Moduli relations between \(\ell\)-adic representations and the regular inverse Galois problem. (English) Zbl 1497.11114
Summary: There are two famous Abel Theorems. Most well-known, is his
description of “abelian (analytic) functions” on a one dimensional compact complex torus. The other collects together those complex tori, with their prime degree isogenies, into one space. Riemann’s generalization of the first features his famous \(\Theta\) functions. His deepest work aimed at extending Abel’s second theorem; he died before he fulfilled this.
That extension is often pictured on complex higher dimension torii. For Riemann, though, it was to spaces of Jacobians of compact Riemann surfaces, \(W\), of genus \(g\), toward studying the functions \(\varphi : W \to \mathbb{P}^1_z\) on them. Data for such pairs \((W, \varphi)\) starts with a monodromy group \(G\) and conjugacy classes \(\mathbb{C}\) in \(G\). Many applications come from putting all such covers attached to \((G, C)\) in natural – Hurwitz – families.
We connect two such applications: The Regular Inverse Galois Problem (RIGP) and Serre’s Open Image Theorem (OIT). We call the connecting device Modular Towers (MTs). Backdrop for the OIT and RIGP uses J.-P. Serre’s books [Abelian \(\ell\)-adic representations and elliptic curves. McGill University Lecture Notes. New York-Amsterdam: W. A. Benjamin, Inc. (1968; Zbl 0186.25701); Topics in Galois theory. Notes written by Henri Darmon. Boston, MA etc.: Jones and Bartlett Publishers (1992; Zbl 0746.12001)]. Serre’s OIT example is the case where MT levels identify as modular curves.
With an example that isn’t modular curves, we explain conjectured MT properties – generalizing a Theorem of Hilbert’s – that would conclude an OIT for all MTs. Solutions of pieces on both ends of these connections are known in significant cases.
That extension is often pictured on complex higher dimension torii. For Riemann, though, it was to spaces of Jacobians of compact Riemann surfaces, \(W\), of genus \(g\), toward studying the functions \(\varphi : W \to \mathbb{P}^1_z\) on them. Data for such pairs \((W, \varphi)\) starts with a monodromy group \(G\) and conjugacy classes \(\mathbb{C}\) in \(G\). Many applications come from putting all such covers attached to \((G, C)\) in natural – Hurwitz – families.
We connect two such applications: The Regular Inverse Galois Problem (RIGP) and Serre’s Open Image Theorem (OIT). We call the connecting device Modular Towers (MTs). Backdrop for the OIT and RIGP uses J.-P. Serre’s books [Abelian \(\ell\)-adic representations and elliptic curves. McGill University Lecture Notes. New York-Amsterdam: W. A. Benjamin, Inc. (1968; Zbl 0186.25701); Topics in Galois theory. Notes written by Henri Darmon. Boston, MA etc.: Jones and Bartlett Publishers (1992; Zbl 0746.12001)]. Serre’s OIT example is the case where MT levels identify as modular curves.
With an example that isn’t modular curves, we explain conjectured MT properties – generalizing a Theorem of Hilbert’s – that would conclude an OIT for all MTs. Solutions of pieces on both ends of these connections are known in significant cases.
MSC:
| 11F32 | Modular correspondences, etc. |
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
| 11R58 | Arithmetic theory of algebraic function fields |
| 14H30 | Coverings of curves, fundamental group |
| 20B05 | General theory for finite permutation groups |
References:
| [1] | MTs, ( 1 2 A 5 , C 3 4 ). (This and Ni(A 4 , C 3 4 ) are still |
| [2] | (5.19a) There are two kinds of cusps, HM and near-HM, with near-HM having a special action un-der the complex conjugation operator. |
| [3] | 19b) The components at level 1 have resp. genuses 12 and 9 (> 1): Faltings kicks in here as in §4.4.2. (5.19c) Apply [14]: the genus 12 (resp. 9) component has one (resp. no) component of real points. |
| [4] | has more examples illustrating the theme of hav-ing cusp types -based on using refined aspects of G -separate components. That includes those in • [48] and and [31] •. Recall the key issue: |
| [5] | When Hurwitz spaces have several compo-nents, identify moduli definition fields geo-metrically so as to recognize the G Q action on the components. |
| [6] | Φ A n ,Spin n : Ni(Spin n , C 3 n−1 ) in → Ni(A n , C 3 n−1 ) in . |
| [7] | Theorem 5.21. [32, Main Thm.] For all n, there is one braid orbit for Ni(A n , C 3 n−1 ) in . For n odd, Φ A n ,Spin n is one-one, and the abelianized MT is nonempty. For n even, Ni(Spin n , C 3 n−1 ) in is empty. Here is the meaning for n even. Lift the entries of g g g ∈ Ni(A n , C 3 n−1 ) in to same order entries in Spin n , to getĝ g g. Then, the result does not satisfy product-one: g 1 , . . . ,ĝ n−1 = −1 (the lift invariant in this case). I used this to test many properties of MTs. Here it showed that there is a nonempty Modular Tower over Ni(A n , C , r ≥ n, there are precisely two components, with one obstructed by the lift invariant, the other not. [27, Thm. 2.8] gives a procedure to describe the -Frattini module for any -perfect G, and therefore of the sequence { k G ab } ∞ k=0 . [27, Thm. 2.8] labels Schur multiplier types, especially those called antecedent. Ex-ample: In MTs where G = A n , the antecedent to the level 0 spin cover affects MT components and cusps at all levels ≥ 1 (as in [29]). [64, I.4.5] extends the classical notion of Poincaré duality to any pro-group. It applied to the pro-com-pletion of π 1 (X) with X any compact Riemann surface. [72] uses the extended notion, intended for groups that have extensions by pro-groups of any finite group. Main Result: The universal -Frattini cover G (and G ab ) is an -Poincaré duality group of dimension 2. The result was Thm. 3.17 [29, Cor. 4.19]. |
| [8] | §3.4.2 and §3.4.3 applied this to the Spin n → A n case, for Ni(A 5 , C 3 4 ) and Ni(A 4 , C +3 2 −3 2 ) for different -Frattini lattice quotients (Def. 3.6). |
| [9] | 3.4 Progress on MT conjectures and the OIT As with modular curves, the actual MT levels come alive by recognizing moduli properties attached to par-ticular (sequences) of cusps. It often happens with MTs that level 0 of the tower has no resemblance to modular curves, though a modular curve resemblance arises at higher levels. Level 0 of alternating group towers illustrate: They have little resemblance to modular curves. Yet, often level 1 starts a subtree of cusps that contains the cusp-tree of modular curves. We can see this from the clas-sification of cusps discussed in • [29, §3] •. We leave much discussion of generalizing Serre’s OIT to [34]. |
| [10] | Still, we make one point here, based on what is in • [58] • and • [21] •. With † either inner or absolute equivalence, it is the interplay of two Nielsen classes that gives a clear picture of the bifurcation between the two types of decomposi-tion groups, CM and GL 2 . Those Nielsen classes are Ni(D , C 2 4 ) †,rd and Ni((Z/ ) 2 × s Z/2, C 2 4 ) †,rd . (5.20) In the example(s) of [34], the same thing happens. Of course, the j values don’t have the same interpre-tation as for Serre’s modular curve case. Further, as in • [36] •, there are nontrivial lift invariants, and more complicated, yet still tractible, braid orbits. |
| [11] | , [38], [12] and [13] in one place. [12] assumes the Main MT Conjecture (5.9) is wrong: for some finite group G satisfying the usual conditions for and C and some number field K, the correspond-ing MT has a K point at every level. Using [71] com-pactifications of the MT levels, for almost all primes p p p of K, this would give a projective system of O K,p p p (integers of K completed at p p p) points on cusps. The results here considered what MTs (and some general-izations) would support such points for almost all p p p us-ing Harbater patching (from [44]) around the Harbater-Mumford cusps. We continue the concluding paragraphs of • [25] • on HM components and cusps. [13] continues the results of [12]. It ties together notions of HM components and the cusps that correspond to them, connecting several threads in the theory. They construct, for every pro- |
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