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Integral Cohomology of the Siegel Modular Variety of Degree Two and Level Three Mustafa Arslan Louisiana State University and Agricultural and Mechanical College

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Integral Cohomology of the Siegel Modular Variety of Degree Two and Level Three Mustafa Arslan Louisiana State University and Agricultural and Mechanical College Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2006 Integral cohomology of the Siegel modular variety of degree two and level three Mustafa Arslan Louisiana State University and Agricultural and Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Applied Mathematics Commons Recommended Citation Arslan, Mustafa, "Integral cohomology of the Siegel modular variety of degree two and level three" (2006). LSU Doctoral Dissertations. 2504. https://digitalcommons.lsu.edu/gradschool_dissertations/2504 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. INTEGRAL COHOMOLOGY OF THE SIEGEL MODULAR VARIETY OF DEGREE TWO AND LEVEL THREE A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial ful¯llment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics by Mustafa Arslan B.S. in Math., Middle East Technical University, 1996 May 2006 Acknowledgements I would like to thank Dr. Jerome W. Ho®man for his guidance and invaluable help, Dr. Leonard Richardson for constant reminders and enormous patience, Dr. Gestur Olafsson for giving his time when needed and committee members Dr. Robert F. Lax, Dr. Richard A. Litherland, Dr. Frank Neubrander, Dr. Carlos Slawson, and Dr. Paul van Wamelen for their service. I would also like to thank my wife Reena Kothari for her encouragement and support. ii Table of Contents Acknowledgements . ii Abstract . iv Introduction . 1 1 The Siegel Modular Variety of Degree Two and Level Three. 5 1.1 Siegel Modular Varieties . 5 1.2 Compacti¯cation of Siegel Modular Varieties . 8 ¤ 1.3 Important Subvarieties of A2(3) ................... 9 1.3.1 The Cohomology of Boundary Components . 14 2 The Integral Cohomology of A2(3) . 16 2.1 The Rational Cohomology of A2(3) . 16 2.2 Deligne's Spectral Sequence over Integers . 18 2.3 The Main Result . 22 References . 26 Appendix: The Maple Code . 35 Vita.............................................................. 36 iii Abstract p;q In this thesis work Deligne's spectral sequence Er with integer coe±cients for the embedding of the Siegel modular variety of degree two and level three, A2(3), into ¤ its Igusa compacti¯cation, A2(3) is investigated. It is shown that E3 = E1 and this information is applied to compute the cohomology groups of A2(3) over the integers. iv Introduction For a positive integer n the Siegel upper-half space of degree n, Sn, is de¯ned to be the space t Sn = f¿ 2 Mn(C)j ¿ = ¿; Im ¿ > 0g: The symplectic group Sp(2n; R) acts on Sn in the following way: 0 1 B AB C γ = @ A : ¿ 7! (A¿ + B)(C¿ + D)¡1 CD where γ 2 Sp(2n; R) and ¿ 2 Sn. A Siegel modular variety An(¡) of degree n is de¯ned as the quotient space ¡nSn of the Siegel upper-half space by the action of an arithmetic subgroup ¡ of symplectic group Sp(2n; Q). Of particular interest are those Siegel modular varieties for which the arithmetic group ¡ is a principal congruence subgroup ¡n(m), in which case the corresponding variety is called the Siegel modular variety of degree n and level m, denoted by An(m). These varieties are important from various perspectives: 1. They naturally occur as the moduli space of principally polarized abelian varieties with level structures. 2. Automorphic forms for the group Sp(2n; R) and its metaplectic covering typically appear as sections of vector bundles over these spaces. 3. One has ¤ » ¤ H (¡nSn; Q) = H (¡; Q): 1 The isomorphism still holds over the integers if the group ¡ is torsion-free. Therefore this is a way to compute the cohomology of certain arithmetic subgroups of Sp(2n; Q). Although there is a substantial amount of information on the Siegel modular varieties, these spaces are still poorly understood. In fact we know the rational (co)homology of only a few of them. The known cases are the following: - degree 1: These are better known as modular curves (the symplectic group of degree 1 is just the special linear group). The topological properties of these curves is studied in the nineteenth century and the arithmetic of them is a current topic of research. - degree 2, levels 1 and 2: Results are due to R. Lee and S. Weintraub ([10],[11]). They also computed the Hodge numbers for the Igusa compacti- ¯cation of A2(4) in [13]. - degree 2, levels 3 and 4: J. Ho®man and S. Weintraub computed the rational cohomology of A2(3) in [7] and A2(4) in [6]. - degree 3, level 1: The rational cohomology is computed by R. Hain in [5]. In this work, the main result is the determination of the integral cohomology of A2(3). Much of the work in this direction is done in [7] and [8]. The latter paper contains information about integral cohomology of A2(3), and in fact the only 3 cases left open by this work is the determination of torsion parts in H (A2(3); Z) 4 and H (A2(3); Z). The types of the torsion parts of these groups are also given in 3 [8]. According to this the torsion part of H (A2(3); Z) may have elements of order 4 divisible by 2 or 3 and the torsion of H (A2(3); Z) may have elements of order 2 3 only. Our computations agrees also with the existing results on other integral cohomology groups. We can summarize our main result in this work as follows: 8 > > Z q = 0; > > > Z21 © Z=2 © (Z=3)10 q = 2; <> Hq(A (3); Z) = 139 15 35 (1) 2 > Z © (Z=2) © (Z=3) q = 3; > > 81 > Z q = 4; > > : 0 otherwise. The method we use to determine the integral cohomology groups is speci¯c to the case A2(3). In this special case, Deligne's integral spectral sequence p;q p [q] p+q E2 = H (D ; Z) ) H (A2(3); Z) [0] ¤ degenerates at E3, where D = A2(3) is the Igusa compacti¯cation of A2(3) and D[q] is the disjoint union of q by q intersections of the components of the boundary ¤ A2(3) ¡A2(3). Once we know this fact the computations of cohomology groups of A2(3) are fairly easy, because the di®erentials of this spectral sequence, which are Gysin homomorphisms, can be easily implemented, as matrices of intersection numbers of cycle classes, in a software. One has to ¯nd, of course, a set of gener- ¤ ¤ ators for H (A2(3) ; Z), in terms of dual cycles, and has to know the intersection numbers of certain cycle classes to do this. Fortunately this tedious task is carried p ¤ out by J. Ho®man and S. Weintraub [7]: the cohomology groups H (A2(3) ; Z) are free of ranks 1; 0; 61; 0; 61; 0 and 1 for p = 0; 1;:::; 6 ([7, theorem 1.1]) and one can choose a generator sets for H2 and H4 using cycle classes of components of the boundary and Humbert surfaces. This is another factor making the computa- tions easier because the incidence geometry of these subvarieties is explained by a combinatorial topology called Tits building with sca®olding. 3 The article [7] sets as a background for this thesis work and much of the infor- mation is taken from it. In the next chapter we go over briefly the basic de¯nitions about Siegel modular varieties and basic facts about A2(3). Second chapter in- cludes the proof of the degeneracy of the Deligne's integral spectral sequence and the main result. 4 Chapter 1 The Siegel Modular Variety of Degree Two and Level Three The background material for this chapter is [7], [8] [9] and [14]. All of the informa- tion related to the Siegel modular variety of degree 2 and level 3 is coming from ¯rst two. 1.1 Siegel Modular Varieties The set of n by n complex symmetric matrices with the positive-de¯nite imaginary part, is called the Siegel upper-half space of degree n denoted by Sn which is the n-dimensional version of the upper-half plane S1: t Sn = f¿ 2 Mn(C)j ¿ = ¿; Im ¿ > 0g: The real symplectic group of degree n, Sp(2n; R) consists of all real square matrices X of dimension 2n satisfying 0 1 0 1 B 0 In C B 0 In C tX @ A X = @ A ¡In 0 ¡In 0 where In is the identity matrix of dimension n. This means that if 0 1 B AB C X = @ A CD then the square-blocks, A; B; C and D must satisfy the following relations: t t t t t t AC = CA; BD = DB and AD ¡ CB = In The group Sp(2n; R) acts on Siegel upper-half space Sn (see [14]), the action is given by 0 1 B AB C X ¢ ¿ = @ A ¢ ¿ = (A¿ + B)(C¿ + D)¡1 (1.1) CD 5 µ ¶ AB for ¿ in S and for X = in Sp(2n; R) where A, B, C and D are the n n CD by n portions of X. Actually the more is true: Proposition 1.1. 1. The action of Sp(n; R) on Sn is transitive 2. The group Aut(Sn) of biholomorphic automorphisms of Sn is isomorphic to Sp(2n; R)= § 1. The proof can be found in [14].
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