the Hilbert modular fourfolds determined by the 210 totally real quartic 
fields of smallest discriminant. 
2. THE CUSP TERM 
In this section, we develop the tools needed for computing the cusp term, 
1 
L(]k,•), K 
for an arbitrary quartic field, k. We begin by defining notation. 
Let Ck denote the group of ideal classes and 8k the group of units of (Ok. It is 
well known that the set of cusps of Fk \7j4 is in one-to-one correspondence with Ck. 
Letting [a] be the ideal class corresponding to the cusp n, we have 
L(Fk, [a]) = L(Fk, ) 1 sign (N(a)) 16r d IN(a) 
with d(a-2) the discriminant of a-2 [4]. In this notation, the cusp term is given by 
Z[a]EC, L(rk, [a]). 
We begin with a lemma proved in [6]. 
Lemma 1. If (Ok contains a unit of norm -1 or an integral ideal b for which 
b2 = (3) is a principal ideal with N(O) < O0, then Z- L(rk, K) = 0. 
If the hypothesis of Lemma 1 is not satisfied by k, then there exists a real 
character, ) : Ik -+ +1, on the group of nonzero fractional ideals, satisfying 
4((a)) = sign(N(a)) for all nonzero a E Ok. It follows that b(a2) = 1 and so, 
letting x vary over integral ideals in Ok, 
L(rk,[a])-v lm S S ) 74 g j(x)s 
[a]ECk [a]ECk xcE[a2] 
lim() 
iCk/C21 [7r4 ---*lE N (x) [[xc]Ek 
COMPUTING THE ARITHMETIC GENUS OF HILBERT MODULAR FOURFOLDS 1555 
As in [6], let D be the set of characters : Ik- induced by a complete set 
of characters on the group Ck/C'. Then, 
L(Fk, [a]) = -- N(x) 5 (x) [a] CCk 7;x 
- 
4E-limE - 7r4 
_s--l 
N(z 
(X)s 
74Q 
Now, since each 0 is trivial on principal ideals, each 00 is a quadratic character 
on Ik satisfying 00((a)) = sign(N(a)) for all nonzero a E Ok. Hence, each 4)q 
corresponds, via class field theory, to a quadratic extension k' of k, unramified at 
each finite place and ramified at each infinite place. In fact, as q ranges through 
D, the k' range through all such field extensions of k. 
Since each k' is a totally complex quadratic extension of the totally real number 
field k, we have 
7•4 kR Wk' k/- (1) L(1, I?) = L(1, k'/k) = Rk 
hkRkwk, 
' Idk' 
where hk, hk', Rk, Rk', dk, dk' are the class numbers, regulators, and absolute values 
of the discriminants of Ok and Ok', respectively, and Wk, Wk' are the numbers of 
roots of unity in these rings. 
Combining the above with the fact that wk = 2 yields the following theorem. 
Theorem 2. If 0(k contains a unit of norm -1 or an integral ideal b for which 
b2 = (3) is a principal ideal with N(0) < O0, then - L(Fk, K) = 0. Otherwise, 
)L(Fk 2 
:hk'Rk, (2) L( ,) hk 'k k w / Ii; 
where k' ranges through the set of unramified, totally complex, quadratic extensions 
of k. 
To compute the sum in equation (2), we use the computer algebra package 
KASH [3]. We determine the set of quadratic fields, k', as follows. RayClassField 
and RayClassFieldAuto are used to determine the narrow class field of k (that is, 
the field corresponding to the group of positive principal divisors). Then 
OrderSubfield is used to find all degree 8 subfields. OrderSig is used to de- 
termine which of these subfields are totally complex, and OrderIsSubfield is used 
to determine which of the totally complex subfields contain k. Once the appropriate 
quadratic extensions of k have been determined, the commands OrderClassGroup, 
OrderReg, and OrderTorsionUnitRank can be used to evaluate each summand. 
This computation is carried out only when Ok contains no unit of norm -1 and 
no integral ideal b for which b2 = (P) is a principal ideal with N(O) < 0. We check 
for a unit of norm -1 by using the commands OrderUnitsFund and EltNorm. If 
none exists, we determine whether there is an integral ideal b for which 62 = (/) 
is a principal ideal with N(0) < 0, by using OrderClassGroupCyclicFactors. 
Viewing the class group as a product of cyclic groups, this command provides, for 
each cyclic factor, the order of the factor and an ideal in a class that generates it. 
1556 H. G. GRUNDMAN AND L. E. LIPPINCOTT 
For each cyclic factor of even order, we raise the representative ideal to the order 
of the factor, thus producing a principal ideal. Finally, we use IdealIsPrincipal 
to obtain a generator for the ideal and EltNorm to determine whether the norm of 
the ideal's generator is negative. 
If any of these generators is of negative norm, then setting it equal to 3 gives 
us that by Theorem 2, E, L(Fk, K) = 0. We now show that if no such generator 
exists, then no such 3 exists. 
Lemma 3. If the above process fails to yield a generator of negative norm, then 
every generator of any principal ideal equal to the square of an integral ideal in Ok has positive norm. 
Proof. Suppose that b is an integral ideal such that 62 = (0) and N(0) < 0. 
Since k is totally real and contains no unit of norm -1, b is not principal and so is 
in an ideal class of order 2. 
Let cl, c2,... ,cn be representative ideals for generators for the cyclic factors of 
the class group of k. Let el, e2,...,e, be minimal nonnegative integers such that 
c"' ce2... ? en is in the same ideal class as b. So we have 
c1 e2 = b, 
for some a, 
-7 E Ok. Thus 
(3) 22C2e, C2e2 .2en , 2 1 2 
"''n 
- (13) 
Since each side of equation (3) is a principal ideal, the hypothesis gives us that each 
C2ei has a generator of positive norm. (Note that ei = 0 for any cyclic factor of the 
class group of odd order, since each c2ei is principal.) Hence 7y2 (0) and therefore (13) 
have generators of positive norm. Since Ok has no units of norm -1, all generators 
of (0) have positive norms. This contradicts our supposition and concludes the 
proof. 
3. THE VOLUME AND FIXED POINT TERMS 
As noted in [5], vol(Fk\Kn) =4k (-1), which is easily computed using the 
command zetak in the software package PARI [1]. It follows from the work of 
Siegel [11, 13] that 15k (-1) is a rational integer. Although PARI does not provide 
specific error-bounds for zetak, all of the outputs were within 10-25 of an integral 
multiple of 15 
Computing the fixed point term is more involved. As defined in [4], letting rj be 
the equivalence class of a fixed point z, this term is given by 
(4) E(rFk, [z= r 1 
- MErz i= 
MIO 
where Fz is the isotropy subgroup of z and ( is the rotation factor of M. We 
summarize our computational formula for this term in the following theorem, which 
is easily derived from [6, Theorem 4.2]. 
For any field k', let hk' and Rk' be the class number and regulator of Ok' re- 
spectively. For r G {2, 3, 4, 5, 6} let 
m, 
= (Rk( 
• 
)h r)/(Rkhk), 
and let 6 be the relative discriminant of k(e ) over k. 
COMPUTING THE ARITHMETIC GENUS OF HILBERT MODULAR FOURFOLDS 1557 
Theorem 4. Let k be a totally real quartic number field. 
If dk = 1125, then ZE E(Fk, r) = 1 
If dk = 2000, then •• E(Fk, 7) = 151 
If dk = 2048, then 
Zn 
E(Fk, 7) = *4 
If dk = 2304, then  E(EFk, r) = 61 
If dk 4 {1125, 2000, 2048, 2304}, then 
m2 m3 m4 m5 m6 (5) ZE(rk,7)= 9f2 + 
-f3 
+ f4 + 
-f5 
+ 6f6, 6 216 64 40 864 
where the fr are as given in Table 1. 
The following corollary simplifies the computation. 
TABLE 1. Values of fr for r = 2, 3, 4, 5, 6 
v/- E k, V3 
- 
k 
20k 20k' T f2 f4 f6 
p- 
- - 6 3 0 
p2 2 - 
- 
12 3 0 
p4 54 - 54 38 0 
p4 54 4 
- 
18 20 0 
p4 58 
- 
18 9 0 
v2 0k, v E k 
20k 20k' f2 f4 f6 
p s2 - 25 0 209 
p2 52t2 
- 15 0 209 
p2q2 - - 99 0 209 
p4 
- - 45 0 209 
/ V k, / • k 
20k 20k 6S f2 f4 f6 
p - - 3 0 0 
pq - - 3 0 0 
pqr - - 3 0 0 
pqrs - - 3 0 0 
p2 52 - 78 0 0 
p2 52t2 
- 48 0 0 
p12 54 
- 15 0 0 
p2q s2t2 - 30 0 0 
p 2q s2t2U2 
- 12 0 0 
p2q 54t2 
- 9 0 0 
p2qr 52t2u2 - 30 0 0 
p2qr 52A2U212 
- 12 0 0 
p2qr s54t22 - 9 0 0 
p2q2 52t2 - 300 0 0 
p2q2 s2t2u2 - 120 0 0 
p2q2 52t2u202 
- 48 0 0 
p2q2 q 542 - 90 0 0 
p2q2 54t2U2 
- 36 0 0 
p2q2 54t4 - 27 0 0 
p3q 56t2 U2t2 21 0 0 
p3q 56t2 U402 9 0 0 
p4 54 
- 138 0 0 
p4 54t4 - 48 0 0 
p4 58 U2 45 0 0 
p4 58 U4 21 0 0 
V•k 
30k 30k/ f3 
p - 9 
pq - 9 
pqr - 9 
pqrs - 9 
p2 52 54 
p2 S2t2 90 
p2q 52t2 45 
p2 q 52t2u2 27 
p 2qt 52t2U2 45 
p2qr 52t2u2,2 27 
p2q2 52t2 234 
p2q2 52t2U2 144 
p2q2 
2 52t2u2v2 90 
p3q - 36 
p4 54 162 
p4 54t4 90 
V3- E k 
30k 30k/ f3 
p2 - 40 
p22 - 112 
p4 - 76 
V/E k 
50k 50k' f5 
p4 4 16 
p4 $ q4 12 
:p4 
_ 
2 
i~5 
1558 H. G. GRUNDMAN AND L. E. LIPPINCOTT 
Corollary 5. Let k be a totally real quartic field. 
If V2 V k, then f4 = 0. 
If V5 ? k, then fs = 0. 
If v3• k, then f6 = 0. 
To compute the values of ZE E(Fk, r ) we first use the KASH function OrderDisc 
to check if k is any of the four special fields listed in Theorem 4. If not, we evaluate 
equation (5) as follows: EltRoot determines which, if any, of v2, V'5, and 4/3 are 
in k; then applying Corollary 5 as appropriate determines the values of some of 
the fr. For the remaining fr, we use Table 1 to determine what ideals need to 
be factored and use OrderDisc to define any needed relative discriminants. The 
function Factor is used to factor the ideals, and the exponents of the factorization 
are extracted for use with the table. The values of the mr are computed directly 
using the KASH functions OrderReg and OrderClassGroup for the regulators and 
class numbers, respectively. 
TABLE 2. Arithmetic genera of Hilbert modular fourfolds 
dk X dk X dk X dk X dk X dk X 
725 1 7625 5 13068 7 17609 5 22221 7 26125 12 
1125 2 8000 5 13448 6 17725 7 22545 9 26176 10 
1600 2 8069 3 13525 6 17989 5 22592 9 26224 9 
1957 1 8112 4 13625 7 18097 7 22676 9 26225 10 
2000 3 8468 3 13676 6 18432 8 22784 11 26525 10 
2048 2 8525 4 13725 7 18496 14 22896 8 26541 9 
2225 2 8725 4 13768 5 18625 9 23252 8 26569 12 
2304 3 8768 4 13824 7 18688 10 23297 7 26825 9 
2525 2 8789 3 13888 6 18736 6 23301 9 26873 8 
2624 2 8957 3 13968 9 19025 8 23377 7 27004 10 
2777 1 9225 5 14013 5 19225 8 23525 9 27225 14 
3600 4 9248 5 14197 5 19429 6 23552 9 27329 8 
3981 2 9301 3 14272 5 19525 8 23600 9 27472 9 
4205 2 9792 5 14336 6 19600 10 23665 7 27648 13 
4225 3 9909 5 14400 8 19664 12 23724 11 27725 10 
4352 4 10025 5 14656 7 19773 10 24197 9 27792 14 
4400 3 10273 3 14725 6 19796 12 24336 13 28025 11 
4525 3 10304 5 15125 8 19821 7 24400 12 28224 13 
4752 4 10309 4 15188 5 20032 8 24417 9 28224 14 
4913 3 10512 7 15317 5 20225 8 24437 8 28400 11 
5125 4 10816 5 15529 5 20308 7 24525 11 28473 9 
5225 3 10889 3 15952 5 20808 9 24749 7 28669 9 
5725 3 11025 7 16225 7 21025 9 24832 15 28677 9 
5744 3 11197 3 16317 7 21056 8 24917 8 28749 10 
6125 4 11324 5 16357 5 21200 8 25088 11 29237 8 
6224 3 11344 4 16400 9 21208 7 25225 10 29248 11 
6809 2 11348 4 16448 7 21308 8 25488 14 29268 13 
7053 3 11525 5 16448 7 21312 10 25492 9 29813 9 
7056 5 11661 4 16609 5 21469 7 25525 10 29952 13 
7168 4 12197 4 16997 6 21568 11 25717 7 30056 18 
7225 4 12357 6 17069 5 21725 8 25808 9 30056 16 
7232 4 12400 6 17417 6 21737 8 25857 14 30125 14 
7488 5 12544 9 17424 10 21801 9 25893 8 30273 10 
7537 2 12725 5 17428 6 21964 10 25961 8 30400 12 
7600 4 13025 6 17600 7 22000 11 26032 9 30512 11 
COMPUTING THE ARITHMETIC GENUS OF HILBERT MODULAR FOURFOLDS 1559 
4. IMPLEMENTATION AND RESULTS 
Our implementation consists of a shell script calling two programs: a simple 
PARI program that produces a table of values of (k(-1), and a KASH program 
that performs the remainder of the calculations. As input, the script takes a file of 
fields in a form output by PARI as found at the University of Bordeaux's number 
fields web site [2], with one field per line. Our PARI program creates a new file of 
(k(-1) values which our KASH program reads along with the original input file. 
For each field in the input file, the KASH program computes the volume term, the 
fixed point term, and the cusp term as described in this paper and combines them 
to compute the arithmetic genus. 
We ran the programs using PARI 2.1.4 and KASH 2.2 on a SUN Ultra-60, with 
a 450MHz Sun UltraSPARC-II processor, running the Solaris 5.8 operating system. 
Results for the Hilbert modular varieties defined by the 210 totally real quartic 
fields of smallest discriminant are given in Table 2. As an indication of the time 
involved in these calculations, it takes approximately 40 minutes to compute the 
first 100 cases, 100 minutes to compute the 210 given in the table, and 13 hours to 
compute the first 1000. 
As a final note, running the program using KASH 2.4 resulted in errors, appar- 
ently arising from a problem with the OrderSubfield command. 
REFERENCES 
[1] C. Batut, K. Belabas, D. Benardi, H. Cohen, and M. Olivier. User's Guide to PARI-GP, 
1998. (ftp://megrez .math. u-bordeaux. fr/pub/pari). 
[2] J. Buchmann, F. Diaz y Diaz, D. Ford, P. Letard, M. Olivier, M. Pohst, and A. Schwarz. 
Tables of number fields of low degree, (ftp://megrez.math.u-bordeaux.fr/pub/ 
numberfields/). 
[3] M. Daberkow, C. Fieker, J. Kliiners, M. Pohst, K. Roegner, and K. Wildanger. Kant v4. 
J. Symbolic Comp., 24:267-283, 1997. (http://www.math.TU-Berlin.DE/-kant/kash.html). 
MR1484479 (99g:11150) 
[4] E. Freitag. Hilbert Modular Forms. Springer-Verlag, Berlin, 1980. MR1050763 (91c:11025) 
[5] H. G. Grundman. Hilbert modular varieties over Galois quartic fields. J. Number Theory, 63 
(1997), 47-58. MR1438648 (98c:11040) 
[6] H. G. Grundman and L. E. Lippincott. Hilbert modular fourfolds of arithmetic genus one. 
In High Primes and Misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie 
Williams, Fields Inst. Commun., 41, pp. 217-226. Amer, Math. Soc., 2004. MR2076248 
(2005d:11065) 
[7] F. Hirzebruch. Hilbert modular surfaces. Enseign. Math., II. Ser., 19 (1973), 183-281. 
MR0393045 (52:13856) 
[8] F. Hirzebruch. Topological Methods in Algebraic Geometry. Springer-Verlag, Berlin, 3rd edi- 
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[10] F. Hirzebruch and D. Zagier. Classification of Hilbert modular surfaces. In Complex Analysis 
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MR0480356 (58:524) 
[11] C. L. Siegel. Berechnung von Zetafunktionen an ganzzahligen Stellen. Nachr. Akad. Wiss. 
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[12] G. van de Geer. Hilbert Modular Surfaces. Springer-Verlag, Berlin, 1988. MR0930101 
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Enseign. Math, 22 (1976), 55-95. MR0406957 (53:10742) 
1560 H. G. GRUNDMAN AND L. E. LIPPINCOTT 
BRYN MAWR COLLEGE, 101 N. MERION AVE., BRYN MAWR, PENNSYLVANIA 19010 
E-mail address: grundman@brynmawr. edu 
BRYN MAWR COLLEGE, 101 N. MERION AVE., BRYN MAWR, PENNSYLVANIA 19010 
E-mail address: llippincObrynmawr. edu 


Computing the Arithmetic Genus of Hilbert Modular Fourfolds

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Bryn Mawr CollegeScholarship, Research, and Creative Work at Bryn MawrCollegeMathematics Faculty Research and Scholarship Mathematics2006Computing the Arithmetic Genus of HilbertModular FourfoldsHelen G. GrundmanBryn Mawr College, hgrundma@brynmawr.eduL. E. LippincottLet us know how access to this document benefits you.Follow this and additional works at: http://repository.brynmawr.edu/math_pubsPart of the Mathematics CommonsThis paper is posted at Scholarship, Research, and Creative Work at Bryn Mawr College. http://repository.brynmawr.edu/math_pubs/7For more information, please contact repository@brynmawr.edu.Custom CitationGrundman, H.G., and L.E. Lippincott. "Computing the Arithmetic Genus of Hilbert Modular Fourfolds." Math. Comp. 75 (2006):1553-1560.MATHEMATICS OF COMPUTATION Volume 75, Number 255, July 2006, Pages 1553-1560 S 0025-5718(06)01842-4 Article electronically published on March 21, 2006 COMPUTING THE ARITHMETIC GENUS OF HILBERT MODULAR FOURFOLDS H. G. GRUNDMAN AND L. E. LIPPINCOTT ABSTRACT. The Hilbert modular fourfold determined by the totally real quar- tic number field k is a desingularization of a natural compactification of the quotient space Fk\H4, where Fk = PSL2(Ok) acts on H4 by fractional linear transformations via the four embeddings of k into R. The arithmetic genus, equal to one plus the dimension of the space of Hilbert modular cusp forms of weight (2, 2, 2, 2), is a birational invariant useful in the classification of these varieties. In this work, we describe an algorithm allowing for the automated computation of the arithmetic genus and give sample results. 1. INTRODUCTION Let k be a totally real algebraic number field of degree n over Q with ring of integers Ok and discriminant dk. For i = 1,..., n and a E k, let a - a(i) denote the ith embedding of k into the real numbers. The group SL2 (0k) acts on tln, the product of n copies of the complex upper half plane, by ( c b ( , ( (1)Zl ) (2)(1 c (2) 2 a(2) a"(n)zn+ ~2b(n) c d) (ZlZ2 Zn)= c(l)zl + d(l) ' C(2)z2 + d(2),'.' c(n)z + d(n) The Hilbert modular group of the field k is Fk = PSL2(0k). The Hilbert modular variety of k is a desingularization of the natural compactification of Fk\7knR. For more detailed discussions of the construction of these varieties, see [7] and [12]. The arithmetic genus, in the sense of Hirzebruch [8], is a birational invariant useful in the classification of nonsingular varieties. Specifically, it is equal to one plus the dimension of the space of Hilbert modular cusp forms of weight (2, 2, 2, 2). By [4], the arithmetic genus of the Hilbert modular variety of k is given by (-1)"vol(Fk\THn) + C E(rk, ) + L(rk, 4), where the first summation is over a complete set of representatives of the I7k- equivalence classes of fixed points of Fk and the second is over the set of cusps. Precise definitions of E(rk, 7) and L(rk, K) are given below. Hirzebruch [7] provided a formula for computing the arithmetic genus in the cases where the ring Ok contains a unit of norm -1. This has been widely used for computing the arithmetic genus of Hilbert modular varieties of odd dimension, since -1 is a unit of norm -1 in any field of odd degree. For varieties of even Received by the editor April 23, 2004 and, in revised form, May 10, 2005. 2000 Mathematics Subject Classification. Primary 11F41, 14E08; Secondary 14J10, 14J35. The first author wishes to acknowledge the support of the Faculty Research Fund of Bryn Mawr College. (2006 American Mathematical Society Reverts to public domain 28 years from publication 1553 1554 H. G. GRUNDMAN AND L. E. LIPPINCOTT dimension, the computation is more complicated. In the 1970's, Hirzebruch, van de Ven, and Zagier [9, 10] demonstrated how to carry out these computations for surfaces, but until recently little progress was made with higher degrees. It must be said that part of this is due to the earlier lack of computational tools. For example, a challenge as little as a decade ago, the computation of the volume term, (-1)nvol(Fk\ n), is all but trivialized by now-available software packages. In [6], the current authors focused on the degree four case and determined how to compute the fixed point term, E. E(Fk, ri). This left the cusp term, , L(rk, ), as the final roadblock to completing the computations. In this paper, we offer a now complete solution to the problem, providing an algorithm for computing the arithmetic genus of the Hilbert modular variety of an arbitrary quartic field k, and describing our implementation thereof. We develop the necessary mathematics and describe our algorithm for computing the cusp term in Section 2. In Section 3, we review the needed results from [6] and describe our algorithms for computing the volume and fixed point terms. Finally, in Section 4, we describe our specific implementation and give a table of the arithmetic genera of the Hilbert modular fourfolds determined by the 210 totally real quartic fields of smallest discriminant. 2. THE CUSP TERM In this section, we develop the tools needed for computing the cusp term, 1 L(]k,•), K for an arbitrary quartic field, k. We begin by defining notation. Let Ck denote the group of ideal classes and 8k the group of units of (Ok. It is well known that the set of cusps of Fk \7j4 is in one-to-one correspondence with Ck. Letting [a] be the ideal class corresponding to the cusp n, we have L(Fk, [a]) = L(Fk, ) 1 sign (N(a)) 16r d IN(a) with d(a-2) the discriminant of a-2 [4]. In this notation, the cusp term is given by Z[a]EC, L(rk, [a]). We begin with a lemma proved in [6]. Lemma 1. If (Ok contains a unit of norm -1 or an integral ideal b for which b2 = (3) is a principal ideal with N(O) < O0, then Z- L(rk, K) = 0. If the hypothesis of Lemma 1 is not satisfied by k, then there exists a real character, ) : Ik -+ +1, on the group of nonzero fractional ideals, satisfying 4((a)) = sign(N(a)) for all nonzero a E Ok. It follows that b(a2) = 1 and so, letting x vary over integral ideals in Ok, L(rk,[a])-v lm S S ) 74 g j(x)s [a]ECk [a]ECk xcE[a2] lim() iCk/C21 [7r4 ---*lE N (x) [[xc]Ek COMPUTING THE ARITHMETIC GENUS OF HILBERT MODULAR FOURFOLDS 1555 As in [6], let D be the set of characters : Ik- induced by a complete set of characters on the group Ck/C'. Then, L(Fk, [a]) = -- N(x) 5 (x) [a] CCk 7;x - 4E-limE - 7r4 _s--l N(z (X)s 74Q Now, since each 0 is trivial on principal ideals, each 00 is a quadratic character on Ik satisfying 00((a)) = sign(N(a)) for all nonzero a E Ok. Hence, each 4)q corresponds, via class field theory, to a quadratic extension k' of k, unramified at each finite place and ramified at each infinite place. In fact, as q ranges through D, the k' range through all such field extensions of k. Since each k' is a totally complex quadratic extension of the totally real number field k, we have 7•4 kR Wk' k/- (1) L(1, I?) = L(1, k'/k) = Rk hkRkwk, ' Idk' where hk, hk', Rk, Rk', dk, dk' are the class numbers, regulators, and absolute values of the discriminants of Ok and Ok', respectively, and Wk, Wk' are the numbers of roots of unity in these rings. Combining the above with the fact that wk = 2 yields the following theorem. Theorem 2. If 0(k contains a unit of norm -1 or an integral ideal b for which b2 = (3) is a principal ideal with N(0) < O0, then - L(Fk, K) = 0. Otherwise, )L(Fk 2 :hk'Rk, (2) L( ,) hk 'k k w / Ii; where k' ranges through the set of unramified, totally complex, quadratic extensions of k. To compute the sum in equation (2), we use the computer algebra package KASH [3]. We determine the set of quadratic fields, k', as follows. RayClassField and RayClassFieldAuto are used to determine the narrow class field of k (that is, the field corresponding to the group of positive principal divisors). Then OrderSubfield is used to find all degree 8 subfields. OrderSig is used to de- termine which of these subfields are totally complex, and OrderIsSubfield is used to determine which of the totally complex subfields contain k. Once the appropriate quadratic extensions of k have been determined, the commands OrderClassGroup, OrderReg, and OrderTorsionUnitRank can be used to evaluate each summand. This computation is carried out only when Ok contains no unit of norm -1 and no integral ideal b for which b2 = (P) is a principal ideal with N(O) < 0. We check for a unit of norm -1 by using the commands OrderUnitsFund and EltNorm. If none exists, we determine whether there is an integral ideal b for which 62 = (/) is a principal ideal with N(0) < 0, by using OrderClassGroupCyclicFactors. Viewing the class group as a product of cyclic groups, this command provides, for each cyclic factor, the order of the factor and an ideal in a class that generates it. 1556 H. G. GRUNDMAN AND L. E. LIPPINCOTT For each cyclic factor of even order, we raise the representative ideal to the order of the factor, thus producing a principal ideal. Finally, we use IdealIsPrincipal to obtain a generator for the ideal and EltNorm to determine whether the norm of the ideal's generator is negative. If any of these generators is of negative norm, then setting it equal to 3 gives us that by Theorem 2, E, L(Fk, K) = 0. We now show that if no such generator exists, then no such 3 exists. Lemma 3. If the above process fails to yield a generator of negative norm, then every generator of any principal ideal equal to the square of an integral ideal in Ok has positive norm. Proof. Suppose that b is an integral ideal such that 62 = (0) and N(0) < 0. Since k is totally real and contains no unit of norm -1, b is not principal and so is in an ideal class of order 2. Let cl, c2,... ,cn be representative ideals for generators for the cyclic factors of the class group of k. Let el, e2,...,e, be minimal nonnegative integers such that c"' ce2... ? en is in the same ideal class as b. So we have c1 e2 = b, for some a, -7 E Ok. Thus (3) 22C2e, C2e2 .2en , 2 1 2 "''n - (13) Since each side of equation (3) is a principal ideal, the hypothesis gives us that each C2ei has a generator of positive norm. (Note that ei = 0 for any cyclic factor of the class group of odd order, since each c2ei is principal.) Hence 7y2 (0) and therefore (13) have generators of positive norm. Since Ok has no units of norm -1, all generators of (0) have positive norms. This contradicts our supposition and concludes the proof. 3. THE VOLUME AND FIXED POINT TERMS As noted in [5], vol(Fk\Kn) =4k (-1), which is easily computed using the command zetak in the software package PARI [1]. It follows from the work of Siegel [11, 13] that 15k (-1) is a rational integer. Although PARI does not provide specific error-bounds for zetak, all of the outputs were within 10-25 of an integral multiple of 15 Computing the fixed point term is more involved. As defined in [4], letting rj be the equivalence class of a fixed point z, this term is given by (4) E(rFk, [z= r 1 - MErz i= MIO where Fz is the isotropy subgroup of z and ( is the rotation factor of M. We summarize our computational formula for this term in the following theorem, which is easily derived from [6, Theorem 4.2]. For any field k', let hk' and Rk' be the class number and regulator of Ok' re- spectively. For r G {2, 3, 4, 5, 6} let m, = (Rk( • )h r)/(Rkhk), and let 6 be the relative discriminant of k(e ) over k. COMPUTING THE ARITHMETIC GENUS OF HILBERT MODULAR FOURFOLDS 1557 Theorem 4. Let k be a totally real quartic number field. If dk = 1125, then ZE E(Fk, r) = 1 If dk = 2000, then •• E(Fk, 7) = 151 If dk = 2048, then Zn E(Fk, 7) = *4 If dk = 2304, then  E(EFk, r) = 61 If dk 4 {1125, 2000, 2048, 2304}, then m2 m3 m4 m5 m6 (5) ZE(rk,7)= 9f2 + -f3 + f4 + -f5 + 6f6, 6 216 64 40 864 where the fr are as given in Table 1. The following corollary simplifies the computation. TABLE 1. Values of fr for r = 2, 3, 4, 5, 6 v/- E k, V3 - k 20k 20k' T f2 f4 f6 p- - - 6 3 0 p2 2 - - 12 3 0 p4 54 - 54 38 0 p4 54 4 - 18 20 0 p4 58 - 18 9 0 v2 0k, v E k 20k 20k' f2 f4 f6 p s2 - 25 0 209 p2 52t2 - 15 0 209 p2q2 - - 99 0 209 p4 - - 45 0 209 / V k, / • k 20k 20k 6S f2 f4 f6 p - - 3 0 0 pq - - 3 0 0 pqr - - 3 0 0 pqrs - - 3 0 0 p2 52 - 78 0 0 p2 52t2 - 48 0 0 p12 54 - 15 0 0 p2q s2t2 - 30 0 0 p 2q s2t2U2 - 12 0 0 p2q 54t2 - 9 0 0 p2qr 52t2u2 - 30 0 0 p2qr 52A2U212 - 12 0 0 p2qr s54t22 - 9 0 0 p2q2 52t2 - 300 0 0 p2q2 s2t2u2 - 120 0 0 p2q2 52t2u202 - 48 0 0 p2q2 q 542 - 90 0 0 p2q2 54t2U2 - 36 0 0 p2q2 54t4 - 27 0 0 p3q 56t2 U2t2 21 0 0 p3q 56t2 U402 9 0 0 p4 54 - 138 0 0 p4 54t4 - 48 0 0 p4 58 U2 45 0 0 p4 58 U4 21 0 0 V•k 30k 30k/ f3 p - 9 pq - 9 pqr - 9 pqrs - 9 p2 52 54 p2 S2t2 90 p2q 52t2 45 p2 q 52t2u2 27 p 2qt 52t2U2 45 p2qr 52t2u2,2 27 p2q2 52t2 234 p2q2 52t2U2 144 p2q2 2 52t2u2v2 90 p3q - 36 p4 54 162 p4 54t4 90 V3- E k 30k 30k/ f3 p2 - 40 p22 - 112 p4 - 76 V/E k 50k 50k' f5 p4 4 16 p4 $ q4 12 :p4 _ 2 i~5 1558 H. G. GRUNDMAN AND L. E. LIPPINCOTT Corollary 5. Let k be a totally real quartic field. If V2 V k, then f4 = 0. If V5 ? k, then fs = 0. If v3• k, then f6 = 0. To compute the values of ZE E(Fk, r ) we first use the KASH function OrderDisc to check if k is any of the four special fields listed in Theorem 4. If not, we evaluate equation (5) as follows: EltRoot determines which, if any, of v2, V'5, and 4/3 are in k; then applying Corollary 5 as appropriate determines the values of some of the fr. For the remaining fr, we use Table 1 to determine what ideals need to be factored and use OrderDisc to define any needed relative discriminants. The function Factor is used to factor the ideals, and the exponents of the factorization are extracted for use with the table. The values of the mr are computed directly using the KASH functions OrderReg and OrderClassGroup for the regulators and class numbers, respectively. TABLE 2. Arithmetic genera of Hilbert modular fourfolds dk X dk X dk X dk X dk X dk X 725 1 7625 5 13068 7 17609 5 22221 7 26125 12 1125 2 8000 5 13448 6 17725 7 22545 9 26176 10 1600 2 8069 3 13525 6 17989 5 22592 9 26224 9 1957 1 8112 4 13625 7 18097 7 22676 9 26225 10 2000 3 8468 3 13676 6 18432 8 22784 11 26525 10 2048 2 8525 4 13725 7 18496 14 22896 8 26541 9 2225 2 8725 4 13768 5 18625 9 23252 8 26569 12 2304 3 8768 4 13824 7 18688 10 23297 7 26825 9 2525 2 8789 3 13888 6 18736 6 23301 9 26873 8 2624 2 8957 3 13968 9 19025 8 23377 7 27004 10 2777 1 9225 5 14013 5 19225 8 23525 9 27225 14 3600 4 9248 5 14197 5 19429 6 23552 9 27329 8 3981 2 9301 3 14272 5 19525 8 23600 9 27472 9 4205 2 9792 5 14336 6 19600 10 23665 7 27648 13 4225 3 9909 5 14400 8 19664 12 23724 11 27725 10 4352 4 10025 5 14656 7 19773 10 24197 9 27792 14 4400 3 10273 3 14725 6 19796 12 24336 13 28025 11 4525 3 10304 5 15125 8 19821 7 24400 12 28224 13 4752 4 10309 4 15188 5 20032 8 24417 9 28224 14 4913 3 10512 7 15317 5 20225 8 24437 8 28400 11 5125 4 10816 5 15529 5 20308 7 24525 11 28473 9 5225 3 10889 3 15952 5 20808 9 24749 7 28669 9 5725 3 11025 7 16225 7 21025 9 24832 15 28677 9 5744 3 11197 3 16317 7 21056 8 24917 8 28749 10 6125 4 11324 5 16357 5 21200 8 25088 11 29237 8 6224 3 11344 4 16400 9 21208 7 25225 10 29248 11 6809 2 11348 4 16448 7 21308 8 25488 14 29268 13 7053 3 11525 5 16448 7 21312 10 25492 9 29813 9 7056 5 11661 4 16609 5 21469 7 25525 10 29952 13 7168 4 12197 4 16997 6 21568 11 25717 7 30056 18 7225 4 12357 6 17069 5 21725 8 25808 9 30056 16 7232 4 12400 6 17417 6 21737 8 25857 14 30125 14 7488 5 12544 9 17424 10 21801 9 25893 8 30273 10 7537 2 12725 5 17428 6 21964 10 25961 8 30400 12 7600 4 13025 6 17600 7 22000 11 26032 9 30512 11 COMPUTING THE ARITHMETIC GENUS OF HILBERT MODULAR FOURFOLDS 1559 4. IMPLEMENTATION AND RESULTS Our implementation consists of a shell script calling two programs: a simple PARI program that produces a table of values of (k(-1), and a KASH program that performs the remainder of the calculations. As input, the script takes a file of fields in a form output by PARI as found at the University of Bordeaux's number fields web site [2], with one field per line. Our PARI program creates a new file of (k(-1) values which our KASH program reads along with the original input file. For each field in the input file, the KASH program computes the volume term, the fixed point term, and the cusp term as described in this paper and combines them to compute the arithmetic genus. We ran the programs using PARI 2.1.4 and KASH 2.2 on a SUN Ultra-60, with a 450MHz Sun UltraSPARC-II processor, running the Solaris 5.8 operating system. Results for the Hilbert modular varieties defined by the 210 totally real quartic fields of smallest discriminant are given in Table 2. As an indication of the time involved in these calculations, it takes approximately 40 minutes to compute the first 100 cases, 100 minutes to compute the 210 given in the table, and 13 hours to compute the first 1000. As a final note, running the program using KASH 2.4 resulted in errors, appar- ently arising from a problem with the OrderSubfield command. REFERENCES [1] C. Batut, K. Belabas, D. Benardi, H. Cohen, and M. Olivier. User's Guide to PARI-GP, 1998. (ftp://megrez .math. u-bordeaux. fr/pub/pari). [2] J. Buchmann, F. Diaz y Diaz, D. Ford, P. Letard, M. Olivier, M. Pohst, and A. Schwarz. Tables of number fields of low degree, (ftp://megrez.math.u-bordeaux.fr/pub/ numberfields/). [3] M. Daberkow, C. Fieker, J. Kliiners, M. Pohst, K. Roegner, and K. Wildanger. Kant v4. J. Symbolic Comp., 24:267-283, 1997. (http://www.math.TU-Berlin.DE/-kant/kash.html). MR1484479 (99g:11150) [4] E. Freitag. Hilbert Modular Forms. Springer-Verlag, Berlin, 1980. MR1050763 (91c:11025) [5] H. G. Grundman. Hilbert modular varieties over Galois quartic fields. J. Number Theory, 63 (1997), 47-58. MR1438648 (98c:11040) [6] H. G. Grundman and L. E. Lippincott. Hilbert modular fourfolds of arithmetic genus one. In High Primes and Misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, Fields Inst. Commun., 41, pp. 217-226. Amer, Math. Soc., 2004. MR2076248 (2005d:11065) [7] F. Hirzebruch. Hilbert modular surfaces. Enseign. Math., II. Ser., 19 (1973), 183-281. MR0393045 (52:13856) [8] F. Hirzebruch. Topological Methods in Algebraic Geometry. Springer-Verlag, Berlin, 3rd edi- tion, 1978. MR1335917 (96c:57002) [9] F. Hirzebruch and A. Van de Ven. Hilbert modular surfaces and the classification of algebraic surfaces. Invent. Math., 23 (1974), 1-29. MR0364262 (51:517) [10] F. Hirzebruch and D. Zagier. Classification of Hilbert modular surfaces. In Complex Analysis and Algebraic Geometry, pp. 43-77. University Press and Iwanami Shoten, Cambridge, 1977. MR0480356 (58:524) [11] C. L. Siegel. Berechnung von Zetafunktionen an ganzzahligen Stellen. Nachr. Akad. Wiss. Gottingen II. Math. Phys. Kl., pages 87-102, 1969. MR0252349 (40:5570) [12] G. van de Geer. Hilbert Modular Surfaces. Springer-Verlag, Berlin, 1988. MR0930101 (89c:11073) [13] D. Zagier. On the values at negative integers of the zeta-function of a real quadratic field. Enseign. Math, 22 (1976), 55-95. MR0406957 (53:10742) 1560 H. G. GRUNDMAN AND L. E. LIPPINCOTT BRYN MAWR COLLEGE, 101 N. MERION AVE., BRYN MAWR, PENNSYLVANIA 19010 E-mail address: grundman@brynmawr. edu BRYN MAWR COLLEGE, 101 N. MERION AVE., BRYN MAWR, PENNSYLVANIA 19010 E-mail address: llippincObrynmawr. edu 

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Computing the Arithmetic Genus of Hilbert Modular Fourfolds

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