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1. Talk at UCI Irvine in 1995, Weil's Decomposition Theorem, Siegel's Theorem and Néron Divisors, Based on an English translation of Siegel's famous theorem bounding integral points on an affine curve. I use this to motivate Néron's improvement of Weil's distructions – from Weil's Thesis proving the Mordell-Weil generalization of Poincare's conjecture on rational points on an abelian variety. Like Hilbert's irreducibility theorem, the distributions are a general conceptual tool for translating geometry to arithmetic. Since, however, Lang avoided them in his Diophantine Geometry book just about no one older than me knows about them. weildecomp.pdf

2. Oberwolfach 02/08/02 on Field Arithmetic, Talk on Configuration spaces for wildly ramified covers, Arithmetic fundamental groups and noncommutative algebra, PSPUM vol. of the American Math. Society (2002), 223–247. Generalizes to wild ramification one half of Grothendieck's famous deformation approach to the tame fundamental group of an curve in positive characteristic. Does not assume Galois covers, a non-trivial point to go beyond tame ramification. The main construction is the configuration space of covers having a specific type of ramification data – generalization of Herbrand's higher ramification group upper numbering. Main device: explicit computation of all tamely ramified embeddings of a finite extension of a power series in one variable over the algebraic closure of a finite field. oberwolf02-08-02.pdf

3. Idaho State U., 10/02/08, Separated variable equations and Finite Simple Groups, On how equations of form f(x)-g(y)=0 led to the monodromy method, the B(ranch)C(ycle)L(emma) and the formulation of the genus 0 problem. Goes with an essay on solving Davenport's Problem to explain using group theory to study algebraic equations. Includes how to apply Hurwitz spaces. I lighten the essay with comments on my education at University of Michigan math. dept. during my graduate years '64-'67. ISU-Dav-Thomp10-02-08.html %-%-% ISU-Dav-Thomp10-02-08.pdf

4. Combinatorics of Sphere Covers and the sh-incidence Matrix, The narrative (bibliography is at CSCshIncBib.pdf) of a 2008 grant proposal. It concentrates on how the sh-incidence matrix – a pairing on cusps – graphically displays pertinent information about Hurwitz spaces. §4 shows infinitely many sh-incidence matrices for Alternating group (reduced) Hurwitz spaces of 4 branch point covers. Each space is level 0 of a collection of Modular Towers, each as rich as the classical collection of modular curve towers. The matrix gives the genus of the Hurwitz spaces, as compactified upper-half plane quotients. For measure §4 includes the sh-incidence matrix – something new – of the modular curve called X₀(p²). The resemblances: At high enough tower levels there are sufficient p cusps on the the Alternating group towers (thanks to a Fried-Serre formula) to attack a version of Serre's Open Image Theorem for modular curve towers. Differences start with level 0 (but not higher) of the alternating group towers having cusps types never occuring on modular curve towers. CSCshIncNar.pdf

5. Variables Separated Equations and Finite Simple Groups: (unabridged) a more complete version of UMStoryShort.html. Davenport's problem is to figure out the nature of two polynomials over a number field having the same ranges on almost all residue class fields of the number field. Solving this problem initiated the monodromy method. That included two new tools: the B(ranch)C(ycle)L(emma) and the Hurwitz monodromy group. By walking through Davenport's problem with hindsight, variables separated equations let us simplify lessons on using these tools. We attend to these general questions:

1. What allows us to produce branch cycles, and what was their effect on the Genus 0 Problem (of Guralnick/Thompson)?
2. What is in the kernel of the Chow motive map, and how much is it captured by using (algebraic) covers?
3. What groups arise in 'nature' (a 'la a paper by R. Solomon)?

Each phrase addresses formulating problems based on equations. We seem to need explicit algebraic equations. Yet why, and how much do we lose/gain in using more easily manipulated surrogates for them? To make this clear we consider the difference in the result for Davenport's Problem and that for its formulation over finite fields, using a technique of R. Abhyankar. UCIVarSepEq.pdf

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