HTML and/or PDF files in the folder deflist-mt
For an html and pdf (or ppt) file with the same name, the html is an exposition. Click on any of the [ 8] items below.
An unclickable "Pending" is still in construction. Use [ Comment on ...] buttons to respond to each item, or to the whole page at the bottom.

✺ Overview connecting the Main Conjecture on M(odular) T(ower)s, the R(egular) I(nverse) G(alois) P(roblem) and the S(trong) T(orsion) C(onjecture): MTs incorporates much of what remains mysterious about the RIGP. The STC implies the MT Main Conjecture. Conversely the Main Conjecture directly implies much about the STC. Recent progress on the Main Conjecture makes it a different route to proving results on the STC. mt-rigp-stc.html

✺ Cusp and component trees on a Modular Tower: Especially the role of g-p' cusp branches. Cusp-Comp-Tree.html

✺ Fried-Serre Lifting Invariant test: Tests for existence of a projective sequence of components on a Modular Tower. Translates as a projective sequence of cohomological lifting invariants. The source technique: p-Poincare duality of mapping class group that defines all projective systems of components. Explicitness depends on identifying types of cusps that lie on a given component. FS-Lift-Inv.html

✺ Main Modular Tower Conjecture: The Luminy 2006 characterization of the only possible exceptions to the Main MT Conjecture when r=4. Then, an explanation of Cadoret's result that the Strong Torsion Conjecture implies the (general) Main MT Conjecture. Pending 01/20/07: Main-MT-Conj.html

✺ Modular Towers: A special case of projective sequences of Hurwitz spaces. Restrictive conditions are designed to provide spaces that show the RIGP generalizes a famous result on modular curves, the Mazur-Merel result. The generalizing conjecture the Main Conjecture on MTs: We expect no rational points at high tower levels. Modular-Towers.html

✺ Universal Frattini cover of a finite group: Each (pro)finite group G has a universal Frattini cover, versal for embedding problems. It is projective in the profinite group category. The most mysterious p-group extensions of G are quotients of this group. It is the seeding ingredient to form the spaces that comprise the levels of a Modular Tower. Pending 01/28/07: p-Frattini-cov.html

✺ p-Poincare Duality: Includes a discussion of Weigel's p-Poincare duality result and how Frattini Principle 3 from the 2006 Luminy paper uses it produce p cusps on Modular Tower levels. Pending 01/20/07: p-Poincare-Dual.html

✺ A Moduli Approach to Generalizing Serre's Open Image Theorem: The O(pen)I(mage)T(heorem) starts with this: j' an algebraic number, and Y₀(p) ={Y₀(p^k+1)}_k=1^∞, the p-elementary modular curves covering the classical j-line minus ∞. The 1st part says the Galois closure of the projective system of points over j' has, as (Galois) group, an open subgroup of either GL₂(Z/p^k+1) (GL₂-case) or, for j' complex quadratic, the j'-analog of Z^*_p. The 2nd part of Serre's result: For almost all p, you can replace “ open subgroup” with “ equals.” The html file explains further:

1. Why Serre's is the dihedral group case of a conjectured result for all Modular Towers.
2. How Serre's result divides into three pieces: Frattini Step 1; p-adically-near-∞ Step 2; and integrality Step 3.
3. Modular Tower progress on Step 1 and Step 2, based on the notions of p- and g(roup)p' cusps.

Open-Image-Th.html

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