HTML or PDF (or PPT) files in ./paplist-cov

HTML and/or PDF files in the folder paplist-cov
paplist-cov: R(egular)I(nverse)G(alois)P(roblem) and Arithmetic of Covers (outside Modular Towers) For an html and pdf (or ppt) file with the same name, the html is an exposition. Click on any of the [ 28] 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.

1. On a Conjecture of Schur, Michigan Math. J. Volume 17, Issue 1 (1970), 41–55 (pdf also on-line at the Michigan Math Journal). My first paper, though not first in print. It gives the classification of exceptional polynomials – those that map one-one on infinitely many residue fields – of a number field. They are up to (very precise) linear change over the algebraic closure compositions of cyclic (like xⁿ) and Chebychev polynomials. Schur's 1921 Conjecture generated much literature: at its solution Charles Wells sent me a bibliography of over 550 papers, most showing certain families of polynomials – given by the form of their coefficients – contained none with the exceptionality property. An essential step was recognizing and using a reduction to polynomials with primitive monodromy group.

Includes the first serious use of R(iemann)'s E(xistence) T(heorem) on a problem of this type, a start of the monodromy method. Tchebychev covering groups are dihedral and easy to characterize. So, RET was quick, but not essential here. Yet, Schur's Conjecture was special within Davenport's problem, and RET has proved essential for that. Further, using RET opened the territory to many other problems (see the html file). Schur's original conjecture was technically easier than Davenport's Problem. Still, by considering its analog for rational functions, the monodromy method connected to Serre's O(pen)I(mage)T(heorem) (UMStoryExc-OIT.html and GCMTAMS78.pdf) and, so, to modular curves. SchurConj70.html %-%-% SchurConj70.pdf