Polynomials with PSL(2) monodromy
Abstract
Let K be a field of characteristic p>0, and let q be a power of p. We determine all polynomials f in K[t]\K[t^p] of degree q(q1)/2 such that the Galois group of f(t)u over K(u) has a transitive normal subgroup isomorphic to PSL_2(q), subject to a certain ramification hypothesis. As a consequence, we describe all polynomials f in K[t], of degree not a power of p, such that f is functionally indecomposable over K but f decomposes over an extension of K. Moreover, except for one ramification setup (which is treated in the companion paper arxiv:0707.1837), we describe all indecomposable polynomials f in K[t] of nonppower degree which are exceptional, in the sense that xy is the only absolutely irreducible factor of f(x)f(y) which lies in K[x,y]. It is known that, when K is finite, a polynomial f is exceptional if and only if it induces a bijection on infinitely many finite extensions of K.
 Publication:

arXiv eprints
 Pub Date:
 July 2007
 DOI:
 10.48550/arXiv.0707.1835
 arXiv:
 arXiv:0707.1835
 Bibcode:
 2007arXiv0707.1835G
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Number Theory;
 12F12;
 14G27
 EPrint:
 44 pages