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Hurwitz Monodromy, Spin Separation and Higher Levels of a Modular Tower

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Hurwitz Monodromy, Spin Separation and Higher Levels of a Modular Tower Hurwitz monodromy, spin separation and higher levels of a Modular Tower Paul Bailey and Michael D. Fried Abstract. Each finite p-perfect group G (p a prime) has a universal central p-extension coming from the p part of its Schur multiplier. Serre gave a Stiefel- Whitney class approach to analyzing spin covers of alternating groups (p = 2) aimed at geometric covering space problems that included their regular realization for the Inverse Galois Problem. A special case of a general result is that any finite simple group with a nontrivial p part to its Schur multiplier has an infinite string of perfect centerless group covers exhibiting nontrivial Schur multipliers for the prime p. Sequences of moduli spaces of curves attached to G and p, called Modular Towers, capture the geometry of these many appearances of Schur multipliers in degeneration phenomena of Harbater-Mumford cover representatives. These are modular curve tower generalizations. So, they inspire conjectures akin to Serre's open image theorem, including that at suitably high levels we expect no rational points. Guided by two papers of Serre's, these cases reveal common appearance of spin structures producing θ-nulls on these moduli spaces. The results immedi- ately apply to all the expected Inverse Galois topics. This includes systematic exposure of moduli spaces having points where the field of moduli is a field of definition and other points where it is not. Contents List of Tables 81 1. Overview, related fundamental groups and preliminaries 81 1.1. Quick summary of results 82 1.2. Classical paths on Uz 85 1.3. Results of this paper 86 1.4. General results and divergences with modular curves 89 2. Presenting H4 as an extension 95 2.1. Notation and the groups Br, Hr and Mr 95 2.2. Geometrically interpreting M4 and M¯ 4 96 2.3. Proof of Prop. 2.3 98 2000 Mathematics Subject Classification. Primary 11F32, 11G18, 11R58; Secondary 20B05, 20C25, 20D25, 20E18, 20F34. Thm. 2.9, from a 1987 preprint of M. Fried, includes observations of J. Thompson. D. Semmen corrected a faulty first version of Prop. 5.4 and made the computations of Lem. 9.17. Support for Fried came from NSF #DMS-9970676 and a senior research Alexander von Humboldt award. 79 80 P. BAILEY AND M. FRIED 2.4. Statement of the main presentation theorem 100 2.5. Start of Theorem 2.9 proof 101 2.6. Properties of Q; combinatorics of (2.11) 103 2.7. The shift γ1 = q1q2q1 and the middle twist γ1 = q2 106 2.8. General context with modular curve illustrations 107 2.9. p-lifting property and using sh-incidence 110 2.10. The sh-incidence matrix 111 3. Nielsen classes and the Inverse Galois Problem 112 1 3.1. Equivalences on covers of Pz 113 3.2. Motivation from the Inverse Galois Problem 114 3.3. Frattini group covers 116 3.4. One prime at a time 118 3.5. Refined s-equivalence from NSn (G; C) 120 3.6. p-perfect groups and fine moduli 122 3.7. SL2(C) action on Hurwitz spaces and covers of Λr and Jr 124 rd rd 3.8. Monodromy groups of : H! Ur and : H ! Jr 126 4. Moduli and reduced Modular Towers 129 4.1. Reduced Modular Towers 129 4.2. j-line covers when r = 4 130 4.3. Reduced moduli spaces: b-fine and fine 131 4.4. Points on a Modular Tower 133 5. Group theory of an A5 Modular Tower 136 5.1. Frattini curve covers 137 5.2. The normalizer of a p-Sylow and Loewy display 137 5.3. Producing 2A~5 140 5.4. Cusps and a theorem of Serre 143 5.5. Further Modular Towers for A5, p = 2 and r = 4 144 5.6. Modular Towers from p-split data 147 5.7. Criteria for Gk faithful on Mk and appearance of 1Gk 147 6. Real points on Hurwitz spaces 149 6.1. Diophantine properties of a Modular Tower 149 6.2. Bounding levels with K points 150 6.3. Remainder of Thm. 6.1 proof 151 6.4. Real points when r = 4 153 6.5. Level 0 R points on the (A5; C34 ) Modular Tower 156 7. Cusps on a Modular Tower 158 7.1. Fundamental domains and cusp data 158 7.2. Cusp notation 160 7.3. H-M and near H-M reps. in Ni(G1; C34 ) 163 8. Cusp widths and the genus of components 165 8.1. Orbit genus and fixed points of γ0 and γ1 166 8.2. Subtle conjugations 169 8.3. Length two and four cusp widths 171 8.4. Cusp widths of length 6 and 10 173 8.5. The sh-incidence matrix at level 1 176 8.6. Real points on level 1 of the (A5; C34 ) Modular Tower 178 9. Completing level 1 of an A5 Modular Tower 181 9.1. Diophantine implications 181 HURWITZ MONODROMY AND MODULAR TOWERS 81 9.2. Working with [S+95] 182 9.3. Schur multipliers detect two H4 orbits on Ni1 = Ni(G1; C34 ) 184 ¯ rd 9.4. Spin separation and two M4 orbits on Ni1 188 9.5. Understanding spin separation 193 9.6. Inductive steps in going to higher Modular Tower levels 196 9.7. Generalizing the notion of H-M reps. 197 rd Appendix A. Coset description of a fundamental domain for Hk 199 Appendix B. Serre's half-canonical invariant 199 B.1. Half-canonical classes from odd ramification 200 B.2. Coordinates from θ functions 200 B.3. θ functions along the moduli space 201 B.4. Monodromy, characteristics and Pryms 204 Appendix C. Covers and lifting invariants 207 C.1. The lifting invariant and Nielsen separation 207 C.2. Frattini central extensions 208 Appendix D. The Open Image Theorem 208 D.1. Interpreting and extending the Open Image Theorem 208 D.2. p-part of the outer monodromy group 209 D.3. Finding the Frobenius 210 Appendix E. Simple branching and j-awareness 212 E.1. More complicated Hurwitz spaces for non-p-perfect G0 212 E.2. Modular curve quotients from a Modular Tower 212 ¯ + − Appendix F. A representation difference between M4 on Ni1 and Ni1 215 Appendix G. Where are those regular realizations? 216 G.1. Reminder of the setup 216 G.2. From Chevalley realizations to the whole p-Frattini tower 217 References 218 List of Tables abs 1 List Ni(A5; C34 ) 105 2 sh-Incidence Matrix for Ni0 112 3 Constellation of spaces H(An; C3r ) 122 + 4 mp s 10 × 10 part of sh-Incidence Matrix for O1 178 5 The configuration of P1 s at a special fiber 210 1. Overview, related fundamental groups and preliminaries Here is one corollary of a result from the early 1990s. There is an exact sequence ([FV92], discussed in [MM99, Cor. 6.14] and [V¨o96, Cor. 10.30]): 1 ~ Y (1.1) 1 ! F! ! GQ ! Sn ! 1: n=2 The group on the left is the profree group on a countable number of generators. The group on the right is the direct product of the symmetric groups, one copy for each integer: (1.1) catches the absolute Galois group GQ of Q between two 82 P. BAILEY AND M. FRIED ¯ known groups. Suppose a subfield K of Q has GK a projective (profinite) group. Further, [FV92] conjectures: Then GK is pro-free (on countably many generators) if and only if K is Hilbertian. The conjecture generalizes (1.1) and Shafarevich's conjecture that the cyclotomic closure of Q has a pro-free absolute Galois group. So, (1.1) is a positive statement about the Inverse Galois Problem. Still, it works by maneuvering around the nonprojectiveness of GQ as a projective profinite group. Modular Towers captures, in moduli space properties, implications of this nonprojective nature of GQ. Its stems from constructing from each finite group G, a projective profinite group G~ whose quotients naturally entangle G with many classical spaces: modular curves (x2.8) and spaces of Prym varieties (xB.4 and xE.2), to mention just two special cases. This G~ started its arithmetic geometry life on a different brand of problem [FJ86, Chap. 21]. [Fri95a] gave definitions, motivations and applications surrounding Modular Towers starting from the special case of modular curve towers. This paper contin- ues that, with a direct attack on properties of higher Modular Tower levels. Our models for applications include the Open Image Theorem on modular curve tow- ers in [Ser68] (details in xD.2). In Modular Tower's language this is the (four) dihedral group involution realizations case ([Fri94] and [Fri95c] have very elemen- tary explanations). Modular Towers comes from profinite group theory attached to any finite group, a prime dividing its order and a choice of conjugacy classes in the group. This moduli approach uses a different type of group theory | modular representations | than the homogeneous space approach typical of modular curves or Siegel upper half space. Still, these spaces appear here with a more elementary look and a new set of applications. It shows especially when we compare the monodromy groups of the tower levels with those for modular curves. A key property used in [Ser68]: Levels of a modular curve tower attached to the prime p (p ≥ 5) have a projective sequence of Frattini covers as monodromy groups. Results similar to this should hold also for Modular Towers (Ques. D.1). That is part of our suggestion that there is an Open Image Theorem for Modular Towers. We first briefly summarize results; then set up pre- liminaries to explain them in detail.
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